I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's easy, then it means you already know this material, and you're wasting your time.
> I agree with the sentiment of this. I think our obsession with innate ~~mathematical~~ skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.
The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
I think we do our children a disservice by convincing them that some of their peers are just “born with it”, because it discourages them from continuing to try.
What we should be teaching children is HOW to learn. At the moment it’s a by-product of learning about some topic. If we look at the old adage “feed a man a fish”, the same is true of learning.
“Teach someone mathematics and they will learn mathematics. Teach someone to learn and they will learn anything”.
Caveat here is that "talent" and "dedication" is linked to speed at least in the beginning. For instance, any student can learn calculus given enough time and advice even starting from scratch. However, the syllabus wants all this to happen in one semester.
This gives you vicious and virtuous cycles: Students' learning speed increases with time and past success. So "talented" students learn quickly and have extra time to further explore and improve, leading to further success. Students who struggle with the time constraint are forced to take shortcuts like memorizing "magic formulas" without having time to really understand. Trying to close that gap is very hard work.
Thank you for the insight that academic (in a very broad sense) bulk-fixed-time approach does in fact produce both of the cycles, and the gap indeed only widens with time (speaking from personal experience, especially from my life as an undergrad student).
Reminds me of my personal peeve that "studying" should not be "being taught", studying is pursuit of understanding, "being taught" is what happens in primary school (and I'm aware I'm simplifying here).
I would say that you could generalize this even further outside of education. A few early successes in life can greatly accelerate one's trajectory, while early failures could set one many years back. And this happens independently of whether those events are due to skill or luck.
Indeed, speed is often read as "smarts" whereas I would maintain it's much more often "preparation". We can't on one hand believe in the plasticity and retrainability of the mind, while simultaneously believing that speed is something only a few are born with. On the nature/nurture scale, I think it's 20/80 or so - but prodigies and geniuses have an interest that keeps them thinking and learning 10x or 100x more than other kids, and a little bump that lets them get started easier and therefore much earlier.
This sets them up for fantastic success very quickly. [1] shows a great example of this.
I'm fond of saying "You can do anything you want, but wanting is the hard part", because to truly be a grandmaster, genius-level mathematician, olympic athlete, etc, requires a dedication and amount of preparation that almost nobody can manage. Starting late, with emotional baggage, kids, and having to spend 5 years relearning how to learn? Forget it.
> I'm fond of saying "You can do anything you want, but wanting is the hard part", because to truly be a grandmaster, genius-level mathematician, olympic athlete, etc, requires a dedication and
I was having a problem agreeing with this subthread, and I have you to thank for putting it into words that I can finally formulate my disagreement against.
Have you never met one of those people for whom they did not need to "want"? They could literally phone it in and still do better than anyone else, no matter how dedicated they were. Even should practice/study be necessary for them, they benefited from it to some absurd proportion that I couldn't even guess to quantify. I've known more than one of these people.
I think most believe they don't exist for two reasons. The first is the ridiculous number of television shows and movies that depict motivation as being the key to success. We're just inundated with the (unsupported by evidence) that this is the means to extraordinary genius. Second, I would say that this is the most comforting theory. "Why yes, I could have been a gifted whatever or a talented something-or-other if I had put the time in, but I chose this other thing instead."
Maybe some would say we all need to believe this, that a society that doesn't believe in it is harsher or more unkind.
I think I have met those folks. Maybe not. And you're welcome!
They're just quick. But the ones I've met, at least, are quick to make associations. When I really dig and ask them to explain themselves or a concept, they usually make analogies to things they know, but I don't. Then I have to go learn that thing. Then they try the analogy again, but I haven't fully learned it from years of making analogies about it.
Years of grad school experience was painful like this, until I got to a point 10 years after grad school, after a PhD, and well into research, that I "just got" things (in my subfield) as well. It's these experiences that made me feel that it's 80% preparation and perspiration (both of which are dominated by time), and 20% "other" mythology. Don't get me wrong, that 20% is what makes a 2 year old read earlier than others, and getting started reading at 2 (and continuing it!) for 4 years before starting school will make you light years ahead of your peers. The same goes for chess, math, etc etc. There is something legendary about Oppenheimer learning enough dutch in 6 weeks to deliver a lecture. Or perhaps learning to translate his lecture and memorizing it. Who knows.
Do we really believe there's a magical "genius" such that they can do anything? No, so what are the limits to their genius? The limits are defined by what they are a genius at. This is a tautological definition.
I'm not saying "Anyone at any time can become a genius at anything". I'm saying "If you take a kid, start early, and cultivate them just right so that you have time to realize compounding effects, - you can let them grow into basically anything" (probablistically speaking - there are learning disabilities and physical issues etc).
> I think most believe they don't exist for two reasons.
I add third (okay, 2b) - because the pain of coming up with the fact other people are better than you at a deep, fundamental level is too overwhelming.
Bobby Fisher won his first US Championships at 14 against people who had been playing chess longer than he had been alive. Suggesting they didn't want it more, or practice more than some kid is silly.
"We can't on one hand believe in the plasticity and retrainability of the mind, while simultaneously believing that speed is something only a few are born with."
Sure we can, the initial orientation of neurons differs between people, so some people need less "plasticity and retrainability" to be good at a task. Plasticity is physical characteristic like height and varies between people.
Initial speed usually isn't that important, but speed of learning is important and makes the difference between possible and impossible within a human lifetime.
I think there's a probabalistic argument I'm making that's more in line with the article.
Yes - there will be 10x-ers. And that group will have a 10x-er iside it, and so on given exponential dropoff of frequency of talent. Bobby Fisher is a few std dev above even the best, perhaps.
Generally speaking, "You can do anything you want, but wanting (enough, and naturally) is the hardest part" might need a three standard deviation limit.
Have you heard the phrase: Being average among those who practice makes you 9X% among the population? I think that's what I'm saying - you can be a top performer if you dedicate yourself, especially early enough, but almost nobody will.
I agree with you. I don’t think I’m naturally gifted at much (I’m just average), but I was taught stubborn hard work pretty early on. Unfortunately it took me until my 20s to figure out I could be athletic if I applied that hard work. I could also be good at programming doing the same. I’ve met people who are truly gifted and it’s amazing, but I’m pretty decent at the things I worked hard at.
"Initial speed usually isn't that important, but speed of learning is important and makes the difference between possible and impossible within a human lifetime."
Likely so, but is suggest that personality, drive and motivation are also very important factors. I know from experience that stuff I had little interest in as a youngster and that I've still little in I still know little about.
Yes, my interests have grown and broadened over the years but simply I regard some stuff so irrelevant to my life that it's not worth a second thought and I am much better off applying my limited number of neurons to matters of greater importance and enjoyment.
Of course, no one has the luxury of just learning about what one finds interesting and or enjoyable, life's knocks and experiences along with utilitarian-like imperatives force one to learn stuff they'd rather not know about.
I find it is good to go back to things you struggled with in the past and come at them with a new and broader understanding.
> I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus.
It might be different for intellectual skills but I am not that sure.
Almost anyone can become decent at almost anything though. Which is good already!
> I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus.
Being tall isn't a skill. I suspect you could be skillful enough at basketball to overcome the hight disadvantage. However, I think most people who might become that skillful see the high disadvantage (plus the general difficulty of becoming a pro basketball player) and take a different path through life. It's also possible that the amount of time that would be needed to grow your skill past the height disadvantage is too long, so it's not feasible to do it to gain a position in the NBA.
Intelligence is also not a skill, but the thing that makes you skillful in all cognitive tasks. Just like what height does to basketball players.
>Intelligence is also not skill, but the thing that makes you skillful in all cognitive tasks.
Careful with that "all", even the most highly intelligent humans still have peaks and deficits in different domains.
It's a matter of the definition. The general factor of intelligence, which is measured through various somewhat lossy proxies like IQ tests, is exactly the degree to which someone exceeds expectation on all cognitive tasks (or vice versa).
The interesting finding is that this universal correlation is strong, real, and durable. Of course people in general have cognitive domains where they function better or worse than their g factor indicates, and that's before we get into the fact that intellectual task performance is strongly predicated on knowledge and practice, which is difficult to control for outside of tests designed (successfully, I must add) to do so.
Height is one physical attribute that helps, and professional players are mostly above average height for a reason. But also hand-eye coordination and fast-twitch muscles help even more. Many basketball players are very explosive athletes, because it's a sport with a relatively small play area and lots of quick movements are needed.
Track and swimming are where innate physical attributes have the most obvious benefits. Michael Phelphs had the perfect body for swimming. There is no amount of trainingg that 99.999% of the population could do to get close to what Usain Bolt ran. Most humans could not train to run under 4 minutes in a mile or under 2:30 in a marathon. They just don't have the right muscular and cardiovascular physiology.
Team sports are of course more complicated as other qualities come into play that aren't as directly physiological.
> Most humans could not train to run under 4 minutes in a mile or under 2:30 in a marathon.
Of course, but I don't think anyone was seriously suggesting that. The vast majority of humans can become pretty good at swimming though. And that was my interpretation of the original claim about cognitive tasks, mathematics, etc.
Most NBA players are under 2 meters tall. The average height is 1.99 meters.
Since we are being pedantic, your statement may be true but it is unsupported by the data you presented. To make it simple, let's talk about the imaginary basketball league with four players, of unit less heights of 4, 4, 4, and 1. The average height is 3.25, yet 3/4 the players are taller than average.
A paid promotion of International Median is not Average Association.
Seems like he has more than the average number of legs as well.
The fact that he has a wiki page, and that many folks with born without or who have lost legs (~500,000/year Americans experience limb loss or are born with a limb difference https://amputee-coalition.org/resources/limb-loss-statistics...) do not, suggest that the number of people with < 2 is far greater than the number of people with > 2. So the average is still less than 2.
For better or worse, number of legs (or number of arms) is canonical example people use to demonstrate the statistical principal a significant majority of a population can be above average of some metric.
Simpson's Paradox[0] is the reason people are so easily seduced by the tempting, but dead wrong, illusion that humans are in any sense equal in their innate capacity for anything.
Because it turns out that, in the NBA, height does not correspond with ability! This of course makes sense, because all the players are filtered by being NBA professional basketballers. A shorter player simply has more exceptional ability in another dimension, be that dodging reflex, ability to visualize and then hit a ball trajectory from the three point line, and so on. Conversely, a very tall player is inherently useful for blocking, and doesn't have to be as objectively good at basketball in order to be a valuable teammate.
Despite this lack of correlation, when you look at an NBA team you see a bunch of very tall fellows indeed. Simpson's Paradox.
We see the same thing in intellectual pursuits. "I'm not nearly as smart as the smartest programmer I know, but I get promoted at work so I must be doing something right. Therefore anyone could do this, they just have to work hard like I did". Nope. You've already been selected into "professional programmer", this logic doesn't work.
So you're saying success at maths isn't an inbuilt ability. Instead, it depends on an (inbuilt) ability to hyper focus... Which you are just born with?
Not even that. It depends on the learned ability to stop pushing yourself when your focus is wavering. That's how you develop aversion towards the topic. Let your natural curiosity draw you to particular topics (that's why you might have a winding road through the subject).
parent comment was a bit tounge-in-cheek but I'll continue the sentiment: You're saying that the curiosity is "natural" hence one is either born with it or not. I think that there is no way around the fact that it will be hard and uncomfortable to mimic the progress of someone that has an innate inclination towards a subject (be it talent or focus or curiosity) artificially.
Hey, that doesn't have to be what "natural curiosity" means. Besides which it makes no sense to say people are born with complex interests. I mean, OK, your genes might incline you a certain way, but that's not the same thing.
Being interested in a subject is massively helpful to learning it. But interest arises circumstantially, it's an emotion. The grim reality that it would be really useful to you to learn a certain subject does not necessarily make you interested in the subject, unfortunately. (Perhaps "financially interested", but that's something else.)
I think there is some natural inclination towards abstract thinking versus more grounded in reality, just judging based on kids I know. Some of them really enjoy playing with ideas in their heads, some enjoy playing with things they can touch more. It seems likely that those different attractions would express themselves in how much they practice different things as time goes on.
I was talking about curiosity in general not curiosity about something in particular. We are naturally inquisitive to the point we have to be restrained by our parents. The problem is some of the restraints are based on the fears of our parents and not on actual dangers. Also, it's hard to develop an appreciation for something when it's forced fed to you.
> You're saying that the curiosity is "natural" hence one is either born with it or not.
Why does curiosity being natural necessarily mean some people are born without it? It could also mean everyone (or every average human) is born with it, and overtime it gets pushed out of people.
Some infants explore vastly more than others.
So the minimum might not be zero, but it isn’t some fixed quantity.
That's not because it doesn't exist, it's because it's restrained
How?
People do studies on weeks old babies, it seems unlikely curiosity has been restrained at that point.
>I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
And this also gives the proponent (you in this case) an excuse to blame a person for not focusing hard enough or not being dedicated enough if they don't grasp the basics, let alone excel.
I don’t know if I agree. Grad school was profoundly humbling to me because it really showed me that there are a LOT of people out there that are just much much better than me at math. There are different levels of innate talent.
>The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
Nonsense!
The brain you are born with materially dictates the ceiling of your talent. A person with average ability can with dedication and focus over many years become reasonably good, but a genius can do the same in 1 year and at a young age.
We have an education system which gives an A Grade if you pass the course, but 1 person may put on 5 hours a week and the other works day and night.
What makes you think that "genius" is nature and not nurture? I'd love to see the evidence for this; i'm deeply skeptical.
Edit: I don't mean to argue that there aren't genetics involved in determining aptitude on certain tasks, of course, but the assumption that genius is born and never made feels like a very shallow understanding of the capacity of man.
> I'd love to see the evidence for this; i'm deeply skeptical.
Cool, come and have a coffee with me :) I have older and younger siblings and was the one randomly blessed.
Whereas most recognised talents are associated with hard work and so there is then this visible link, I am a good example as I did the bare minimum throughout education (and beyond...).
The way my brain processes and selectively discards/stores the information it receives is very different to majority of the population. I have no control over it.
I take zero credit for any of my achievments - I regularly meet intelligent people near to retirement who have been to a tier 1 university, may have PHDs, worked 60 hours a week since they were born, been on course and what not and cannot reach the levels I can.
My nurturing was no different to siblings/peers (and was terrible!)
Note: I have my weaknesses too, but as a whole, I am exceptional. Not through effort!! Completely random - neither of my parents are intelligent and nothing up the ancestary tree as far as I know.
I am also exceptional in many ways, (some of them negative), and some of this is clearly inherited and likely genetic. I share too many innate strengths with my father and, to a lesser extent, my siblings to disagree with this. But I just don't know how you could preclude developmental factors like "when you started reading as a child", "what sort of puzzles and games you played as a child", "lack of trauma as a child", etc.
The boostrap skill is the ability to obsess over something. To focus and self-reward on anything is a heaven sent. Good thing we do not medicate that if we are unable to get that energy on the road, that base skill.
"What we should be teaching children is HOW to learn."
Absolutely correct. And that begins with getting their interest, thus their attention; and that's a whole subject in and of itself.
I've had some success converting people by telling them others had convinced them they were stupid. They usually have one or two things they are actually good at, like a domain they flee to. I simply point out how everything else is exactly like [say] playing the guitar. Eventually you will be good enough to sing at the same time. Clearly you already are a genius. I cant even remember the most basic cords or lyrics because I've never bothered with it.
I met the guitar guy a few years later outside his house. He always had just one guitar but now owned something like 20, something like a hundred books about music. Quite the composer. It looked and sounded highly sophisticated. The dumb guy didn't exist anymore.
But also, some people are stupid, right?
Intellect is like a gas, it will expand to fill its container. The container, in humans, is epigenetic and social — genetics only determines how hot or cold your gas is, ie how fast and how fluidly it expands, but you’re taught your limits — it’s best to see stupid as not how limited you are relative to other but what limits you have now and may abandon in the future.
That said, some people received a smaller starting container, and might need some help cracking it. That’s the work of those who think they’ve found a bigger one.
The inborn part is how quickly you get results (good or bad). Stupidity is the results.
If we spent 50% of time thinking productively - inborn thinking speed would matter. But in my estimate even 5% is generous.
So it matters far more what kind of feedback you have to filter out the wrong results, and how much time you spend thinking - than how quickly you can do it.
Also practice helps with speed.
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> The belief that some people have an innate skill that they are born with is deeply unhelpful.
In practice your views result in stinting access of non-existent (in your opinion) talented children to a faster education track. They don't exist therefore they don't need different treatment (finer points get lost when the idea disseminates). Quite a hot theme in American education two (or so) years ago.
Math is a stepping stone to critical thinking skills. And while one can probably learn those skills in any way, math forces you to learn those skills by learning the method of writing proofs. No other field forces you to push your critical thinking skills to the limit that math does.
> Whilst some people (mostly spectrum) do seem have an innate talent
I think the only thing in autism that I'd call an innate talent is detail-oriented thinking by default. It'd be the same type of "innate talent" as, say, synesthesia, or schizophrenia: a side effect of experiencing the world differently.
> a side effect of experiencing the world differently
A side effect for which there is a substantial, lifelong, and most importantly wide cost, even if it occasionally confers usually small, usually fleeting, and most importantly narrow advantage.
At such cost with such narrow advantage, why has it persisted so pervasively? I would counter that the advantage is wider and the cost narrower than your current value system is allowing you to accept.
Natural selection doesn't care about cost or advantage, only reproduction.
It is the sum of costs and advantages that lead to reproductive success. The trait is still here and still prevalent meaning people are still getting laid and starting families and presumably leading fulfilling lives.
I'm not sure what you are trying to say.
> I'm not sure what you are trying to say.
I'm saying, if it doesn't ruin lives to the point of preventing reproduction, then it stays in the gene pool.
Basically, I'm saying this:
> The trait is still here and still prevalent meaning people are still getting laid and starting families and presumably leading fulfilling lives.
As long as an organism isn't performing too badly, it stays in the gene pool. It can persist and even share its genes more broadly, if in diluted form, to the other more successful organisms. And then some of those mixed-genes organisms may occasionally express more strongly, but again not enough to affect reproductive success across the population.
Yes, there is a significant cost to being built differently regardless of perceived advantages (by one's self or others). For example, as an autistic, I have to cope with finding interaction with non-autistics quite difficult for me, even if detail-oriented thinking can make certain tasks seem easier to me.
> The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.
My first thought when the article got to the dialog between logic and intuition bit was that the same is true for school level physics.
> an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
I see you're familiar with Ryan North.
> I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
I would argue something different. The "skill" angle is just thinly veiled ladder-pulling.
Sure, math is hard work, and there's a degree of prerequisites that need to be met to have things click, but to the mindset embodied by the cliche "X is left as an exercise for the reader" is just people rejoicing on the idea they can needlessly make life hard for the reader for no reason at all.
Everyone is familiar with the "Ivory tower" cliche, but what is not immediately obvious is how the tower aspect originates as a self-promotion and self-defense mechanism to sell the idea their particular role is critical and everyone who wishes to know something is obligated to go through them to reach their goals. This mindset trickles down from the top towards lower levels. And that's what ultimately makes math hard.
Case in point: linear algebra. The bulk of the material on the topic has been around for many decades, and the bulk of the course material,l used to teach that stuff, from beginner to advanced levels, is extraordinarily cryptic and mostly indecipherable. But then machine learning field started to take off and suddenly we started to see content addressing even advanced topics like dimensionality reduction using all kinds of subspace decomposition methods as someting clear and trivial. What changed? Only the type of people covering the topic.
I saw a lot of this when I went to college for engineering, some professors had this ability (or willingness) to make hard things simple, and others did the opposite, it was the same with the books, I dreaded the "exercise for the reader" shit, I don't think I ever did any of those, so those were all proofs I never got.
I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
> I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
I don't agree. First of all, ladder-pulling in math is observed at all levels, not only cutting-edge stuff. Secondly, it's in applied mathematics where pure math takes a queue onto where to focus effort. See how physics drives research into pure math.
> A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's easy, then it means you already know this material, and you're wasting your time.
It’s a well-established effect in pedagogics that learning vs. difficulty has a non-monotonic relationship, where you don’t learn efficiently if the material is either too hard or too easy compared to your current level. There is an optimum learning point somewhere in-between where the material is “challenging” – but neither “trivial” nor “insurmountable” – to put it that way.
It's funny because I've had the opposite heuristic most of my line: the things I want to do most are whatever is hardest. This worked great for building my maths and physics skills and knowledge.
But when I started focusing on making money I've come to believe it's a bad heuristic for that purpose.
> If it's easy, then it means you already know this material, and you're wasting your time
I think that's also a trap. Even professional athletes spend a little bit of their time doing simple drills: shooting free throws, fielding fly balls, hitting easy groundstrokes.
Sometimes your daily work keeps up the "easy" skills, but if you haven't used a skill in a while, it's not a bad idea to do some easy reps before you try to combine it with other skills in difficult ways.
I cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains.
I've taught math to psychology students, and they just don't get it. I remember the frustration of the section's head when a student asked "what's a square root?" We all know how many of our fellow pupils struggled with maths. I'm not saying they all lacked the capability to learn it, but it can't be the case they all were capable but "it was the teacher's fault". Even then, how do you explain the difference between those who struggled and those who breezed through the material?
Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so.
So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.
Not the original poster, but I want to push back on one thing -- being capable of something and being one of the best in the world at something are hugely different. Forgive me if I'm putting words in your math -- you mentioned "placing the bar for mathematical skill pretty" low but also mentioned running a sub-10s 100m. If, correspondingly, your notion of mathematical success is being Terence Tao, then I envy your ambition.
I do broadly agree with your position that some people are going to excel where others fail. We know there trivially exist people with significant disabilities that will never excel in certain activities. What the variance is on "other people" (a crude distinction) I hesitate to say. And whatever the solution is, if there is even a solution, I'd at least like for the null hypothesis to be "this is possible, we just may need to change our approach or put more time in".
On a slightly more philosophical note, I firmly believe that it is important to believe some things that are not necessarily true -- let's call this "feel-good thinking". If someone is truly putting significant dedicated effort in and not getting results, that is a tragedy. I would, however, greatly prefer that scenario to the one in which people are regularly told, "well, you could just be stupid." That is a self-fulfilling prophecy.
> cannot agree. It's just "feel-good thinking."
Not really. There's nothing inherently special about people who dedicated enough time to learn a subject.
> "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained.
What a bad comparison. So far in human history there were less than 200 people who ran 100m in less than 10s.
I think you're just reflecting an inflated sense of self worth.
> Not really. There's nothing inherently special about people who dedicated enough time to learn a subject.
"You didn't work hard enough." People really blame you for that, not for lacking talent.
> So far in human history there were less than 200 people who ran 100m in less than 10s.
And many millions have tried. There may be 200 people who can run it under 10s, but there are thousands that can run it under 11s, and hundreds of thousands that can run it under 12s. Unless you've got clear evidence that those people can actually run 100m in less than 10s if they simply try harder, I think the experience of practically every athlete is that they hit a performance wall. And it isn't different for maths, languages, music, sculpting (did you ever try that?), etc. Where there are geniuses, there also absolute blockheads.
That's not to say that people won't perform better when they work harder, but the limits are just not the same for everyone. So 'capable of mathematical reasoning' either is some common denominator barely worth mentioning, or the statement simply isn't true. Clickbait, if you will.
I'm the author of what you've just described as clickbait.
Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.
The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.
Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).
When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.
As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.
In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.
There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.
I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.
Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.
A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.
In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.
This power law argument immediately reminds me of education studies literature that (contrary to the math teachers in this thread) emphasize that mathematical ability is learned cumulatively, that a student's success feeds on itself and advances their ability to grasp more difficult material.
As for my own half-baked opinion, I want to say that the Church-Turing Thesis and Chomsky's innate theory of cognition have something to add to the picture. Homo sapiens as a species essentially has the capacity to think analytically and mathematically; I want to argue this is a universal capacity loosely analogous to the theory of universal Turing machines. So what matters is people's early experiences where they learn how to both practice and, critically, to play, when they learn difficult ideas and skills.
Also, as an amateur pianist, most people don't know that modern piano teaching emphasizes not fixed limits of the student but that many students learn the wrong techniques even from well-meaning piano coaches. Just the other day I was watching a recent YouTube Julliard-level masterclass where the teacher was correcting a student on her finger playing technique, presumably this student had been taught the wrong technique since childhood. With music or sports a coach can visually see many such technique problems; with math teaching it of course harder.
This beats TFA. Interesting relation between cumulativeness and distribution ("Yule process"). But how does this explain variation is how quickly children pick up maths - would you argue it's due to prior exposure e.g. parental tutoring?
Any comments on the "10x programmer"?
There is math the abstract field and math the concrete example you're working on.
Current education is _extremely_ biased to concrete arithmetic and a bit of algebra. If you have a predisposition to either you will do extremely well. If you don't you won't.
Those have little to do with how math is done by mathematicians.
In short: education needs to catch up to what's happened since the 1920s in maths. Parents are conservative and don't want their kids to learn something they themselves don't understand, so we're stuck with what we have until enough generations pass and 20th century math is absorbed by osmosis into the curriculum.
> document the secret "oral tradition" of top mathematician
> A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.
Sounds like people mostly living in different bubbles? What do they know about the world?
They aren't hanging out with the kids who fail in school because maths and reading and writing is to hard, and then start selling drugs instead and get guns and start killing each other.
> [they] don't think it's because of their genes
Do you think someone would tell you, if he/she thought it was?
I mean, that can come off as arrogant? Wouldn't they rather tend to say "it was hard work, anyone can do it" and prioritize being liked by others
> Pareto-like distribution like what we see in math
Unclear to me what you have in mind. If there's a graph it'd be interesting to have a look? I wonder whats on the different axis, and how you arrived at the numbers and data points
> Sounds like people mostly living in different bubbles? What do they know about the world?
Well, they do know something about math — in particular that it requires a certain "attitude", something that no-one told them about in school and they felt they only discovered by chance.
Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence". They viewed intelligence as a by-product of their method, not the other way around. They even wrote books as an attempt to share this method (which is quite hard to achieve, for reasons I explain in my book.)
Why do you bring "kids who fail in school" and "start selling drugs" into this conversation? What does it have to do with whether math genius is driven by genetics or idiosyncratic cognitive development?
And why would a mathematician be disqualified from discussing the specifics of math just because they're not hanging out with lost kids? Are you better qualified? Did you sequence the DNA of those kids and identified the genes responsible for their learning difficulties?
>> [they] don't think it's because of their genes
> Do you think someone would tell you, if he/she thought it was?
Well, an example I know quite well is mine. I was certainly "gifted" in math — something like in the top 1% of my generation, but not much above and definitely nowhere near the IMO gold medallists whom I met early in my studies.
A number of random events happened to me, including the chance discovery of certain ways to mentally engage with mathematical objects. This propelled me onto an entirely different trajectory, and I ended up solving tough conjectures & publishing in Inventiones & Annals of Math (an entirely different planet from the top 1% I started from)
My relative position wrt my peer group went through a series of well-delineated spikes from 17yo (when I started as an undergrad) to 35yo (when I quit academia), associated with specific methodological & psychological breakthroughs. I'm pretty confident that my genes stayed the same during this entire period.
And as to why I was initially "gifted", I do have some very plausible non-genetic factors that might be the explanation.
I don't claim this proves anything. But I see no reason why my account should be disqualified on the grounds that I'm good at math.
Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates. Why should math be the exception?
> they do know something about math ... that it requires a certain "attitude"
Of course. That does not mean that intelligence doesn't play a (big) role.
> Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence"
That doesn't make sense. Back when they were active, intelligence, IQ tests and the heritability of intelligence hadn't been well studied. They didn't have enough information, like we do today: https://en.wikipedia.org/wiki/Heritability_of_IQ#Estimates"Various studies have estimated the heritability of IQ to be between 0.7 and 0.8 in adults and 0.45 in childhood in the United States."
And, evolution and genetics weren't these peolpe's domains. Does it make sense to assume they were authorities in genetics and inheritance, because were good at maths and physics?
Sometimes they were wrong about their own domains. Einstein did say "Genius is 1% talent and 99% hard work" (I can understand how it makes sense from his own perspective, although he didn't know enough about this animal species, to say that).
But he also said "God does not play dice" and was wrong about his own domain.
> Why do you bring "kids who fail in school" and "start selling drugs" into this conversation?
It was an example showing that the researchers live in bubbles.
That they're forming their believes about humans, based on small skewed samples of people. There's billions of people out there vastly different from themselves, whom they would have left out, if thinking about about others' abilities to learn.
In fact, now it seems to me that you too live in a bubble, I hope you don't mind.
> Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates.
1) Maths and 2) evolution, DNA, genetics, intelligence, learning and inheritability are not the same domains.
Anyway, best wishes with the book and I hope it'll be helpful to people who want to study mathematics.
Current estimates of the "heritability" of intelligence are far, far lower than "0.7 or 0.8"; they're probably below 0.1, and that's before digging into what "heritability" means, which is not generally what people think it does.
I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.
> Current estimates of the "heritability" of intelligence are far, far lower than "0.7 or 0.8"; they're probably below 0.1
Sources please, if you have time? I tried to find something supporting what you wrote, but wasn't able to (this far).
They've studied men in Sweden during 40 years. From the abstract:
"We found that high intelligence is familial, heritable, and caused by the same genetic and environmental factors responsible for the normal distribution of intelligence."
"... 360,000 sibling pairs and 9000 twin pairs from 3 million 18-year-old males with cognitive assessments administered as part of conscription to military service in Sweden between 1968 and 2010 ..."
Looking at Figure 3, in that pager, about identical twins and non-identical (two-egg) twins -- I think that settles it for me.
Seems they arrive at a bit above 0.4 as heritability. Yes that's less than 0.7 - 0.8 but I wouldn't say "far far lower", and more than 0.1. Also, they're 18 years old, not adults.
> I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.
Well, in his reply to me, he was sort of name dropping and appealing to (the wrong) authorities, didn't make a good impression on me. Plus writing about himself, but he's a single person. -- I would have preferred links to research on large numbers of people.
> what "heritability" means, which is not generally what people think it does.
That sounds interesting. Can I guess: You mean that people believe that heritability means how likely a trait is to get inherited from parent to child? When in fact it means: (https://en.wikipedia.org/wiki/Heritability)
"What is the proportion of the variation in a given trait within a population that is not explained by the environment or random chance?"
Dear cutemonster,
I know this reply may not suffice to convince you, but unfortunately I won't be able to argue forever.
Did you ever consider the possibility that you might be the one living in a bubble?
FYI, the concept of innate talent predated IQ tests and twin studies by many millenia. Two of the authors I'm citing in my book (Descartes and Grothendieck) believed that innate talent existed and they both declared they would have loved to be naturally gifted like these or these people they knew.
You're declaring that these incredibly smart people were wrong about their own domains, which is a pretty bold claim to make. What do you have in support of this claim? A fake Einstein quote?
It's a sad fact of life that most quotes attributed to Einstein are fabricated. Next time, please check "The Ultimate Quotable Einstein", compiled by Alice Calaprice.
This may come as a shock to you, but Google page 1 isn't always a reliable resource. Nor is Wikipedia, even though it's quite often correct. As it happens, there's a pretty large "Heritability of IQ" bubble on the internet. It's active and vocal, but it's also quite weak scientifically — the page you're citing is a typical symptom, and it absolutely doesn't reflect the current scientific knowledge.
The IQ heritability claims that you're citing are based on twin studies and they have taken in serious beating in the past decade, especially in light of GWAS.
It's true that a number of people have been fooled by twin studies, most notably Steven Pinker, in Chapter 19 of the Blank Slate (did you read it?)
You see, Pinker is a linguist and apparently he isn't mathematically equipped to fully comprehend the intrinsic limitations of Bouchard's approach. Did you read Bouchard's 1990 paper on twins reared apart? Do you find it convincing? Are you aware that even The Bell Curve's Charles Murray thinks that this approach, abundantly cited by Pinker, is structurally flawed? Are you aware of the fundamental instability of IQ estimates based on twins reared together? Aren't you concerned that even a mild violation of Equal Environment Assumption, plugged into Falconer's equation, would drastically reduce the estimates?
If you don't understand what I'm talking about, if you've never read the authors and the primary research I'm citing, then it's quite likely that you're the one living in a social media bubble.
Feel free to also subscribe to my own Substack, where I plan to cover these topics in the coming months:
https://davidbessis.substack.com
All the best,
David.
Some of the stuff on Gusev's substack is pretty startling, and I highly recommend it.
Thank you for taking the time to comment here!
> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.
Sure they don't. Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work. It runs completely contrary to the findings of modern biology and psychology, most of our intellectual potential at adulthood is genetic.
The floor and ceiling you will operate on in your life is decided the moment of chromosomal crossover.
> Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work
I suppose that makes sense from their own personal perspectives (but that doesn't make them right), in that they had to put in lots of time and work, but didn't do anything to become bright people.
> The floor and ceiling you will operate on in your life
Interesting that what you wrote got downvoted. Lots of flat-earthers here? (figuratively speaking)
So, for starters: you don't have any evidence, if I understood it properly. None whatsoever. That's really not the basis for arguing "become 1000x better." If only because your operationalization is missing. If you can't measure someone math's skills, how can you say they can become 1000x better? I think the whole article manages not to even speak about what "math" actually is supposed to be. Symbol manipulation according to axioms?
Your starting point is the way elite mathematicians think about themselves. But people don't understand themselves. They don't understand their own motivations, their own capabilities, their own logic. You know who are best at explaining what/how other people think? Average people. Hence the success of mediocrity in certain types of quizzes and politics.
I'm sure you're right about the mixture of logic and intuition. I've had the thought myself, mainly about designing systems, but there is some analogy: you've got to "see through" the way from the top to the bottom, how it connects, and then fill the layers in between. But that intuition is about a very, very specific domain. And it's not given that is a priori equally distributed. More likely than not, it's isn't.
Your whole argument then is based in naive psychology. E.g., this
> What can someone gain by improving their mathematical thinking?
> Joy, clarity and self-confidence.
> Children do this all the time. That’s why they learn so fast.
Are there no other reasons children learn so fast? It's not even given that joy and clarity makes children learn faster. What is known is that children do learn fast under pressure. Have you seen the skills of child soldiers? It's amazing, but it comes of course at great cost. But they did learn. Children pick up languages at a relatively high speed (note: learning a new language is still very well possible at later ages, certainly until middle age), but that's got nothing to do with joy, clarity and self-confidence. They also do it under the dreariest of circumstances.
So I'd say: your argument, or at least the quanta article, is at odds with common sense, and with psychological research, and doesn't provide concrete evidence.
You might have ideas for teaching maths better. But beware there's a long tradition of people who've tried to improve the maths curriculum, and basically all failed.
I'll give you one more point for thought (if you ever read this): intuition can also be a negative. I've practiced with my daughter for her unprepared math exam (she dropped it at one point, and then wanted to have it on her grade list anyway). One thing that I clearly remember, and it's not just her, is that she had very weird ideas about the meaning of e.g. x, even in simple equations. They were nearly magical. It was hard to get her to treat x like she would treat any other term. At one point, she failed to see that e.g. 1/3 = x^-1 is easy to solve, even when she had written down 1/x = x^-1 right next to it. Her intuition blocked her logic. My conclusion is that it's certainly easy to frak up someone's understanding of maths, unless you're really teaching, tutoring and monitoring 1-on-1. There's no solution for maths but good teachers, and a lot of fast feedback. Quite an old lesson.
