
The Mind of a Mathematician - lordgrenville
https://paw.princeton.edu/article/mind-mathematician
======
usgroup
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 the point of that?

I think we’d be better off if we took the time to understand great work and
let our appreciation of great people live through that.

~~~
knzhou
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 least technical thing he's done.

~~~
mncharity
> 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.

~~~
knzhou
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.

~~~
chongli
Human brains (and bodies) are relentless optimizers. "Use it or lose it"
applies to everything from muscle tone and flexibility up to higher cognitive
stuff like mathematics. The reason nothing "sticks" is because kids stop using
it the moment they stop being tested on it.

We treat education like an assembly line that's supposed to turn every kid
into a productive member of society, using a one-size-fits-all curriculum.
That's a wrong-headed approach. We should treat it like a mix between olympic
trials and career fair. Identify the precocious and the enthusiastic in each
subject area. Let teens decide what they want to do.

It won't work, however, until we give teens the right to self-determination.
Helicopter parents trying to force their kids to be engineers or doctors,
ability be damned, is one of the most pernicious factors in the sorry state of
education. The other, perhaps equally pernicious, is the parents who can't or
won't help their kids with anything despite their eagerness to learn.

------
imbusy111
What strikes me the most is the amount of support he received and the people
he was able to meet when he was a child. I don't want to dismiss his talent,
but this level of support is NOT available to 99%+ of population. His parents
have done a wonderful job.

To recall my days as an undergraduate at a well-known UK university studying
mathematics - large classes (300+) where everyone is disposable, nothing to do
but sleep on the table when you work through the problems quickly at a
workshop, no interesting projects or opportunities to latch on to develop
yourself, nothing to do for days but party, waste talent and occasionally read
a math book to catch up. Essentially you are hammered to be average until
maybe you enter a PhD course. At that point it is too late.

~~~
knzhou
Sure, Tao's an outlier among outliers, but in general nurturing a prodigy does
_not_ require a lot of parental time or money. I've met a number of such
prodigies in the course of my work, and generally their parental involvement
amounts to buying them a book every few months, at their kid's insistence. The
kids then teach themselves, often by sneaking glances at their books or math
problems while hiding out at the back of a standard school classroom. They are
not more privileged or supported than the average middle class child in a
first world country.

The world's best educational resources are dropping rapidly in price; with
some searching, you can even get an mathematical education up to what Tao knew
in graduate school completely for free.

~~~
lonesword
> I've met a number of such prodigies in the course of my work

If you don't mind, can you elaborate more on what kind of work you do? Do you
run a school for the gifted (assuming such a thing exists)?

~~~
knzhou
I've taught at the U.S. Physics Olympiad's training camp, and also tutor high
school students who want to learn a ton of physics quickly. (These kids
usually find me themselves, through Google.)

My most obnoxiously smart student was this 12 year old who would interrupt my
lectures on classical mechanics with tricky questions I couldn't solve. The
next year he came back, doing the same thing but for quantum mechanics, and
the year after that he got bored of this and enrolled in an elite college. I
want to stress that none of this required great wealth or privilege. The cost
for his books was a small fraction of what other middle class parents would
pay for, say, music lessons or SAT tutoring.

~~~
77pt77
Any examples of those questions?

~~~
knzhou
When covering Huygen's principle: the wavelets always give a backwards-moving
wave in addition to the forwards-moving wave. In other words, Huygen's
principle is time symmetric. So in real life, why don't you get the backwards
wave?

When covering the principle of least action: often, applying it will give
infinitely many solutions, or none at all. An example with no solutions would
be the harmonic oscillator with x(0) = 0, x(t) = 0, and t not a multiple of
half the period. An example with infinitely many solutions would be the
principle of least time applied between the two foci of an ellipse with
reflecting walls. So what happens in these cases? Does Lagrangian mechanics
just not work?

------
Jun8
Tao's _Solving Mathematical Problems: A Personal Perspective_
([https://www.amazon.com/Solving-Mathematical-Problems-
Persona...](https://www.amazon.com/Solving-Mathematical-Problems-Personal-
Perspective/dp/0199205604)) mentioned here is an excellent book that provides
keys to mathematical approach to problems and is now a classic ranking there
with Polya's _How to Solve It_.

One thing the article doesn't touch on is Tao's prominence in massively
collaborative math research through his blog (latest such work from there was
discussed on HN a day ago
[https://news.ycombinator.com/item?id=21542054](https://news.ycombinator.com/item?id=21542054)).
This new approach, first proposed by Tom Gowers
([https://gowers.wordpress.com/2009/01/27/is-massively-
collabo...](https://gowers.wordpress.com/2009/01/27/is-massively-
collaborative-mathematics-possible/)) in 2009 has successfully solved a number
of open problems in the past ten years
([https://polymathprojects.org/2019/02/03/ten-years-of-
polymat...](https://polymathprojects.org/2019/02/03/ten-years-of-polymath/)).

