Everyone is capable of, and can benefit from, mathematical thinking

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'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 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.

"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.

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.

> 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

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?

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.