You'll want to get your daughter to start using Math Academy for at least 30 minutes per day [0].
You don't need to be able to run 100m in less than 10 seconds. But almost everyone probably could run a marathon in three and a half hours. How many people do you think have actualized their physical potential, or how far is the average person removed from it?
If someone's smart enough to get into a psychology class they are smart enough to be thought basic undergrad math. It wasn't your teaching failure necessarily, but it was someone's teaching failure at some point.
Not everyone can play Rachmaninov like Lugansky or do math like Terence Tao, but there is absolutely no doubt that almost all people are magnitudes away from their latent potential in almost all domains. I'm fairly certain you could teach any average person how to play Rachmaninov decently. You could bring any person to a reasonably high mathematical level. You can get any person to lift a few hundred pounds.
Most people today read at a 7th grade level, can't do basic math, and are out of air after 3 flights of stairs. "Everyone can do everything" is maybe not literally right but directionally right given how utterly far removed we are from developing practically anyone's potential.
There's a difference between being able to memorize what a square root is and being able to do math - which to mathematicians means being able to organize a proof.
I've found that the people who most believe in math being a genetic ability are the ones who do not work in the symbolic world of modern math, but in the semantic world of whatever the field the math describes is.
The two are rather different.
Square roots are not some "mathematical trivia", they're amongst the most fundamental operations in mathematics.
In arithmetic. There is a lot more to math than arithmetic.
Square roots are fundamental to (real and complex) analysis and to algebra (in the study of polynomials), so the two major branches of modern mathematics.
They are fundamental to the arithmetic used in analysis and algebra.
One can get along quite well without them once you replace the arithmetic expressions with more general objects.
Just come on. The square root of 2 is the easiest example of an irrational number, this has been known since Ancient Greece. You can't compute distances in Euclidean spaces without the square root. "Solving equations by roots" is the bread and butter of algebra. Adjoining roots to a field is how you get Galois Theory. Several algorithms related to number theory have complexity O(sqrt(n)). And so on.
You chose an extremely poor example and now you're trying to die on that hill. Please don't die on that hill.
Strangely enough, you'd be hard pressed to find a mathematician who doesn't know what a square root is.
And I didn't mention genetics. Nature is complicated.
You'd also be hard pressed to find one who doesn't know how to flush a toilet.
Neither trivia has anything to do with being good at mathematics as done by mathematicians.
Are you an LLM? You brought up the point of mathematicians not knowing what a square root is yourself. Anyway, the square root is is so many levels below maths as done by mathematicians, it's laughable.
Pretty sure you followed the thread wrong. They only mentioned the square root in response to you?
It certainly is suspicious that their name starts, "llm"
This is mostly correct. Working memory plays a huge component in grokking more complicated mathematical components, and IQ itself is separated into performance and verbal IQ (which together constitute your IQ score) and its demonstrably robust. Some people find this easier than others and that is OK.
I dont disagree with the premise that mathematical thinking can benefit anybody, but its absurd notion that everything abstract is teachable and learnable to all is a fantasy of a distinctly left-wing variety, who would have you believe that everything is just social conditioning and human beings dont differ from one-another.
Imagine our world was extremely similar to how it is now in any way you'd care to imagine, except two things were different.
1. Everyone (young, old, poor, rich) thinks that maths is interesting and fun and beautiful and important. Not important "to get a good job" or "to go to a good college" or "to be an impressive person", but rather important because it's fun and interesting. And maybe it also helps you think clearly and get a good job and all these practical things, but they're secondary to the tremendous beauty and wondrousness of the domain.
2. Everyone believes that barring actual brain injuries people can learn mathematics to a pretty high level. Not Ramanujan level, not Terrence Tao, not even a research mathematician at one of the smaller universities, but a level of extreme comfort, let's say a minimum level of being able to confidently ace the typical types of exams 17 and 18 year olds face to finish secondary school in various countries.
Would you claim that in that world - people think maths is great, and that anyone can learn it - we'd see similar levels of ability and enjoyment of mathematics?
My claim is that we don't live in "Math-World", as described above, but "Anti-Math-World". And further, that anyone suggesting things have to be the way they are in Anti-Math-World is not only wrong, but also fundamentally lacking imagination and courage.
Kids are told week in week out that maths is stupid, that they are stupid, that their parents themselves are stupid, that the parents hated maths, that the teachers are stupid, and then when they end up doing poorly, people say: "ahhh, some kids just aren't bright!"
Parents who like things like learning and maths and reading and so on, have kids that tend to like those things. And parents that don't, usually don't. Saying that this somehow tells us something concrete and inalterable about the nature of the human brain is preposterous.
It's a card that's used by grown-ups who are terrified by the idea that our education systems are fundamentally broken.
"Kids are told week in week out that maths is stupid, that they are stupid. …."
Come on, how often are kids exposed to such stupid talk? I suspect very infrequently.
My grandmother, who wasn't stupid by any means but who knew only basic arithmetic, would never have uttered such nonsense.
And I'd stress, like many of her generation and background, her knowledge of mathematics was minimal, if she'd been ask what calculus was she'd likely have been perplexed and probably have guessed it to be some kind of growth on one's foot.
In math classrooms filled with 30 students on average for 12 years? With usually all 30 of those students having mathematically illiterate parents who can't help with their homework?
Every student hears it enough for it to be a widespread sentiment. It spreads like an infection.
I see it a lot on the internet, mostly I'm the young people places. People talk shit on math all the time, just like they do in person. Just slights and jabs that they've never needed the pythagorean theorem or to do an integral, and thus it was all wasted time and effort. You might just not hang around young people much, or at least their online congregations (i don't blame you). The idea that math education is dumb and useless is very much alive in the young adult to preteen tiktokified spaces online
> how often are kids exposed to such stupid talk? I suspect very infrequently
Pretty often, by other kids. And kids listen to kids, not well-meaning, mealy-mouthed adults.
I think most people can become fairly skilled in useful fields if educated properly, and the people who can't are a small group that can be cared for. I agree that even in a better education system, people aren't all going to be equally skilled in the same fields, just that most people can contribute something of value.
Anybody can do everything if we restrict everything to things everyone can do.
> Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so.
Bad example, it's much more likely to create a musical prodigy by providing early and appropriate guidance. Of course this is not easy as it assumes already ideal teaching methods and adequate motivation to the youngling, but even those with some learning difficulties have the potential to excel. The subtypes of intellect required to play complex music and absord advanced abstract math subjects are quite different, former requiring strong short-term memory (sightreading) the latter fluid intelligence -I think almost everyone is familiar with these terms by now and knows that one can score high/low on certain subtypes of an IQ test affecting the total score-.
BTW IDK if the Rachmaninoff choice was deliberate to imply that even the most capable who lack the hand size won't be able to perform his works well yeah, but there are like 1000s of others composers accessible that the audiences appreciate even more. Attempting to equate music with sports in such manner is heavily Americanized and therefore completely absurd. Tons of great pianists who didn't have the hand size to interpret his most majestic works and of others. Tons of others who could but never bothered. There have been winners of large competitions who barely played any of his works during all stages of audition or generally music requiring immense bodily advantage. Besides, it's almost 100% not a hand size issue when there are 5 year old kids playing La Campanella with remarkable fluidity.
And even in this case this isn't even the point. Most conservatory alumni today are 100x skilled than the pianists of previous generations... yet they all sound the exact same, their playing lacks character/variability, deepness, elegance to the point where the composers ideas end up distorted. And those can be very skilled but just have poor understanding of the art, which is what music is, not the fast trills/runs, clean arpeggios, very strict metronomic pulse.
> So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.
Well the vast majority of people in the Soviet Union were very math literate, regardless of what they ended up working as (although indeed most became engineers) and in quite advanced subjects. This is obviously a product of the extensive focus of primary and secondary education on the sciences back then.
So the point isn't to make everyone have PhD level math background and I heavily dislike the dork undertones/culture that everyone should love doing abstract math on their freetime or have to have some mathematical temperament' . But let's not go the other way and claim that those not coming close to achieving the knowledge those in the top % of the fields possess, they are chumps.
> the vast majority of people in the Soviet Union were very math literate
I doubt that.
> although indeed most became engineers
And that is demonstrably false.
Anyway, most of your argument boils down to: there's a bunch of people that can't do a certain task, there's a bunch that has mediocre skills, a few that are good, and a handful that's really good. There's no argument, just observation, not even related to effort, which is what this discussion is about.
And maths is no different from sports or music in that sense. Most people suck at math, and will always suck at it. The things described in the article are personal reflections of elite mathematicians. They have no bearing on development of knowledge and skills of us mortals, if only because those reflections have no truth value. It's all just "feel good" thoughts, no data, nothing provable, etc.
Yes, effort improves skill, but everyone has a limit, and the ones with the high limits we call talented.
> I doubt that.
Not even the most vicious polemicists of communist states deny that i.e. the superiority of the Soviet-era education system.
> Most people suck at math, and will always suck at it.
is strictly US-centric.
> Anyway, most of your argument boils down to: there's a bunch of people that can't do a certain task, there's a bunch that has mediocre skills, a few that are good, and a handful that's really good. There's no argument, just observation, not even related to effort, which is what this discussion is about.
No, you came up with that cause you have a very poor understanding of what constitutes a good musician which, like any other typical HNer believes, is another LeetCode type of thing where the more problems like a good little monkey you can solve, the smarter you become. And I already stated that there are people striving in the arts, even more than the ones with the supposedly 'better' skills according to absurd and clueless standards you set, i.e. no, those who can't access specific repertoire easily are not people that can't do a certain task or a bunch that has mediocre skills.
> And maths is no different from sports or music in that sense.
Let me guess, you also think that if painters can't draw photorealistically they're not deserving the artist title and lack talent? Or that the opera is all about who can sing the highest note?
Anyone who lumps every discipline within one other like that without realizing they require completely different things to be considered successful at and believe everything boils down to some supposedly 'objective' absurd video-game like character strategy always serves to remind that the US today has nothing to offer other than hi-tech bombing technology and horrific subculture.
Nowhere did I deny the existence of some people having the innate ability to absorb skills faster and better than the others and of course this is an interdisciplinary fact. But it definitely doesn't hold the same weight for every single discipline for one to strive.
> If it's easy, then it means you already know this material, and you're wasting your time.
One thing I'm anticipating from LLM-based tutoring is an adaptive test that locates someone's frontier of knowledge, and plots an efficient route toward any capability goal through the required intermediate skills.
Trying to find the places where math starts getting difficult by skimming through textbooks takes too long; especially for those of us who were last in school decades ago.
LLMs hallucinate and are completely the wrong tool for this. People need to learn about the history of AI.
Expert-systems using techniques from the 80s are doing this right now [0].
>and plots an efficient route toward any capability goal through the required intermediate skills.
LLMs currently can't find efficient paths longer than 5 hops when given a simple itinerary. Expecting them to do anything but a tactical explanation of issues they have seen in training is extremely naive with something as high dimensional as math.
I am trying to stress pushing through these barriers with my kid right now. The second her brain encounters something beyond its current sphere she just shuts down.
I have heard it is the ego protecting itself by rejecting something outright rather than admitting you can't do it. It still happens to me all the time. My favorite technique was one I heard from a college professor. He starts reading while filling a notepad with sloppy notes, once a page is filled he just throws it away. He claimed it was the fastest way to "condition his brain to the problem space". More than the exercise I like the idea that your brain cannot even function in that space until it has been conditioned.
As a kid I was also terrible at maths, then later became obsessed with it as an adult because I didn't understand it, just like OP. It was the (second) best thing I've ever done! The world becomes a lot more interesting.
I haven't been able to grasp maths as a kid nor as an adult.
I've tried night classes, tutors, activities. Nothing sticks.
Even the standard 12x tables I struggle at. I want to understand it but my brain just can't understand the non-practicality side of things.
My best friend was like that. Couldn't see the practicality until he got bit by a geology and water science bug. He went from calling me to get help figuring out percentages to doing chemistry equations in his head because he "got" the applicability.
My brother's mom tutors math. One of her insights with a former student was that they were in need of forming some number sense. She started by walking them both out to the street: "how many tires are there on this street of parked cars?" The student, already flummoxed, started panic guessing. So she started with counting.
For times tables, have you developed any intuition around it? For me, times tables are rectangles composed of unit squares and that helps with my intuition. Modern Common Core standards in the US focuses a lot on exposing different mental models to students. And after seeing the same 4x6 enough times your brain will automatically associate that with its solution. Instead of calculating, it is memorized.
My brain doesn't require car tires, geology, or other practical needs: it likes puzzles. I struggle with medical stuff and I can feel my brain switching to meh-mode and hardly anything sticks. I don't know how many times I have been told about the different kinds of sugar and how your body uses that energy and I would still have to look it up.
> For times tables, have you developed any intuition around it?
I've tried different approaches.
4x3 being 3x4
But somehow I end up miscounting and giving the answer for 4x4 or be off a digit every-time.
A good example was that I was at a brewery last night.
They didn't do pint glasses but glasses of: 1/4, 1/2 and 2/3rd's.
I thought 1/2 was more than 2/3rd's so I ordered a 1/2 rather than thinking I was getting more.
However I was unable to visual how 2/3 is more than 1/2 when 1/2 is half a pint, or half a glass.
My visual capabilities are great at others but just couldn't formulate the equation of 2/3 is more than 1/2.
Very simple stuff, but it just doesn't meld.
What _is_ a fraction to you? How do you visualize it? I didn't calculate to decimal to compare the size of those fractions (I couldn't tell you intuitively if 5/7 is more than 9/13, I would have to convert the denominator or calculate the decimal).
For me, a progress bar or a pizza is the default. And because I cook rice daily and we were talking volume, my "progress bar" was like a measuring cup. Mentally, I can stand two measuring cups next one another, and filling one of two parts is less than filling two of three parts.
Maybe there are ways to be more aware of or insert ways to force the use of fractions in physical space. More cooking, more building, etc. The more comfortable you get the more intuition you should build
Famously the A&W restaurant chain failed when introducing a 1/3 lb hamburger because many customers thought it was smaller than a 1/4 lb hamburger.
> However I was unable to visual how 2/3 is more than 1/2 when 1/2 is half a pint, or half a glass.
Maybe visualize splitting a pint with a friend. If you split the pint into 2 equal parts and each of you gets 1 of those 2 parts you each get the same amount.
Then visualize splitting it instead into 3 equal parts. You get 1 of those parts and your buddy gets 2. There's no fractions there so it should be easier to visualize that your buddy got twice as much as you did.
Comparing those two visualizations might make it easier to see that someone who gets 2/3 of a pint gets more than someone who gets 1/2 of a pint.
> I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
Absolutely. There's also a pernicious idea that only young people can master complex maths or music. This is a self-fulfilling prophecy - why bother try if you're going to fail due to being old? Or perhaps it's an elitist psy-op, giving the children of wealthy parents further advantage because of course no-one can catch up.
I grow increasingly convinced that the difference in “verbal” and “mathematical” intelligence is in many ways a matter of presentation.
While it’s indisputable that terse symbolic formalisms have great utility, one can capture all the same information verbally.
This is perhaps most evident in formal logic. It’s not hard to imagine a restricted formalized subset of natural language that is amenable to mechanical manipulation that is isomorphic to say modal logic.
And finally, for logic at least, there is something of a third way. Diagrammatic logical systems such as Existential Graphs capture the full power of propositional, predicate, and modal logic in a way that is neither verbal nor conventionally symbolic.
Amazingly, I believe that today, with the myriad of tools available, anyone can advance in sciences like mathematics at their own pace by combining black-box and white-box approaches. Computers, in this context, could serve as your personal “Batcomputer” [1]. That said, I would always recommend engaging in social sciences with others, not working alone.
Who knows? You might also contribute meaningfully to these fields as you embrace your own unique path.
I'd like to improve my mathematical knowledge (which has degraded to basically zero).
Can you talk about the path or resources you are using to improve here?
If you know how to multiply and divide, Math Academy will take you to college level math pretty quickly.
This perspective has discouraged so many people from exploring their potential
How have you been working on it? Asking for a friend ;)
I'd strongly recommend Math Academy for at least 30 minutes per day.
easy_things -> comfort_zone
I took an online electronics tech course 15 years ago and what got me was my math skills were atrocious. Not shocking since like learning a new language or music use it or lose it is the obvious answer to why I sucked. I spent half my time re-learning math just so I could complete the course.
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Most people don't try hard at the gym. Most people don't try hard at music. Most people aren't comedians.
This reads like the foreword to the incel handbook.
And that's obviously bad because incels are not real people with feelings.
It is bad because it suffers from misattribution error, ultimately not leading to any solution and often making the situation worse. A downward spiral of misinterpreted signal
> ...my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
As a child of the 80s and 90s, "getting girls as a programmer" made me snort. Nerds do seem to have it a bit better now; the money/financial security of software development helps. But as a whole, we developers are still less socially capable than our sales/hr/marketing counterparts
"A loser in societal view"... What does that objectively mean? That only reads like you had or have a low sense of self worth. It must've been your perceived definition of what society is because how could you have come to such a conclusion? I think I'd actually subconsciously tend more to viewing someone as "a loser" if they made such a statement because it comes off as self victimization (without an apparent explanation to an outside observer).
And what's the shtick about girls? What are and were you looking for, love and a genuine relationship or attention to compensate for something? Personally I think your values and personality are what matter most and personality is usually what people fall in love with. Though charisma can help a lot to get the ball rolling. Most of what it takes is to treat people normally and nicely and you will have as much of a chance to find love as most people.
Though respect from peers and attention from women ideally shouldn't be your driving force. I think curiosity and passion are much better driving forces that don't involve such external factors and possibilities for insecurities.
Your post reads as if it expresses a frustration and a sense of entitlement. You may not be intrinsically entitled to the things you think you are. Think about that for a bit and try to be rational.
I think he lives in or are from India, society and family expectations are different there, not his "fault".
@faangguyindia, I hope everything will be ok one day :-) Or maybe it already is? You wrote "was".
I might be wrong but I think if you (@faangguy) manage to create a life that makes you feel happy, the women will notice that you're happy, and that goes a long way. But you'll also need to be somewhere where there are some women around? (obviously) If you're in a FAANG in SF maybe for the moment that's not so easy
You will stay a loser as long as you care about what some fictional mystical society thinks of you.
Do the stuff you're good at, provide for your family, earn the respect of your peers and forget about the rest.
I assume OP is an Indian. And from what I've observed, Indian society is highly paternalistic and status-seeking in nature. Parents demand marriage and grand-children as soon as their offspring hit a certain age and success.
This just demonstrates that you dont understand how sexual selection works. For men, yes, aesthetic appearance is a considerable (main?) component in initial attraction, which is further tempered by compatible personality after that initial connection. For women, social value is the principle signifier, which is then tempered by facial symmetry, not demonstrating socially unacceptable habits and having sone degree of physical security, but the latter is the most variable across cultures.
Handsome men are largely irrelevant.
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I've been reading the author's book, Mathematica, and it's awesome. The title of this post doesn't do it justice.
He shows that math skill is almost more like a sports talent than it is knowledge talent. He claims this based on the way people have to learn how to manipulate different math objects in their heads, whether treating them as rotated shapes, slot machines, or origami. It's like an imagination sport.
Also, he inspired me to relearn a lot of fundamental math on MathAcademy.com which has been super fun and stressful. I feel like I have the tetris effect but with polynomials now.
Sounds really cool.
It reminds me of programming, that moment when new code starts to really sync up and code goes from being a bunch of text to more intuitive structures. When really in the zone it feels like each function has its own shape and vibe. Like a clean little box function or a big ugly angry urchin function or a useless little circle that doesn't do anything and you make a note to get rid of. I can kinda see the whole graph connected by the data that flows through them.
There's a lot of interesting discrete math that can supercharge programming at different levels of scale. What's pretty cool is that it reveals a layer of understanding when I watch my toddlers learn math from counting.
One of the interesting things is being able to exactly describe how something is an anti-pattern, because you have a precise language for describing constraints.
I would love to learn about some of these anti-pattern proofs if you have any examples or references you can share!
> rotated shapes, slot machines, or origami
Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out!
Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not.
I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day informing me of its arrival. Thank you for confirming that it's worth it!
I've had a MathAcademy subscription for some time and it's quite good. I'd say it's best at generating problems and using spaced repetition to reinforce learning, but I think it falls short in explaining why something is useful or applicable. I don't know, most math education seems to be "here's an equation and this is how you solve it" and MathAcademy is undoubtedly the best at that, but I wish there were resources that were more like "here's how we discovered this, what we used to do before, why it's useful, and here's some scenarios where you'd use it."
I have so wanted such resources for years. I have found some and should make a list.
The first time the difference between understanding some math, and understanding what the math meant, was after high school Trig. The moment I started manually programming graphics from scratch, the circle as a series of dots, trigonometry transformed in my mind. I can't even say what the difference was - the math was exactly the same - but some larger area of my brain suddenly connected with all the concepts I had already learned.
While ordering the "Mathematica: A Secret World of Intuition and Curiosity" I came across these books, which looked very promising in the "learning formal math by expanding intuition" theme, so I bought them too:
Field Theory For The Non-Physicist, by Ville Hirvonen [0]
Lagrangian Mechanics For The Non-Physicist, by Ville Hirvonen [1]
The Gravity of Math: How Geometry Rules the Universe, by Steve Nadis, Shing-Tung Yau [2]
Vector: A Surprising Story of Space, Time, and Mathematical Transformation, by Robyn Arianrhod [3]
If you're interested in how vector calculus developed, and who was instrumental, all the way from Newton/Leibnitz to Dirac or so, by way of Hamilton, Maxwell, Einstein and others, then Robyn Arianrhod's 'Vector' is brilliant.
But be warned, it gets progressively harder, along with the concepts, so unless you're conversant with tensors, at some point you will have to put on your thinking cap.
I really want to try MathAcademy.com. How quickly do you think someone doing light study could move from a Calc 1 -> advanced stuff using that site? In my case I could put in at least 30 minutes to an hour a day.
I can't speak to the advanced stuff but here's my stats on Fundamentals I:
Total time on site (gathered from a web extension): 40h 30m
Total days since start: 32
Total XP earned: 1881
Since "1 XP is roughly equivalent to 1 minute of focused work", I "should have" only spent 31 hours. I did the placement test and started at ~30%, and now I'm at 76%. I'd say 75% is stuff I learned in HS but never had a great handle on, 25% I never knew before.
Overall, I'm quite happy with the course. I'm learning a lot every day and feel like I have stronger fundamentals than I did when I was in school. The spaced review is good but I do worry I'll lose it again, so I'm thinking of ways I can integrate this sort of math into my development projects. It's no Duolingo, you really do have to put in effort and aim for a certain number of Xp per day (I try for 60 XP rather than time).
>>It's like an imagination sport.
Honestly speaking I think this is a wrong way to teach people to think about Math. Math is just one of those things which feels hard because people struggle to hold long trials of manipulations in their head. Especially if they are manipulations to something very large, evolved slowly over hundreds of steps. People are not coming short, its just how the human mind works.
IMO, the right way to teach Math is to teach people that its just base axioms, manipulation rules. And after that its how you evolve the base axiom using rules. People need to be taught how to make one valid change at a time. Of course this means tons of paper work and patience. But that is what Math actually is. Its taking truth and rules, to make new ones.
Im teaching this to my kid, and she often goes like this is it?? its really just laborious paper work??
Im using this method and LLM help at times these days to learn Algorithms and Data Structures. When you start working things from base conditions and build from there. A lot of Algos that otherwise seem like the domain of novel inventions just seem to follow from the manual steps you just worked, and then translated into a program.
When you remove all the fluff, Patience and Paper work is all there is to Math.
The author (and Grothendieck, liberally quoted in the book) disagree with you.
I think the reason you disagree is that it sounds like you’re teaching your child to be good at math class (a perfectly valid and good thing to do). Being good at math class requires being good at rational/logical thinking and computation. It also has only glancing similarities to anything that the author would recognise as mathematics.
>>It also has only glancing similarities to anything that the author would recognise as mathematics.
Nah, these are the same things. Trying to make Math look like is for people who are 'geniuses' i.e people with massive capabilities of holding large thought trials and changelogs in their head is how you arrive at making people look stupid doing math and eventually make them hate the subject.
Math is paper work. Approach it that way and all of a sudden doing a 100 page proof is within everyones reach. If you ask people to hold a 100 page proof in their head, and more importantly make changes to that in random places and fix the entire changelog trial, probably 2 - 3 people on earth will be able to do it, and you will just convince everyone else its not for them.
I have a hunch that big mathematical breakthroughs in history have happened around and after renaissance era due to paper getting cheap and ubiquitous. There is only that much you can do in your brain alone.
Through all of this, don't get me wrong, the rigorous application of rationality that it takes to step-by-step construct a proof is very important and an incredibly useful skill. Also, I agree that basically no-one can hold more than 3 things in their head at once.
The book also agrees vehemently that math is NOT restricted to "geniuses" and even argues that those don't really exist in the way that culture thinks they do.
However! His assertion is that the (to him) tedious, laborious, error-prone, paperwork is not the fundamental output of "doing math". For him, symbolic written mathematics is akin to sheet music. It would in principle be possible to teach students to read and write sheet music and even do manipulations like transposing it to different keys, without ever letting them listen to music. It would be hard and boring. Some students would find the memorization and application of rules satisfying but most would struggle.
In such a classroom, there might be one student who by chance figures out for herself that you can kind of "hear" these symbols in your brain and suddenly all the arbitrary rules seem obvious and natural and she doesn't even have to go through the tedious steps at all to answer questions. "Of course this is in a minor key." she might say. "No, I didn't rigorously check each chord, it's just... obvious".
Such a student would be labeled a "prodigy" or "genius", and would struggle to explain to others that no, what she's doing isn't harder than the her classmates laboriously doing the rote work, it's actually much easier.
Of course... this is not to denigrate sheet music. It's a wonderful invention that makes it possible to transmit music out of one person's brain to the brains of an orchestra.
Just like written mathematics.
The author's contention is that, like the contrived example above, no-one ever talks about "the music" of mathematics, just the sheet music, and therefore things are much harder than they need to be.
One of the simple mathematical examples he uses is to ask: Can you imagine a circle in your head (unironically an amazing thing to be able to do!). Then to ask a question like: Can a straight line intersect a circle in 3 places?
You likely have an immediate, intuitive response to this highly non-trivial mathematical problem. That's the music. Now, try to write that down in mathematical language for someone who can't see circles. Oof, it's going to be a slog.
>>Through all of this, don't get me wrong, the rigorous application of rationality
Much of this is just talking to oneself and testing it to see if our idea holds under test conditions.
I was once watching a video on how chess grandmasters think and work. Most of it is-
1. Do we know a pattern of moves, even if done, in series that is known to score some win/check. If so, lets do it.
2. Are any pieces under attack, If gone can effect point 1. eventually? If yes, lets protect them.
3. What can all possible moves of our pieces prevent opponent from having successfully execute their own point 1. And can we force opponent into point 2? Lets do it.
Basically every our move and its possible outcomes(Known through prior study of patterns of previous games seen), every move of our opponent.
A strong internal monologue and testing imaginary moves.
Math is just this except over paper.
Would you say the book ventures more into the practical side of learning this stuff or is it closer to the tone of this article? I found this article hard to gain anything from. A lot of just motivational cliche statements and nothing really groundbreaking or mind altering. If the book is better at that, I'd love to read it. If it's stories and a lot of fluff, I'd rather skip. So I'm curious what you are getting from it and how practical and applicable it feels to you?
Agree. The article turned me off as well. No specific example, felt like an ad.
Yeah, I quit reading it because it didn't talk about the book, it felt like a meta article.
I actually heard about this book very recently, and it's coming up soon on my (never-ending) reading list.
Happy to hear you're enjoying it, makes me even more confident that I should read it :)
+1 for Math Academy. I’ve been using it daily for over a year now (started October 2023). I summarized my experiences after 100 days here in case it helps anyone: https://gmays.com/math
Thanks for sharing this. I was debating buying the book.
This sounds like a book I needed for one of my early comp sci classes in college. It was called something like Think Like a Programmer: An Introduction to Creative Problem Solving. Maybe it was this, maybe it was something like this.
I mean to say, just applied scientific thinking is important. Even if you never get into pure math or computer programming, applying concepts like "variables", "functions" or "proofs" is universally useful.
To my mind, the premature formalization of the math is the principal contributor to gas lighting and alienation of people from maths. The reduction of concepts to symbols and manipulation thereof, is an afterthought. It's misguided for them to be introduced to people right at the outset.
People need to speak in plain English [0]. To some mathematicians' assertion that English is not precise enough, I say, take a hike. One need to walk before they can run.
Motivating examples need to precede mathematical methods; formulae and proofs ought to be reserved for the appendix, not page 1.
[0] I mean natural language
I'm an adult who's been programming computers professionally for 20 years, and went to school for it, and I've lost most mathematical skills past what I'd learned by 6th grade or so, from lack of use.
People who aren't even working in a field that's STEM-adjacent have even less use for stuff past simple algebra and geometry (the latter mainly useful just for crafting hobbies and home-improvement projects) and a handful of finance-related concepts and formulas.
I expect to go to my grave never having found a reason to integrate something, at this rate.
The result is that any time I try to get back into math (because I feel like I should, I guess?) it's not really motivated by an actual need. The only things that don't bore me to tears for sheer lack of application ends up being recreational math problems, and even that... I mean, I'd rather just read a book or do almost anything else.
For nostalgia I keep a copy of Mathematica on my laptop, so I can pretend to be an an actual computer scientist and not just an overpaid button-presser.
Ten years ago I used it to fit a nonlinear model to some performance metrics to predict the behaviour of a disk array past the maximum load level I was allowed to use for testing.
That’s the last time I did something that Excel couldn’t handle.
In university I learned how matrix exponentiation can be used to calculate the maximum throughput of a mesh network. In real life everyone just buys 10x the bandwidth they need.
It’s depressing.
Do you play any games that require mathematical reasoning? You might realize you are using integration without calling it integration i.e. calculating expected values.
Maybe. I also do a lot of complex math when I throw a ball.
Weirdly, accurately throwing a ball never got me a good grade on a math test.
I feel the opposite.
Before high school, math is just a grind of memorization and unmotivated manipulation of numbers.
Many students (ok, me, but I expect the same was true of others), get turned on to math for the first time when they encounter proofs in high school geometry and also actual applications in high school physics.
It's a revelation to students that math can be a way to go from one truth to another and thus find new truths. It's a way of thinking and that can be very exciting.
Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills. Once they develop a gap in their knowledge it becomes difficult to progress unless those gaps are addressed. For lots of students, it all ends with fractions. You'd be surprised how many adults don't really understand fractions. For others it ends with algebra, and for the college bound it ends with calculus.
Only math majors and a minority of engineering/science/CS folks get past the "standard sequence" of math courses in college and gain an appreciation for the really interesting stuff that comes AFTER all that.
> Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills
It's tragic and all, but were there attempts to teach non-preselected children/adolescents interesting stuff first? What are the results?
I don't know.
There is a certain amount of drudgery that unavoidable in learning mathematics, IMHO. Moreover, looking at the math textbooks of my niece and the students that I've tutored over the years, there's no effort spared in trying to make math "fun". If anything there's perhaps too much distraction caused by attempting to make a connection with the students and a failure to clearly and logically explain the math.
I don't have the answers, but believe that it's incredibly helpful to track students, intervene when they show a lack of understanding and not let them slip through the cracks. I do agree with the OP that everyone is capable of math but it's a long journey and it takes a lot of practice.
Mathematics is the conversion of a large number of object languages in to a single meta language that lets us talk about all of them.
The sin of modern mathematics is that it's meta language is so ill define that you need towers of software to manipulate it without contradiction. Rewriting all of it into s-expressions with a term rewriting system for proofs under a sequent calculus is an excellent first step to making it accessible.
We do not need to go back to the 16th century when men were men, an numbers positive. If people want to look at what math talks about instead of how it talks about let them pick up stamp collecting.
Weirdly enough, mathematicians have been manipulating expressions and writing proofs for centuries without (routinely) stumbling into contradictions all without the need of formal proof calculi or s-expressions.
I have nothing but admiration for projects like Lean and Coq and working in them can be a lot of fun (coupled with a lot of frustration when "obvious" things sometimes take an inordinate amount of time to prove), but Wiles' proof of FLT (the corrected version) was published in 1995. We're almost 30 years later and people are just now working on a formalisation which could take many years (https://leanprover-community.github.io/blog/posts/FLT-announ...). Mathematicians can't afford to be waiting for proof systems to catch up, at least not right now.
It wasn't until the 1930s that people realised second order logic with arithmetic will always lead to contradictions without guardrails. Before then all mathematics was done in the object language of whatever the field in question was and only translated to the meta language for succinctness after the fact.
The magic of modern maths is that we can now work only with the meta language and get results free from contradiction. For this we absolutely need a modern notation to do the new type of maths since we are no longer grounded by the reality of the object language.
None of what you wrote is true.
Feel free to explain why.
Your very first sentence is simply wrong, I don't know what more there is to say other than that you clearly don't understand what Gödel's results are about (hint: they're called "incompleteness theorems", not "contradiction theorems"). Maybe read a textbook? I could recommend several good ones.
The other sentences basically fall in the category of "not even wrong", i.e. basically nonsensical.
Godel starts by the assumption that the system is free from contradictions which makes his papers largely irrelevant here.
Again, feel free to explain how you can have a second order logical system without axiomatic limitations that doesn't contain contradictions.
Hell, go by the more popular version of first order logic + set theory if you're more comfortable with that.
Nobody has found a contradiction in (first or second order) ZFC in over a hundred years. You're out of your depth.
>Nobody has found a contradiction in (first or second order) ZFC in over a hundred years.
Yes, which is exactly why we have guardrails like ZFC in place. We got all sorts of exciting results without them. We of course have to keep adding axioms to ZFC - or replacing them all together - because there is a lot of math out there which is outside it's purview.
>You're out of your depth.
You're being incredibly abrasive for no reason.
What does premature formalization mean and when does it occur? Do you mean formal in the sense of using formulaic, rote manipulations or formal in the sense of proofs and rigor?
As someone who went on to study mathematics at the graduate level, I was bored out of my mind in high school math and most subjects. What's missing from a lot of primary and secondary school education is context, and that's what makes it boring. Math wasn't easy because I was particularly good at it. It was easy because it was just blindly following formulas and basic logic.
Something is very wrong with our educational system because almost all math at the primary and secondary levels is basic logic. So when people with this maximum level of mathematics education say they're "bad at math" or "don't get math", it means that they lack extremely basic logic and reasoning skills.
In my mind, we need to teach mathematics in a contextual way (note that I don't necessarily mean applications) in a way to enrich the reasoning and exploring of it. This should include applications, yes, but not be fully concentrated on applications. Sometimes one needs to just learn and think without being tied to some arbitrary standard of it being applied.
This sentiment comes up all the time here. Mathematics uses formalism because it's easier.
It's easier to read "a(b+c+d) = ab+ac+ad" than
> If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.
It's well known that good notation is exactly the one that elevates good intuition. For example, the Legendre symbol has the property that (a/p)*(b/p)=(a*b/p), an important visual cue that you wouldn't get from writing down (in way too many words) what the Legendre symbol actually means.
Also, most actually good mathematical textbooks aren't just dumps of formulae and proofs and they do contain motivation, examples, pictures, etc. You're attacking a strawman. But you can't just relegate the formalism and proofs to the appendices, that's crazy.
