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The Mind of a Mathematician

paw.princeton.edu

51–60 of 68 posts

Re: The Mind of a Mathematician

#51
post #18

I very much like that people like Tao are famous. It’s relatively good for the intellectual spirit and hopefully it inspires others. However this fixation with his “extraordinary capacity”, scores, medals and accolades; I wish we could skip it. Let’s talk about the awesome work and why it’s awesome. Let’s play through some of it: I’m relatively sure most people know nothing more about Tao than “he’s a genius”. What’s…

i've said this same thing (on r/math), because it feels like blind hero worship and fetishization, but i got pushback that "it's only human". oh well. most people don't care about the math but only about hero worship and math is just another opportunity.

I think our world would be in better shape if people like Terry Tao were the heroes more people worshipped.

Re: The Mind of a Mathematician

#52
post #49
post #43

Earlier quoted context omitted.

I'd (conservatively) speculate most of us here are in the global 1% in terms of "ability to understand math". Have tried multiple times to follow his work (from the blog). But I'm not math-fluent enough to understand or otherwise appreciate the content. Possibly I could but it requires so much effort. It's much easier to enjoy Bach's or Messi's genius :)

This (Bach) is a parallel that comes up often in both lay and professional music on the nature of "genius". I've had some success with the following analogy. Music is not math, but the conversation between research mathematicians is a bit like the conversation between composers. In order for them to understand each other well they must understand the notation, but also the instruments/tools, the history, etc. However…

its often said around school, that the higher level mathematicians cant sit down and talk to each other about their work much .. things go too far into specialization..

Re: The Mind of a Mathematician

#53
post #49

Earlier quoted context omitted.

This (Bach) is a parallel that comes up often in both lay and professional music on the nature of "genius". I've had some success with the following analogy. Music is not math, but the conversation between research mathematicians is a bit like the conversation between composers. In order for them to understand each other well they must understand the notation, but also the instruments/tools, the history, etc. However…

its often said around school, that the higher level mathematicians cant sit down and talk to each other about their work much .. things go too far into specialization..

This is true, that there are lot of silos in mathematics which make communication difficult. In this sense the analogy breaks down a bit.

Re: The Mind of a Mathematician

#55

Earlier quoted context omitted.

It's easy to say you don't need to be a genius to be a mathematician when you're a genius celebrity mathematician. Personally I gave up on math as a career once I realized all the people who got into top grad schools for math were like international IMO participants, USAMO or equivalent at least, started taking classes beyond calculus in 10th grade, etc. Even though I love it and even have an undergrad degree in it,…

Math isn't a competition. You can study and research math as a hobby, just like hundreds of thousands of people do with computer programming

True, but if you intend to make a career in math, which is what opportune gave up on, it is very much a competition. The odds are better than succeeding in basketball, but only because there are a lot more minor league teams.

Re: The Mind of a Mathematician

#56
Could anyone here recommend me something that actually talks about how genius mathematicians and others go about thinking about problems? I've read Hamming's "You and Your Research" and really liked it, but I would love to see similar things by other thinkers who might have a different take than someone like Hamming.

Re: The Mind of a Mathematician

#57
The author notes "Let us hope there are no real-world applications" when describing the Navier-Stokes regularity and smoothness problem. It looks like he meant to write the opposite and this was an error. However, in some sense, the statement may also be true. That is, people have been doing fluid experiments and numerical experiments (numerical fluid mechanics is a mature discipline) without relying on whether the conjecture is true. I mean, flight development was not waiting for an answer!

My point is engineering moved on without an answer to this particular question. At the same time, answering questions such as these (and the ones on number theory two of which were mentioned in the article) that occupy pure mathematicians provide more than artistic pleasure -- they provide the giant leaps that spur new technology. The often quoted example from Tao's research is compressed sensing but even bigger splashes like the idea of computers and the stored program, that can be traced back to Turing's seminal paper, comes from pure mathematics.

Charles Fefferman, mentioned in the article, makes 3 very cogent arguments[1] for the applicability of pure mathematics to the society, and in fact, says very pertinently that the line between pure and applied math is very blurry in the first place: [1]: https://www.youtube.com/watch?v=3LgjMjVA4sY

1. that unanticipated applications show up from purely theoretical questions

2. a rigourous study of math provides a way of thinking that prepares students to work in a range of fields that require quantitative or analytical thinking.

3. math is capable of revealing those ground breaking discoveries that happen rarely

Re: The Mind of a Mathematician

#58
post #35
post #18

I very much like that people like Tao are famous. It’s relatively good for the intellectual spirit and hopefully it inspires others. However this fixation with his “extraordinary capacity”, scores, medals and accolades; I wish we could skip it. Let’s talk about the awesome work and why it’s awesome. Let’s play through some of it: I’m relatively sure most people know nothing more about Tao than “he’s a genius”. What’s…

This is an unsolvable problem. There already is plenty of expository content about what Tao actually does, e.g. his excellent blog. It's just that it can't get popular because it's technical; that's the iron law of STEM popularization, you can only choose one. The most press he's gotten recently has been from this "eigenvectors from eigenvalues" thing that he solved three ways in two hours, probably because it's the…

> This is an unsolvable problem. [...] it can't get popular because it's technical; that's the iron law of STEM popularization, you can only choose one.

