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

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 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.

> 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.

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.