The ironic thing is, I swear that this must have been how math (at least more advanced math) was taught a century ago. Or at least, nowadays I've taken to relying on textbooks from the early-to-mid 20th century to learn new math. Maybe it's survivor bias and the only textbooks from back then that anyone remembers are the good ones.
I hate new textbooks because they're so built around instant gratification. They just come out and tell you how to solve the problem without building the solution up in any way. Maybe afterward they take a swipe at telling you how it works, but that's just completely the wrong way around IMO. It robs me of the chance to mull things over, try to anticipate how this will all come together in the end, and generally have my own "aha" moments along the way.
But, getting back to what you say, I think that it also engenders this tendency to reduce math to symbol manipulation. Because if they give you the formula in the first paragraph, then all subsequent explanation is going to end up being anchored to that formula. And IMO that's just completely wrong. Mathematical notation is at its best when it's a formalizing tool and mnemonic device for cementing concepts you already mostly understand. It's at its worst when it's being used as the primary communication channel.
(It's also an essential tool for actually performing any kind of symbolic reasoning such as algebraic manipulation, of course, but I'm mainly thinking about pedagogical uses here.)
I believe you're right, even though I don't have any evidence except for my own experience.
This issue becomes very clear when you see how many ways there are to express a simple concept like linear regression. I've had the chance to see that for myself in university when I pursued a bunch of classes from different domains.
The fact that introductory statistics (y = a + bx), econometrics (Y = beta_0 + beta_1 * X) and machine learning (theta = epsilon * x, incl. matrix notation) talk about the same formula with quite different notation can definitely be confusing. All of them have their historical or logical reasons for formulating it that way, but I believe it's an unnecessary source of friction.
If we go back to basic maths, I believe it's the same issue. Early in my elementary school, the pedagogical approach was this:
0. only work with numbers until some level
1. introduce the first few letters of the alphabet as variables (a, b, c) - despite no one ever explaining why "variable" and "constant" are nouns all of a sudden
2. abruptly switch to the last letters of the alphabet (x, y, z), two of which don't exist in my native language
3. reintroduce (a,b,c) as sometimes free variables, and sometimes very specific things (e.g., discriminant of a quadratic equation)
4. and so on for greek letters, etc.
It's not something that's too difficult to grasp after some time, but I think it's a waste to introduce this friction to kids when they're also dealing with completely unrelated courses, social problems, biological differences, etc. If you're confused by "why" variables are useful, why does the notation change all the time, and why it sometimes doesn't - and who gets to decide - this never gets resolved.
Not to mention how arbitrarily things are presented, no explanation of how things came to be or why we learn them, and every other problem that schools haven't tackled since my grandparents were kids.
I have never disagreed more with a comment. You can fully decide that you're not interested in mathematics, after having taken all of the math classes that you could possibly be offered before university, without ever encountering a proof, or even a mathematical definition.
I do agree that explaining why mathematical concepts are useful is something that's often lacking in mathematical curricula, but not that the problem is premature abstraction. Like another commenter said, the opposite is true. The way children are first introduced to (and therefore soured to) something that adults call "math" is by performing pointless computation that has as much to do with actual math as lensmaking has to do with astronomy.
A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it. So is there some scientific study backing up the claim that all these people could easily learn it or are we just making it up because its a nice egalitarian thesis for a math popularization book?
The same goes for language skills, by the way. In the US, 21% of adults are illiterate, and 54% of adults have literacy below sixth-grade level.[1] This is higher than in other developed countries. For example, in Germany, 10% are illiterate, and 32% have literacy below fifth-grade level.[2]
General intelligence also seems to have been trending downward since the 1970s (the reverse Flynn Effect)[3]. It has been measured in the US and Europe.
So, while it is true that the education system and other factors have an influence, the idea that "everybody is capable of X" is wrong and harmful. It's the equivalent of "nobody needs a wheelchair" or "everybody can see perfectly." People are different. A lot of nerds only hang out with other nerds, which screws up their perception of society.
What a weird comment. Are you trying to argue by analogy that a decent fraction of the population are not capable of literacy? It seems self-evident that low literacy rates have nothing to do with innate ability. I see no evidence to suggest that math is any different.
The idea that 21% of US adults are "illiterate" doesn't mean they can't read, by the way:
> English literacy test results from 2014 suggest that 21% of U.S. adults ages 16 to 65 score at or below PIAAC literacy level 1, meaning they have difficulty "[completing] tasks that require comparing and contrasting information, paraphrasing, or making low-level inferences." Included in that 21% is the 4.2% of respondents who were unable to be assessed due to language barriers, cognitive disability, or physical disability.
A lot of people aren't aware that the academic definition of "literacy" was changed around 1950 to no longer refer to "alphabetical literacy", which is still what most people think literacy means (in lay usage).
And this particular survey ignored people who didn't speak English but were entirely literate in their native language. Which obviously has nothing to do with the educational system or people's intelligence.
That certain countries both now and in the past have had significantly higher mathematical ability among the general population and much higher proportions going on to further study suggests that ability isn’t innate but that people don’t choose it. In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun.
> have had significantly higher mathematical ability among the general population
This is not really true is it? There were not that many standardized testing globally to measure such claims. Many people were in poverty and did not get tested, did not go to schools, or finished schools very early (5, 9 years). Many more kids go to school these days.
> In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun
It is just wrong. It was the same as now, except it was critical for people to show results because otherwise you had grim perspectives in the life, there was little "fun". People wanted to get into universities to get better jobs and to get better apartments, to be able to leave their parents. You could not just buy places, but a good position in some public body would guarantee you a nice place. FYI engineers could earn more in comparison to other jobs, not to mention if you could get into defense industry.
Why would ability not be innate just because some people with the ability don't use it?
Or more specifically, two of my friends teach special needs children in the 50 to 70 IQ band. Who are we going to blame for them not becoming mathematicians? The teachers, for not unlocking their hidden potential? The kids, for not trying hard enough? Claiming that the only thing holding them back is choice seems as cruel as it is wrong, to me.
Yeah, we're probably not cultivating anywhere near the potential that we could, but I personally guarantee you I am not Ramanujan or Terence Tao.
Well, I guess what I mean is that most people have some level of general intelligence that when applied correctly can generally give good results in most subjects. In general the people who do well in school do well in everything, even if they have a preference, and as such could do well in most of those subjects if they went on to further study. The evidence tends to be that in lower income countries people push towards subjects more likely to bring financial stability than those they prefer which bears this out somewhat.
There are some extreme cases of course but I’m not sure the general public needs to worry too much about those, most of us aren’t an Einstein nor do we have learning disabilities.
The extreme case does not imply a binary scenario ie that there are those that can those that cannot.
Rather, learning ability is a continuum. people have varying degrees of ability to learn mathematics. Couple this with environmental factors and society generates a huge variability in mathematical ability that crosses income levels and other demographics.
This view is rejected by many because it is against the push for equality.
You get a huge variability if you consider the absolute extreme outliers. Most people should be able to reach a level of competence where they can understand mathematical concepts abstractly and apply that same reasoning to other areas, and not feel a visceral rejection at the mere idea. I think that's a modest enough standard that a good portion of any given population should be able to reach, and yet education is failing at achieving that.
Your statement is not backed up by data and simply wishing it should happen isn’t a strong argument.
You probably have a narrow definition of “most people” (probably some motivated high school or undergraduate student) and too loose with what it means to “understand mathematical concepts abstractly”.
Take an analogy: imagine professional musicians saying that most people should be able to take a piece of music and understand its harmonic structure, then apply it to a new setting to generate a new piece. Most people will reject this idea as absurd.
Where's the data backing up what you said?
>You probably have a narrow definition of “most people” (probably some motivated high school or undergraduate student)
I was thinking "3-4 out of 5 people you pick on the street at random".
>too loose with what it means to “understand mathematical concepts abstractly”.
Enough that they could recognize whether a mathematical concept is applied correctly (e.g. if I have a 2% monthly interest, should I multiply it by 12 to get the annual interest? Why, or why not?) and conversely to correctly apply concepts they already understand to new situations, as well as to leverage those concepts to potentially learn new ones that depend on them.
>imagine professional musicians saying that most people should be able to take a piece of music and understand its harmonic structure, then apply it to a new setting to generate a new piece. Most people will reject this idea as absurd.
Okay, but we're arguing about what is the case, not about which idea has more popular support. Since most people don't understand thing 1 about composition, why should their opinion matter? A skilled composer's opinion on the matter should have more bearing than a million laymen's.
> Where's the data backing up what you said?
What he is saying is the default hypothesis based on our understanding of biology and psychology. If you have variability in genes you'll get variability in characteristics that are connected to them - height, bone structure, mental capacity, etc.
It is on you to prove that there is an arbitrary cut-off when it comes to this variance from which point it doesn't matter in regards to e.g. cognitive and mathematical ability.
> Enough that they could recognize whether a mathematical concept is applied correctly (e.g. if I have a 2% monthly interest, should I multiply it by 12 to get the annual interest? Why, or why not?) and conversely to correctly apply concepts they already understand to new situations, as well as to leverage those concepts to potentially learn new ones that depend on them.
No it doesn't if they do not have the abilities to comprehend it. I think you're living in a bubble of at least average-smart people and don't get that probably millions if not billions of people around the globe (based on average IQs) won't really get that concept.
>If you have variability in genes you'll get variability in characteristics that are connected to them - height, bone structure, mental capacity, etc.
Then you're agreeing with me. The thing all of those have in common is that they follow normal distributions. The shortest recorded adult and the tallest recorded adult are quite far apart, yet the vast majority of adults are between 150-200 cm tall. That's precisely what I was saying; the outliers of mathematical skill are very very far apart, but most people are roughly equally capable.
>I think you're living in a bubble of at least average-smart people and don't get that probably millions if not billions of people around the globe (based on average IQs) won't really get that concept.
What I'm saying it that it even someone with below-average IQ could do it, if taught properly. Mathematics is less about being smart and more about being rigorous.
>A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it
Any healthy/able individual could learn to deadlift twice their bodyweight with sufficient training, but the vast majority of people never reach this basic fitness milestone, because they don't put any time into achieving it. There's a very large gap between what people are capable of theoretically and what they achieve in practice.
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> So is there some scientific study backing up the claim that all these people could easily learn it [emphasis added]
Who said it would be easy?
It is easy to learn for some.
I do not think that Bessis's argument is entirely "made up"
We are not really taught (thought) to think, we are taught to memorize. Until one actually tries to think, you really can't tell if they're able to do it.
I'm not a math teacher, but I do enjoy math, and I have helped several family members and friends with math courses.
I've long thought that almost all have the capability to learn roughly high school level math, though it will take more effort for some than for others. And a key factor to keep up a sustained effort is motivation. A lot of people who end up hating math or think they're terrible at it just haven't had the right motivation. Once they do, and they feel things start to make sense and they're able to solve problems, things get a lot easier.
Personally I also feel that learning math, especially a bit higher-level stuff where you go into derivations and low-level proofs, has helped me a lot in many non-math areas. It changed the way I thought about other stuff, to the better.
Though, helping my family members and friends taught me that different people might need quite different approaches to start to understand new material. Some have an easier time approaching things from a geometrical or graph perspective, others really thrive on digging into the formulas early on etc. One size does not fit all.
One size doesn't fit all is what I believe Common Core math is attempting. The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all
> The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all
I cannot overstate enough how consistently and extremely this has turned my kids off from math. 3-for-3 on absolutely hating this. Having to solve the same thing five different ways just pisses them off, and, like... yeah, of course it does. They want to finish the work and go play and it feels like you're just fucking with them and disrespecting their time by making them solve the same problem several times, even if that's not the intent.
I also have long felt that anyone who has the ability to read and write at a high-school level in their primary language also has the mental capacity to learn to do math at a high school level. As a pedagogical challenge, I think the main stumbling block for most people with math is not the complexity of it but rather the dryness of how its taught. The rules of language are at least as complex, but many more people learn language at a high-school level than math. There are lots of reasons for that, most obviously language is used more in people's every day lives.
Effort, combined with the right motivation, can overcome most perceived barriers
It sounds like trivial insight, but at least in my experience many adults and even teachers have this "it's hard so it's ok to not want to do it" attitude towards math. And I think that is very detrimental.
Well, isn't that a summary of most things? Most things worth learning are hard, but many things not worth learning are also hard. So we have to prioritize what hard things are worth learning. Math is low on the list for many people for (I think) understandable reasons.
math is low on the list then they bitch that they’re unemployable with a soft skill degree doing middle school level work. i get this is ironic as a pure math student is also fairly unemployable without extraneous skills, but they also tend to shine the brightest once they make it in.
What I meant was I think it's detrimental to be priming the kids with a negative view, or nurturing any negative views.
There is this element of abstract mathematical thinking that many young people get exposed to at some point in the educational system but just never "get it" and they disconnect. This is where it goes awry as the gap only widens later on and its a pity.
Working with symbols, equations etc. feels like it should be more widely accessible. Its almost a game-like pursuit, it should not be alienating.
It might be a failure of educators recognizing what are the pathways to get the brain to adopt these more abstract modes of representing and operating.
NB: mathematicians are not particularly interested in solving this, many seem to derive a silly pleasure of making math as exclusive as possible. Typical example is to refuse to use visual representation, which is imprecise but helps build intuition.
Lots of people seem to get permanently lost right around when operations on fractions are introduced. Other places, too, but that seems like the earliest one where a lot of people get lost and never really find their way back.
Factoring was another that lost a lot of folks in my class. Lots of frustration around it seeming both totally pointless and the process involving lots of guessing, several classmates were like "well, fuck math forever I guess" at that point, like if they'd been asked to dig a ditch with a spoon and then fill it back in.
I don't know how widespread this phenomenon is but in the book Do Not Erase [0] it seems that there are quite a few prominent mathematicians who do use visual representations in their work.
There is this long-running (and quite fascinating I think) debate about the different "types" of mathematical thinking. Logical vs Intuitive, Geometric vs Algebraic etc. Can't recall where exactly but I remember reading about a 19th century mathematician that crowed they had not a single figure in their masterpiece.
Visualization is probably not a silver bullet but a lot of people are visual thinkers so maybe it would help a few more to reach a higher level in mathematical thinking.
I’m far from being any kind of serious mathematician, but I’ve learned more in the last couple years of taking that seriously as an ambition than in decades of relegating myself to inferiority on it.
One of the highly generous mentors who dragged me kicking and screaming into the world of even making an attempt told me: “There are no bad math students. There are only bad math teachers who themselves had bad math teachers.”
Sadly, when I was a postdoc, an eminent mathematician I was working under once shared a story that he found amusing that one of his colleagues was once asked a question in the form: "This might be a stupid question, but..." and the response was "There are no stupid questions, only stupid people."
Run into too many people like that, who I daresay are common in the field, and it's easy to see how people become dispirited and give up.
Isn't that a positive statement, that you can ask questions without worry since they aren't stupid?
Maybe, I guess it's easily misinterpreted though O.O
I think we can recognize Pauli for his identification of one of the few magic gadgets we accept around spin statistics without accepting his educational philosophy: “Das ist nicht einmal falsch.”
He was right on the nature of the universe, he was wrong on making a better world. I for one forgive him on the basis of time served.
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How much of math aversion stems from a chain reaction of ineffective instruction
According to an excellent mentor: all of it minus epsilon.
Wouldn't it then follow that all students of the same teachers end up with the same skill level in math? Not sure that's the case.
Cantor gave his life to the Continuum Hypothesis, Hilbert gave much of his life to similar goals.
You’re making an argument somewhat along those lines, but given that I didn’t stipulate a convergence condition your conclusions can be dismissed by me.
If it were a valid argument then we’d need Gödel.
Did you mean to reply to another post? I don't follow at all.
Doesn't follow. Bell curve in, shifted bell curve out. Ideally this also tweaks the variance a bit.
In other words: Some students flourish despite their teachers, some flourish because of them.
And how would you call the students in the left tail of the Bell curve if not bad students?
Below average students and as long as the average is high enough they are still very competent.
A bad teacher instead gives you a bimodal distribution and just doesn't bother teaching those students.
I think for most people the issue is that they never even get to the fun stuff. I remember not really liking math right until university where we had set theory in the first semester, defined the number sets from scratch went on to monoids, groups, rings etc. That "starting from scratch" and defining everything was extremely satisfying!
Interesting, I somewhat of an opposite reaction, although I am certainly not a mathematician. Once everything became definitions, my eyes glazed over - in most cases the rationale for the definitions was not clear and the definitions appeared over-complicated.
It took me some time, but now it's a lot better -- like a little game I somewhat know the rules of. I now accept that mathematicians are often worrying about maximal abstraction or addressing odd pathological corner cases. This allows me to wade through the complexity without getting overwhelmed like I used to.
My dad always told me growing up today math was like a game and a puzzle, and I hated that. I also hated math at the time. It felt more like torture than a game.
I didn't fall in love with math until Statistics, Discrete Math, Set Theory and Logic.
It was the realization that math is a language that can be used to describe all the patterns of real world, and help cut through bullshit and reckon real truths about the world.
Yes, I agree! And also that a lot of the fun stuff is hidden behind historically opaque terminology. Although I'm also sympathetic to the fact that writing accessible explanations is a separate and hard to master skill. Once you understand something it can be really hard to step back into the mindset of not understanding it and figuring out an explanation that would make the idea "click".
I think a lot of maths is secretly a lot easier than it appears, but just missing an explanation that makes it easy to get the core idea to build upon.
For example, I've been meaning to write an explorable[0] for explaining positional notation in any integer base (so binary, hexadecimal, etc) in a way that any child who can read clocks should be able to follow. Possibly teaching multiplication along the way.
Conceptually it's quite simple: imagine a counter that looks like an analog clock, but with the digits 0 to 9 and a +1 and -1 button. We can use it to count between zero and nine, but if we add one to nine, we step back to zero. Oh no! Ok, but we can solve this by adding a second counter. Whenever the first counter does a full circle, we increase it by one. A full circle on the first counter is ten steps, so each step on the second counter represents ten steps. But what if the second counter wants to count ten steps? No problem, just add a third! And so on.
So then the natural question is... what if we have fewer digits than 0 to 9? Like 0 to 7? Oh, we get octal numbers. 0 and 1 is binary. Adding more digits using letters from the alphabet?
The core approach is just a very physical representation of base-10 positional, which hopefully it makes it easy to do the counting and follow what is happening. No "advanced" concepts like "base" or "exponentiation" needed, but those are abstractions that are easy to put on top when they get older.
I've asked around with friends who have kids - most of them learn to read clocks somewhere between four and six, and by the time they're eight they can all count to 100. So I would expect that in theory this approach would make the idea of binary and hexadecimal numbers understandable at that age already.
EDIT: funny enough the article also mentions that precisely thanks to positional notation, almost every adult can immediately answer the question "what is one billion minus one".
If you’re interested in computer science, have you ever looked at the Software Foundations course by UPenn? It follows a similar approach of having you build all sorts of fascinating math principles and constructions from the ground up. But then it keeps going, all the way up to formal methods of software analysis and verification.
totally agree! in high school, lots of things were vaguely defined. I remember, I didn't fully understand what "f o g" was until I was given the definition of a monoid. Also limits and derivations, once you get the proper definition, you can pretty easily derive all the formulas and theorems you use in high school. Also in high school, we mostly did calculations and simple deductions, but at university we were proving everything. Nice change of perspective.
In college I took Formal Logic II as it fulfilled requirements in both my Comp Sci and Phil major. It turned out that PHIL 104 was cross listed as MATH 562, because the professor who taught Logic I was allowed to teach whatever he wanted for the followup class. I had technically taken the prereq, which was a basic CS logic course, but I was in way over my head. It was one of the most fun courses I took in college.
We were given the exact text of the final exam weeks in advance, and were allowed to do anything at all to prepare, including collaborating with the other students or asking other professors (who couldn't make heads or tails of it). The goal was to be able to answer 1 or 2 out of the 10 questions on the exam, and even if you couldn't you got a B+ at minimum.
I wish I had a better memory, but I believe one of the questions I successfully answered was to prove Post's Theorem using Turing machines? The problem is, I never used the knowledge from that class again, but to this day I still think about it. It would be amazing to go back and learn more about that fascinating intersection of philosophy and computer science.
What I loved the most was that it combined hard math with the kind of esoteric metaphysical questions about mathematics which many practitioners despise because they feel like it undermines their work. It turns out, when you go that deep it's impossible not to touch on the headier stuff.
In my high school we were basically only instructed to get good at applied math. Calculus. Which more often than not was simply "plugging it in". Most of that work is trivially automatible through Mathematica. When I reached a university, I took number theory and abstract algebra and it blew my mind that math was actually so beautiful in a way that defied explanation. When I took real analysis I finally saw the side of calculus that didn't seem like a waste of time.
One day, I went back to my high school and spoke to my computer science mentor back then [1]. I passionately asked him why we were never exposed to group theory. The answer, he said, was the SAT. None of that stuff is on the SAT, so it can't be justified teaching.
[1] The great Andrew Merrill
Calculus isn't on the standard SAT or even the Math Subject II tests [0], at least not in the early 2000s.
They taught us calculus because it's a prereq for engineering and physics and that's been important since the space race.
Eh, I mean it's not on the SATs, but why isn't it?
In Canada, we had a similar calculus based curriculum up to the first year of university. A little bit of linear algebra thrown into the mix. Why is that? Well you need calculus to do any form of engineering, physics, certain domains of chem/bio, stats, certain domains of economics, etc etc. Math in society is first and foremost a tool. I say this as a person who majored in Pure Math and focused on algebra and number theory. For the vast majority of students, it truly is about the practicality. Math just has the layer of abstraction that makes it hard to enjoy without deliberate framing unlike the sciences or humanities.
Many people dismiss mathematics because they aren't interested in it. I definitely wasn't interested in doing dozens and dozens of error-prone problem sets that mainly boiled down to performing arithmetic. I don't need to do that.
My number theory professor was a brilliant mind, someone who had spent lots of time at the Institute for Advanced Study, and he absolutely sucked at adding/subtracting/multiplying numbers. And that was something he freely admitted. It wasn't important to his work, and it isn't important to mine.
If someone still on X could send them the hn@ycombinator.com support address, that'd be useful.
> the provocative claim
Leibniz made that claim centuries ago in his critical remarks on John Locke's Essay on Human Understanding. Leibniz specifically said that Locke's lack of mathematical knowledge led him to (per Leibniz) his philosophical errors regarding the nature of 'substance'.
I haven't read his Human Understanding, but his Second Treatise is really weak in ways that can't really be blamed on lack of mathematical training (unless we're going with "all rigorous thinking is math") so there may be more to it in his case than just "he didn't study math enough".
Leibniz wasn't saying that "rigorous thinking" is only available to mathematically trained or that Locke's reasoning was not "rigorous".
His critique of Locke was that one can not have a correct model of human understanding (or world model) based on purely philosophical means, and that the lack of exposure to certain aspects of modern mathematics (that was emerging at that time) was the basis of Locke's misunderstandings.
Last year I read How to Solve It and the first half of one of Polya's other books - Mathematics and Plausible Reasoning. I certainly didn't commit them to memory, and I never systematically tried to apply them during self-study, but they do sometimes help give me a pointer in the right direction (i.e. trying to think of auxiliary problems to solve, trying to find a way to make the known & unknown closer together... etc.).
Auxiliary problems are something that always screwed me in college, when we were doing Baby Rudin, if a proof required a lemma or something first I usually couldn't figure out the lemma. Or in general, if I didn't quickly find the 'insight' needed to prove something, I often got frustrated and gave up.
This material seems like it would be good to actually teach in school, just like a general 'how to think and approach mathematical problems'. Feels kinda weird that I had to seek out the material as an adult...
One other thing I got out of the Polya books, was I realized how little I remember about geometry. So many of their examples are geometrical and that made them harder for me to grok. That's something I wish I could revisit.
"It's the economy stupid" is what I would say. Mental capacity is capacity. Most of us don't study math not because we don't want to but because we can't.
I bet if you asked in a survey of people that if you were given a UBI that covered all your expenses and needs what would you do? It would be perfection of the self or art. Both of these are what is practicing and learning math.
One thing I can agree with is that if one is stressed out and have poor psychological habits you will suffer and be miserable regardless.
I would say focusing on mindfullness(like vipasaana) can go a long way in this. But mindfulness is not an intellectual exercise, one has to live it. Do multiple hours of meditation a day and it gets you somewhere good in a few months.
I can actually tell you how to do this right now. Take a 10 day vipassana course and practice the recommended two hours a day. If you use only therapy, seriously follow the therapist, but do some meditation too.
Once you do this, you will soon develop Attention(Concentration) and Equanimity(Inner Calmness in the face of tremendous external circumstances).
Soon you start realizing some inescapable facts:
1. Your current moment is ever changing.
2. One attaches more of the self(the ego, I me and mine) with the past and tries to predict his state in the future and ends up miserable. Don't think with the I, you are bound to suffer. The right action is timeless and free of the I. This is the reduction of the ego.
3. There is tremendous joy in focusing on being present in the moment. If you are into running and all you are doing is taking joy in your feet, breath and posture all the time for 5,10 miles you know what I mean.
This is the key to everything. No amount of book reading on self improvement can get this to you
Didn’t we have that experiment during Covid? A bunch of people got paid to stay at home, sometimes for 2 years. How many Grammy nominations since then have gotten to musicians that came out of that? A new face at the Oscars? MOMA exhibiting an artist that was a barista before the pandemic?
At least anecdotally, many people around me now have more children, on the other hand.
I must disagree. I consider art same as entertainment to the most part. I would want to be good at math and I also disagree that it has anything to do with mental capacity. It's not a competition, I don't need to be better at math than others but my pursuit of other things like cryptography, better algorithms, and understanding physics is limited by my primitive understanding of mathematics.
If I was a multi-millionaire, learning lots of math on my free time would be one of the things I would pursue while chilling at my beach house.
Thanks.
> art same as entertainment
Could you volunteer me how much time you spend on it? And how is your day job?
> my pursuit of other things like cryptography, better algorithms, and understanding physics is limited by my primitive understanding of mathematics.
Could you volunteer why you would want to learn these subjects? Is it your day job or is it something you would like to pursue in the future.
> If I was a multi-millionaire, learning lots of math on my free time would be one of the things I would pursue while chilling at my beach house.
I said UBI
I don't think UBI is feasible, it is anathema to the human condition to be content with the bare minimum. if not for our own selves, we would want the best life possible for our loved ones (present or future). my "UBI" would be a couple of million dollars.
I want to learn those subjects because I enjoy learning and understanding. Life should be lived with knowledge applied through wisdom.
> Could you volunteer me how much time you spend on it? And how is your day job?
On entertainment? I can't tell you, I like to watch a movie or a tv show whenever I have time for it. There are more enjoyable pursuits in life, and most worthy pursuits involve adversity and require perseverance.
> I don't think UBI is feasible, it is anathema to the human condition to be content with the bare minimum.
Hm? That is exactly why UBI works, no? As the name indicates, the bare minimum is taken care so that you can work towards "the best life possible for our loved ones" without worrying about starving, sickness or homelessness.
In contrast, if you were wrong and people would be content with the bare minimum, then UBI would be a bad idea. Though then they could just commit some felonies and be content with having a cell, bread and water for the rest of their lives, no?
I posted a more detailed sibling comment on this thread, but that's not why UBI works. it just shifts what the "bare minimum" is. Most Americans aren't fighting to get the best tent spot while out on the streets because they can't afford housing or begging for food on the streets. UBI isn't solving that, except as a welfare replacement for a small percentage of the population (and not a great replacement either). maybe with UBI, everyone who lives in a crappy apartment can now afford a nicer apartment, but costs for the nicer apartment would naturally go up as well. In other words, most people won't quit their jobs because of UBI, they would just temporarily afford nicer things. Those that do quit their jobs can not work and not worry about starving, but that's not a new condition. if you don't want to work in America, you won't starve. maybe housing would be a problem but the people for whom housing would be a problem if they stopped working are not typically the same people who would be content with the cheapest/worst livable condition (UBI).
In short, it is silly to expect UBI to be a means by which people would work only if they want to work, and they would pursue their passions and dreams instead. That kind of a society I think is possible, but it would also have to reach a level of wealth where money itself is not required (think star trek).
> In short, it is silly to expect UBI to be a means by which people would work only if they want to work
I think it is misleading to call that scenario UBI and that the rest of your post also shares barely any resemblance to actual UBI plans, goals or side effects.
if that is the case, then I must apologize for my ignorance.
> my "UBI" would be a couple of million dollars.
There's a difference between what your human brain would become accustomed to (which you'd be right, it'd scale up and up forever) versus what would allow the base level of health and opportunity. As in, not having to worry about eating the next day.
And because you're right that human brains strive for more wealth, UBI should grant you the opportunity to pursue it without fear of failure.
UBI can be used as a replacement for existing welfare programs, but you're not pursuing arts or starting a business on it. My point was, people will still prioritize earning more money when on UBI instead of pursue their passions because it won't be enough. UBI is not a safety net, if a middle class salary person fails, they would have to work hourly lower wage work, that's why they keep working their middle class job, it isn't because they fear starvation or losing their shelter.
UBI would relieve stress for the lowest earning people, but it won't result in pursuit of passion for most people. economically speaking, because most people can afford certain things (like rent) the price of those things will go up, things are priced based on what potential consumers are willing to pay. If rent costs $1000 for a specific type of unit, but suddenly everyone on UBI can afford that easily, the landlord would raise the rent, the cost of things won't remain static when wages rise for a large portion of the population. Increased demand without increased cost is loss of potential revenue. The quality of life for people on UBI would be barely surviving, and UBI would need to increase constantly to keep people from becoming homeless or starving.
This is the "Cobra effect" embodied. It provides a perverse incentive. healthcare in the US is out of control for this reason. health care providers keep increasing cost, because the patient is not the client, the insurance company is, so long as everyone is getting insurance, the cost of care is the maximum reliably predictable pay out by the insurance company. Not increasing cost is just bad business. You will have to also force all kinds of businesses from raising prices if it can work, and even when it works UBI will result in subsidizing low-wager workers for businesses, because they'll still have to work some job to afford anything outside of food,shelter and the basics.
A practical alternative to UBI is a local tax on businesses, kind of like a property tax but this tax is based on an inverse of an assessment of wages, rent, welfare pay out and other social conditions in the area. the higher wages are, lower rent is,etc.. the lower the tax is, it might even result in a credit. An inverse of a perverse incentive like UBI. Unemployment would also be partially funded through this, the unemployed would forever get a UBI like pay out so long as they are pursuing education or work of some kind based on what is in demand in their area. Businesses get a healthy talent pool to choose from and cost of living is balanced.
UBI scoped for self-actualization would be nice, imagine all the mighty works people would make if not for The Markets making us organize around badware.
Come on, get real. If people had all of their needs covered a lot more of them would sit around getting high and playing video games than perfecting their art.
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Perhaps some, but why is that a problem? There are already people who do this.
I would perfect my video gaming art. Teenagers in their basement would fear me after school.
And how far would that get one doing that? 1, 5, 10, 40 years?
Is that not better than the current alternative where they're forced to become grifters?
No one is forced to become a grifter.
Yeah, but we are all in software so for us economy is not a concern. All of us should learn maths.
If the article is in any way representative of the book, then I'm not sure what there is to be learned from the book. That mathematical skills can be honed through practice? That it happens at an intuitive, pre-rigorous level before it is ready to be written down on paper? How surprising. And I doubt he can disprove the genetical component of intelligence, only show that there are other components to mathematical productivity as well.
At least I know that David Bessis's mathematical work is not as shallow as this. His twitter thread on the process https://x.com/davidbessis/status/1849442592519286899 is actually quite insightful. I would guess this also made it into the book in some longform version, but I don't know whether I would buy the book just for that.
This topic is probably the worst possible topic for Quanta.
The book, as I understand it, is about the life changing power of mathematical thinking. Quanta's mission is to make deep mathematics and mathematicians a "sexy" "human interest" topic, by making it as non-mathematical as possible while keeping a superficial veneer of mathematics.
Agree. I’ve been trying to learn ML and data for a few years now and, around 2021 I guess, realised Maths was the real block.
I’ve tried a bunch of courses (MIT linalg, Coursera ICL Maths for ML, Khan etc etc) but what I eventually realised is my foundations were so, so weak being mid 30s and having essentially stopped learning in HS (apart from a business stats paper at Uni).
Enter a post on reddit about Mathacademy (https://www.mathacademy.com/). It’s truly incredible. I’m doing around 60-90 minutes a day and properly understanding and developing an intuition for things. They’ve got 3 pre-uni courses and I’ve now nearly finished the first one. It’s truly a revelation to be able to intuit and solve even simple problems and, having skipped ahead so far in my previous study, see fuzzy links to what’s coming.
Cannot recommend it enough. I’m serious about enrolling in a Dip Grad once I’ve finished the Uni level stuff. Maybe even into an MA eventually.
Too often people think of learning as accumulating knowledge and believe blockers are about not enough knowledge stored.
That would be like strength training by carrying stuff home and believing that the point is to have a lot of stuff at home.
Intelligence is about being able to frame and analyze things on the fly and that ability comes from framing and analyzing lots of different things, not from memorizing the results of past (or common forms of) analysis.
I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao.
As anti-intellectual as it sounds, you could imagine someone asking, is it worth devoting years of your life to study this subject which becomes increasingly esoteric and not obviously of specific benefit the further you go, at least prima facie? Many people wind up advocating for mathematics via aesthetics, saying: well it’s very beautiful out there in the weeds, you just have to spend dozens of years studying to see the view. That marketing pitch has never been the most persuasive for me.
Pure math is probably not worth the squeeze. I think more important to everyday life is systems thinking and a bit of probability/stats, mainly bayesian updates. "Superforecasting" was an eye-opening book to me, I could see how most people would benefit massively by it.
Similar to systems thinking, just the ability to play out scenarios in your head given a set of rules is a very useful skill, one which programmers tend to either be good at because of genetics or because we do it every day (i.e. simulate code in our head). You can tell when someone lacks this ability when discussing something like evolutionary psychology. Someone with a systems thinking mindset and an ability to simulate evolution tend to understand it as obvious how evolutionary pressures tend to, and really must, create certain behavior patterns (on average), while people without this skill tend to think humans are a blank slate because it's easier to think about, and also is congruent with modern sensibilities.
This skill applies in everyday life, especially when you need to understand economics (even basic things like supply and demand seems elusive to many), politics etc.
Abstract Algebra, Combinatorics, and Discrete Mathematics are all definitely worth the squeeze; and incidentally something that could easily be taught to middle- and high-schoolers with the right examples.
Is it worth it to be able to think better, have a growth mindset and learn how to learn? Yes. Everyone can benefit from that. Pushing on into higher math? No, very few people can benefit from that.
Math doesn’t seem to me the only source of thinking clearly, or learning how to learn, etc. And if I’m searching for an aesthetic high, there are definitely better places to look — and ones that don’t require such a long runway.
It doesn't need to be for me to be right. These are false constraints you're trying to put on it. Mathematics in moderation can benefit everyone. This claim stands.
I'll second guerrilla - you can absolutely benefit from mathematical thinking without pushing into territory higher than undergaduate studies.
You can even benefit from the thinking taught in good high school coursework (or studying online).
At an arithmetic, bookkeeping level you can better appreciate handling finances and the seductive pitfalls surrounding wagers (gambling, betting, risk taking).