I'm reminded of education research of the form "We tried to teach topic T to students in grade G. We taught it really, really badly. Surprisingly, that didn't work! We draw the obvious conclusion... students in grade G are developmentally unready to understand topic T."

So yes, it might truly be unsolvable. It's certainly difficult. Non-interactive defusing of misconceptions is dauntingly hard - even harder than remedial filling of foundational gaps. Then you add constraints on article length, and the medium may just not be adequate.

But I would be more comfortable with such an argument, if there was wider recognition of how wretched our current science education content is, and how badly it's failing students. How poor the stories we tell about the physical world. If your five-year old wants to know what finger-paint color to use for the Sun, don't ask first-tier astronomy graduate students - there's a wide-spread misconception, so they'll mostly get it wrong. On HN a few days back, there was something like a 'best 2019 astronomy books for children'... and the books were something vaguely like 0 of 10 on getting it right. We're just not set up to teach misconception-free transferable broad understanding.

Seeing research talks, I frequently think "Oh nifty - that concept/description/graphic/video is awesome: accessible and clarifyingly insightful. It should be part of every introduction to this topic. Down to primary school even." ... and won't be any year soon. The pipeline from researcher conversation, to talk, to paper and professional tome, down and down to education and popular content, is regrettably also a gradient from accessible/insightful/transferable/correct to confused/superficial/unusable/nonsense. There just hasn't been the incentives and infrastructure to do better.

In another comment you mention intro physics and olympiads. So take friction. We now know how friction works, down to nanoscale. But last I saw, we don't even try to teach that. Instead it's the decades-old plug-and-chug on Amontons' Laws of large objects sliding on pig fat. Sure, there's some nice training on system decomposition. But if we cared about actual understanding of the physical world, rather than the educational artifact of "Introductory Physics", our teaching focus would need to be different. Similarly, professors complain about PhD candidates lacking a rough quantitative feel for the field, and the current educational focus is far from fixing that. But, perhaps I'm out of date, and things are more-recently improving? Fermi problems are becoming much more common, for instance.

Ending on an upbeat note. An MIT project to create introductory cell-biology VR, obtained domain expertise by pulling in and interviewing researchers. One challenge was apparently... getting them to leave. Such was their enthusiasm. Suggesting that if infrastructure has the right shape, the massive and scarce expertise needed for better content might actually be plausibly obtained. And personalized education via XR might be sufficient, and sufficiently disruptive, to deliver it. So perhaps there's hope to do transformatively better than we have been.

Re: The Mind of a Mathematician

#59
post #35

Earlier quoted context omitted.

This is an unsolvable problem. There already is plenty of expository content about what Tao actually does, e.g. his excellent blog. It's just that it can't get popular because it's technical; that's the iron law of STEM popularization, you can only choose one. The most press he's gotten recently has been from this "eigenvectors from eigenvalues" thing that he solved three ways in two hours, probably because it's the…

> This is an unsolvable problem. [...] it can't get popular because it's technical; that's the iron law of STEM popularization, you can only choose one. I'm reminded of education research of the form "We tried to teach topic T to students in grade G. We taught it really, really badly. Surprisingly, that didn't work! We draw the obvious conclusion... students in grade G are developmentally unready to understand topic…

Sorry, but this is idealistic to the extreme. You are completely dismissing the results of education research by saying that every educator is incompetent, despite almost all of these people having far more education experience than both of us combined. Your proposal to fix education is apparently to hyper-focus on a few technical points, like friction and the Sun's color. I don't see how this will help; have you actually spent time teaching intro physics?

In practice, high school and even college introductory classes have a hard time making the basics stick -- stuff as simple as just F = ma. Redirecting half the time you spend on that towards an exquisitely detailed model of friction is not going to produce better physicists. On the average, it's going to produce students that understand F = ma even less, but can recite a couple friction-related buzzwords.

There is no magic curriculum fix that will suddenly make mass education easy. Everybody who thinks about education starts by imagining there is, then reality hits.

Re: The Mind of a Mathematician

#60
post #28

> Yitang Zhang, a mathematician at the University of New Hampshire, proved that there are an infinite number of primes that are separated by, at most, 70 million. > To date, they have managed to prove that there are an infinite number of primes separated by, at most, 246 I'd like to ask a very noob question. What kind of an approach would give us such an exact upper bound, when primes as a concept (in a layman's intu…

Analytic number theorist here. I can answer your question. The method doesn't really yield "exactly 70,000,000". If you traced through his method, and worked out each step in more detail, you'd probably get a bound of (say) 64,189,288 -- or some random number of like that. If you read through it still more carefully, and tried to introduce genuine improvements , you'd improve this further. Indeed, the bound has been…

Thank you!
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