My claim isn’t really that there’s no benefit or utility to math — that’s obviously false — but that maybe its benefits to regular people are more modest than the cheerleaders want to admit.
What are the costs (in your estimation at least) to "regular people" (regular by your metric) of not engaging in easy bake low level "mathematical thinking".
* How many have a lower return on { X } through not understanding compound interest, tax brackets, leveraging assets, etc.
* How many have steady net losses through "magical thinking" wrt gambling, betting, hot stock tips.
You’re sort of making my point — there are people out there who think math education sets the mind free and opens the gates of higher cognition, and then others talking about hum drum stuff like tax brackets and compound interest. If the benefits really just amount to a few units of pre-algebra content, that would be disappointing.
> If the benefits really just amount to a few units of pre-algebra content, that would be disappointing.
They don't though - there are benefits all the way up.
Giving a few examples of benefits from low order mathematical thinking (understanding concepts of compounding, etc) does not equal a statement that these are the only benefits.
Great, so what are those benefits all the way up?
At least a finite countable lists worth, there's something for everybody really.
Some of us are satisfied with, say, dimensional reductions of spectral geophysical surveys to improve imaging and train that against existing production records to highlight potential mineral leases.
Others might like to leverage symbolic algebra systems to crack quantum encryption candidates.
A good number like to get rich via high frequency trading, some like to sail satellutes against the magnetic currents of the planet, there's a world of optimisations in logistics that have been implemented and are still being fine tuned to improve throughput efficiency and fuel consumption.
But you're not really asking in good faith here, are you?
i’ve always found that mathematical thinkers do better, no matter the role. it’s difficult to express how, it’s kind of intangible, but it’s just a mental process that helps break down tasks and come up with unique solutions.
that being said, my marketing pitch for math was always, nothing else is even close to as intellectually stimulating as math.
yeah, I wish we could be more specific here. Like I appreciate the things you’re saying, and I don’t doubt the truth or sincerity of that opinion — but I feel like I’ve encountered some math PhD students who were surprisingly inflexible in the way they thought, and very parochial in their intellectual interests. I guess I just haven’t (anecdotally) found the transference of math training to other domains as much as others seem to.
i think that’s a characteristic of a lot of PhD’s though - they become so specialized they become inflexible. i’m more talking about applied math into industry types. in insurance, tech, etc, i always tend to find the mathy people are faster, come up with unique solutions, and are more self motivated.
I've lately been struck by people having a life difficulty due to "missing a clue" absent some experience. The person poorly conceptualizing and executing physical therapy, for lack of athletic experiences. The person poorly handling cognitive decline, for lack of a grasp of work processes. The person variously failing from having physical discomfort as an abort criteria, for lack of experiences normalizing its deferral.
"I have this clue at hand" can have broad impacts. Software development's emphasis on clarity, naming, and communication protocols, helped me a lot with infant conversation. Math done well, can be a rich source of clues, especially around thinking clearly.
There's an idea that education should provide more life skills (like personal finance). And another, that education should have a punch list (as in construction), of "everyone at least leaves with these". Now AIish personalized instruction will perhaps permit delivering a massive implicit curriculum, far larger than we usually think of as a reasonable set of learning objectives. Just as a story can teach far more than the obstensible moral/punchline of the story, so too might each description, example, question and problem, dynamically tuned in concert. So perhaps it's time to start exploring how to use that? In the past, we worked by indirection - "do literary criticism, and probabilisticly obtain various skills". And here, from math. Perhaps there's a near-term opportunity to be more explicit, and thorough, about the cluefulnesses we'd like to provide?
I studied math hard for several years in college and graduate school—purely out of interest and enjoyment, not for any practical purpose. That was more than forty years ago, but Bessis's description of the role of intuition in learning and doing math matches my recollection of my subjective experience of it.
Whether that youthful immersion in math in fact benefitted me in later life and whether that kind of thinking is actually desirable for everyone as he seems to suggest—I don't know. But it is a thought-provoking interview.
I also studied it and got several degrees, but I don't think that it actually benefited me. I think high school math is incredibly important to be able to think clearly in a quantitative way, and one university-level statistics course, but all the other university math... I dont think it helped me at all. I am disappointed by it because I feel that I was misled to believe that it would be useful and helpful.
Have you ever ascribed numbers to real life personal problems?
I find that managing to frame something bothersome into a converging limit somehow, really dissolves stress.. A few times at least.
That’s an interesting approach. I don’t think I’ve done that myself, but I can see how it could be helpful.
One positive effect of having studied pure mathematics when young might have been that I became comfortable with thinking in multiple layers of abstraction. In topology and analysis, for example, you have points, then you have sets of points, then you have properties of those sets of points (openness, compactness, discreteness, etc.), then you have functions defining the relations among those sets of points and their properties, then you have sets of functions and the properties of those sets, etc.
I never used mathematical abstraction hierarchies directly in my later life, but having thought in those terms when young might have helped me get my head around multilayered issues in other fields, like the humanities and social sciences.
But a possible negative effect of spending too much time thinking about mathematics when young was overexposure to issues with a limited set of truth values. In mainstream mathematics, if my understanding is correct, every well-formed statement is either true or false (or undecided or undecidable). Spending too much time focusing on true/false dichotomies in my youth might have made it harder for me to get used to the fuzziness of other human endeavors later. I think I eventually did, though.
Thanks for sharing. The reverse direction here, I'm trying to go from fuzziness to the exactness of those true/false dichotomies, haha. The way I've been attacking mathematics, it's like a tree in the forest, one could start with the axioms and from the base reach each branch and the leaves and fruits. But I've just been walking around the tree, looking at the leaves and fruits and branches from different directions to see ways of climbing without doing a whole lot of climbing. What I mean is I've been thinking and reading in an imprecise way a whole lot without actually juggling symbols with pen and paper, haha. Or a roadtrip analogy, I've done little driving and a lot of map ogling. At least I won't miss the turns when I pick up some speed.
I want to say yes, but I have two counters. One is that math nerds at school insisted on intimidating for the win and I just hated it.
The second is notation. I had a snob teacher who insisted on using Newton not Leibniz and at school in the 1970s this is just fucked. One term of weirdness contradicting what everyone else in the field did. Likewise failure to explain notation, it's hazing behaviour.
So yes, everyone benefits from maths. But no, it's not a level playing field. Some maths people, are just toxic.
The different notations in calculus are unfortunately a historical accident and you're going to see all of them if you do anything with it (it's certainly not true that everyone in the field uses the same notation). I agree that it's frustrating but so are irregular verbs in Spanish/French/other languages - you can't really change it.
> One is that math nerds at school insisted on intimidating for the win and I just hated it.
Only an adult can look and see what that was - immature, insecure little boys, desperately trying to show off as bigger/more mature or kick down anybody showing any weakness or mistake. Often issues from home manifesting hard. Its trivial to look back without emotions, but going through it... not so much.
If my kids ever go through something similar (for any reasons, math nerds are just one instance of bigger issue) I'll try reasoning above, not sure if it will help though.
> Only an adult can look and see what that was - immature, insecure little boys, desperately trying to show off as bigger/more mature or kick down anybody showing any weakness or mistake. Often issues from home manifesting hard. Its trivial to look back without emotions, but going through it... not so much.
I'm not so sure that adults always get it or rise about this. This happens in the workplace all the time.
> One is that math nerds at school insisted on intimidating for the win and I just hated it.
For me the worst part was the teachers that encouraged that behavior and did the same.
"mathematics is a game of back-and-forth between intuition and logic" I teach/guide Math at our school (we run a small school and currently have kids under age 10) and this is so so true.
This was a very tiring blog post for me. And I have a quip about posts that open with questions but close without obvious definite answer, no matter how simple it is.
Tiring because the answer wasn't revealed? That was the whole point :) It's the path, not the summit.
Does the quip have to do with someone asking what the quip is?
Not sure this article captured it for me.
Plato's Meno has Socrates showing that even a slave can reason mathematically.
It's not really math alone but modeling more generally that activates people's reasoning. Math and logic are just those models that are continuous+topological and discrete+logic-operation variants, both based in dimension/orthogonality. But all modeling is over attribution - facts, opinions, etc., and there's a lot of modeling with a healthy dose of salience - heuristics, emotions, practice, etc. Math by design is salience-free (though it incorporates goals and weights), so it's the perspective and practice that liberates people from bias and assumptions. In that respect it can be beautiful, and makes other more conditioned reasoning seem tainted (but it has to work harder to be relevant).
However, experts can project mathematical models onto reality. Hogwash about quantum observer effects and effervescent quantum fields stem from projecting the assumptions required to do the math (or adopt the simplifying forms). Yes, the model is great at predictions. No, it doesn't say what else is possible, or even what we're seeing (throwing baseballs at the barn, horses run out, so barns are made of horses...). Something similar happens with AI math: it can generate neat output, so it must be intelligent. The impulse is so strong that adherents declare that non-symbolic thinking is not thinking at all, and discount anything unquantifiable (in discourse at least). Assuming what you're trying to prove is rarely helpful, but very easy to do accidentally when tracking structured thinking.
Has anyone here self-taught themselves math in later life?
I studied up to A level (aged 19) but honestly started hating math aged 16 after previously loving it.
It’s a big regret of mine that I fell out of love with it.
I self taught myself coding and Spanish and much enjoy self study if I can find the right material.
It started off as a bunch of non-math literate folks teaching themselves math from scratch, including trigonometry, calculus etc, and ending in Fourier series. It is a very approachable and fun book.
I was the same In high school.
2 weeks ago I hired a professor to help me learn math again so I can attend University computer science.
I can tell you, you can and should.
I'm totally addicted to math, I work as a programmer once I finish my work for the day I spend all my free time learning math again.
I'm still going over the very basics like 9 th grade stuff but I can see already it's going to go fine! I'm enjoying it so much!
Math Academy and (optionally for a deeper study) the Art of Problem Solving books.
- Intermediate Algebra for College Students - Blitzer (ISBN-13 978-0134178943 )
- College Algebra - Blitzer (ISBN-13 978-0321782281)
- Precalculus - Blitzer (ISBN-13 978-0321559845)
- Precalculus - Stewart (ISBN-13 978-1305071759)
- Thomas' Calculus: Early Transcendentals (ISBN-13 978-0134439020)
- Calculus - Stewart (ISBN-13 978-1285740621)
The main goal of learning is to understand the ideas and concepts at hand as “deeply” as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By “deeply” we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no “perfect” state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: “Young man, in mathematics you don't understand things. You just get used to them”, which I think really means that getting “used to” some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination.
Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap).
Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding.
Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory.
Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note.
Cross-reference. Don't read linearly. Instead, have multiple textbooks, and “dig deep” into concepts. If you learn about something new (say, linear combinations) — look them up in two textbooks. Watch a video about them. Read the Wikipedia page. Then write down in your notes what a linear combination is.
Learning is a social activity, so maybe enroll in a community college course or find a local study group. I find it's especially important to have someone to discuss things with when learning math. I also recommend finding good public spaces to work in—libraries and coffee shops are timeless math spaces.
Pay graduate students at your local university to tutor you.
Take walks, they're essential for learning math.
Khan Academy is not enough. It has broad enough coverage, I think, but not enough diversity of exercises. College Algebra basically is a combination of Algebra 1, Algebra 2, relevant Geometry, and a touch of Pre-Calculus. College Algebra, however, is more difficult than High School Algebra 1 and 2. I would tend to agree that you should start with either Introductory Algebra for College Students by Blitzer or, if your foundations are solid enough (meaning something like at or above High School Algebra 2 level), Intermediate Algebra by Blitzer. Basically, Introductory Algebra by Blitzer is like Pre-Algebra, Algebra 1, and Algebra 2 all rolled into one. It's meant for people that don't have a good foundation from High School. I would just add, if it is still too hard (which I doubt it will be for you, based on your comment), then I would go back and do Fearon's Pre-Algebra (maybe the best non-rigorous Math textbook I've ever seen). Intermediate Algebra is like College Algebra but more simple. College Algebra is basically like High School Algebra 1 and 2 on steroids plus some Pre-Calculus. The things that are really special about Blitzer is that he keeps math fun, he writes in a more engaging way than most, he gives super clear—and numerous—examples, his books have tons of exercises, and there are answers to tons of the exercises in the back of the book (I can't remember if it's all the odds, or what). By the time you go through Introductory, Intermediate, and College Algebra, you will have a more solid foundation in Algebra than many, if not most, students. If you plan to move on to Calculus, you'll need it. There's a saying that Calculus class is where students go to fail Algebra, because it's easy to pass Algebra classes without a solid foundation in it, but that foundation is necessary for Calculus. Blitzer has a Pre-Calculus book, too, if you want to proceed to Calculus. It's basically like College Algebra on steroids with relevant Trigonometry. Don't get the ones that say “Essentials”, though. Those are basically the same as the standard version but with the more advanced stuff cut out.
Given the sentiment in these comments, I figure this crowd might be interested in the book "Measurement" by Paul Lockhart (the guy who wrote "A Mathematician's Lament")
He's of the opinion that math should be taught not as jumping through hoops for "reasons", but as an art, enjoyable for its own sake, and that this would actually produce more confident and capable thinkers than the current approach. (I think the argument applies to almost all education but his focus is just on math.)
> everyone can, and should, try to improve their mathematical thinking — not necessarily to solve math problems, but as a general self-help technique
Agreed with the above. Almost everyone can probably expand their mathematical thinking abilities with deliberate practice.
> But I do not think this is innate, even though it often manifests in early childhood. Genius is not an essence. It’s a state. It’s a state that you build by doing a certain job.
Though his opinion on mathematical geniuses above, I somewhat disagree with. IMO everyone has a ceiling when it comes to math.
> IMO everyone has a ceiling when it comes to math.
Reading most of the answers here, I can only conclude most of you were home schooled or went to some fancy schools for gifted children.
An average human is unable to even write properly. Even basic mathematical operations like multiplication and division are too complicated from their perspective.
God that's one of the most pessimistic views of humanity I've seen on the web, and I'm pretty pessimistic. Developed nations have a 99.2% literacy rate per wikipedia. 69% of OECD students meet "level 2" mathematical proficiency (https://www.oecd.org/en/topics/sub-issues/mathematics-litera...), which indicates an understanding above the basic mathematical operators like multiplication and division.
FWIW I did not go to any fancy school, I was educated in the state system.
Not sure about schools for gifted, but I think lots of people here a bit live in bubbles.
Maybe industrialized countries, a bit better off neighborhoods, STEM classes in high school, and then university? (Not everyone but ... Most here?)
As a young child your brain is much more suitable to learn languages. You can make kids learn 4 languages effortlessly in the right context. When you grow up, slowly shift the focus to abstract thinking. And that shift can rely on building intuition using visualisation and experience.
I totally agree! The barriers many of us face with math are less about ability and more about how we've been taught to approach it. All it took was for me to change my math teacher at school, and boom. Love, but at second sight. And curiosity and persistence can unlock more than just numbers
Statistical (Bayesian) thinking is an extremely underrated way of thinking of almost everything.
Frankly, its overrated. Now you can adjust your priors.
I'm pretty down on it just by association. So many people who are super-into it seem to be doing a lot of that rigorously-wrong engineer-brain thing.
lol yea.
bayesian thinking doesnt come to me naturally.. i have no intuition for it. seems forced. believe me - i have tried. but there are those who are swearing by it.
Two thoughts
1) It's tragic that being "bad at math" is often positioned as some kind of badge of honour.
2) It's definitely not the case that everyone is capable of mathematical thinking. Having spent a certain amount of time trying to teach one of my kids some semblance of mathematical thinking, I can report confidently that his ability in this area is almost non-existent. His undeniable skills lie in music and writing, but definitely not in maths.
Yes, music and maths have some things in common. But musical thinking is not mathematical thinking.
I have an autistic friend with dyscalculia. They see numeric digits as individual characters (as in a story), each with their own personalities. Each digit has its own color, its own feelings. But they are not quantities; they don't make up quantities. Numbers are very nearly opaque to them. I wonder how this theory would apply to them. Do they still perform mathematical thinking? They're still capable of nearly all the same logic that I am, and even some that I'm not (their synesthesia gives them some color/pattern/vibes logic that I don't have)... just not math.
I don't have dyscalculia but behind the numbers I have my own intuitive system(s) that I jump to sometimes when doing arithmetic. I think we all do since the early days arithmetic in school or what not, perhaps the dyscalculia folk missed making some connection at some point. I feel that arithmetic with numbers without that intuitive system is rote memory..
Indeed, I do also have an intuitive system for this; that's what I mean by "make up a quantity". When something makes up a quantity for me, then I can perform math on it. I have to wonder if my friend with dyscalculia doesn't have a "quantity" system like I do. They can't really perform multiplication or division but can technically perform basic addition and subtraction, though it takes them a while. I think it may be the rote memory you mention doing that for them.
Great to see so many reactions to my interview, thanks!
I see that many people are confused by the interview's title, and also by my take that math talent isn't primarily a matter of genes. It may sound like naive egalitarianism, but it's not. It's a statement about the nature of math as a cognitive activity.
For the sake of clarity, let me repost my reply to someone who had objected that my take was "clickbait".
This person's comment began with a nice metaphor:
'I cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains.'
Here was my reply (copy-pasted from my post buried somewhere deep in the discussion):
I'm the author of what you've just described as clickbait.
Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.
The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.
Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).
When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.
As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.
In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.
There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.
I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.
Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.
A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.
In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.
> There are actual techniques that career mathematicians know about.
Your best example is the decimal system in contrast to roman numerals. I think that explains the point well. The zero is one of those tricks, and most people know it now, but that wasn't true until very recently.
I think you've simply redefined genius. Many years ago I read an article on youth football, if I remember correctly, and in it there was a bit about the writers visit to the Ajax Youth Academy. In it he writes of a moment during practice when a plane flies over and all the 7(?) year olds on the pitch look up to see it except for one kid who keeps his eyes on the ball. That kid (of course) grows up to be a very good midfielder for Real (I'm forgetting the exact details, I think its Wesley Sniejder?). My point is: whatever that motive energy is that manifests as the single minded pre-occupation with math at an age when everybody else's attention is all over the place is that inherent thing that people call genius. I have read many of Thurston's non-mathematical writings about himself and in it this sort of singular pre-occupation is also clear -- which is why he developed his preternatural geometric vision.
Indeed, it does involve redefining genius as a "state", or "flow", or "trajectory".
When I say it's not primarily genetic, many people wrongly assume there's an entirely explainable and replicable way of accessing this state. There isn't.
The 20,000 hours rule is a bit misleading, because who gets to invest 20,000 hours into something? How do you create this drive, this trajectory? You must have a good hope that it'll yield something worth the effort.
This is why the injunction to "work harder" so often misses the mark.
However, even if only a tiny fraction of the population will end up becoming a "genius", it's very important to debunk the myth, because the real story has valuable lessons for everyone: it gives concrete and pragmatic indications on what one should be on lookout for.
It's not fully teachable up to genius level, but the directionality is teachable and extremely valuable.
> I see that many people are confused [...] by my take that math talent isn't primarily a matter of genes
Speaking only for myself, I'm not confused at all. Rather I vigorously disagree with this statement, and think that stumping for this counterfactual premise leads to cruel behavior towards children (in particular) who plainly do not have what it takes to learn, for example and in particular, algebra.
> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.
This is not their subject of expertise, and they are simply wrong. Why? Simpson's Paradox, ironically.
I think you really are confused. You are mistakenly equating "non-primarily genetic" with "easily teachable".
The story is much more complex than "if it's not genetic then everybody should get it". It's quite cruel to assume that if you don't get math today you'll never get it, and there are tons of documented counter-examples of kids who didn't get it at all who end up becoming way above average.
If you think that Descartes, Newton, Einstein, Feynman, Grothendieck (to just cite a few) are all equally misled on their own account because of Simpson's Paradox, which statistical result will to bring to the table to justify that YOU are right?
By the way, Stanislas Dehaene, one of the leading researchers on the neuroscience of mathematical cognition, is also on my side.
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When you write a comment here, you're talking not simply (or even primarily) to the person you're responding to, but also to the rest of the community. Have you considered what value there might be, or not be, in a comment like this?
We are extraordinarily fortunate to be taken seriously enough as a community for the primary sources on stories like this to come talk to us directly. I'd ask, as a neighbor of yours in this community, that you not (rhetorically) chase those people away.
Thanks!
That's a fair ask, that post was not one of my better moments. As much as I dislike the rhetorical tactic wielded with "everyone who doesn't agree with me is simply confused", a one-sentence tell off does nothing to counter it, and just makes me look like a jerk. Which I was.
No problem! You'd be doing me a favor if you noted any time you see me in the same situation. Thanks for hearing me out!
The evidence of your confusion is that you cited one sentence in their comment but ignored the next sentence which provide the justification in abstract for that sentence. Therefore your basic reading comprehension skills are confused. It easier to disagree vigorously but when you do that it clouds your analytic ability to follow the actual arguments given in that way.
For example finance is such an important aspect of our lifes and you just need some understanding of math principles to understand how to make good financial choices.
Ironically, the ones who don't do well at school (inc maths) are the ones who then become trades like builders/plumbers etc or run small businesses like a shop. And so are regularly working with numbers via estimates and billing.
Shouldn’t that headline read “or can benefit from”. Not a shining example of good mathematical thinking. Oh the irony.
This is pretty interesting. I did reasonably well in maths up to the A-levels, and then absolutely collapsed in university. I never got a grade better than B- in any maths-adjacent class. Discrete maths was my worst topic, I barely scraped a pass. And the irony was that I majored in CS and physics.
I should probably find a time machine and re-do everything.
- Hey teach, will I really need all these logarithms, derivatives and vectors in my adult life?
- No, but the smarter kids might.
I used to get very frustrated that others could not intuit information the way I could. I have a lot of experience trying to express quantities to leaders and policymakers.
At the very minimum, I ask people to always think of the distribution of whatever figure they are given.
Just that is far more than so many are willing to do.
Waste of time. Just talk in terms of what they want to hear. They are just interested in the payoffs (not in the details).
As info explodes and specialists dive deeper into their niches, info asymmetry between ppl increases. There are thousands of specialists running in different directions at different speeds. Leaders can't keep up.
Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.
And while most specialists think this "sync" process happens through "education" and getting everyone to understand a complex ever changing universe, the truth is large diverse groups are kept in sync via status signalling, carrot/stick etc. This is why leaders will pay attention when you talk in terms of what increases clout/status/wealth/security/followers etc. Cause thats their biggest tool to prevent schisms and collapse.
> Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.
This is overthinking it. People with power tend to be interested in outcomes. They can't evaluate all the reasoning of all their reports. It comes down to building credibility with a track record and articulating outcomes, when you want to advise decision makers.
I believe charisma, confidence and looks also play a huge role.
But, is that profitable?
I'm both being sarcastic and real with this question.
If I can earn an extra 1 million being 'dumb' and thus ensure quality healthcare, education, housing, is it smart to try to be smart?
This is the true tragedy of the commons (or the reverse tragedy, to be precise).
there's thinking mathematically and then there's being able to fluently read math articles on wikipedia as if they're easier than ernest hemingway. I can do the former and the latter I will insist until my grave is impossible for me.
I used to judge myself for not understanding everything in math articles on Wikipedia, but as time has gone on I've realized that their purpose isn't really to be an introduction, but a reference. Especially as the topics become more esoteric. So they're not really there for you to learn things from scratch, but for people who already understand them to look things up. Which is why you'll sometimes see random obscure & difficult factoids in articles about common mathematical concepts.
(don't have any examples on-hand atm, this is just my general perception after years of occasionally looking things up there)
I've heard that and I think it's silly. They handwave away why nothing should ever be explained. Wikipedia doesn't work like that for any other topic.
You'll see something like a mathematical proof with no explanation and it's end of article. The edit history will have explanations aggressively removed.
The equivalent would be the article for say, splay tree, to have no diagrams and just a block of code - feeling no obligation to explain what it is or if you looked up a chemical and it would just give you some chemical equation, some properties and feel no obligation to tell you its use, whether it's hazardous or where you might find it... Or imagine a European aristocrat and all that is allowed is their heraldry and genealogy. Explanations of what the person did or why they're important are forbidden because, it's just a reference after all.
Nope, these math people are a special kind of bird and I'm not one of them.
I don't know anybody who first learns about new mathematical ideas from Wikipedia. Mathematics is a body of knowledge, not just simple isolated theorems or definitions. You learn new mathematics from textbooks.
Even for reference purposes there are often better resources. E.g. proofwiki is usually better for looking up proofs because the proofs and definitions are interconnected.
If I run across a term like "bialgebra" while doing work, I don't always have the leisure time to derail my life and sign up for a 4 month class at a local university. Sometimes, I just need to move on with my task at hand and get something working.
I'm familiar with the mathematician response to this, I've heard it before and I fundamentally disagree with it. At work last week I gave someone a crash course in the simplex method and linear programming in about 30 minutes and it was a good-enough explanation that I came back in a few hours and the code was right.
This isn't impossible. There's just some wild apprehension that I'll never understand which insists everything is a grueling 1,000 hour journey to some kind of valhalla of enlightenment so you can bask in some aesthetic beauty of how perfect math is, as tears drip down from your cheeks, or something like that.
I mean come on now. Sometimes all you want is the cliff notes.
Well, you're comparing a concrete algorithm from applied mathematics (simplex) to a term from abstract algebra. I'm not exactly sure how you'd expect such a general concept to be described. Where in your work do terms like "bialgebra" regularly come up and is "it's some sort of algebraic structure" not enough of an understanding to continue without digging into the details? Maybe your problem is with people who put abstract mathematics into applied material without motivation or explanation and not with research mathematicians themselves?
Would you expect to be able to read the "cliff notes" on French and then be able to read Camus in the original? That's what I mean by "body of knowledge" as opposed to individual facts.
I have a lot of trouble reading math formulas, implemented as code I understand most stuff though. Is there a good math book or something similar that teaches things using code or helps translating formulas to code?
I realized some time in middle age that I have to convert formulas and equations to steps and things happening to something "passing through" each step—to algorithms. It's painful and slow and also the only way I stand a chance in hell of reading mathematical writing.
That's probably why math writing largely makes me feel dyslexic, while programming came naturally. And why I hate Haskell and find it painful to read even though I understand the "hard" concepts behind it just fine—it's the form of it I can't deal with, not the ideas.
I conducted an interview with leetcode 2 years ago while not doing any leetcoding before. Surprisingly, by just applying some math tricks i finished them and got into later rounds. So yes math tricks are helpful.
This guy is unbelievably French (I mean in his intellectual character). Here I was expecting a kind of rehash of the 20th century movements of pure math and high modernism[0], but instead we get a frankly Hegelian concept of math or at least a Hegel filtered through 20th and 21st century French philosophy.
there is a debate between the intuitionists, formalists, and the symbolists nicely captured in the intro chapter of Heyting's Intuitionism.
constructive mathematics is close to computation and programming. and many including myself have a natural feel or intuition for it. A majority of euclids elements, and galois's original proof are constructive in nature.
I was actually thinking Jean Paul Satre when I read his answers
That's a very important thougt and I belive the world would be better, if more people would connect with their mathematical side.
To use an absolutist statement about human well bieng tied to the authors pet ideals,is a classic example of magical thinking, and a lack of understanding in general,tied to a character assination of the many many excellent non numerative humans living and thriving everywhere.
Not nice at all.
But what is the difference between math talent and plug-n-chug math talent? That seems to be the most significant filter.
And yet it is flame-bait to suggest that programmers benefit from mathematical thinking. I've not met a more passionate and divided crowd on the issue. Most traditional engineers wouldn't disagree that they use and benefit from mathematical thinking. Programmers though?
I don't think there's a single answer as to why many dislike it so much. Some folks view it as a way to gate-keep programming. Others view it as useless ("I've been a successful programmer all my life and I've never used math").
On the other side of the coin there are many who view our craft as a branch of applied mathematics -- informatics if you will.
It's a shame the title begins with 'Mathematica', makes one think the book is about the Wolfram software. That's the first thing I thought of when I saw the title. Hopefully Wolfram doesn't sue him for copyright violation or some such infringement.
I guess I assumed it was a reference to Alfred Whitehead's and Bertrand Russell's book Principia Mathematica, which predates Wolfram himself by several decades.
This interplay between intuition and logic is exactly what makes the magic happen. You need intuition to feel your way forward, and then logic to solidify your progress so far, and also for ideas maybe not directly accessible via intuition only. I've experienced that myself, and it is even well-documented, because I wrote technical reports and such at each stage. My discovery of Abstraction Logic went through various stages:
1) First, I had a vague vision of how I want to do mathematics on a computer, based on my experience in interactive theorem proving, and what I didn't like about the current state of affairs: https://doi.org/10.47757/practal.1
2) Then, I had a big breakthrough. It was still quite confused, but what I called back then "first-order abstract syntax" already contained the basic idea: https://obua.com/publications/practical-types/1/
3) I tried to make sense of this then by developing abstraction logic: https://doi.org/10.47757/abstraction.logic.1 .
After a while I realized that this version only allowed universes consisting of two elements, because I didn't
distinguish between equality and logical equality, which then led to a revised version: https://doi.org/10.47757/abstraction.logic.2
4) My work so far was dominated by intuition based on syntax, and I slowly understood the semantic structures behind this:
the mathematical universe consisting of values, and operations and operators on top of that: https://obua.com/publications/philosophy-of-abstraction-logi...
5) I started to play around with this version of abstraction logic by experimenting with automating it, giving a
talk about it at a conference, (unsuccessfully) trying to publish a paper about it, and implementing a VSCode
plugin for it. As a result of using that plugin I realized that my understanding until now of what axioms are was too narrow:
https://practal.com/press/aair/1/
6) As a consequence of my new understanding, I realized that besides terms, templates are also essential: https://arxiv.org/abs/2304.00358
7) I decided to consolidate my understanding through a book. By taking templates seriously from the start when writing,
I realized their true importance, which led to a better syntax for terms as well, and to a clearer presentation of Abstraction Algebra. It also opened up my thinking
of how Abstraction Algebra is turned into Abstraction Logic: https://practal.com/abstractionlogic/
8) Still lots of stuff to do ...
I would not be surprised if that is exactly the way forward for AIs as well. They clearly have cracked (some sort of) intuition now, and we now need to add that interplay between logic and intuition to the mix.
What is 13% of 91?
I don't know. Do you?
But I now 10% of 91 is 9.1
I got somewhere eh?
Hey also I know that 1% of 91 is 0.91 Duh! lets triple that.
0.91 x 3 = 0.9x3 + 0.01x3 = 2.7 + 0.03 = 2.73
Now lets add 9.1..
11.83 Weee!
(Now my date is rolling her eyes and the waiter is stone faced)
"High school students are often unhappy with math, because they think it requires some innate things that they don’t have,” Bessis said. “But that’s not true; really it relies on the same type of intuition we use every day."
Agreed, but from my observation mathematics is often taught with a rigor that's more suited to students with a highly mathematical and or scientific aptitude (and with the assumption that students will progress to university-level mathematics), thus this approach often alienates those who've a more practical outlook towards the subject.
Mathematics syllabuses are set by those with high mathematical knowledge and it seems they often lose sight of the fact that they are trying to teach students who may not have the aptitude or skills in the subject to the degree that they have.
From, say, mid highschool onwards students are confronted with a plethora of mathematical expressions that seemingly have no connection their daily lives or their existence per se. For example students are expected to remember the many dozens of trigonometrical identities that litter textbooks (or they did when I was at school), and for some that's difficult and or very tedious. I know, I recall forgetting a few important identities at crucial moments such as in the middle of an exam.
Perhaps a better approach—at least for those who are seemingly disinterested in (or with a phobia about) mathematics—would be to spend more time on both the historical and practical side of mathematics.
Providing students with instances of why earlier mathematicians (earlier because the examples are simpler) struggled with mathematical problems and why many mathematical ideas and concepts not only preceded but were later found to be essential for engineering, physics and the sciences generally to advance would, I reckon, go a long way towards easing the furtive more gently into world of mathematics and of mathematical thinking.
Dozens of names come to mind, Euclid, Descartes, Fermat, Lagrange, Galois, Hamilton and so on. And I'd wager that telling students the story of how the young head-strong Évariste Galois met his unfortunate end—unfortunate for both him and mathematics—would never be forgotten by students even if they weren't familiar with his mathematics—which of course they wouldn't be. That said, the moment Galois' name was mentioned in university maths they'd sit up and take instant note.
Yes, I know, teachers will be snorting that there just isn't time in the syllabus for all that stuff, my counterargument is that it makes no sense if you alienate students and turn them off mathematics altogether. Clearly, a balance has to be struck, tailoring the subject matter to suit students would seem the way with the more mathematically inclined being taught deeper theoretical/more advanced material.
I always liked mathematics especially calculus as it immediately made sense to me and I always understood why it was important for a comprehensive understanding of the sciences. Nevertheless, I can't claim that I was a 'natural' mathematician in the more usual context of those words. I struggled with some concepts and some I didn't find interesting such as parts of linear algebra.
Had some teacher taken the time to explain its crucial importance in say physics with some examples then I'm certain my interest would have been piqued and that I'd have showed more interest in learning the subject.
trial
I just published the comment above by mistake and couldn't find a way to delete it.
I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's easy, then it means you already know this material, and you're wasting your time.
> I agree with the sentiment of this. I think our obsession with innate ~~mathematical~~ skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.
The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
I think we do our children a disservice by convincing them that some of their peers are just “born with it”, because it discourages them from continuing to try.
What we should be teaching children is HOW to learn. At the moment it’s a by-product of learning about some topic. If we look at the old adage “feed a man a fish”, the same is true of learning.
“Teach someone mathematics and they will learn mathematics. Teach someone to learn and they will learn anything”.
Caveat here is that "talent" and "dedication" is linked to speed at least in the beginning. For instance, any student can learn calculus given enough time and advice even starting from scratch. However, the syllabus wants all this to happen in one semester.
This gives you vicious and virtuous cycles: Students' learning speed increases with time and past success. So "talented" students learn quickly and have extra time to further explore and improve, leading to further success. Students who struggle with the time constraint are forced to take shortcuts like memorizing "magic formulas" without having time to really understand. Trying to close that gap is very hard work.
Thank you for the insight that academic (in a very broad sense) bulk-fixed-time approach does in fact produce both of the cycles, and the gap indeed only widens with time (speaking from personal experience, especially from my life as an undergrad student).
Reminds me of my personal peeve that "studying" should not be "being taught", studying is pursuit of understanding, "being taught" is what happens in primary school (and I'm aware I'm simplifying here).
I would say that you could generalize this even further outside of education. A few early successes in life can greatly accelerate one's trajectory, while early failures could set one many years back. And this happens independently of whether those events are due to skill or luck.
Indeed, speed is often read as "smarts" whereas I would maintain it's much more often "preparation". We can't on one hand believe in the plasticity and retrainability of the mind, while simultaneously believing that speed is something only a few are born with. On the nature/nurture scale, I think it's 20/80 or so - but prodigies and geniuses have an interest that keeps them thinking and learning 10x or 100x more than other kids, and a little bump that lets them get started easier and therefore much earlier.
This sets them up for fantastic success very quickly. [1] shows a great example of this.
I'm fond of saying "You can do anything you want, but wanting is the hard part", because to truly be a grandmaster, genius-level mathematician, olympic athlete, etc, requires a dedication and amount of preparation that almost nobody can manage. Starting late, with emotional baggage, kids, and having to spend 5 years relearning how to learn? Forget it.
1. https://danielkarim.com/how-to-become-a-genius-the-polgar-ex...
> I'm fond of saying "You can do anything you want, but wanting is the hard part", because to truly be a grandmaster, genius-level mathematician, olympic athlete, etc, requires a dedication and
I was having a problem agreeing with this subthread, and I have you to thank for putting it into words that I can finally formulate my disagreement against.
Have you never met one of those people for whom they did not need to "want"? They could literally phone it in and still do better than anyone else, no matter how dedicated they were. Even should practice/study be necessary for them, they benefited from it to some absurd proportion that I couldn't even guess to quantify. I've known more than one of these people.
I think most believe they don't exist for two reasons. The first is the ridiculous number of television shows and movies that depict motivation as being the key to success. We're just inundated with the (unsupported by evidence) that this is the means to extraordinary genius. Second, I would say that this is the most comforting theory. "Why yes, I could have been a gifted whatever or a talented something-or-other if I had put the time in, but I chose this other thing instead."
Maybe some would say we all need to believe this, that a society that doesn't believe in it is harsher or more unkind.
I think I have met those folks. Maybe not. And you're welcome!
They're just quick. But the ones I've met, at least, are quick to make associations. When I really dig and ask them to explain themselves or a concept, they usually make analogies to things they know, but I don't. Then I have to go learn that thing. Then they try the analogy again, but I haven't fully learned it from years of making analogies about it.
Years of grad school experience was painful like this, until I got to a point 10 years after grad school, after a PhD, and well into research, that I "just got" things (in my subfield) as well. It's these experiences that made me feel that it's 80% preparation and perspiration (both of which are dominated by time), and 20% "other" mythology. Don't get me wrong, that 20% is what makes a 2 year old read earlier than others, and getting started reading at 2 (and continuing it!) for 4 years before starting school will make you light years ahead of your peers. The same goes for chess, math, etc etc. There is something legendary about Oppenheimer learning enough dutch in 6 weeks to deliver a lecture. Or perhaps learning to translate his lecture and memorizing it. Who knows.
Do we really believe there's a magical "genius" such that they can do anything? No, so what are the limits to their genius? The limits are defined by what they are a genius at. This is a tautological definition.
I'm not saying "Anyone at any time can become a genius at anything". I'm saying "If you take a kid, start early, and cultivate them just right so that you have time to realize compounding effects, - you can let them grow into basically anything" (probablistically speaking - there are learning disabilities and physical issues etc).
> I think most believe they don't exist for two reasons.
I add third (okay, 2b) - because the pain of coming up with the fact other people are better than you at a deep, fundamental level is too overwhelming.
Bobby Fisher won his first US Championships at 14 against people who had been playing chess longer than he had been alive. Suggesting they didn't want it more, or practice more than some kid is silly.
"We can't on one hand believe in the plasticity and retrainability of the mind, while simultaneously believing that speed is something only a few are born with."
Sure we can, the initial orientation of neurons differs between people, so some people need less "plasticity and retrainability" to be good at a task. Plasticity is physical characteristic like height and varies between people.
Initial speed usually isn't that important, but speed of learning is important and makes the difference between possible and impossible within a human lifetime.
I think there's a probabalistic argument I'm making that's more in line with the article.
Yes - there will be 10x-ers. And that group will have a 10x-er iside it, and so on given exponential dropoff of frequency of talent. Bobby Fisher is a few std dev above even the best, perhaps.
Generally speaking, "You can do anything you want, but wanting (enough, and naturally) is the hardest part" might need a three standard deviation limit.
Have you heard the phrase: Being average among those who practice makes you 9X% among the population? I think that's what I'm saying - you can be a top performer if you dedicate yourself, especially early enough, but almost nobody will.
I agree with you. I don’t think I’m naturally gifted at much (I’m just average), but I was taught stubborn hard work pretty early on. Unfortunately it took me until my 20s to figure out I could be athletic if I applied that hard work. I could also be good at programming doing the same. I’ve met people who are truly gifted and it’s amazing, but I’m pretty decent at the things I worked hard at.
"Initial speed usually isn't that important, but speed of learning is important and makes the difference between possible and impossible within a human lifetime."
Likely so, but is suggest that personality, drive and motivation are also very important factors. I know from experience that stuff I had little interest in as a youngster and that I've still little in I still know little about.
Yes, my interests have grown and broadened over the years but simply I regard some stuff so irrelevant to my life that it's not worth a second thought and I am much better off applying my limited number of neurons to matters of greater importance and enjoyment.
Of course, no one has the luxury of just learning about what one finds interesting and or enjoyable, life's knocks and experiences along with utilitarian-like imperatives force one to learn stuff they'd rather not know about.
I find it is good to go back to things you struggled with in the past and come at them with a new and broader understanding.
> I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus.
It might be different for intellectual skills but I am not that sure.
Almost anyone can become decent at almost anything though. Which is good already!
> I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus.
Being tall isn't a skill. I suspect you could be skillful enough at basketball to overcome the hight disadvantage. However, I think most people who might become that skillful see the high disadvantage (plus the general difficulty of becoming a pro basketball player) and take a different path through life. It's also possible that the amount of time that would be needed to grow your skill past the height disadvantage is too long, so it's not feasible to do it to gain a position in the NBA.
Intelligence is also not a skill, but the thing that makes you skillful in all cognitive tasks. Just like what height does to basketball players.
>Intelligence is also not skill, but the thing that makes you skillful in all cognitive tasks.
Careful with that "all", even the most highly intelligent humans still have peaks and deficits in different domains.
It's a matter of the definition. The general factor of intelligence, which is measured through various somewhat lossy proxies like IQ tests, is exactly the degree to which someone exceeds expectation on all cognitive tasks (or vice versa).
The interesting finding is that this universal correlation is strong, real, and durable. Of course people in general have cognitive domains where they function better or worse than their g factor indicates, and that's before we get into the fact that intellectual task performance is strongly predicated on knowledge and practice, which is difficult to control for outside of tests designed (successfully, I must add) to do so.
Height is one physical attribute that helps, and professional players are mostly above average height for a reason. But also hand-eye coordination and fast-twitch muscles help even more. Many basketball players are very explosive athletes, because it's a sport with a relatively small play area and lots of quick movements are needed.
Track and swimming are where innate physical attributes have the most obvious benefits. Michael Phelphs had the perfect body for swimming. There is no amount of trainingg that 99.999% of the population could do to get close to what Usain Bolt ran. Most humans could not train to run under 4 minutes in a mile or under 2:30 in a marathon. They just don't have the right muscular and cardiovascular physiology.
Team sports are of course more complicated as other qualities come into play that aren't as directly physiological.
> Most humans could not train to run under 4 minutes in a mile or under 2:30 in a marathon.
Of course, but I don't think anyone was seriously suggesting that. The vast majority of humans can become pretty good at swimming though. And that was my interpretation of the original claim about cognitive tasks, mathematics, etc.
Most NBA players are under 2 meters tall. The average height is 1.99 meters.
https://www.lines.com/guides/average-height-nba-players/1519
Since we are being pedantic, your statement may be true but it is unsupported by the data you presented. To make it simple, let's talk about the imaginary basketball league with four players, of unit less heights of 4, 4, 4, and 1. The average height is 3.25, yet 3/4 the players are taller than average.
A paid promotion of International Median is not Average Association.
Most people have an above-average number of legs.
What's the average number of legs for humans ?
A bit less than two
What about Frank Lentini ?
https://en.m.wikipedia.org/wiki/Frank_Lentini
Seems like he has more than the average number of legs as well.
The fact that he has a wiki page, and that many folks with born without or who have lost legs (~500,000/year Americans experience limb loss or are born with a limb difference https://amputee-coalition.org/resources/limb-loss-statistics...) do not, suggest that the number of people with < 2 is far greater than the number of people with > 2. So the average is still less than 2.
For better or worse, number of legs (or number of arms) is canonical example people use to demonstrate the statistical principal a significant majority of a population can be above average of some metric.
Simpson's Paradox[0] is the reason people are so easily seduced by the tempting, but dead wrong, illusion that humans are in any sense equal in their innate capacity for anything.
Because it turns out that, in the NBA, height does not correspond with ability! This of course makes sense, because all the players are filtered by being NBA professional basketballers. A shorter player simply has more exceptional ability in another dimension, be that dodging reflex, ability to visualize and then hit a ball trajectory from the three point line, and so on. Conversely, a very tall player is inherently useful for blocking, and doesn't have to be as objectively good at basketball in order to be a valuable teammate.
Despite this lack of correlation, when you look at an NBA team you see a bunch of very tall fellows indeed. Simpson's Paradox.
We see the same thing in intellectual pursuits. "I'm not nearly as smart as the smartest programmer I know, but I get promoted at work so I must be doing something right. Therefore anyone could do this, they just have to work hard like I did". Nope. You've already been selected into "professional programmer", this logic doesn't work.
[0]: https://en.wikipedia.org/wiki/Simpson's_paradox
So you're saying success at maths isn't an inbuilt ability. Instead, it depends on an (inbuilt) ability to hyper focus... Which you are just born with?
Not even that. It depends on the learned ability to stop pushing yourself when your focus is wavering. That's how you develop aversion towards the topic. Let your natural curiosity draw you to particular topics (that's why you might have a winding road through the subject).
parent comment was a bit tounge-in-cheek but I'll continue the sentiment: You're saying that the curiosity is "natural" hence one is either born with it or not. I think that there is no way around the fact that it will be hard and uncomfortable to mimic the progress of someone that has an innate inclination towards a subject (be it talent or focus or curiosity) artificially.
Hey, that doesn't have to be what "natural curiosity" means. Besides which it makes no sense to say people are born with complex interests. I mean, OK, your genes might incline you a certain way, but that's not the same thing.
Being interested in a subject is massively helpful to learning it. But interest arises circumstantially, it's an emotion. The grim reality that it would be really useful to you to learn a certain subject does not necessarily make you interested in the subject, unfortunately. (Perhaps "financially interested", but that's something else.)
I think there is some natural inclination towards abstract thinking versus more grounded in reality, just judging based on kids I know. Some of them really enjoy playing with ideas in their heads, some enjoy playing with things they can touch more. It seems likely that those different attractions would express themselves in how much they practice different things as time goes on.
I was talking about curiosity in general not curiosity about something in particular. We are naturally inquisitive to the point we have to be restrained by our parents. The problem is some of the restraints are based on the fears of our parents and not on actual dangers. Also, it's hard to develop an appreciation for something when it's forced fed to you.
> You're saying that the curiosity is "natural" hence one is either born with it or not.
Why does curiosity being natural necessarily mean some people are born without it? It could also mean everyone (or every average human) is born with it, and overtime it gets pushed out of people.
Some infants explore vastly more than others.
So the minimum might not be zero, but it isn’t some fixed quantity.
That's not because it doesn't exist, it's because it's restrained
How?
People do studies on weeks old babies, it seems unlikely curiosity has been restrained at that point.
>I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
And this also gives the proponent (you in this case) an excuse to blame a person for not focusing hard enough or not being dedicated enough if they don't grasp the basics, let alone excel.
I don’t know if I agree. Grad school was profoundly humbling to me because it really showed me that there are a LOT of people out there that are just much much better than me at math. There are different levels of innate talent.
>The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
Nonsense!
The brain you are born with materially dictates the ceiling of your talent. A person with average ability can with dedication and focus over many years become reasonably good, but a genius can do the same in 1 year and at a young age.
We have an education system which gives an A Grade if you pass the course, but 1 person may put on 5 hours a week and the other works day and night.
What makes you think that "genius" is nature and not nurture? I'd love to see the evidence for this; i'm deeply skeptical.
Edit: I don't mean to argue that there aren't genetics involved in determining aptitude on certain tasks, of course, but the assumption that genius is born and never made feels like a very shallow understanding of the capacity of man.
> I'd love to see the evidence for this; i'm deeply skeptical.
Cool, come and have a coffee with me :) I have older and younger siblings and was the one randomly blessed.
Whereas most recognised talents are associated with hard work and so there is then this visible link, I am a good example as I did the bare minimum throughout education (and beyond...).
The way my brain processes and selectively discards/stores the information it receives is very different to majority of the population. I have no control over it.
I take zero credit for any of my achievments - I regularly meet intelligent people near to retirement who have been to a tier 1 university, may have PHDs, worked 60 hours a week since they were born, been on course and what not and cannot reach the levels I can.
My nurturing was no different to siblings/peers (and was terrible!)
Note: I have my weaknesses too, but as a whole, I am exceptional. Not through effort!! Completely random - neither of my parents are intelligent and nothing up the ancestary tree as far as I know.
I am also exceptional in many ways, (some of them negative), and some of this is clearly inherited and likely genetic. I share too many innate strengths with my father and, to a lesser extent, my siblings to disagree with this. But I just don't know how you could preclude developmental factors like "when you started reading as a child", "what sort of puzzles and games you played as a child", "lack of trauma as a child", etc.
The boostrap skill is the ability to obsess over something. To focus and self-reward on anything is a heaven sent. Good thing we do not medicate that if we are unable to get that energy on the road, that base skill.
"What we should be teaching children is HOW to learn."
Absolutely correct. And that begins with getting their interest, thus their attention; and that's a whole subject in and of itself.
I've had some success converting people by telling them others had convinced them they were stupid. They usually have one or two things they are actually good at, like a domain they flee to. I simply point out how everything else is exactly like [say] playing the guitar. Eventually you will be good enough to sing at the same time. Clearly you already are a genius. I cant even remember the most basic cords or lyrics because I've never bothered with it.
I met the guitar guy a few years later outside his house. He always had just one guitar but now owned something like 20, something like a hundred books about music. Quite the composer. It looked and sounded highly sophisticated. The dumb guy didn't exist anymore.
But also, some people are stupid, right?
Intellect is like a gas, it will expand to fill its container. The container, in humans, is epigenetic and social — genetics only determines how hot or cold your gas is, ie how fast and how fluidly it expands, but you’re taught your limits — it’s best to see stupid as not how limited you are relative to other but what limits you have now and may abandon in the future.
That said, some people received a smaller starting container, and might need some help cracking it. That’s the work of those who think they’ve found a bigger one.
The inborn part is how quickly you get results (good or bad). Stupidity is the results.
If we spent 50% of time thinking productively - inborn thinking speed would matter. But in my estimate even 5% is generous.
So it matters far more what kind of feedback you have to filter out the wrong results, and how much time you spend thinking - than how quickly you can do it.
Also practice helps with speed.
> The belief that some people have an innate skill that they are born with is deeply unhelpful.
In practice your views result in stinting access of non-existent (in your opinion) talented children to a faster education track. They don't exist therefore they don't need different treatment (finer points get lost when the idea disseminates). Quite a hot theme in American education two (or so) years ago.
Math is a stepping stone to critical thinking skills. And while one can probably learn those skills in any way, math forces you to learn those skills by learning the method of writing proofs. No other field forces you to push your critical thinking skills to the limit that math does.
> Whilst some people (mostly spectrum) do seem have an innate talent
I think the only thing in autism that I'd call an innate talent is detail-oriented thinking by default. It'd be the same type of "innate talent" as, say, synesthesia, or schizophrenia: a side effect of experiencing the world differently.
> a side effect of experiencing the world differently
A side effect for which there is a substantial, lifelong, and most importantly wide cost, even if it occasionally confers usually small, usually fleeting, and most importantly narrow advantage.
At such cost with such narrow advantage, why has it persisted so pervasively? I would counter that the advantage is wider and the cost narrower than your current value system is allowing you to accept.
Natural selection doesn't care about cost or advantage, only reproduction.
It is the sum of costs and advantages that lead to reproductive success. The trait is still here and still prevalent meaning people are still getting laid and starting families and presumably leading fulfilling lives.
I'm not sure what you are trying to say.
> I'm not sure what you are trying to say.
I'm saying, if it doesn't ruin lives to the point of preventing reproduction, then it stays in the gene pool.
Basically, I'm saying this:
> The trait is still here and still prevalent meaning people are still getting laid and starting families and presumably leading fulfilling lives.
As long as an organism isn't performing too badly, it stays in the gene pool. It can persist and even share its genes more broadly, if in diluted form, to the other more successful organisms. And then some of those mixed-genes organisms may occasionally express more strongly, but again not enough to affect reproductive success across the population.
Yes, there is a significant cost to being built differently regardless of perceived advantages (by one's self or others). For example, as an autistic, I have to cope with finding interaction with non-autistics quite difficult for me, even if detail-oriented thinking can make certain tasks seem easier to me.
> The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.
My first thought when the article got to the dialog between logic and intuition bit was that the same is true for school level physics.
> an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
I see you're familiar with Ryan North.
> I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
I would argue something different. The "skill" angle is just thinly veiled ladder-pulling.
Sure, math is hard work, and there's a degree of prerequisites that need to be met to have things click, but to the mindset embodied by the cliche "X is left as an exercise for the reader" is just people rejoicing on the idea they can needlessly make life hard for the reader for no reason at all.
Everyone is familiar with the "Ivory tower" cliche, but what is not immediately obvious is how the tower aspect originates as a self-promotion and self-defense mechanism to sell the idea their particular role is critical and everyone who wishes to know something is obligated to go through them to reach their goals. This mindset trickles down from the top towards lower levels. And that's what ultimately makes math hard.
Case in point: linear algebra. The bulk of the material on the topic has been around for many decades, and the bulk of the course material,l used to teach that stuff, from beginner to advanced levels, is extraordinarily cryptic and mostly indecipherable. But then machine learning field started to take off and suddenly we started to see content addressing even advanced topics like dimensionality reduction using all kinds of subspace decomposition methods as someting clear and trivial. What changed? Only the type of people covering the topic.
I saw a lot of this when I went to college for engineering, some professors had this ability (or willingness) to make hard things simple, and others did the opposite, it was the same with the books, I dreaded the "exercise for the reader" shit, I don't think I ever did any of those, so those were all proofs I never got.
I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
> I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
I don't agree. First of all, ladder-pulling in math is observed at all levels, not only cutting-edge stuff. Secondly, it's in applied mathematics where pure math takes a queue onto where to focus effort. See how physics drives research into pure math.
> A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's easy, then it means you already know this material, and you're wasting your time.
It’s a well-established effect in pedagogics that learning vs. difficulty has a non-monotonic relationship, where you don’t learn efficiently if the material is either too hard or too easy compared to your current level. There is an optimum learning point somewhere in-between where the material is “challenging” – but neither “trivial” nor “insurmountable” – to put it that way.
It's funny because I've had the opposite heuristic most of my line: the things I want to do most are whatever is hardest. This worked great for building my maths and physics skills and knowledge.
But when I started focusing on making money I've come to believe it's a bad heuristic for that purpose.
> If it's easy, then it means you already know this material, and you're wasting your time
I think that's also a trap. Even professional athletes spend a little bit of their time doing simple drills: shooting free throws, fielding fly balls, hitting easy groundstrokes.
Sometimes your daily work keeps up the "easy" skills, but if you haven't used a skill in a while, it's not a bad idea to do some easy reps before you try to combine it with other skills in difficult ways.
I cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains.
I've taught math to psychology students, and they just don't get it. I remember the frustration of the section's head when a student asked "what's a square root?" We all know how many of our fellow pupils struggled with maths. I'm not saying they all lacked the capability to learn it, but it can't be the case they all were capable but "it was the teacher's fault". Even then, how do you explain the difference between those who struggled and those who breezed through the material?
Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so.
So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.
Not the original poster, but I want to push back on one thing -- being capable of something and being one of the best in the world at something are hugely different. Forgive me if I'm putting words in your math -- you mentioned "placing the bar for mathematical skill pretty" low but also mentioned running a sub-10s 100m. If, correspondingly, your notion of mathematical success is being Terence Tao, then I envy your ambition.
I do broadly agree with your position that some people are going to excel where others fail. We know there trivially exist people with significant disabilities that will never excel in certain activities. What the variance is on "other people" (a crude distinction) I hesitate to say. And whatever the solution is, if there is even a solution, I'd at least like for the null hypothesis to be "this is possible, we just may need to change our approach or put more time in".
On a slightly more philosophical note, I firmly believe that it is important to believe some things that are not necessarily true -- let's call this "feel-good thinking". If someone is truly putting significant dedicated effort in and not getting results, that is a tragedy. I would, however, greatly prefer that scenario to the one in which people are regularly told, "well, you could just be stupid." That is a self-fulfilling prophecy.
> cannot agree. It's just "feel-good thinking."
Not really. There's nothing inherently special about people who dedicated enough time to learn a subject.
> "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained.
What a bad comparison. So far in human history there were less than 200 people who ran 100m in less than 10s.
I think you're just reflecting an inflated sense of self worth.
> Not really. There's nothing inherently special about people who dedicated enough time to learn a subject.
"You didn't work hard enough." People really blame you for that, not for lacking talent.
> So far in human history there were less than 200 people who ran 100m in less than 10s.
And many millions have tried. There may be 200 people who can run it under 10s, but there are thousands that can run it under 11s, and hundreds of thousands that can run it under 12s. Unless you've got clear evidence that those people can actually run 100m in less than 10s if they simply try harder, I think the experience of practically every athlete is that they hit a performance wall. And it isn't different for maths, languages, music, sculpting (did you ever try that?), etc. Where there are geniuses, there also absolute blockheads.
That's not to say that people won't perform better when they work harder, but the limits are just not the same for everyone. So 'capable of mathematical reasoning' either is some common denominator barely worth mentioning, or the statement simply isn't true. Clickbait, if you will.
I'm the author of what you've just described as clickbait.
Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.
The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.
Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).
When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.
As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.
In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.
There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.
I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.
Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.
A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.
In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.
This power law argument immediately reminds me of education studies literature that (contrary to the math teachers in this thread) emphasize that mathematical ability is learned cumulatively, that a student's success feeds on itself and advances their ability to grasp more difficult material.
As for my own half-baked opinion, I want to say that the Church-Turing Thesis and Chomsky's innate theory of cognition have something to add to the picture. Homo sapiens as a species essentially has the capacity to think analytically and mathematically; I want to argue this is a universal capacity loosely analogous to the theory of universal Turing machines. So what matters is people's early experiences where they learn how to both practice and, critically, to play, when they learn difficult ideas and skills.
Also, as an amateur pianist, most people don't know that modern piano teaching emphasizes not fixed limits of the student but that many students learn the wrong techniques even from well-meaning piano coaches. Just the other day I was watching a recent YouTube Julliard-level masterclass where the teacher was correcting a student on her finger playing technique, presumably this student had been taught the wrong technique since childhood. With music or sports a coach can visually see many such technique problems; with math teaching it of course harder.
This beats TFA. Interesting relation between cumulativeness and distribution ("Yule process"). But how does this explain variation is how quickly children pick up maths - would you argue it's due to prior exposure e.g. parental tutoring?
Any comments on the "10x programmer"?
There is math the abstract field and math the concrete example you're working on.
Current education is _extremely_ biased to concrete arithmetic and a bit of algebra. If you have a predisposition to either you will do extremely well. If you don't you won't.
Those have little to do with how math is done by mathematicians.
In short: education needs to catch up to what's happened since the 1920s in maths. Parents are conservative and don't want their kids to learn something they themselves don't understand, so we're stuck with what we have until enough generations pass and 20th century math is absorbed by osmosis into the curriculum.
> document the secret "oral tradition" of top mathematician
> A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.
Sounds like people mostly living in different bubbles? What do they know about the world?
They aren't hanging out with the kids who fail in school because maths and reading and writing is to hard, and then start selling drugs instead and get guns and start killing each other.
> [they] don't think it's because of their genes
Do you think someone would tell you, if he/she thought it was?
I mean, that can come off as arrogant? Wouldn't they rather tend to say "it was hard work, anyone can do it" and prioritize being liked by others
> Pareto-like distribution like what we see in math
Unclear to me what you have in mind. If there's a graph it'd be interesting to have a look? I wonder whats on the different axis, and how you arrived at the numbers and data points
> Sounds like people mostly living in different bubbles? What do they know about the world?
Well, they do know something about math — in particular that it requires a certain "attitude", something that no-one told them about in school and they felt they only discovered by chance.
Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence". They viewed intelligence as a by-product of their method, not the other way around. They even wrote books as an attempt to share this method (which is quite hard to achieve, for reasons I explain in my book.)
Why do you bring "kids who fail in school" and "start selling drugs" into this conversation? What does it have to do with whether math genius is driven by genetics or idiosyncratic cognitive development?
And why would a mathematician be disqualified from discussing the specifics of math just because they're not hanging out with lost kids? Are you better qualified? Did you sequence the DNA of those kids and identified the genes responsible for their learning difficulties?
>> [they] don't think it's because of their genes
> Do you think someone would tell you, if he/she thought it was?
Well, an example I know quite well is mine. I was certainly "gifted" in math — something like in the top 1% of my generation, but not much above and definitely nowhere near the IMO gold medallists whom I met early in my studies.
A number of random events happened to me, including the chance discovery of certain ways to mentally engage with mathematical objects. This propelled me onto an entirely different trajectory, and I ended up solving tough conjectures & publishing in Inventiones & Annals of Math (an entirely different planet from the top 1% I started from)
My relative position wrt my peer group went through a series of well-delineated spikes from 17yo (when I started as an undergrad) to 35yo (when I quit academia), associated with specific methodological & psychological breakthroughs. I'm pretty confident that my genes stayed the same during this entire period.
And as to why I was initially "gifted", I do have some very plausible non-genetic factors that might be the explanation.
I don't claim this proves anything. But I see no reason why my account should be disqualified on the grounds that I'm good at math.
Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates. Why should math be the exception?
> they do know something about math ... that it requires a certain "attitude"
Of course. That does not mean that intelligence doesn't play a (big) role.
> Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence"
That doesn't make sense. Back when they were active, intelligence, IQ tests and the heritability of intelligence hadn't been well studied. They didn't have enough information, like we do today: https://en.wikipedia.org/wiki/Heritability_of_IQ#Estimates "Various studies have estimated the heritability of IQ to be between 0.7 and 0.8 in adults and 0.45 in childhood in the United States."
And, evolution and genetics weren't these peolpe's domains. Does it make sense to assume they were authorities in genetics and inheritance, because were good at maths and physics?
Sometimes they were wrong about their own domains. Einstein did say "Genius is 1% talent and 99% hard work" (I can understand how it makes sense from his own perspective, although he didn't know enough about this animal species, to say that).
But he also said "God does not play dice" and was wrong about his own domain.
> Why do you bring "kids who fail in school" and "start selling drugs" into this conversation?
It was an example showing that the researchers live in bubbles.
That they're forming their believes about humans, based on small skewed samples of people. There's billions of people out there vastly different from themselves, whom they would have left out, if thinking about about others' abilities to learn.
In fact, now it seems to me that you too live in a bubble, I hope you don't mind.
> Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates.
1) Maths and 2) evolution, DNA, genetics, intelligence, learning and inheritability are not the same domains.
Anyway, best wishes with the book and I hope it'll be helpful to people who want to study mathematics.
Current estimates of the "heritability" of intelligence are far, far lower than "0.7 or 0.8"; they're probably below 0.1, and that's before digging into what "heritability" means, which is not generally what people think it does.
I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.
> Current estimates of the "heritability" of intelligence are far, far lower than "0.7 or 0.8"; they're probably below 0.1
Sources please, if you have time? I tried to find something supporting what you wrote, but wasn't able to (this far).
Instead I found this from 2015:
Thinking positively: The genetics of high intelligence, https://pmc.ncbi.nlm.nih.gov/articles/PMC4286575/
They've studied men in Sweden during 40 years. From the abstract:
"We found that high intelligence is familial, heritable, and caused by the same genetic and environmental factors responsible for the normal distribution of intelligence."
"... 360,000 sibling pairs and 9000 twin pairs from 3 million 18-year-old males with cognitive assessments administered as part of conscription to military service in Sweden between 1968 and 2010 ..."
Looking at Figure 3, in that pager, about identical twins and non-identical (two-egg) twins -- I think that settles it for me.
Seems they arrive at a bit above 0.4 as heritability. Yes that's less than 0.7 - 0.8 but I wouldn't say "far far lower", and more than 0.1. Also, they're 18 years old, not adults.
It's been discussed at HN: https://news.ycombinator.com/item?id=10488998
> I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.
Well, in his reply to me, he was sort of name dropping and appealing to (the wrong) authorities, didn't make a good impression on me. Plus writing about himself, but he's a single person. -- I would have preferred links to research on large numbers of people.
> what "heritability" means, which is not generally what people think it does.
That sounds interesting. Can I guess: You mean that people believe that heritability means how likely a trait is to get inherited from parent to child? When in fact it means: (https://en.wikipedia.org/wiki/Heritability) "What is the proportion of the variation in a given trait within a population that is not explained by the environment or random chance?"
Dear cutemonster,
I know this reply may not suffice to convince you, but unfortunately I won't be able to argue forever.
Did you ever consider the possibility that you might be the one living in a bubble?
FYI, the concept of innate talent predated IQ tests and twin studies by many millenia. Two of the authors I'm citing in my book (Descartes and Grothendieck) believed that innate talent existed and they both declared they would have loved to be naturally gifted like these or these people they knew.
You're declaring that these incredibly smart people were wrong about their own domains, which is a pretty bold claim to make. What do you have in support of this claim? A fake Einstein quote?
It's a sad fact of life that most quotes attributed to Einstein are fabricated. Next time, please check "The Ultimate Quotable Einstein", compiled by Alice Calaprice.
This may come as a shock to you, but Google page 1 isn't always a reliable resource. Nor is Wikipedia, even though it's quite often correct. As it happens, there's a pretty large "Heritability of IQ" bubble on the internet. It's active and vocal, but it's also quite weak scientifically — the page you're citing is a typical symptom, and it absolutely doesn't reflect the current scientific knowledge.
The IQ heritability claims that you're citing are based on twin studies and they have taken in serious beating in the past decade, especially in light of GWAS.
It's true that a number of people have been fooled by twin studies, most notably Steven Pinker, in Chapter 19 of the Blank Slate (did you read it?)
You see, Pinker is a linguist and apparently he isn't mathematically equipped to fully comprehend the intrinsic limitations of Bouchard's approach. Did you read Bouchard's 1990 paper on twins reared apart? Do you find it convincing? Are you aware that even The Bell Curve's Charles Murray thinks that this approach, abundantly cited by Pinker, is structurally flawed? Are you aware of the fundamental instability of IQ estimates based on twins reared together? Aren't you concerned that even a mild violation of Equal Environment Assumption, plugged into Falconer's equation, would drastically reduce the estimates?
If you don't understand what I'm talking about, if you've never read the authors and the primary research I'm citing, then it's quite likely that you're the one living in a social media bubble.
If you're interesting in learning more about the actual science of IQ heritability, I recommend using Sasha Gusev's Substack as an entry point: https://theinfinitesimal.substack.com/p/comments-on-no-intel...
Feel free to also subscribe to my own Substack, where I plan to cover these topics in the coming months: https://davidbessis.substack.com
All the best, David.
Some of the stuff on Gusev's substack is pretty startling, and I highly recommend it.
Thank you for taking the time to comment here!
> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.
Sure they don't. Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work. It runs completely contrary to the findings of modern biology and psychology, most of our intellectual potential at adulthood is genetic.
The floor and ceiling you will operate on in your life is decided the moment of chromosomal crossover.
> Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work
I suppose that makes sense from their own personal perspectives (but that doesn't make them right), in that they had to put in lots of time and work, but didn't do anything to become bright people.
> The floor and ceiling you will operate on in your life
Interesting that what you wrote got downvoted. Lots of flat-earthers here? (figuratively speaking)
So, for starters: you don't have any evidence, if I understood it properly. None whatsoever. That's really not the basis for arguing "become 1000x better." If only because your operationalization is missing. If you can't measure someone math's skills, how can you say they can become 1000x better? I think the whole article manages not to even speak about what "math" actually is supposed to be. Symbol manipulation according to axioms?
Your starting point is the way elite mathematicians think about themselves. But people don't understand themselves. They don't understand their own motivations, their own capabilities, their own logic. You know who are best at explaining what/how other people think? Average people. Hence the success of mediocrity in certain types of quizzes and politics.
I'm sure you're right about the mixture of logic and intuition. I've had the thought myself, mainly about designing systems, but there is some analogy: you've got to "see through" the way from the top to the bottom, how it connects, and then fill the layers in between. But that intuition is about a very, very specific domain. And it's not given that is a priori equally distributed. More likely than not, it's isn't.
Your whole argument then is based in naive psychology. E.g., this
> What can someone gain by improving their mathematical thinking?
> Joy, clarity and self-confidence.
> Children do this all the time. That’s why they learn so fast.
Are there no other reasons children learn so fast? It's not even given that joy and clarity makes children learn faster. What is known is that children do learn fast under pressure. Have you seen the skills of child soldiers? It's amazing, but it comes of course at great cost. But they did learn. Children pick up languages at a relatively high speed (note: learning a new language is still very well possible at later ages, certainly until middle age), but that's got nothing to do with joy, clarity and self-confidence. They also do it under the dreariest of circumstances.
So I'd say: your argument, or at least the quanta article, is at odds with common sense, and with psychological research, and doesn't provide concrete evidence.
You might have ideas for teaching maths better. But beware there's a long tradition of people who've tried to improve the maths curriculum, and basically all failed.
I'll give you one more point for thought (if you ever read this): intuition can also be a negative. I've practiced with my daughter for her unprepared math exam (she dropped it at one point, and then wanted to have it on her grade list anyway). One thing that I clearly remember, and it's not just her, is that she had very weird ideas about the meaning of e.g. x, even in simple equations. They were nearly magical. It was hard to get her to treat x like she would treat any other term. At one point, she failed to see that e.g. 1/3 = x^-1 is easy to solve, even when she had written down 1/x = x^-1 right next to it. Her intuition blocked her logic. My conclusion is that it's certainly easy to frak up someone's understanding of maths, unless you're really teaching, tutoring and monitoring 1-on-1. There's no solution for maths but good teachers, and a lot of fast feedback. Quite an old lesson.
You'll want to get your daughter to start using Math Academy for at least 30 minutes per day [0].
New solutions exist for old problems.
[0] https://www.justinmath.com/books/#the-math-academy-way
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You don't need to be able to run 100m in less than 10 seconds. But almost everyone probably could run a marathon in three and a half hours. How many people do you think have actualized their physical potential, or how far is the average person removed from it?
If someone's smart enough to get into a psychology class they are smart enough to be thought basic undergrad math. It wasn't your teaching failure necessarily, but it was someone's teaching failure at some point.
Not everyone can play Rachmaninov like Lugansky or do math like Terence Tao, but there is absolutely no doubt that almost all people are magnitudes away from their latent potential in almost all domains. I'm fairly certain you could teach any average person how to play Rachmaninov decently. You could bring any person to a reasonably high mathematical level. You can get any person to lift a few hundred pounds.
Most people today read at a 7th grade level, can't do basic math, and are out of air after 3 flights of stairs. "Everyone can do everything" is maybe not literally right but directionally right given how utterly far removed we are from developing practically anyone's potential.
There's a difference between being able to memorize what a square root is and being able to do math - which to mathematicians means being able to organize a proof.
I've found that the people who most believe in math being a genetic ability are the ones who do not work in the symbolic world of modern math, but in the semantic world of whatever the field the math describes is.
The two are rather different.
Square roots are not some "mathematical trivia", they're amongst the most fundamental operations in mathematics.
In arithmetic. There is a lot more to math than arithmetic.
Square roots are fundamental to (real and complex) analysis and to algebra (in the study of polynomials), so the two major branches of modern mathematics.
They are fundamental to the arithmetic used in analysis and algebra.
One can get along quite well without them once you replace the arithmetic expressions with more general objects.
Just come on. The square root of 2 is the easiest example of an irrational number, this has been known since Ancient Greece. You can't compute distances in Euclidean spaces without the square root. "Solving equations by roots" is the bread and butter of algebra. Adjoining roots to a field is how you get Galois Theory. Several algorithms related to number theory have complexity O(sqrt(n)). And so on.
You chose an extremely poor example and now you're trying to die on that hill. Please don't die on that hill.
Strangely enough, you'd be hard pressed to find a mathematician who doesn't know what a square root is.
And I didn't mention genetics. Nature is complicated.
You'd also be hard pressed to find one who doesn't know how to flush a toilet.
Neither trivia has anything to do with being good at mathematics as done by mathematicians.
Are you an LLM? You brought up the point of mathematicians not knowing what a square root is yourself. Anyway, the square root is is so many levels below maths as done by mathematicians, it's laughable.
Pretty sure you followed the thread wrong. They only mentioned the square root in response to you?
It certainly is suspicious that their name starts, "llm"
This is mostly correct. Working memory plays a huge component in grokking more complicated mathematical components, and IQ itself is separated into performance and verbal IQ (which together constitute your IQ score) and its demonstrably robust. Some people find this easier than others and that is OK.
I dont disagree with the premise that mathematical thinking can benefit anybody, but its absurd notion that everything abstract is teachable and learnable to all is a fantasy of a distinctly left-wing variety, who would have you believe that everything is just social conditioning and human beings dont differ from one-another.
Imagine our world was extremely similar to how it is now in any way you'd care to imagine, except two things were different.
1. Everyone (young, old, poor, rich) thinks that maths is interesting and fun and beautiful and important. Not important "to get a good job" or "to go to a good college" or "to be an impressive person", but rather important because it's fun and interesting. And maybe it also helps you think clearly and get a good job and all these practical things, but they're secondary to the tremendous beauty and wondrousness of the domain.
2. Everyone believes that barring actual brain injuries people can learn mathematics to a pretty high level. Not Ramanujan level, not Terrence Tao, not even a research mathematician at one of the smaller universities, but a level of extreme comfort, let's say a minimum level of being able to confidently ace the typical types of exams 17 and 18 year olds face to finish secondary school in various countries.
Would you claim that in that world - people think maths is great, and that anyone can learn it - we'd see similar levels of ability and enjoyment of mathematics?
My claim is that we don't live in "Math-World", as described above, but "Anti-Math-World". And further, that anyone suggesting things have to be the way they are in Anti-Math-World is not only wrong, but also fundamentally lacking imagination and courage.
Kids are told week in week out that maths is stupid, that they are stupid, that their parents themselves are stupid, that the parents hated maths, that the teachers are stupid, and then when they end up doing poorly, people say: "ahhh, some kids just aren't bright!"
Parents who like things like learning and maths and reading and so on, have kids that tend to like those things. And parents that don't, usually don't. Saying that this somehow tells us something concrete and inalterable about the nature of the human brain is preposterous.
It's a card that's used by grown-ups who are terrified by the idea that our education systems are fundamentally broken.
"Kids are told week in week out that maths is stupid, that they are stupid. …."
Come on, how often are kids exposed to such stupid talk? I suspect very infrequently.
My grandmother, who wasn't stupid by any means but who knew only basic arithmetic, would never have uttered such nonsense.
And I'd stress, like many of her generation and background, her knowledge of mathematics was minimal, if she'd been ask what calculus was she'd likely have been perplexed and probably have guessed it to be some kind of growth on one's foot.
In math classrooms filled with 30 students on average for 12 years? With usually all 30 of those students having mathematically illiterate parents who can't help with their homework?
Every student hears it enough for it to be a widespread sentiment. It spreads like an infection.
I see it a lot on the internet, mostly I'm the young people places. People talk shit on math all the time, just like they do in person. Just slights and jabs that they've never needed the pythagorean theorem or to do an integral, and thus it was all wasted time and effort. You might just not hang around young people much, or at least their online congregations (i don't blame you). The idea that math education is dumb and useless is very much alive in the young adult to preteen tiktokified spaces online
> how often are kids exposed to such stupid talk? I suspect very infrequently
Pretty often, by other kids. And kids listen to kids, not well-meaning, mealy-mouthed adults.
I think most people can become fairly skilled in useful fields if educated properly, and the people who can't are a small group that can be cared for. I agree that even in a better education system, people aren't all going to be equally skilled in the same fields, just that most people can contribute something of value.
Anybody can do everything if we restrict everything to things everyone can do.
> Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so.
Bad example, it's much more likely to create a musical prodigy by providing early and appropriate guidance. Of course this is not easy as it assumes already ideal teaching methods and adequate motivation to the youngling, but even those with some learning difficulties have the potential to excel. The subtypes of intellect required to play complex music and absord advanced abstract math subjects are quite different, former requiring strong short-term memory (sightreading) the latter fluid intelligence -I think almost everyone is familiar with these terms by now and knows that one can score high/low on certain subtypes of an IQ test affecting the total score-.
BTW IDK if the Rachmaninoff choice was deliberate to imply that even the most capable who lack the hand size won't be able to perform his works well yeah, but there are like 1000s of others composers accessible that the audiences appreciate even more. Attempting to equate music with sports in such manner is heavily Americanized and therefore completely absurd. Tons of great pianists who didn't have the hand size to interpret his most majestic works and of others. Tons of others who could but never bothered. There have been winners of large competitions who barely played any of his works during all stages of audition or generally music requiring immense bodily advantage. Besides, it's almost 100% not a hand size issue when there are 5 year old kids playing La Campanella with remarkable fluidity.
And even in this case this isn't even the point. Most conservatory alumni today are 100x skilled than the pianists of previous generations... yet they all sound the exact same, their playing lacks character/variability, deepness, elegance to the point where the composers ideas end up distorted. And those can be very skilled but just have poor understanding of the art, which is what music is, not the fast trills/runs, clean arpeggios, very strict metronomic pulse.
> So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.
Well the vast majority of people in the Soviet Union were very math literate, regardless of what they ended up working as (although indeed most became engineers) and in quite advanced subjects. This is obviously a product of the extensive focus of primary and secondary education on the sciences back then.
So the point isn't to make everyone have PhD level math background and I heavily dislike the dork undertones/culture that everyone should love doing abstract math on their freetime or have to have some mathematical temperament' . But let's not go the other way and claim that those not coming close to achieving the knowledge those in the top % of the fields possess, they are chumps.
> the vast majority of people in the Soviet Union were very math literate
I doubt that.
> although indeed most became engineers
And that is demonstrably false.
Anyway, most of your argument boils down to: there's a bunch of people that can't do a certain task, there's a bunch that has mediocre skills, a few that are good, and a handful that's really good. There's no argument, just observation, not even related to effort, which is what this discussion is about.
And maths is no different from sports or music in that sense. Most people suck at math, and will always suck at it. The things described in the article are personal reflections of elite mathematicians. They have no bearing on development of knowledge and skills of us mortals, if only because those reflections have no truth value. It's all just "feel good" thoughts, no data, nothing provable, etc.
Yes, effort improves skill, but everyone has a limit, and the ones with the high limits we call talented.
> I doubt that.
Not even the most vicious polemicists of communist states deny that i.e. the superiority of the Soviet-era education system.
https://www.amazon.com/What-Ivan-Knows-Johnny-Doesnt/dp/4871...
which is also why the following statement
> Most people suck at math, and will always suck at it.
is strictly US-centric.
> Anyway, most of your argument boils down to: there's a bunch of people that can't do a certain task, there's a bunch that has mediocre skills, a few that are good, and a handful that's really good. There's no argument, just observation, not even related to effort, which is what this discussion is about.
No, you came up with that cause you have a very poor understanding of what constitutes a good musician which, like any other typical HNer believes, is another LeetCode type of thing where the more problems like a good little monkey you can solve, the smarter you become. And I already stated that there are people striving in the arts, even more than the ones with the supposedly 'better' skills according to absurd and clueless standards you set, i.e. no, those who can't access specific repertoire easily are not people that can't do a certain task or a bunch that has mediocre skills.
> And maths is no different from sports or music in that sense.
Let me guess, you also think that if painters can't draw photorealistically they're not deserving the artist title and lack talent? Or that the opera is all about who can sing the highest note?
Anyone who lumps every discipline within one other like that without realizing they require completely different things to be considered successful at and believe everything boils down to some supposedly 'objective' absurd video-game like character strategy always serves to remind that the US today has nothing to offer other than hi-tech bombing technology and horrific subculture.
Nowhere did I deny the existence of some people having the innate ability to absorb skills faster and better than the others and of course this is an interdisciplinary fact. But it definitely doesn't hold the same weight for every single discipline for one to strive.
> If it's easy, then it means you already know this material, and you're wasting your time.
One thing I'm anticipating from LLM-based tutoring is an adaptive test that locates someone's frontier of knowledge, and plots an efficient route toward any capability goal through the required intermediate skills.
Trying to find the places where math starts getting difficult by skimming through textbooks takes too long; especially for those of us who were last in school decades ago.
LLMs hallucinate and are completely the wrong tool for this. People need to learn about the history of AI.
Expert-systems using techniques from the 80s are doing this right now [0].
[0] https://www.mathacademy.com/how-our-ai-works
>and plots an efficient route toward any capability goal through the required intermediate skills.
LLMs currently can't find efficient paths longer than 5 hops when given a simple itinerary. Expecting them to do anything but a tactical explanation of issues they have seen in training is extremely naive with something as high dimensional as math.
I am trying to stress pushing through these barriers with my kid right now. The second her brain encounters something beyond its current sphere she just shuts down.
I have heard it is the ego protecting itself by rejecting something outright rather than admitting you can't do it. It still happens to me all the time. My favorite technique was one I heard from a college professor. He starts reading while filling a notepad with sloppy notes, once a page is filled he just throws it away. He claimed it was the fastest way to "condition his brain to the problem space". More than the exercise I like the idea that your brain cannot even function in that space until it has been conditioned.
As a kid I was also terrible at maths, then later became obsessed with it as an adult because I didn't understand it, just like OP. It was the (second) best thing I've ever done! The world becomes a lot more interesting.
I haven't been able to grasp maths as a kid nor as an adult.
I've tried night classes, tutors, activities. Nothing sticks.
Even the standard 12x tables I struggle at. I want to understand it but my brain just can't understand the non-practicality side of things.
My best friend was like that. Couldn't see the practicality until he got bit by a geology and water science bug. He went from calling me to get help figuring out percentages to doing chemistry equations in his head because he "got" the applicability.
My brother's mom tutors math. One of her insights with a former student was that they were in need of forming some number sense. She started by walking them both out to the street: "how many tires are there on this street of parked cars?" The student, already flummoxed, started panic guessing. So she started with counting.
For times tables, have you developed any intuition around it? For me, times tables are rectangles composed of unit squares and that helps with my intuition. Modern Common Core standards in the US focuses a lot on exposing different mental models to students. And after seeing the same 4x6 enough times your brain will automatically associate that with its solution. Instead of calculating, it is memorized.
My brain doesn't require car tires, geology, or other practical needs: it likes puzzles. I struggle with medical stuff and I can feel my brain switching to meh-mode and hardly anything sticks. I don't know how many times I have been told about the different kinds of sugar and how your body uses that energy and I would still have to look it up.
> For times tables, have you developed any intuition around it?
I've tried different approaches. 4x3 being 3x4
But somehow I end up miscounting and giving the answer for 4x4 or be off a digit every-time.
A good example was that I was at a brewery last night.
They didn't do pint glasses but glasses of: 1/4, 1/2 and 2/3rd's.
I thought 1/2 was more than 2/3rd's so I ordered a 1/2 rather than thinking I was getting more.
However I was unable to visual how 2/3 is more than 1/2 when 1/2 is half a pint, or half a glass.
My visual capabilities are great at others but just couldn't formulate the equation of 2/3 is more than 1/2.
Very simple stuff, but it just doesn't meld.
What _is_ a fraction to you? How do you visualize it? I didn't calculate to decimal to compare the size of those fractions (I couldn't tell you intuitively if 5/7 is more than 9/13, I would have to convert the denominator or calculate the decimal).
For me, a progress bar or a pizza is the default. And because I cook rice daily and we were talking volume, my "progress bar" was like a measuring cup. Mentally, I can stand two measuring cups next one another, and filling one of two parts is less than filling two of three parts.
Maybe there are ways to be more aware of or insert ways to force the use of fractions in physical space. More cooking, more building, etc. The more comfortable you get the more intuition you should build
Famously the A&W restaurant chain failed when introducing a 1/3 lb hamburger because many customers thought it was smaller than a 1/4 lb hamburger.
https://awrestaurants.com/blog/aw-third-pound-burger-fractio...
> However I was unable to visual how 2/3 is more than 1/2 when 1/2 is half a pint, or half a glass.
Maybe visualize splitting a pint with a friend. If you split the pint into 2 equal parts and each of you gets 1 of those 2 parts you each get the same amount.
Then visualize splitting it instead into 3 equal parts. You get 1 of those parts and your buddy gets 2. There's no fractions there so it should be easier to visualize that your buddy got twice as much as you did.
Comparing those two visualizations might make it easier to see that someone who gets 2/3 of a pint gets more than someone who gets 1/2 of a pint.
> I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
Absolutely. There's also a pernicious idea that only young people can master complex maths or music. This is a self-fulfilling prophecy - why bother try if you're going to fail due to being old? Or perhaps it's an elitist psy-op, giving the children of wealthy parents further advantage because of course no-one can catch up.
I grow increasingly convinced that the difference in “verbal” and “mathematical” intelligence is in many ways a matter of presentation.
While it’s indisputable that terse symbolic formalisms have great utility, one can capture all the same information verbally.
This is perhaps most evident in formal logic. It’s not hard to imagine a restricted formalized subset of natural language that is amenable to mechanical manipulation that is isomorphic to say modal logic.
And finally, for logic at least, there is something of a third way. Diagrammatic logical systems such as Existential Graphs capture the full power of propositional, predicate, and modal logic in a way that is neither verbal nor conventionally symbolic.
Amazingly, I believe that today, with the myriad of tools available, anyone can advance in sciences like mathematics at their own pace by combining black-box and white-box approaches. Computers, in this context, could serve as your personal “Batcomputer” [1]. That said, I would always recommend engaging in social sciences with others, not working alone.
Who knows? You might also contribute meaningfully to these fields as you embrace your own unique path.
[1] https://dc.fandom.com/wiki/Batcomputer
I'd like to improve my mathematical knowledge (which has degraded to basically zero).
Can you talk about the path or resources you are using to improve here?
If you know how to multiply and divide, Math Academy will take you to college level math pretty quickly.
This perspective has discouraged so many people from exploring their potential
How have you been working on it? Asking for a friend ;)
I'd strongly recommend Math Academy for at least 30 minutes per day.
easy_things -> comfort_zone
I took an online electronics tech course 15 years ago and what got me was my math skills were atrocious. Not shocking since like learning a new language or music use it or lose it is the obvious answer to why I sucked. I spent half my time re-learning math just so I could complete the course.
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Most people don't try hard at the gym. Most people don't try hard at music. Most people aren't comedians.
This reads like the foreword to the incel handbook.
And that's obviously bad because incels are not real people with feelings.
It is bad because it suffers from misattribution error, ultimately not leading to any solution and often making the situation worse. A downward spiral of misinterpreted signal
> ...my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
As a child of the 80s and 90s, "getting girls as a programmer" made me snort. Nerds do seem to have it a bit better now; the money/financial security of software development helps. But as a whole, we developers are still less socially capable than our sales/hr/marketing counterparts
"A loser in societal view"... What does that objectively mean? That only reads like you had or have a low sense of self worth. It must've been your perceived definition of what society is because how could you have come to such a conclusion? I think I'd actually subconsciously tend more to viewing someone as "a loser" if they made such a statement because it comes off as self victimization (without an apparent explanation to an outside observer).
And what's the shtick about girls? What are and were you looking for, love and a genuine relationship or attention to compensate for something? Personally I think your values and personality are what matter most and personality is usually what people fall in love with. Though charisma can help a lot to get the ball rolling. Most of what it takes is to treat people normally and nicely and you will have as much of a chance to find love as most people.
Though respect from peers and attention from women ideally shouldn't be your driving force. I think curiosity and passion are much better driving forces that don't involve such external factors and possibilities for insecurities.
Your post reads as if it expresses a frustration and a sense of entitlement. You may not be intrinsically entitled to the things you think you are. Think about that for a bit and try to be rational.
I think he lives in or are from India, society and family expectations are different there, not his "fault".
@faangguyindia, I hope everything will be ok one day :-) Or maybe it already is? You wrote "was".
I might be wrong but I think if you (@faangguy) manage to create a life that makes you feel happy, the women will notice that you're happy, and that goes a long way. But you'll also need to be somewhere where there are some women around? (obviously) If you're in a FAANG in SF maybe for the moment that's not so easy
You will stay a loser as long as you care about what some fictional mystical society thinks of you.
Do the stuff you're good at, provide for your family, earn the respect of your peers and forget about the rest.
I assume OP is an Indian. And from what I've observed, Indian society is highly paternalistic and status-seeking in nature. Parents demand marriage and grand-children as soon as their offspring hit a certain age and success.
This just demonstrates that you dont understand how sexual selection works. For men, yes, aesthetic appearance is a considerable (main?) component in initial attraction, which is further tempered by compatible personality after that initial connection. For women, social value is the principle signifier, which is then tempered by facial symmetry, not demonstrating socially unacceptable habits and having sone degree of physical security, but the latter is the most variable across cultures.
Handsome men are largely irrelevant.
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I've been reading the author's book, Mathematica, and it's awesome. The title of this post doesn't do it justice.
He shows that math skill is almost more like a sports talent than it is knowledge talent. He claims this based on the way people have to learn how to manipulate different math objects in their heads, whether treating them as rotated shapes, slot machines, or origami. It's like an imagination sport.
Also, he inspired me to relearn a lot of fundamental math on MathAcademy.com which has been super fun and stressful. I feel like I have the tetris effect but with polynomials now.
Sounds really cool.
It reminds me of programming, that moment when new code starts to really sync up and code goes from being a bunch of text to more intuitive structures. When really in the zone it feels like each function has its own shape and vibe. Like a clean little box function or a big ugly angry urchin function or a useless little circle that doesn't do anything and you make a note to get rid of. I can kinda see the whole graph connected by the data that flows through them.
There's a lot of interesting discrete math that can supercharge programming at different levels of scale. What's pretty cool is that it reveals a layer of understanding when I watch my toddlers learn math from counting.
One of the interesting things is being able to exactly describe how something is an anti-pattern, because you have a precise language for describing constraints.
I would love to learn about some of these anti-pattern proofs if you have any examples or references you can share!
> rotated shapes, slot machines, or origami
Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out!
Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not.
I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day informing me of its arrival. Thank you for confirming that it's worth it!
I've had a MathAcademy subscription for some time and it's quite good. I'd say it's best at generating problems and using spaced repetition to reinforce learning, but I think it falls short in explaining why something is useful or applicable. I don't know, most math education seems to be "here's an equation and this is how you solve it" and MathAcademy is undoubtedly the best at that, but I wish there were resources that were more like "here's how we discovered this, what we used to do before, why it's useful, and here's some scenarios where you'd use it."
I have so wanted such resources for years. I have found some and should make a list.
The first time the difference between understanding some math, and understanding what the math meant, was after high school Trig. The moment I started manually programming graphics from scratch, the circle as a series of dots, trigonometry transformed in my mind. I can't even say what the difference was - the math was exactly the same - but some larger area of my brain suddenly connected with all the concepts I had already learned.
While ordering the "Mathematica: A Secret World of Intuition and Curiosity" I came across these books, which looked very promising in the "learning formal math by expanding intuition" theme, so I bought them too:
Field Theory For The Non-Physicist, by Ville Hirvonen [0]
Lagrangian Mechanics For The Non-Physicist, by Ville Hirvonen [1]
The Gravity of Math: How Geometry Rules the Universe, by Steve Nadis, Shing-Tung Yau [2]
Vector: A Surprising Story of Space, Time, and Mathematical Transformation, by Robyn Arianrhod [3]
[0] https://www.amazon.com/dp/B0CN7HMTJN
[1] https://www.amazon.com/dp/B0CN7HMK38
[2] https://www.amazon.com/dp/1541604296
[3] https://www.amazon.com/dp/0226821102
Excited to read each (based on their synopses & ratings), and if I will get compounding fluency across both math and physics between all five books.
Burn Math Class follows that tradition, although it starts pretty basic, so it requires some patience.
https://a.co/d/fZnWUU8
If you're interested in how vector calculus developed, and who was instrumental, all the way from Newton/Leibnitz to Dirac or so, by way of Hamilton, Maxwell, Einstein and others, then Robyn Arianrhod's 'Vector' is brilliant.
But be warned, it gets progressively harder, along with the concepts, so unless you're conversant with tensors, at some point you will have to put on your thinking cap.
The reviews on Goodreads – including my own – are worth reading to get a flavour: https://www.goodreads.com/book/show/202104095-vector
I really want to try MathAcademy.com. How quickly do you think someone doing light study could move from a Calc 1 -> advanced stuff using that site? In my case I could put in at least 30 minutes to an hour a day.
Hard to say but this should give you an idea [0].
At that rate, less than a year is reasonable.
[0] https://www.justinmath.com/what-is-the-highest-sustainable-d...
I can't speak to the advanced stuff but here's my stats on Fundamentals I:
Total time on site (gathered from a web extension): 40h 30m Total days since start: 32
Total XP earned: 1881
Since "1 XP is roughly equivalent to 1 minute of focused work", I "should have" only spent 31 hours. I did the placement test and started at ~30%, and now I'm at 76%. I'd say 75% is stuff I learned in HS but never had a great handle on, 25% I never knew before.
Overall, I'm quite happy with the course. I'm learning a lot every day and feel like I have stronger fundamentals than I did when I was in school. The spaced review is good but I do worry I'll lose it again, so I'm thinking of ways I can integrate this sort of math into my development projects. It's no Duolingo, you really do have to put in effort and aim for a certain number of Xp per day (I try for 60 XP rather than time).
>>It's like an imagination sport.
Honestly speaking I think this is a wrong way to teach people to think about Math. Math is just one of those things which feels hard because people struggle to hold long trials of manipulations in their head. Especially if they are manipulations to something very large, evolved slowly over hundreds of steps. People are not coming short, its just how the human mind works.
IMO, the right way to teach Math is to teach people that its just base axioms, manipulation rules. And after that its how you evolve the base axiom using rules. People need to be taught how to make one valid change at a time. Of course this means tons of paper work and patience. But that is what Math actually is. Its taking truth and rules, to make new ones.
Im teaching this to my kid, and she often goes like this is it?? its really just laborious paper work??
Im using this method and LLM help at times these days to learn Algorithms and Data Structures. When you start working things from base conditions and build from there. A lot of Algos that otherwise seem like the domain of novel inventions just seem to follow from the manual steps you just worked, and then translated into a program.
When you remove all the fluff, Patience and Paper work is all there is to Math.
The author (and Grothendieck, liberally quoted in the book) disagree with you.
I think the reason you disagree is that it sounds like you’re teaching your child to be good at math class (a perfectly valid and good thing to do). Being good at math class requires being good at rational/logical thinking and computation. It also has only glancing similarities to anything that the author would recognise as mathematics.
>>It also has only glancing similarities to anything that the author would recognise as mathematics.
Nah, these are the same things. Trying to make Math look like is for people who are 'geniuses' i.e people with massive capabilities of holding large thought trials and changelogs in their head is how you arrive at making people look stupid doing math and eventually make them hate the subject.
Math is paper work. Approach it that way and all of a sudden doing a 100 page proof is within everyones reach. If you ask people to hold a 100 page proof in their head, and more importantly make changes to that in random places and fix the entire changelog trial, probably 2 - 3 people on earth will be able to do it, and you will just convince everyone else its not for them.
I have a hunch that big mathematical breakthroughs in history have happened around and after renaissance era due to paper getting cheap and ubiquitous. There is only that much you can do in your brain alone.
Through all of this, don't get me wrong, the rigorous application of rationality that it takes to step-by-step construct a proof is very important and an incredibly useful skill. Also, I agree that basically no-one can hold more than 3 things in their head at once.
The book also agrees vehemently that math is NOT restricted to "geniuses" and even argues that those don't really exist in the way that culture thinks they do.
However! His assertion is that the (to him) tedious, laborious, error-prone, paperwork is not the fundamental output of "doing math". For him, symbolic written mathematics is akin to sheet music. It would in principle be possible to teach students to read and write sheet music and even do manipulations like transposing it to different keys, without ever letting them listen to music. It would be hard and boring. Some students would find the memorization and application of rules satisfying but most would struggle.
In such a classroom, there might be one student who by chance figures out for herself that you can kind of "hear" these symbols in your brain and suddenly all the arbitrary rules seem obvious and natural and she doesn't even have to go through the tedious steps at all to answer questions. "Of course this is in a minor key." she might say. "No, I didn't rigorously check each chord, it's just... obvious".
Such a student would be labeled a "prodigy" or "genius", and would struggle to explain to others that no, what she's doing isn't harder than the her classmates laboriously doing the rote work, it's actually much easier.
Of course... this is not to denigrate sheet music. It's a wonderful invention that makes it possible to transmit music out of one person's brain to the brains of an orchestra.
Just like written mathematics.
The author's contention is that, like the contrived example above, no-one ever talks about "the music" of mathematics, just the sheet music, and therefore things are much harder than they need to be.
One of the simple mathematical examples he uses is to ask: Can you imagine a circle in your head (unironically an amazing thing to be able to do!). Then to ask a question like: Can a straight line intersect a circle in 3 places?
You likely have an immediate, intuitive response to this highly non-trivial mathematical problem. That's the music. Now, try to write that down in mathematical language for someone who can't see circles. Oof, it's going to be a slog.
>>Through all of this, don't get me wrong, the rigorous application of rationality
Much of this is just talking to oneself and testing it to see if our idea holds under test conditions.
I was once watching a video on how chess grandmasters think and work. Most of it is-
1. Do we know a pattern of moves, even if done, in series that is known to score some win/check. If so, lets do it.
2. Are any pieces under attack, If gone can effect point 1. eventually? If yes, lets protect them.
3. What can all possible moves of our pieces prevent opponent from having successfully execute their own point 1. And can we force opponent into point 2? Lets do it.
Basically every our move and its possible outcomes(Known through prior study of patterns of previous games seen), every move of our opponent.
A strong internal monologue and testing imaginary moves.
Math is just this except over paper.
Would you say the book ventures more into the practical side of learning this stuff or is it closer to the tone of this article? I found this article hard to gain anything from. A lot of just motivational cliche statements and nothing really groundbreaking or mind altering. If the book is better at that, I'd love to read it. If it's stories and a lot of fluff, I'd rather skip. So I'm curious what you are getting from it and how practical and applicable it feels to you?
Agree. The article turned me off as well. No specific example, felt like an ad.
Yeah, I quit reading it because it didn't talk about the book, it felt like a meta article.
I actually heard about this book very recently, and it's coming up soon on my (never-ending) reading list.
Happy to hear you're enjoying it, makes me even more confident that I should read it :)
+1 for Math Academy. I’ve been using it daily for over a year now (started October 2023). I summarized my experiences after 100 days here in case it helps anyone: https://gmays.com/math
Thanks for sharing this. I was debating buying the book.
This sounds like a book I needed for one of my early comp sci classes in college. It was called something like Think Like a Programmer: An Introduction to Creative Problem Solving. Maybe it was this, maybe it was something like this.
I mean to say, just applied scientific thinking is important. Even if you never get into pure math or computer programming, applying concepts like "variables", "functions" or "proofs" is universally useful.
To my mind, the premature formalization of the math is the principal contributor to gas lighting and alienation of people from maths. The reduction of concepts to symbols and manipulation thereof, is an afterthought. It's misguided for them to be introduced to people right at the outset.
People need to speak in plain English [0]. To some mathematicians' assertion that English is not precise enough, I say, take a hike. One need to walk before they can run.
Motivating examples need to precede mathematical methods; formulae and proofs ought to be reserved for the appendix, not page 1.
[0] I mean natural language
I'm an adult who's been programming computers professionally for 20 years, and went to school for it, and I've lost most mathematical skills past what I'd learned by 6th grade or so, from lack of use.
People who aren't even working in a field that's STEM-adjacent have even less use for stuff past simple algebra and geometry (the latter mainly useful just for crafting hobbies and home-improvement projects) and a handful of finance-related concepts and formulas.
I expect to go to my grave never having found a reason to integrate something, at this rate.
The result is that any time I try to get back into math (because I feel like I should, I guess?) it's not really motivated by an actual need. The only things that don't bore me to tears for sheer lack of application ends up being recreational math problems, and even that... I mean, I'd rather just read a book or do almost anything else.
For nostalgia I keep a copy of Mathematica on my laptop, so I can pretend to be an an actual computer scientist and not just an overpaid button-presser.
Ten years ago I used it to fit a nonlinear model to some performance metrics to predict the behaviour of a disk array past the maximum load level I was allowed to use for testing.
That’s the last time I did something that Excel couldn’t handle.
In university I learned how matrix exponentiation can be used to calculate the maximum throughput of a mesh network. In real life everyone just buys 10x the bandwidth they need.
It’s depressing.
Do you play any games that require mathematical reasoning? You might realize you are using integration without calling it integration i.e. calculating expected values.
Maybe. I also do a lot of complex math when I throw a ball.
Weirdly, accurately throwing a ball never got me a good grade on a math test.
I feel the opposite.
Before high school, math is just a grind of memorization and unmotivated manipulation of numbers.
Many students (ok, me, but I expect the same was true of others), get turned on to math for the first time when they encounter proofs in high school geometry and also actual applications in high school physics.
It's a revelation to students that math can be a way to go from one truth to another and thus find new truths. It's a way of thinking and that can be very exciting.
Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills. Once they develop a gap in their knowledge it becomes difficult to progress unless those gaps are addressed. For lots of students, it all ends with fractions. You'd be surprised how many adults don't really understand fractions. For others it ends with algebra, and for the college bound it ends with calculus.
Only math majors and a minority of engineering/science/CS folks get past the "standard sequence" of math courses in college and gain an appreciation for the really interesting stuff that comes AFTER all that.
> Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills
It's tragic and all, but were there attempts to teach non-preselected children/adolescents interesting stuff first? What are the results?
I don't know.
There is a certain amount of drudgery that unavoidable in learning mathematics, IMHO. Moreover, looking at the math textbooks of my niece and the students that I've tutored over the years, there's no effort spared in trying to make math "fun". If anything there's perhaps too much distraction caused by attempting to make a connection with the students and a failure to clearly and logically explain the math.
I don't have the answers, but believe that it's incredibly helpful to track students, intervene when they show a lack of understanding and not let them slip through the cracks. I do agree with the OP that everyone is capable of math but it's a long journey and it takes a lot of practice.
Mathematics is the conversion of a large number of object languages in to a single meta language that lets us talk about all of them.
The sin of modern mathematics is that it's meta language is so ill define that you need towers of software to manipulate it without contradiction. Rewriting all of it into s-expressions with a term rewriting system for proofs under a sequent calculus is an excellent first step to making it accessible.
We do not need to go back to the 16th century when men were men, an numbers positive. If people want to look at what math talks about instead of how it talks about let them pick up stamp collecting.
Weirdly enough, mathematicians have been manipulating expressions and writing proofs for centuries without (routinely) stumbling into contradictions all without the need of formal proof calculi or s-expressions.
I have nothing but admiration for projects like Lean and Coq and working in them can be a lot of fun (coupled with a lot of frustration when "obvious" things sometimes take an inordinate amount of time to prove), but Wiles' proof of FLT (the corrected version) was published in 1995. We're almost 30 years later and people are just now working on a formalisation which could take many years (https://leanprover-community.github.io/blog/posts/FLT-announ...). Mathematicians can't afford to be waiting for proof systems to catch up, at least not right now.
It wasn't until the 1930s that people realised second order logic with arithmetic will always lead to contradictions without guardrails. Before then all mathematics was done in the object language of whatever the field in question was and only translated to the meta language for succinctness after the fact.
The magic of modern maths is that we can now work only with the meta language and get results free from contradiction. For this we absolutely need a modern notation to do the new type of maths since we are no longer grounded by the reality of the object language.
None of what you wrote is true.
Feel free to explain why.
Your very first sentence is simply wrong, I don't know what more there is to say other than that you clearly don't understand what Gödel's results are about (hint: they're called "incompleteness theorems", not "contradiction theorems"). Maybe read a textbook? I could recommend several good ones.
The other sentences basically fall in the category of "not even wrong", i.e. basically nonsensical.
Godel starts by the assumption that the system is free from contradictions which makes his papers largely irrelevant here.
Again, feel free to explain how you can have a second order logical system without axiomatic limitations that doesn't contain contradictions.
Hell, go by the more popular version of first order logic + set theory if you're more comfortable with that.
Nobody has found a contradiction in (first or second order) ZFC in over a hundred years. You're out of your depth.
>Nobody has found a contradiction in (first or second order) ZFC in over a hundred years.
Yes, which is exactly why we have guardrails like ZFC in place. We got all sorts of exciting results without them. We of course have to keep adding axioms to ZFC - or replacing them all together - because there is a lot of math out there which is outside it's purview.
>You're out of your depth.
You're being incredibly abrasive for no reason.
What does premature formalization mean and when does it occur? Do you mean formal in the sense of using formulaic, rote manipulations or formal in the sense of proofs and rigor?
As someone who went on to study mathematics at the graduate level, I was bored out of my mind in high school math and most subjects. What's missing from a lot of primary and secondary school education is context, and that's what makes it boring. Math wasn't easy because I was particularly good at it. It was easy because it was just blindly following formulas and basic logic.
Something is very wrong with our educational system because almost all math at the primary and secondary levels is basic logic. So when people with this maximum level of mathematics education say they're "bad at math" or "don't get math", it means that they lack extremely basic logic and reasoning skills.
In my mind, we need to teach mathematics in a contextual way (note that I don't necessarily mean applications) in a way to enrich the reasoning and exploring of it. This should include applications, yes, but not be fully concentrated on applications. Sometimes one needs to just learn and think without being tied to some arbitrary standard of it being applied.
This sentiment comes up all the time here. Mathematics uses formalism because it's easier.
It's easier to read "a(b+c+d) = ab+ac+ad" than
> If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.
It's well known that good notation is exactly the one that elevates good intuition. For example, the Legendre symbol has the property that (a/p)*(b/p)=(a*b/p), an important visual cue that you wouldn't get from writing down (in way too many words) what the Legendre symbol actually means.
Also, most actually good mathematical textbooks aren't just dumps of formulae and proofs and they do contain motivation, examples, pictures, etc. You're attacking a strawman. But you can't just relegate the formalism and proofs to the appendices, that's crazy.
The ironic thing is, I swear that this must have been how math (at least more advanced math) was taught a century ago. Or at least, nowadays I've taken to relying on textbooks from the early-to-mid 20th century to learn new math. Maybe it's survivor bias and the only textbooks from back then that anyone remembers are the good ones.
I hate new textbooks because they're so built around instant gratification. They just come out and tell you how to solve the problem without building the solution up in any way. Maybe afterward they take a swipe at telling you how it works, but that's just completely the wrong way around IMO. It robs me of the chance to mull things over, try to anticipate how this will all come together in the end, and generally have my own "aha" moments along the way.
But, getting back to what you say, I think that it also engenders this tendency to reduce math to symbol manipulation. Because if they give you the formula in the first paragraph, then all subsequent explanation is going to end up being anchored to that formula. And IMO that's just completely wrong. Mathematical notation is at its best when it's a formalizing tool and mnemonic device for cementing concepts you already mostly understand. It's at its worst when it's being used as the primary communication channel.
(It's also an essential tool for actually performing any kind of symbolic reasoning such as algebraic manipulation, of course, but I'm mainly thinking about pedagogical uses here.)
I believe you're right, even though I don't have any evidence except for my own experience.
This issue becomes very clear when you see how many ways there are to express a simple concept like linear regression. I've had the chance to see that for myself in university when I pursued a bunch of classes from different domains.
The fact that introductory statistics (y = a + bx), econometrics (Y = beta_0 + beta_1 * X) and machine learning (theta = epsilon * x, incl. matrix notation) talk about the same formula with quite different notation can definitely be confusing. All of them have their historical or logical reasons for formulating it that way, but I believe it's an unnecessary source of friction.
If we go back to basic maths, I believe it's the same issue. Early in my elementary school, the pedagogical approach was this: 0. only work with numbers until some level 1. introduce the first few letters of the alphabet as variables (a, b, c) - despite no one ever explaining why "variable" and "constant" are nouns all of a sudden 2. abruptly switch to the last letters of the alphabet (x, y, z), two of which don't exist in my native language 3. reintroduce (a,b,c) as sometimes free variables, and sometimes very specific things (e.g., discriminant of a quadratic equation) 4. and so on for greek letters, etc.
It's not something that's too difficult to grasp after some time, but I think it's a waste to introduce this friction to kids when they're also dealing with completely unrelated courses, social problems, biological differences, etc. If you're confused by "why" variables are useful, why does the notation change all the time, and why it sometimes doesn't - and who gets to decide - this never gets resolved.
Not to mention how arbitrarily things are presented, no explanation of how things came to be or why we learn them, and every other problem that schools haven't tackled since my grandparents were kids.
I have never disagreed more with a comment. You can fully decide that you're not interested in mathematics, after having taken all of the math classes that you could possibly be offered before university, without ever encountering a proof, or even a mathematical definition.
I do agree that explaining why mathematical concepts are useful is something that's often lacking in mathematical curricula, but not that the problem is premature abstraction. Like another commenter said, the opposite is true. The way children are first introduced to (and therefore soured to) something that adults call "math" is by performing pointless computation that has as much to do with actual math as lensmaking has to do with astronomy.
A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it. So is there some scientific study backing up the claim that all these people could easily learn it or are we just making it up because its a nice egalitarian thesis for a math popularization book?
The same goes for language skills, by the way. In the US, 21% of adults are illiterate, and 54% of adults have literacy below sixth-grade level.[1] This is higher than in other developed countries. For example, in Germany, 10% are illiterate, and 32% have literacy below fifth-grade level.[2]
General intelligence also seems to have been trending downward since the 1970s (the reverse Flynn Effect)[3]. It has been measured in the US and Europe.
So, while it is true that the education system and other factors have an influence, the idea that "everybody is capable of X" is wrong and harmful. It's the equivalent of "nobody needs a wheelchair" or "everybody can see perfectly." People are different. A lot of nerds only hang out with other nerds, which screws up their perception of society.
[1]: https://www.thenationalliteracyinstitute.com/post/literacy-s... [2]: https://leo.blogs.uni-hamburg.de [3]: https://www.sciencedirect.com/science/article/pii/S016028962...
What a weird comment. Are you trying to argue by analogy that a decent fraction of the population are not capable of literacy? It seems self-evident that low literacy rates have nothing to do with innate ability. I see no evidence to suggest that math is any different.
The idea that 21% of US adults are "illiterate" doesn't mean they can't read, by the way:
> English literacy test results from 2014 suggest that 21% of U.S. adults ages 16 to 65 score at or below PIAAC literacy level 1, meaning they have difficulty "[completing] tasks that require comparing and contrasting information, paraphrasing, or making low-level inferences." Included in that 21% is the 4.2% of respondents who were unable to be assessed due to language barriers, cognitive disability, or physical disability.
https://en.wikipedia.org/wiki/Literacy_in_the_United_States
A lot of people aren't aware that the academic definition of "literacy" was changed around 1950 to no longer refer to "alphabetical literacy", which is still what most people think literacy means (in lay usage).
https://en.wikipedia.org/wiki/Literacy
And this particular survey ignored people who didn't speak English but were entirely literate in their native language. Which obviously has nothing to do with the educational system or people's intelligence.
That certain countries both now and in the past have had significantly higher mathematical ability among the general population and much higher proportions going on to further study suggests that ability isn’t innate but that people don’t choose it. In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun.
> have had significantly higher mathematical ability among the general population
This is not really true is it? There were not that many standardized testing globally to measure such claims. Many people were in poverty and did not get tested, did not go to schools, or finished schools very early (5, 9 years). Many more kids go to school these days.
> In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun
It is just wrong. It was the same as now, except it was critical for people to show results because otherwise you had grim perspectives in the life, there was little "fun". People wanted to get into universities to get better jobs and to get better apartments, to be able to leave their parents. You could not just buy places, but a good position in some public body would guarantee you a nice place. FYI engineers could earn more in comparison to other jobs, not to mention if you could get into defense industry.
Why would ability not be innate just because some people with the ability don't use it?
Or more specifically, two of my friends teach special needs children in the 50 to 70 IQ band. Who are we going to blame for them not becoming mathematicians? The teachers, for not unlocking their hidden potential? The kids, for not trying hard enough? Claiming that the only thing holding them back is choice seems as cruel as it is wrong, to me.
Yeah, we're probably not cultivating anywhere near the potential that we could, but I personally guarantee you I am not Ramanujan or Terence Tao.
Well, I guess what I mean is that most people have some level of general intelligence that when applied correctly can generally give good results in most subjects. In general the people who do well in school do well in everything, even if they have a preference, and as such could do well in most of those subjects if they went on to further study. The evidence tends to be that in lower income countries people push towards subjects more likely to bring financial stability than those they prefer which bears this out somewhat.
There are some extreme cases of course but I’m not sure the general public needs to worry too much about those, most of us aren’t an Einstein nor do we have learning disabilities.
The extreme case does not imply a binary scenario ie that there are those that can those that cannot.
Rather, learning ability is a continuum. people have varying degrees of ability to learn mathematics. Couple this with environmental factors and society generates a huge variability in mathematical ability that crosses income levels and other demographics.
This view is rejected by many because it is against the push for equality.
You get a huge variability if you consider the absolute extreme outliers. Most people should be able to reach a level of competence where they can understand mathematical concepts abstractly and apply that same reasoning to other areas, and not feel a visceral rejection at the mere idea. I think that's a modest enough standard that a good portion of any given population should be able to reach, and yet education is failing at achieving that.
Your statement is not backed up by data and simply wishing it should happen isn’t a strong argument.
You probably have a narrow definition of “most people” (probably some motivated high school or undergraduate student) and too loose with what it means to “understand mathematical concepts abstractly”.
Take an analogy: imagine professional musicians saying that most people should be able to take a piece of music and understand its harmonic structure, then apply it to a new setting to generate a new piece. Most people will reject this idea as absurd.
Where's the data backing up what you said?
>You probably have a narrow definition of “most people” (probably some motivated high school or undergraduate student)
I was thinking "3-4 out of 5 people you pick on the street at random".
>too loose with what it means to “understand mathematical concepts abstractly”.
Enough that they could recognize whether a mathematical concept is applied correctly (e.g. if I have a 2% monthly interest, should I multiply it by 12 to get the annual interest? Why, or why not?) and conversely to correctly apply concepts they already understand to new situations, as well as to leverage those concepts to potentially learn new ones that depend on them.
>imagine professional musicians saying that most people should be able to take a piece of music and understand its harmonic structure, then apply it to a new setting to generate a new piece. Most people will reject this idea as absurd.
Okay, but we're arguing about what is the case, not about which idea has more popular support. Since most people don't understand thing 1 about composition, why should their opinion matter? A skilled composer's opinion on the matter should have more bearing than a million laymen's.
> Where's the data backing up what you said?
What he is saying is the default hypothesis based on our understanding of biology and psychology. If you have variability in genes you'll get variability in characteristics that are connected to them - height, bone structure, mental capacity, etc.
It is on you to prove that there is an arbitrary cut-off when it comes to this variance from which point it doesn't matter in regards to e.g. cognitive and mathematical ability.
> Enough that they could recognize whether a mathematical concept is applied correctly (e.g. if I have a 2% monthly interest, should I multiply it by 12 to get the annual interest? Why, or why not?) and conversely to correctly apply concepts they already understand to new situations, as well as to leverage those concepts to potentially learn new ones that depend on them.
No it doesn't if they do not have the abilities to comprehend it. I think you're living in a bubble of at least average-smart people and don't get that probably millions if not billions of people around the globe (based on average IQs) won't really get that concept.
>If you have variability in genes you'll get variability in characteristics that are connected to them - height, bone structure, mental capacity, etc.
Then you're agreeing with me. The thing all of those have in common is that they follow normal distributions. The shortest recorded adult and the tallest recorded adult are quite far apart, yet the vast majority of adults are between 150-200 cm tall. That's precisely what I was saying; the outliers of mathematical skill are very very far apart, but most people are roughly equally capable.
>I think you're living in a bubble of at least average-smart people and don't get that probably millions if not billions of people around the globe (based on average IQs) won't really get that concept.
What I'm saying it that it even someone with below-average IQ could do it, if taught properly. Mathematics is less about being smart and more about being rigorous.
>A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it
Any healthy/able individual could learn to deadlift twice their bodyweight with sufficient training, but the vast majority of people never reach this basic fitness milestone, because they don't put any time into achieving it. There's a very large gap between what people are capable of theoretically and what they achieve in practice.
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> So is there some scientific study backing up the claim that all these people could easily learn it [emphasis added]
Who said it would be easy?
It is easy to learn for some.
I do not think that Bessis's argument is entirely "made up"
We are not really taught (thought) to think, we are taught to memorize. Until one actually tries to think, you really can't tell if they're able to do it.
I'm not a math teacher, but I do enjoy math, and I have helped several family members and friends with math courses.
I've long thought that almost all have the capability to learn roughly high school level math, though it will take more effort for some than for others. And a key factor to keep up a sustained effort is motivation. A lot of people who end up hating math or think they're terrible at it just haven't had the right motivation. Once they do, and they feel things start to make sense and they're able to solve problems, things get a lot easier.
Personally I also feel that learning math, especially a bit higher-level stuff where you go into derivations and low-level proofs, has helped me a lot in many non-math areas. It changed the way I thought about other stuff, to the better.
Though, helping my family members and friends taught me that different people might need quite different approaches to start to understand new material. Some have an easier time approaching things from a geometrical or graph perspective, others really thrive on digging into the formulas early on etc. One size does not fit all.
One size doesn't fit all is what I believe Common Core math is attempting. The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all
> The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all
I cannot overstate enough how consistently and extremely this has turned my kids off from math. 3-for-3 on absolutely hating this. Having to solve the same thing five different ways just pisses them off, and, like... yeah, of course it does. They want to finish the work and go play and it feels like you're just fucking with them and disrespecting their time by making them solve the same problem several times, even if that's not the intent.
I also have long felt that anyone who has the ability to read and write at a high-school level in their primary language also has the mental capacity to learn to do math at a high school level. As a pedagogical challenge, I think the main stumbling block for most people with math is not the complexity of it but rather the dryness of how its taught. The rules of language are at least as complex, but many more people learn language at a high-school level than math. There are lots of reasons for that, most obviously language is used more in people's every day lives.
Effort, combined with the right motivation, can overcome most perceived barriers
It sounds like trivial insight, but at least in my experience many adults and even teachers have this "it's hard so it's ok to not want to do it" attitude towards math. And I think that is very detrimental.
Well, isn't that a summary of most things? Most things worth learning are hard, but many things not worth learning are also hard. So we have to prioritize what hard things are worth learning. Math is low on the list for many people for (I think) understandable reasons.
math is low on the list then they bitch that they’re unemployable with a soft skill degree doing middle school level work. i get this is ironic as a pure math student is also fairly unemployable without extraneous skills, but they also tend to shine the brightest once they make it in.
What I meant was I think it's detrimental to be priming the kids with a negative view, or nurturing any negative views.
There is this element of abstract mathematical thinking that many young people get exposed to at some point in the educational system but just never "get it" and they disconnect. This is where it goes awry as the gap only widens later on and its a pity.
Working with symbols, equations etc. feels like it should be more widely accessible. Its almost a game-like pursuit, it should not be alienating.
It might be a failure of educators recognizing what are the pathways to get the brain to adopt these more abstract modes of representing and operating.
NB: mathematicians are not particularly interested in solving this, many seem to derive a silly pleasure of making math as exclusive as possible. Typical example is to refuse to use visual representation, which is imprecise but helps build intuition.
Lots of people seem to get permanently lost right around when operations on fractions are introduced. Other places, too, but that seems like the earliest one where a lot of people get lost and never really find their way back.
Factoring was another that lost a lot of folks in my class. Lots of frustration around it seeming both totally pointless and the process involving lots of guessing, several classmates were like "well, fuck math forever I guess" at that point, like if they'd been asked to dig a ditch with a spoon and then fill it back in.
I don't know how widespread this phenomenon is but in the book Do Not Erase [0] it seems that there are quite a few prominent mathematicians who do use visual representations in their work.
[0] https://en.wikipedia.org/wiki/Do_Not_Erase:_Mathematicians_a...
There is this long-running (and quite fascinating I think) debate about the different "types" of mathematical thinking. Logical vs Intuitive, Geometric vs Algebraic etc. Can't recall where exactly but I remember reading about a 19th century mathematician that crowed they had not a single figure in their masterpiece.
Visualization is probably not a silver bullet but a lot of people are visual thinkers so maybe it would help a few more to reach a higher level in mathematical thinking.
I’m far from being any kind of serious mathematician, but I’ve learned more in the last couple years of taking that seriously as an ambition than in decades of relegating myself to inferiority on it.
One of the highly generous mentors who dragged me kicking and screaming into the world of even making an attempt told me: “There are no bad math students. There are only bad math teachers who themselves had bad math teachers.”
Sadly, when I was a postdoc, an eminent mathematician I was working under once shared a story that he found amusing that one of his colleagues was once asked a question in the form: "This might be a stupid question, but..." and the response was "There are no stupid questions, only stupid people."
Run into too many people like that, who I daresay are common in the field, and it's easy to see how people become dispirited and give up.
That’s a south park quote: https://youtu.be/wWfacULP1o0?si=ddBWXFuQMoxxY-Yu
That was a belly laugh on a tough day.
Thank you Sir or Madame.
Isn't that a positive statement, that you can ask questions without worry since they aren't stupid?
Maybe, I guess it's easily misinterpreted though O.O
I think we can recognize Pauli for his identification of one of the few magic gadgets we accept around spin statistics without accepting his educational philosophy: “Das ist nicht einmal falsch.”
He was right on the nature of the universe, he was wrong on making a better world. I for one forgive him on the basis of time served.
[dead]
How much of math aversion stems from a chain reaction of ineffective instruction
According to an excellent mentor: all of it minus epsilon.
Wouldn't it then follow that all students of the same teachers end up with the same skill level in math? Not sure that's the case.
Cantor gave his life to the Continuum Hypothesis, Hilbert gave much of his life to similar goals.
You’re making an argument somewhat along those lines, but given that I didn’t stipulate a convergence condition your conclusions can be dismissed by me.
If it were a valid argument then we’d need Gödel.
Did you mean to reply to another post? I don't follow at all.
Doesn't follow. Bell curve in, shifted bell curve out. Ideally this also tweaks the variance a bit.
In other words: Some students flourish despite their teachers, some flourish because of them.
And how would you call the students in the left tail of the Bell curve if not bad students?
Below average students and as long as the average is high enough they are still very competent.
A bad teacher instead gives you a bimodal distribution and just doesn't bother teaching those students.
I think for most people the issue is that they never even get to the fun stuff. I remember not really liking math right until university where we had set theory in the first semester, defined the number sets from scratch went on to monoids, groups, rings etc. That "starting from scratch" and defining everything was extremely satisfying!
Interesting, I somewhat of an opposite reaction, although I am certainly not a mathematician. Once everything became definitions, my eyes glazed over - in most cases the rationale for the definitions was not clear and the definitions appeared over-complicated.
It took me some time, but now it's a lot better -- like a little game I somewhat know the rules of. I now accept that mathematicians are often worrying about maximal abstraction or addressing odd pathological corner cases. This allows me to wade through the complexity without getting overwhelmed like I used to.
My dad always told me growing up today math was like a game and a puzzle, and I hated that. I also hated math at the time. It felt more like torture than a game.
I didn't fall in love with math until Statistics, Discrete Math, Set Theory and Logic.
It was the realization that math is a language that can be used to describe all the patterns of real world, and help cut through bullshit and reckon real truths about the world.
Yes, I agree! And also that a lot of the fun stuff is hidden behind historically opaque terminology. Although I'm also sympathetic to the fact that writing accessible explanations is a separate and hard to master skill. Once you understand something it can be really hard to step back into the mindset of not understanding it and figuring out an explanation that would make the idea "click".
I think a lot of maths is secretly a lot easier than it appears, but just missing an explanation that makes it easy to get the core idea to build upon.
For example, I've been meaning to write an explorable[0] for explaining positional notation in any integer base (so binary, hexadecimal, etc) in a way that any child who can read clocks should be able to follow. Possibly teaching multiplication along the way.
Conceptually it's quite simple: imagine a counter that looks like an analog clock, but with the digits 0 to 9 and a +1 and -1 button. We can use it to count between zero and nine, but if we add one to nine, we step back to zero. Oh no! Ok, but we can solve this by adding a second counter. Whenever the first counter does a full circle, we increase it by one. A full circle on the first counter is ten steps, so each step on the second counter represents ten steps. But what if the second counter wants to count ten steps? No problem, just add a third! And so on.
So then the natural question is... what if we have fewer digits than 0 to 9? Like 0 to 7? Oh, we get octal numbers. 0 and 1 is binary. Adding more digits using letters from the alphabet?
The core approach is just a very physical representation of base-10 positional, which hopefully it makes it easy to do the counting and follow what is happening. No "advanced" concepts like "base" or "exponentiation" needed, but those are abstractions that are easy to put on top when they get older.
I've asked around with friends who have kids - most of them learn to read clocks somewhere between four and six, and by the time they're eight they can all count to 100. So I would expect that in theory this approach would make the idea of binary and hexadecimal numbers understandable at that age already.
EDIT: funny enough the article also mentions that precisely thanks to positional notation, almost every adult can immediately answer the question "what is one billion minus one".
[0] https://explorabl.es/
If you’re interested in computer science, have you ever looked at the Software Foundations course by UPenn? It follows a similar approach of having you build all sorts of fascinating math principles and constructions from the ground up. But then it keeps going, all the way up to formal methods of software analysis and verification.
https://softwarefoundations.cis.upenn.edu/
totally agree! in high school, lots of things were vaguely defined. I remember, I didn't fully understand what "f o g" was until I was given the definition of a monoid. Also limits and derivations, once you get the proper definition, you can pretty easily derive all the formulas and theorems you use in high school. Also in high school, we mostly did calculations and simple deductions, but at university we were proving everything. Nice change of perspective.
In college I took Formal Logic II as it fulfilled requirements in both my Comp Sci and Phil major. It turned out that PHIL 104 was cross listed as MATH 562, because the professor who taught Logic I was allowed to teach whatever he wanted for the followup class. I had technically taken the prereq, which was a basic CS logic course, but I was in way over my head. It was one of the most fun courses I took in college.
We were given the exact text of the final exam weeks in advance, and were allowed to do anything at all to prepare, including collaborating with the other students or asking other professors (who couldn't make heads or tails of it). The goal was to be able to answer 1 or 2 out of the 10 questions on the exam, and even if you couldn't you got a B+ at minimum.
I wish I had a better memory, but I believe one of the questions I successfully answered was to prove Post's Theorem using Turing machines? The problem is, I never used the knowledge from that class again, but to this day I still think about it. It would be amazing to go back and learn more about that fascinating intersection of philosophy and computer science.
What I loved the most was that it combined hard math with the kind of esoteric metaphysical questions about mathematics which many practitioners despise because they feel like it undermines their work. It turns out, when you go that deep it's impossible not to touch on the headier stuff.
In my high school we were basically only instructed to get good at applied math. Calculus. Which more often than not was simply "plugging it in". Most of that work is trivially automatible through Mathematica. When I reached a university, I took number theory and abstract algebra and it blew my mind that math was actually so beautiful in a way that defied explanation. When I took real analysis I finally saw the side of calculus that didn't seem like a waste of time.
One day, I went back to my high school and spoke to my computer science mentor back then [1]. I passionately asked him why we were never exposed to group theory. The answer, he said, was the SAT. None of that stuff is on the SAT, so it can't be justified teaching.
[1] The great Andrew Merrill
Calculus isn't on the standard SAT or even the Math Subject II tests [0], at least not in the early 2000s.
They taught us calculus because it's a prereq for engineering and physics and that's been important since the space race.
[0] https://en.wikipedia.org/wiki/SAT_Subject_Tests
Eh, I mean it's not on the SATs, but why isn't it? In Canada, we had a similar calculus based curriculum up to the first year of university. A little bit of linear algebra thrown into the mix. Why is that? Well you need calculus to do any form of engineering, physics, certain domains of chem/bio, stats, certain domains of economics, etc etc. Math in society is first and foremost a tool. I say this as a person who majored in Pure Math and focused on algebra and number theory. For the vast majority of students, it truly is about the practicality. Math just has the layer of abstraction that makes it hard to enjoy without deliberate framing unlike the sciences or humanities.
Many people dismiss mathematics because they aren't interested in it. I definitely wasn't interested in doing dozens and dozens of error-prone problem sets that mainly boiled down to performing arithmetic. I don't need to do that.
My number theory professor was a brilliant mind, someone who had spent lots of time at the Institute for Advanced Study, and he absolutely sucked at adding/subtracting/multiplying numbers. And that was something he freely admitted. It wasn't important to his work, and it isn't important to mine.
The author: Would love to participate but account creation seems to be broken on Hacker News https://x.com/davidbessis/status/1859561768915173466
If someone still on X could send them the hn@ycombinator.com support address, that'd be useful.
> the provocative claim
Leibniz made that claim centuries ago in his critical remarks on John Locke's Essay on Human Understanding. Leibniz specifically said that Locke's lack of mathematical knowledge led him to (per Leibniz) his philosophical errors regarding the nature of 'substance'.
https://www.earlymoderntexts.com/assets/pdfs/leibniz1705book...
I haven't read his Human Understanding, but his Second Treatise is really weak in ways that can't really be blamed on lack of mathematical training (unless we're going with "all rigorous thinking is math") so there may be more to it in his case than just "he didn't study math enough".
Leibniz wasn't saying that "rigorous thinking" is only available to mathematically trained or that Locke's reasoning was not "rigorous".
His critique of Locke was that one can not have a correct model of human understanding (or world model) based on purely philosophical means, and that the lack of exposure to certain aspects of modern mathematics (that was emerging at that time) was the basis of Locke's misunderstandings.
Last year I read How to Solve It and the first half of one of Polya's other books - Mathematics and Plausible Reasoning. I certainly didn't commit them to memory, and I never systematically tried to apply them during self-study, but they do sometimes help give me a pointer in the right direction (i.e. trying to think of auxiliary problems to solve, trying to find a way to make the known & unknown closer together... etc.).
Auxiliary problems are something that always screwed me in college, when we were doing Baby Rudin, if a proof required a lemma or something first I usually couldn't figure out the lemma. Or in general, if I didn't quickly find the 'insight' needed to prove something, I often got frustrated and gave up.
This material seems like it would be good to actually teach in school, just like a general 'how to think and approach mathematical problems'. Feels kinda weird that I had to seek out the material as an adult...
One other thing I got out of the Polya books, was I realized how little I remember about geometry. So many of their examples are geometrical and that made them harder for me to grok. That's something I wish I could revisit.
"It's the economy stupid" is what I would say. Mental capacity is capacity. Most of us don't study math not because we don't want to but because we can't.
I bet if you asked in a survey of people that if you were given a UBI that covered all your expenses and needs what would you do? It would be perfection of the self or art. Both of these are what is practicing and learning math.
One thing I can agree with is that if one is stressed out and have poor psychological habits you will suffer and be miserable regardless.
I would say focusing on mindfullness(like vipasaana) can go a long way in this. But mindfulness is not an intellectual exercise, one has to live it. Do multiple hours of meditation a day and it gets you somewhere good in a few months.
I can actually tell you how to do this right now. Take a 10 day vipassana course and practice the recommended two hours a day. If you use only therapy, seriously follow the therapist, but do some meditation too.
Once you do this, you will soon develop Attention(Concentration) and Equanimity(Inner Calmness in the face of tremendous external circumstances).
Soon you start realizing some inescapable facts:
1. Your current moment is ever changing.
2. One attaches more of the self(the ego, I me and mine) with the past and tries to predict his state in the future and ends up miserable. Don't think with the I, you are bound to suffer. The right action is timeless and free of the I. This is the reduction of the ego.
3. There is tremendous joy in focusing on being present in the moment. If you are into running and all you are doing is taking joy in your feet, breath and posture all the time for 5,10 miles you know what I mean.
This is the key to everything. No amount of book reading on self improvement can get this to you
Didn’t we have that experiment during Covid? A bunch of people got paid to stay at home, sometimes for 2 years. How many Grammy nominations since then have gotten to musicians that came out of that? A new face at the Oscars? MOMA exhibiting an artist that was a barista before the pandemic?
At least anecdotally, many people around me now have more children, on the other hand.
I must disagree. I consider art same as entertainment to the most part. I would want to be good at math and I also disagree that it has anything to do with mental capacity. It's not a competition, I don't need to be better at math than others but my pursuit of other things like cryptography, better algorithms, and understanding physics is limited by my primitive understanding of mathematics.
If I was a multi-millionaire, learning lots of math on my free time would be one of the things I would pursue while chilling at my beach house.
Thanks.
> art same as entertainment
Could you volunteer me how much time you spend on it? And how is your day job?
> my pursuit of other things like cryptography, better algorithms, and understanding physics is limited by my primitive understanding of mathematics.
Could you volunteer why you would want to learn these subjects? Is it your day job or is it something you would like to pursue in the future.
> If I was a multi-millionaire, learning lots of math on my free time would be one of the things I would pursue while chilling at my beach house.
I said UBI
I don't think UBI is feasible, it is anathema to the human condition to be content with the bare minimum. if not for our own selves, we would want the best life possible for our loved ones (present or future). my "UBI" would be a couple of million dollars.
I want to learn those subjects because I enjoy learning and understanding. Life should be lived with knowledge applied through wisdom.
> Could you volunteer me how much time you spend on it? And how is your day job?
On entertainment? I can't tell you, I like to watch a movie or a tv show whenever I have time for it. There are more enjoyable pursuits in life, and most worthy pursuits involve adversity and require perseverance.
> I don't think UBI is feasible, it is anathema to the human condition to be content with the bare minimum.
Hm? That is exactly why UBI works, no? As the name indicates, the bare minimum is taken care so that you can work towards "the best life possible for our loved ones" without worrying about starving, sickness or homelessness.
In contrast, if you were wrong and people would be content with the bare minimum, then UBI would be a bad idea. Though then they could just commit some felonies and be content with having a cell, bread and water for the rest of their lives, no?
I posted a more detailed sibling comment on this thread, but that's not why UBI works. it just shifts what the "bare minimum" is. Most Americans aren't fighting to get the best tent spot while out on the streets because they can't afford housing or begging for food on the streets. UBI isn't solving that, except as a welfare replacement for a small percentage of the population (and not a great replacement either). maybe with UBI, everyone who lives in a crappy apartment can now afford a nicer apartment, but costs for the nicer apartment would naturally go up as well. In other words, most people won't quit their jobs because of UBI, they would just temporarily afford nicer things. Those that do quit their jobs can not work and not worry about starving, but that's not a new condition. if you don't want to work in America, you won't starve. maybe housing would be a problem but the people for whom housing would be a problem if they stopped working are not typically the same people who would be content with the cheapest/worst livable condition (UBI).
In short, it is silly to expect UBI to be a means by which people would work only if they want to work, and they would pursue their passions and dreams instead. That kind of a society I think is possible, but it would also have to reach a level of wealth where money itself is not required (think star trek).
> In short, it is silly to expect UBI to be a means by which people would work only if they want to work
I think it is misleading to call that scenario UBI and that the rest of your post also shares barely any resemblance to actual UBI plans, goals or side effects.
if that is the case, then I must apologize for my ignorance.
> my "UBI" would be a couple of million dollars.
There's a difference between what your human brain would become accustomed to (which you'd be right, it'd scale up and up forever) versus what would allow the base level of health and opportunity. As in, not having to worry about eating the next day.
And because you're right that human brains strive for more wealth, UBI should grant you the opportunity to pursue it without fear of failure.
UBI can be used as a replacement for existing welfare programs, but you're not pursuing arts or starting a business on it. My point was, people will still prioritize earning more money when on UBI instead of pursue their passions because it won't be enough. UBI is not a safety net, if a middle class salary person fails, they would have to work hourly lower wage work, that's why they keep working their middle class job, it isn't because they fear starvation or losing their shelter.
UBI would relieve stress for the lowest earning people, but it won't result in pursuit of passion for most people. economically speaking, because most people can afford certain things (like rent) the price of those things will go up, things are priced based on what potential consumers are willing to pay. If rent costs $1000 for a specific type of unit, but suddenly everyone on UBI can afford that easily, the landlord would raise the rent, the cost of things won't remain static when wages rise for a large portion of the population. Increased demand without increased cost is loss of potential revenue. The quality of life for people on UBI would be barely surviving, and UBI would need to increase constantly to keep people from becoming homeless or starving.
This is the "Cobra effect" embodied. It provides a perverse incentive. healthcare in the US is out of control for this reason. health care providers keep increasing cost, because the patient is not the client, the insurance company is, so long as everyone is getting insurance, the cost of care is the maximum reliably predictable pay out by the insurance company. Not increasing cost is just bad business. You will have to also force all kinds of businesses from raising prices if it can work, and even when it works UBI will result in subsidizing low-wager workers for businesses, because they'll still have to work some job to afford anything outside of food,shelter and the basics.
A practical alternative to UBI is a local tax on businesses, kind of like a property tax but this tax is based on an inverse of an assessment of wages, rent, welfare pay out and other social conditions in the area. the higher wages are, lower rent is,etc.. the lower the tax is, it might even result in a credit. An inverse of a perverse incentive like UBI. Unemployment would also be partially funded through this, the unemployed would forever get a UBI like pay out so long as they are pursuing education or work of some kind based on what is in demand in their area. Businesses get a healthy talent pool to choose from and cost of living is balanced.
UBI scoped for self-actualization would be nice, imagine all the mighty works people would make if not for The Markets making us organize around badware.
Come on, get real. If people had all of their needs covered a lot more of them would sit around getting high and playing video games than perfecting their art.
Perhaps some, but why is that a problem? There are already people who do this.
I would perfect my video gaming art. Teenagers in their basement would fear me after school.
And how far would that get one doing that? 1, 5, 10, 40 years?
Is that not better than the current alternative where they're forced to become grifters?
No one is forced to become a grifter.
Yeah, but we are all in software so for us economy is not a concern. All of us should learn maths.
If the article is in any way representative of the book, then I'm not sure what there is to be learned from the book. That mathematical skills can be honed through practice? That it happens at an intuitive, pre-rigorous level before it is ready to be written down on paper? How surprising. And I doubt he can disprove the genetical component of intelligence, only show that there are other components to mathematical productivity as well.
At least I know that David Bessis's mathematical work is not as shallow as this. His twitter thread on the process https://x.com/davidbessis/status/1849442592519286899 is actually quite insightful. I would guess this also made it into the book in some longform version, but I don't know whether I would buy the book just for that.
This topic is probably the worst possible topic for Quanta.
The book, as I understand it, is about the life changing power of mathematical thinking. Quanta's mission is to make deep mathematics and mathematicians a "sexy" "human interest" topic, by making it as non-mathematical as possible while keeping a superficial veneer of mathematics.
Agree. I’ve been trying to learn ML and data for a few years now and, around 2021 I guess, realised Maths was the real block.
I’ve tried a bunch of courses (MIT linalg, Coursera ICL Maths for ML, Khan etc etc) but what I eventually realised is my foundations were so, so weak being mid 30s and having essentially stopped learning in HS (apart from a business stats paper at Uni).
Enter a post on reddit about Mathacademy (https://www.mathacademy.com/). It’s truly incredible. I’m doing around 60-90 minutes a day and properly understanding and developing an intuition for things. They’ve got 3 pre-uni courses and I’ve now nearly finished the first one. It’s truly a revelation to be able to intuit and solve even simple problems and, having skipped ahead so far in my previous study, see fuzzy links to what’s coming.
Cannot recommend it enough. I’m serious about enrolling in a Dip Grad once I’ve finished the Uni level stuff. Maybe even into an MA eventually.
Too often people think of learning as accumulating knowledge and believe blockers are about not enough knowledge stored.
That would be like strength training by carrying stuff home and believing that the point is to have a lot of stuff at home.
Intelligence is about being able to frame and analyze things on the fly and that ability comes from framing and analyzing lots of different things, not from memorizing the results of past (or common forms of) analysis.
I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao.
As anti-intellectual as it sounds, you could imagine someone asking, is it worth devoting years of your life to study this subject which becomes increasingly esoteric and not obviously of specific benefit the further you go, at least prima facie? Many people wind up advocating for mathematics via aesthetics, saying: well it’s very beautiful out there in the weeds, you just have to spend dozens of years studying to see the view. That marketing pitch has never been the most persuasive for me.
Pure math is probably not worth the squeeze. I think more important to everyday life is systems thinking and a bit of probability/stats, mainly bayesian updates. "Superforecasting" was an eye-opening book to me, I could see how most people would benefit massively by it.
Similar to systems thinking, just the ability to play out scenarios in your head given a set of rules is a very useful skill, one which programmers tend to either be good at because of genetics or because we do it every day (i.e. simulate code in our head). You can tell when someone lacks this ability when discussing something like evolutionary psychology. Someone with a systems thinking mindset and an ability to simulate evolution tend to understand it as obvious how evolutionary pressures tend to, and really must, create certain behavior patterns (on average), while people without this skill tend to think humans are a blank slate because it's easier to think about, and also is congruent with modern sensibilities.
This skill applies in everyday life, especially when you need to understand economics (even basic things like supply and demand seems elusive to many), politics etc.
Abstract Algebra, Combinatorics, and Discrete Mathematics are all definitely worth the squeeze; and incidentally something that could easily be taught to middle- and high-schoolers with the right examples.
Is it worth it to be able to think better, have a growth mindset and learn how to learn? Yes. Everyone can benefit from that. Pushing on into higher math? No, very few people can benefit from that.
Math doesn’t seem to me the only source of thinking clearly, or learning how to learn, etc. And if I’m searching for an aesthetic high, there are definitely better places to look — and ones that don’t require such a long runway.
It doesn't need to be for me to be right. These are false constraints you're trying to put on it. Mathematics in moderation can benefit everyone. This claim stands.
I'll second guerrilla - you can absolutely benefit from mathematical thinking without pushing into territory higher than undergaduate studies.
You can even benefit from the thinking taught in good high school coursework (or studying online).
At an arithmetic, bookkeeping level you can better appreciate handling finances and the seductive pitfalls surrounding wagers (gambling, betting, risk taking).
My claim isn’t really that there’s no benefit or utility to math — that’s obviously false — but that maybe its benefits to regular people are more modest than the cheerleaders want to admit.
What are the costs (in your estimation at least) to "regular people" (regular by your metric) of not engaging in easy bake low level "mathematical thinking".
* How many have a lower return on { X } through not understanding compound interest, tax brackets, leveraging assets, etc.
* How many have steady net losses through "magical thinking" wrt gambling, betting, hot stock tips.
You’re sort of making my point — there are people out there who think math education sets the mind free and opens the gates of higher cognition, and then others talking about hum drum stuff like tax brackets and compound interest. If the benefits really just amount to a few units of pre-algebra content, that would be disappointing.
> If the benefits really just amount to a few units of pre-algebra content, that would be disappointing.
They don't though - there are benefits all the way up.
Giving a few examples of benefits from low order mathematical thinking (understanding concepts of compounding, etc) does not equal a statement that these are the only benefits.
Great, so what are those benefits all the way up?
At least a finite countable lists worth, there's something for everybody really.
Some of us are satisfied with, say, dimensional reductions of spectral geophysical surveys to improve imaging and train that against existing production records to highlight potential mineral leases.
Others might like to leverage symbolic algebra systems to crack quantum encryption candidates.
A good number like to get rich via high frequency trading, some like to sail satellutes against the magnetic currents of the planet, there's a world of optimisations in logistics that have been implemented and are still being fine tuned to improve throughput efficiency and fuel consumption.
But you're not really asking in good faith here, are you?
i’ve always found that mathematical thinkers do better, no matter the role. it’s difficult to express how, it’s kind of intangible, but it’s just a mental process that helps break down tasks and come up with unique solutions.
that being said, my marketing pitch for math was always, nothing else is even close to as intellectually stimulating as math.
yeah, I wish we could be more specific here. Like I appreciate the things you’re saying, and I don’t doubt the truth or sincerity of that opinion — but I feel like I’ve encountered some math PhD students who were surprisingly inflexible in the way they thought, and very parochial in their intellectual interests. I guess I just haven’t (anecdotally) found the transference of math training to other domains as much as others seem to.
i think that’s a characteristic of a lot of PhD’s though - they become so specialized they become inflexible. i’m more talking about applied math into industry types. in insurance, tech, etc, i always tend to find the mathy people are faster, come up with unique solutions, and are more self motivated.
I've lately been struck by people having a life difficulty due to "missing a clue" absent some experience. The person poorly conceptualizing and executing physical therapy, for lack of athletic experiences. The person poorly handling cognitive decline, for lack of a grasp of work processes. The person variously failing from having physical discomfort as an abort criteria, for lack of experiences normalizing its deferral.
"I have this clue at hand" can have broad impacts. Software development's emphasis on clarity, naming, and communication protocols, helped me a lot with infant conversation. Math done well, can be a rich source of clues, especially around thinking clearly.
There's an idea that education should provide more life skills (like personal finance). And another, that education should have a punch list (as in construction), of "everyone at least leaves with these". Now AIish personalized instruction will perhaps permit delivering a massive implicit curriculum, far larger than we usually think of as a reasonable set of learning objectives. Just as a story can teach far more than the obstensible moral/punchline of the story, so too might each description, example, question and problem, dynamically tuned in concert. So perhaps it's time to start exploring how to use that? In the past, we worked by indirection - "do literary criticism, and probabilisticly obtain various skills". And here, from math. Perhaps there's a near-term opportunity to be more explicit, and thorough, about the cluefulnesses we'd like to provide?
I studied math hard for several years in college and graduate school—purely out of interest and enjoyment, not for any practical purpose. That was more than forty years ago, but Bessis's description of the role of intuition in learning and doing math matches my recollection of my subjective experience of it.
Whether that youthful immersion in math in fact benefitted me in later life and whether that kind of thinking is actually desirable for everyone as he seems to suggest—I don't know. But it is a thought-provoking interview.
I also studied it and got several degrees, but I don't think that it actually benefited me. I think high school math is incredibly important to be able to think clearly in a quantitative way, and one university-level statistics course, but all the other university math... I dont think it helped me at all. I am disappointed by it because I feel that I was misled to believe that it would be useful and helpful.
Have you ever ascribed numbers to real life personal problems? I find that managing to frame something bothersome into a converging limit somehow, really dissolves stress.. A few times at least.
That’s an interesting approach. I don’t think I’ve done that myself, but I can see how it could be helpful.
One positive effect of having studied pure mathematics when young might have been that I became comfortable with thinking in multiple layers of abstraction. In topology and analysis, for example, you have points, then you have sets of points, then you have properties of those sets of points (openness, compactness, discreteness, etc.), then you have functions defining the relations among those sets of points and their properties, then you have sets of functions and the properties of those sets, etc.
I never used mathematical abstraction hierarchies directly in my later life, but having thought in those terms when young might have helped me get my head around multilayered issues in other fields, like the humanities and social sciences.
But a possible negative effect of spending too much time thinking about mathematics when young was overexposure to issues with a limited set of truth values. In mainstream mathematics, if my understanding is correct, every well-formed statement is either true or false (or undecided or undecidable). Spending too much time focusing on true/false dichotomies in my youth might have made it harder for me to get used to the fuzziness of other human endeavors later. I think I eventually did, though.
Thanks for sharing. The reverse direction here, I'm trying to go from fuzziness to the exactness of those true/false dichotomies, haha. The way I've been attacking mathematics, it's like a tree in the forest, one could start with the axioms and from the base reach each branch and the leaves and fruits. But I've just been walking around the tree, looking at the leaves and fruits and branches from different directions to see ways of climbing without doing a whole lot of climbing. What I mean is I've been thinking and reading in an imprecise way a whole lot without actually juggling symbols with pen and paper, haha. Or a roadtrip analogy, I've done little driving and a lot of map ogling. At least I won't miss the turns when I pick up some speed.
I want to say yes, but I have two counters. One is that math nerds at school insisted on intimidating for the win and I just hated it.
The second is notation. I had a snob teacher who insisted on using Newton not Leibniz and at school in the 1970s this is just fucked. One term of weirdness contradicting what everyone else in the field did. Likewise failure to explain notation, it's hazing behaviour.
So yes, everyone benefits from maths. But no, it's not a level playing field. Some maths people, are just toxic.
The different notations in calculus are unfortunately a historical accident and you're going to see all of them if you do anything with it (it's certainly not true that everyone in the field uses the same notation). I agree that it's frustrating but so are irregular verbs in Spanish/French/other languages - you can't really change it.
> One is that math nerds at school insisted on intimidating for the win and I just hated it.
Only an adult can look and see what that was - immature, insecure little boys, desperately trying to show off as bigger/more mature or kick down anybody showing any weakness or mistake. Often issues from home manifesting hard. Its trivial to look back without emotions, but going through it... not so much.
If my kids ever go through something similar (for any reasons, math nerds are just one instance of bigger issue) I'll try reasoning above, not sure if it will help though.
> Only an adult can look and see what that was - immature, insecure little boys, desperately trying to show off as bigger/more mature or kick down anybody showing any weakness or mistake. Often issues from home manifesting hard. Its trivial to look back without emotions, but going through it... not so much.
I'm not so sure that adults always get it or rise about this. This happens in the workplace all the time.
> One is that math nerds at school insisted on intimidating for the win and I just hated it.
For me the worst part was the teachers that encouraged that behavior and did the same.
"mathematics is a game of back-and-forth between intuition and logic" I teach/guide Math at our school (we run a small school and currently have kids under age 10) and this is so so true.
I just wrote about this. In fact, you can even see this at play in the video of the kids talking https://blog.comini.in/p/what-happens-in-math-class
This was a very tiring blog post for me. And I have a quip about posts that open with questions but close without obvious definite answer, no matter how simple it is.
Tiring because the answer wasn't revealed? That was the whole point :) It's the path, not the summit.
Does the quip have to do with someone asking what the quip is?
Not sure this article captured it for me.
Plato's Meno has Socrates showing that even a slave can reason mathematically.
It's not really math alone but modeling more generally that activates people's reasoning. Math and logic are just those models that are continuous+topological and discrete+logic-operation variants, both based in dimension/orthogonality. But all modeling is over attribution - facts, opinions, etc., and there's a lot of modeling with a healthy dose of salience - heuristics, emotions, practice, etc. Math by design is salience-free (though it incorporates goals and weights), so it's the perspective and practice that liberates people from bias and assumptions. In that respect it can be beautiful, and makes other more conditioned reasoning seem tainted (but it has to work harder to be relevant).
However, experts can project mathematical models onto reality. Hogwash about quantum observer effects and effervescent quantum fields stem from projecting the assumptions required to do the math (or adopt the simplifying forms). Yes, the model is great at predictions. No, it doesn't say what else is possible, or even what we're seeing (throwing baseballs at the barn, horses run out, so barns are made of horses...). Something similar happens with AI math: it can generate neat output, so it must be intelligent. The impulse is so strong that adherents declare that non-symbolic thinking is not thinking at all, and discount anything unquantifiable (in discourse at least). Assuming what you're trying to prove is rarely helpful, but very easy to do accidentally when tracking structured thinking.
Has anyone here self-taught themselves math in later life?
I studied up to A level (aged 19) but honestly started hating math aged 16 after previously loving it.
It’s a big regret of mine that I fell out of love with it.
I self taught myself coding and Spanish and much enjoy self study if I can find the right material.
Any suggestions?
Try this remarkable book:
Who Is Fourier?: A Mathematical Adventure
https://www.amazon.com/Who-Fourier-Mathematical-Transnationa...
It started off as a bunch of non-math literate folks teaching themselves math from scratch, including trigonometry, calculus etc, and ending in Fourier series. It is a very approachable and fun book.
I was the same In high school.
2 weeks ago I hired a professor to help me learn math again so I can attend University computer science.
I can tell you, you can and should.
I'm totally addicted to math, I work as a programmer once I finish my work for the day I spend all my free time learning math again.
I'm still going over the very basics like 9 th grade stuff but I can see already it's going to go fine! I'm enjoying it so much!
Math Academy and (optionally for a deeper study) the Art of Problem Solving books.
Check out Susan Rigetti's guide: https://www.susanrigetti.com/math
Possibly making a small, focused game!
This is how I got back into learning maths... through necessity and immediate application.
List of good books, sorted by difficulty:
- Maths: A Student's Survival Guide (ISBN-13 978-0521017077)
- Review Text in Preliminary Mathematics - Dressler (ISBN-13 978-0877202035)
- Fearon's Pre-Algebra (ISBN-13 978-0835934534)
- Introductory Algebra for College Students - Blitzer (ISBN-13 978-0134178059)
- Geometry - Jacobs ( 2nd ed, ISBN-13 978-0716717454)
- Intermediate Algebra for College Students - Blitzer (ISBN-13 978-0134178943 )
- College Algebra - Blitzer (ISBN-13 978-0321782281)
- Precalculus - Blitzer (ISBN-13 978-0321559845)
- Precalculus - Stewart (ISBN-13 978-1305071759)
- Thomas' Calculus: Early Transcendentals (ISBN-13 978-0134439020)
- Calculus - Stewart (ISBN-13 978-1285740621)
The main goal of learning is to understand the ideas and concepts at hand as “deeply” as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By “deeply” we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no “perfect” state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: “Young man, in mathematics you don't understand things. You just get used to them”, which I think really means that getting “used to” some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination.
Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap).
Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding.
Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory.
Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note.
Cross-reference. Don't read linearly. Instead, have multiple textbooks, and “dig deep” into concepts. If you learn about something new (say, linear combinations) — look them up in two textbooks. Watch a video about them. Read the Wikipedia page. Then write down in your notes what a linear combination is.
Learning is a social activity, so maybe enroll in a community college course or find a local study group. I find it's especially important to have someone to discuss things with when learning math. I also recommend finding good public spaces to work in—libraries and coffee shops are timeless math spaces.
Pay graduate students at your local university to tutor you.
Take walks, they're essential for learning math.
Khan Academy is not enough. It has broad enough coverage, I think, but not enough diversity of exercises. College Algebra basically is a combination of Algebra 1, Algebra 2, relevant Geometry, and a touch of Pre-Calculus. College Algebra, however, is more difficult than High School Algebra 1 and 2. I would tend to agree that you should start with either Introductory Algebra for College Students by Blitzer or, if your foundations are solid enough (meaning something like at or above High School Algebra 2 level), Intermediate Algebra by Blitzer. Basically, Introductory Algebra by Blitzer is like Pre-Algebra, Algebra 1, and Algebra 2 all rolled into one. It's meant for people that don't have a good foundation from High School. I would just add, if it is still too hard (which I doubt it will be for you, based on your comment), then I would go back and do Fearon's Pre-Algebra (maybe the best non-rigorous Math textbook I've ever seen). Intermediate Algebra is like College Algebra but more simple. College Algebra is basically like High School Algebra 1 and 2 on steroids plus some Pre-Calculus. The things that are really special about Blitzer is that he keeps math fun, he writes in a more engaging way than most, he gives super clear—and numerous—examples, his books have tons of exercises, and there are answers to tons of the exercises in the back of the book (I can't remember if it's all the odds, or what). By the time you go through Introductory, Intermediate, and College Algebra, you will have a more solid foundation in Algebra than many, if not most, students. If you plan to move on to Calculus, you'll need it. There's a saying that Calculus class is where students go to fail Algebra, because it's easy to pass Algebra classes without a solid foundation in it, but that foundation is necessary for Calculus. Blitzer has a Pre-Calculus book, too, if you want to proceed to Calculus. It's basically like College Algebra on steroids with relevant Trigonometry. Don't get the ones that say “Essentials”, though. Those are basically the same as the standard version but with the more advanced stuff cut out.
Given the sentiment in these comments, I figure this crowd might be interested in the book "Measurement" by Paul Lockhart (the guy who wrote "A Mathematician's Lament")
He's of the opinion that math should be taught not as jumping through hoops for "reasons", but as an art, enjoyable for its own sake, and that this would actually produce more confident and capable thinkers than the current approach. (I think the argument applies to almost all education but his focus is just on math.)
> everyone can, and should, try to improve their mathematical thinking — not necessarily to solve math problems, but as a general self-help technique
Agreed with the above. Almost everyone can probably expand their mathematical thinking abilities with deliberate practice.
> But I do not think this is innate, even though it often manifests in early childhood. Genius is not an essence. It’s a state. It’s a state that you build by doing a certain job.
Though his opinion on mathematical geniuses above, I somewhat disagree with. IMO everyone has a ceiling when it comes to math.
> IMO everyone has a ceiling when it comes to math.
Yes, but it's higher than you think: https://www.justinmath.com/your-mathematical-potential-has-a...
Reading most of the answers here, I can only conclude most of you were home schooled or went to some fancy schools for gifted children.
An average human is unable to even write properly. Even basic mathematical operations like multiplication and division are too complicated from their perspective.
God that's one of the most pessimistic views of humanity I've seen on the web, and I'm pretty pessimistic. Developed nations have a 99.2% literacy rate per wikipedia. 69% of OECD students meet "level 2" mathematical proficiency (https://www.oecd.org/en/topics/sub-issues/mathematics-litera...), which indicates an understanding above the basic mathematical operators like multiplication and division.
FWIW I did not go to any fancy school, I was educated in the state system.
Not sure about schools for gifted, but I think lots of people here a bit live in bubbles.
Maybe industrialized countries, a bit better off neighborhoods, STEM classes in high school, and then university? (Not everyone but ... Most here?)
As a young child your brain is much more suitable to learn languages. You can make kids learn 4 languages effortlessly in the right context. When you grow up, slowly shift the focus to abstract thinking. And that shift can rely on building intuition using visualisation and experience.
I totally agree! The barriers many of us face with math are less about ability and more about how we've been taught to approach it. All it took was for me to change my math teacher at school, and boom. Love, but at second sight. And curiosity and persistence can unlock more than just numbers
Some very nice related works that dispel widespread math myths: (1) What Is Mathematics, Really? Hersh, 1997. https://books.google.com/books/about/What_is_Mathematics_Rea... (2) Where Mathematics Comes From. Lakoff and Nunez, 2000. https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
Statistical (Bayesian) thinking is an extremely underrated way of thinking of almost everything.
Frankly, its overrated. Now you can adjust your priors.
I'm pretty down on it just by association. So many people who are super-into it seem to be doing a lot of that rigorously-wrong engineer-brain thing.
lol yea.
bayesian thinking doesnt come to me naturally.. i have no intuition for it. seems forced. believe me - i have tried. but there are those who are swearing by it.
Two thoughts
1) It's tragic that being "bad at math" is often positioned as some kind of badge of honour.
2) It's definitely not the case that everyone is capable of mathematical thinking. Having spent a certain amount of time trying to teach one of my kids some semblance of mathematical thinking, I can report confidently that his ability in this area is almost non-existent. His undeniable skills lie in music and writing, but definitely not in maths.
Yes, music and maths have some things in common. But musical thinking is not mathematical thinking.
I have an autistic friend with dyscalculia. They see numeric digits as individual characters (as in a story), each with their own personalities. Each digit has its own color, its own feelings. But they are not quantities; they don't make up quantities. Numbers are very nearly opaque to them. I wonder how this theory would apply to them. Do they still perform mathematical thinking? They're still capable of nearly all the same logic that I am, and even some that I'm not (their synesthesia gives them some color/pattern/vibes logic that I don't have)... just not math.
I don't have dyscalculia but behind the numbers I have my own intuitive system(s) that I jump to sometimes when doing arithmetic. I think we all do since the early days arithmetic in school or what not, perhaps the dyscalculia folk missed making some connection at some point. I feel that arithmetic with numbers without that intuitive system is rote memory..
Indeed, I do also have an intuitive system for this; that's what I mean by "make up a quantity". When something makes up a quantity for me, then I can perform math on it. I have to wonder if my friend with dyscalculia doesn't have a "quantity" system like I do. They can't really perform multiplication or division but can technically perform basic addition and subtraction, though it takes them a while. I think it may be the rote memory you mention doing that for them.
Great to see so many reactions to my interview, thanks!
I see that many people are confused by the interview's title, and also by my take that math talent isn't primarily a matter of genes. It may sound like naive egalitarianism, but it's not. It's a statement about the nature of math as a cognitive activity.
For the sake of clarity, let me repost my reply to someone who had objected that my take was "clickbait".
This person's comment began with a nice metaphor: 'I cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains.'
Here was my reply (copy-pasted from my post buried somewhere deep in the discussion):
I'm the author of what you've just described as clickbait.
Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.
The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.
Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).
When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.
As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.
In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.
There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.
I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.
Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.
A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.
In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.
> There are actual techniques that career mathematicians know about.
Your best example is the decimal system in contrast to roman numerals. I think that explains the point well. The zero is one of those tricks, and most people know it now, but that wasn't true until very recently.
I think you've simply redefined genius. Many years ago I read an article on youth football, if I remember correctly, and in it there was a bit about the writers visit to the Ajax Youth Academy. In it he writes of a moment during practice when a plane flies over and all the 7(?) year olds on the pitch look up to see it except for one kid who keeps his eyes on the ball. That kid (of course) grows up to be a very good midfielder for Real (I'm forgetting the exact details, I think its Wesley Sniejder?). My point is: whatever that motive energy is that manifests as the single minded pre-occupation with math at an age when everybody else's attention is all over the place is that inherent thing that people call genius. I have read many of Thurston's non-mathematical writings about himself and in it this sort of singular pre-occupation is also clear -- which is why he developed his preternatural geometric vision.
Indeed, it does involve redefining genius as a "state", or "flow", or "trajectory".
When I say it's not primarily genetic, many people wrongly assume there's an entirely explainable and replicable way of accessing this state. There isn't.
The 20,000 hours rule is a bit misleading, because who gets to invest 20,000 hours into something? How do you create this drive, this trajectory? You must have a good hope that it'll yield something worth the effort.
This is why the injunction to "work harder" so often misses the mark.
However, even if only a tiny fraction of the population will end up becoming a "genius", it's very important to debunk the myth, because the real story has valuable lessons for everyone: it gives concrete and pragmatic indications on what one should be on lookout for.
It's not fully teachable up to genius level, but the directionality is teachable and extremely valuable.
> I see that many people are confused [...] by my take that math talent isn't primarily a matter of genes
Speaking only for myself, I'm not confused at all. Rather I vigorously disagree with this statement, and think that stumping for this counterfactual premise leads to cruel behavior towards children (in particular) who plainly do not have what it takes to learn, for example and in particular, algebra.
> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.
This is not their subject of expertise, and they are simply wrong. Why? Simpson's Paradox, ironically.
I think you really are confused. You are mistakenly equating "non-primarily genetic" with "easily teachable".
The story is much more complex than "if it's not genetic then everybody should get it". It's quite cruel to assume that if you don't get math today you'll never get it, and there are tons of documented counter-examples of kids who didn't get it at all who end up becoming way above average.
If you think that Descartes, Newton, Einstein, Feynman, Grothendieck (to just cite a few) are all equally misled on their own account because of Simpson's Paradox, which statistical result will to bring to the table to justify that YOU are right?
By the way, Stanislas Dehaene, one of the leading researchers on the neuroscience of mathematical cognition, is also on my side.
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When you write a comment here, you're talking not simply (or even primarily) to the person you're responding to, but also to the rest of the community. Have you considered what value there might be, or not be, in a comment like this?
We are extraordinarily fortunate to be taken seriously enough as a community for the primary sources on stories like this to come talk to us directly. I'd ask, as a neighbor of yours in this community, that you not (rhetorically) chase those people away.
Thanks!
That's a fair ask, that post was not one of my better moments. As much as I dislike the rhetorical tactic wielded with "everyone who doesn't agree with me is simply confused", a one-sentence tell off does nothing to counter it, and just makes me look like a jerk. Which I was.
No problem! You'd be doing me a favor if you noted any time you see me in the same situation. Thanks for hearing me out!
The evidence of your confusion is that you cited one sentence in their comment but ignored the next sentence which provide the justification in abstract for that sentence. Therefore your basic reading comprehension skills are confused. It easier to disagree vigorously but when you do that it clouds your analytic ability to follow the actual arguments given in that way.
For example finance is such an important aspect of our lifes and you just need some understanding of math principles to understand how to make good financial choices.
Ironically, the ones who don't do well at school (inc maths) are the ones who then become trades like builders/plumbers etc or run small businesses like a shop. And so are regularly working with numbers via estimates and billing.
Shouldn’t that headline read “or can benefit from”. Not a shining example of good mathematical thinking. Oh the irony.
This is pretty interesting. I did reasonably well in maths up to the A-levels, and then absolutely collapsed in university. I never got a grade better than B- in any maths-adjacent class. Discrete maths was my worst topic, I barely scraped a pass. And the irony was that I majored in CS and physics.
I should probably find a time machine and re-do everything.
- Hey teach, will I really need all these logarithms, derivatives and vectors in my adult life?
- No, but the smarter kids might.
I used to get very frustrated that others could not intuit information the way I could. I have a lot of experience trying to express quantities to leaders and policymakers.
At the very minimum, I ask people to always think of the distribution of whatever figure they are given.
Just that is far more than so many are willing to do.
Waste of time. Just talk in terms of what they want to hear. They are just interested in the payoffs (not in the details).
As info explodes and specialists dive deeper into their niches, info asymmetry between ppl increases. There are thousands of specialists running in different directions at different speeds. Leaders can't keep up.
Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.
And while most specialists think this "sync" process happens through "education" and getting everyone to understand a complex ever changing universe, the truth is large diverse groups are kept in sync via status signalling, carrot/stick etc. This is why leaders will pay attention when you talk in terms of what increases clout/status/wealth/security/followers etc. Cause thats their biggest tool to prevent schisms and collapse.
> Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.
This is overthinking it. People with power tend to be interested in outcomes. They can't evaluate all the reasoning of all their reports. It comes down to building credibility with a track record and articulating outcomes, when you want to advise decision makers.
I believe charisma, confidence and looks also play a huge role.
But, is that profitable? I'm both being sarcastic and real with this question.
If I can earn an extra 1 million being 'dumb' and thus ensure quality healthcare, education, housing, is it smart to try to be smart?
This is the true tragedy of the commons (or the reverse tragedy, to be precise).
there's thinking mathematically and then there's being able to fluently read math articles on wikipedia as if they're easier than ernest hemingway. I can do the former and the latter I will insist until my grave is impossible for me.
I used to judge myself for not understanding everything in math articles on Wikipedia, but as time has gone on I've realized that their purpose isn't really to be an introduction, but a reference. Especially as the topics become more esoteric. So they're not really there for you to learn things from scratch, but for people who already understand them to look things up. Which is why you'll sometimes see random obscure & difficult factoids in articles about common mathematical concepts.
(don't have any examples on-hand atm, this is just my general perception after years of occasionally looking things up there)
I've heard that and I think it's silly. They handwave away why nothing should ever be explained. Wikipedia doesn't work like that for any other topic.
You'll see something like a mathematical proof with no explanation and it's end of article. The edit history will have explanations aggressively removed.
The equivalent would be the article for say, splay tree, to have no diagrams and just a block of code - feeling no obligation to explain what it is or if you looked up a chemical and it would just give you some chemical equation, some properties and feel no obligation to tell you its use, whether it's hazardous or where you might find it... Or imagine a European aristocrat and all that is allowed is their heraldry and genealogy. Explanations of what the person did or why they're important are forbidden because, it's just a reference after all.
Nope, these math people are a special kind of bird and I'm not one of them.
I don't know anybody who first learns about new mathematical ideas from Wikipedia. Mathematics is a body of knowledge, not just simple isolated theorems or definitions. You learn new mathematics from textbooks.
Even for reference purposes there are often better resources. E.g. proofwiki is usually better for looking up proofs because the proofs and definitions are interconnected.
If I run across a term like "bialgebra" while doing work, I don't always have the leisure time to derail my life and sign up for a 4 month class at a local university. Sometimes, I just need to move on with my task at hand and get something working.
I'm familiar with the mathematician response to this, I've heard it before and I fundamentally disagree with it. At work last week I gave someone a crash course in the simplex method and linear programming in about 30 minutes and it was a good-enough explanation that I came back in a few hours and the code was right.
This isn't impossible. There's just some wild apprehension that I'll never understand which insists everything is a grueling 1,000 hour journey to some kind of valhalla of enlightenment so you can bask in some aesthetic beauty of how perfect math is, as tears drip down from your cheeks, or something like that.
I mean come on now. Sometimes all you want is the cliff notes.
Well, you're comparing a concrete algorithm from applied mathematics (simplex) to a term from abstract algebra. I'm not exactly sure how you'd expect such a general concept to be described. Where in your work do terms like "bialgebra" regularly come up and is "it's some sort of algebraic structure" not enough of an understanding to continue without digging into the details? Maybe your problem is with people who put abstract mathematics into applied material without motivation or explanation and not with research mathematicians themselves?
Would you expect to be able to read the "cliff notes" on French and then be able to read Camus in the original? That's what I mean by "body of knowledge" as opposed to individual facts.
I have a lot of trouble reading math formulas, implemented as code I understand most stuff though. Is there a good math book or something similar that teaches things using code or helps translating formulas to code?
I realized some time in middle age that I have to convert formulas and equations to steps and things happening to something "passing through" each step—to algorithms. It's painful and slow and also the only way I stand a chance in hell of reading mathematical writing.
That's probably why math writing largely makes me feel dyslexic, while programming came naturally. And why I hate Haskell and find it painful to read even though I understand the "hard" concepts behind it just fine—it's the form of it I can't deal with, not the ideas.
I conducted an interview with leetcode 2 years ago while not doing any leetcoding before. Surprisingly, by just applying some math tricks i finished them and got into later rounds. So yes math tricks are helpful.
This guy is unbelievably French (I mean in his intellectual character). Here I was expecting a kind of rehash of the 20th century movements of pure math and high modernism[0], but instead we get a frankly Hegelian concept of math or at least a Hegel filtered through 20th and 21st century French philosophy.
[0]https://news.ycombinator.com/item?id=41962944
there is a debate between the intuitionists, formalists, and the symbolists nicely captured in the intro chapter of Heyting's Intuitionism.
constructive mathematics is close to computation and programming. and many including myself have a natural feel or intuition for it. A majority of euclids elements, and galois's original proof are constructive in nature.
I was actually thinking Jean Paul Satre when I read his answers
That's a very important thougt and I belive the world would be better, if more people would connect with their mathematical side.
To use an absolutist statement about human well bieng tied to the authors pet ideals,is a classic example of magical thinking, and a lack of understanding in general,tied to a character assination of the many many excellent non numerative humans living and thriving everywhere. Not nice at all.
But what is the difference between math talent and plug-n-chug math talent? That seems to be the most significant filter.
And yet it is flame-bait to suggest that programmers benefit from mathematical thinking. I've not met a more passionate and divided crowd on the issue. Most traditional engineers wouldn't disagree that they use and benefit from mathematical thinking. Programmers though?
I don't think there's a single answer as to why many dislike it so much. Some folks view it as a way to gate-keep programming. Others view it as useless ("I've been a successful programmer all my life and I've never used math").
On the other side of the coin there are many who view our craft as a branch of applied mathematics -- informatics if you will.
It's a shame the title begins with 'Mathematica', makes one think the book is about the Wolfram software. That's the first thing I thought of when I saw the title. Hopefully Wolfram doesn't sue him for copyright violation or some such infringement.
I guess I assumed it was a reference to Alfred Whitehead's and Bertrand Russell's book Principia Mathematica, which predates Wolfram himself by several decades.
This interplay between intuition and logic is exactly what makes the magic happen. You need intuition to feel your way forward, and then logic to solidify your progress so far, and also for ideas maybe not directly accessible via intuition only. I've experienced that myself, and it is even well-documented, because I wrote technical reports and such at each stage. My discovery of Abstraction Logic went through various stages:
1) First, I had a vague vision of how I want to do mathematics on a computer, based on my experience in interactive theorem proving, and what I didn't like about the current state of affairs: https://doi.org/10.47757/practal.1
2) Then, I had a big breakthrough. It was still quite confused, but what I called back then "first-order abstract syntax" already contained the basic idea: https://obua.com/publications/practical-types/1/
3) I tried to make sense of this then by developing abstraction logic: https://doi.org/10.47757/abstraction.logic.1 . After a while I realized that this version only allowed universes consisting of two elements, because I didn't distinguish between equality and logical equality, which then led to a revised version: https://doi.org/10.47757/abstraction.logic.2
4) My work so far was dominated by intuition based on syntax, and I slowly understood the semantic structures behind this: the mathematical universe consisting of values, and operations and operators on top of that: https://obua.com/publications/philosophy-of-abstraction-logi...
5) I started to play around with this version of abstraction logic by experimenting with automating it, giving a talk about it at a conference, (unsuccessfully) trying to publish a paper about it, and implementing a VSCode plugin for it. As a result of using that plugin I realized that my understanding until now of what axioms are was too narrow: https://practal.com/press/aair/1/
6) As a consequence of my new understanding, I realized that besides terms, templates are also essential: https://arxiv.org/abs/2304.00358
7) I decided to consolidate my understanding through a book. By taking templates seriously from the start when writing, I realized their true importance, which led to a better syntax for terms as well, and to a clearer presentation of Abstraction Algebra. It also opened up my thinking of how Abstraction Algebra is turned into Abstraction Logic: https://practal.com/abstractionlogic/
8) Still lots of stuff to do ...
I would not be surprised if that is exactly the way forward for AIs as well. They clearly have cracked (some sort of) intuition now, and we now need to add that interplay between logic and intuition to the mix.
What is 13% of 91? I don't know. Do you? But I now 10% of 91 is 9.1 I got somewhere eh? Hey also I know that 1% of 91 is 0.91 Duh! lets triple that. 0.91 x 3 = 0.9x3 + 0.01x3 = 2.7 + 0.03 = 2.73 Now lets add 9.1.. 11.83 Weee! (Now my date is rolling her eyes and the waiter is stone faced)
What an incredibly neuro-typical thing to claim
Gentle Reminder that the author of this article used to have a wonderful math channel: https://www.youtube.com/c/pbsinfiniteseries
???
"Mathematician Tai-Danae Bradley and physicist Gabe Perez-Giz offer ambitious content ... Previous host Kelsey Houston-Edwards "
Careful there. They'll start voting logically..
https://www.smbc-comics.com/comic/grind
"High school students are often unhappy with math, because they think it requires some innate things that they don’t have,” Bessis said. “But that’s not true; really it relies on the same type of intuition we use every day."
Agreed, but from my observation mathematics is often taught with a rigor that's more suited to students with a highly mathematical and or scientific aptitude (and with the assumption that students will progress to university-level mathematics), thus this approach often alienates those who've a more practical outlook towards the subject.
Mathematics syllabuses are set by those with high mathematical knowledge and it seems they often lose sight of the fact that they are trying to teach students who may not have the aptitude or skills in the subject to the degree that they have.
From, say, mid highschool onwards students are confronted with a plethora of mathematical expressions that seemingly have no connection their daily lives or their existence per se. For example students are expected to remember the many dozens of trigonometrical identities that litter textbooks (or they did when I was at school), and for some that's difficult and or very tedious. I know, I recall forgetting a few important identities at crucial moments such as in the middle of an exam.
Perhaps a better approach—at least for those who are seemingly disinterested in (or with a phobia about) mathematics—would be to spend more time on both the historical and practical side of mathematics.
Providing students with instances of why earlier mathematicians (earlier because the examples are simpler) struggled with mathematical problems and why many mathematical ideas and concepts not only preceded but were later found to be essential for engineering, physics and the sciences generally to advance would, I reckon, go a long way towards easing the furtive more gently into world of mathematics and of mathematical thinking.
Dozens of names come to mind, Euclid, Descartes, Fermat, Lagrange, Galois, Hamilton and so on. And I'd wager that telling students the story of how the young head-strong Évariste Galois met his unfortunate end—unfortunate for both him and mathematics—would never be forgotten by students even if they weren't familiar with his mathematics—which of course they wouldn't be. That said, the moment Galois' name was mentioned in university maths they'd sit up and take instant note.
Yes, I know, teachers will be snorting that there just isn't time in the syllabus for all that stuff, my counterargument is that it makes no sense if you alienate students and turn them off mathematics altogether. Clearly, a balance has to be struck, tailoring the subject matter to suit students would seem the way with the more mathematically inclined being taught deeper theoretical/more advanced material.
I always liked mathematics especially calculus as it immediately made sense to me and I always understood why it was important for a comprehensive understanding of the sciences. Nevertheless, I can't claim that I was a 'natural' mathematician in the more usual context of those words. I struggled with some concepts and some I didn't find interesting such as parts of linear algebra.
Had some teacher taken the time to explain its crucial importance in say physics with some examples then I'm certain my interest would have been piqued and that I'd have showed more interest in learning the subject.
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