~~~
gjm11
I'll quote my own Amazon review of Tao's book:

\----

First, the bad news.

The front cover of this book gives its author as "Terence Tao, Fields Medal
winner 2006". Well ... yes, and no. The thing is, this book was written when
Tao was 15 years old. It reflects the precocious skill and insight of an
outstandingly gifted 15-year-old, who had won a gold medal at the
International Mathematical Olympiad at age 13 (most participants are 17-ish),
but not really those of the outstandingly gifted 31-year-old who won the
Fields.

It's only about 100 pages long; the problems it discusses are mostly
relatively easy (meaning, say, national high-school mathematics competition
level or thereabouts, rather than IMO, so not _that_ easy). It doesn't give
away any very deep secrets (if there are any) about how to solve such
problems. Write down what you know, look for symmetries, simplify step by
step, etc.; the real rocket science, as it were, is hidden away in the bits of
Tao's brain that instinctively know what symmetries to look for, what steps
are likely to lead in the right direction, and so on.

The good news: You wouldn't know it was written by a 15-year-old if the
preface didn't tell you. It _is_ a book about mathematics written by someone
with a Fields-medal-quality brain, and a book about Olympiad-style problems
written by one of the greatest-ever exponents of that art. It contains some
nice problems, with solutions by (I repeat myself) one of the finest minds in
the business. It's also quite cheap.

If you're interested in this, you should also look at Paul Zeitz's "The art
and craft of problem-solving"; it has more pages and more substance to it, but
it's twice the price and wasn't written by a Fields medalist.

\----

Pedantic note: _Tim_ Gowers, not Tom.

------
jeswin
> 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 intuition)
seem abstract and unconcerned with such specific figures?

I hope the question made sense.

~~~
impendia
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 improved to 246. You can see the(long!) proof here:

[https://www.dropbox.com/s/85pt6mvzf5ghukw/newergap-
submitted...](https://www.dropbox.com/s/85pt6mvzf5ghukw/newergap-
submitted.pdf)

Zhang stated 70 million because it's easy to remember.

Anyway, the basic strategy of the proof is to attempt to prove the twin prime
conjecture, and not fall _too_ short.

Where do these numbers come from? You end up needing to do lots of random
computations in the course of the proof. For example, can you compute a
constant C such that the bound

e^(.1x^2 + .4x) + 5cos(pi * x) - sqrt(x^2 + 1) < Cx + 4

holds for all x between 3 and 5? If this appeared in the course of an analytic
number theory proof, you probably wouldn't try to compute the absolute best
value C. It would be tedious and boring, and nobody would want to read it.
You'd just compute _something_ that was good enough, and move on.

~~~
MauranKilom
I know you just threw together a random formula for illustrative purposes, but
I was curious:

[https://www.wolframalpha.com/input/?i=maximize+%7B+%28e%5E%2...](https://www.wolframalpha.com/input/?i=maximize+%7B+%28e%5E%28.1x%5E2+%2B+.4x%29+%2B+5cos%28pi+*+x%29+-+sqrt%28x%5E2+%2B+1%29+-+4%29+%2F+x+%7D+for+3+%3C+x+%3C+5)

-> You can find C by imposing equality for x = 5.

------
amilein7minutes
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](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

------
new2628
Can someone explain what the author meant with the following passage?

"... Navier-Stokes equations, which govern the flow of fluids, including air
currents. In this case, let us hope that it does not have a real-world
application."

~~~
kkylin
One of Terry Tao's recent results shows that an equation that he calls the
"averaged Navier-Stokes equation" can have solutions that "blow up" in finite
time, starting from a perfectly nice initial condition [0]. Presumably the
writer is referring to this, as such solutions would not be very nice to have
if they are actually physically realizable, for example by an airfoil.

[0] [https://terrytao.wordpress.com/2014/02/04/finite-time-
blowup...](https://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-
an-averaged-three-dimensional-navier-stokes-equation/)

------
threwawasy1228
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.

------
JoeSmithson
I always use the Green-Tao Theorem[0] as my go-to example that very
understandable problems are still being solved in modern maths research.

[0]
[https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem)

------
75dvtwin
Celebrating Tao's successes through media, movies and and other pop-culture
vehicles, is very good, and I very happy this is happening.

Shining the light into the continent of Mathematics, and its celebrity
inhabitants, brings attention and, just may be a bit of the appreciation, of
complex and demanding it is to live there...

But, I do think, that to bring more and more people into the field, we have to
also celebrate the diversity and range in the abilities, of the people who
have, and who will have, contributed to the science of Mathematics.

I am sure that not every accomplished, and well respected mathematician need
to have the cognitive brilliancy of Tao or Nash.

I find inspiration and, to a degree, comfort, in this quote by Alexander
Grothendieck (specifically, the point where he felt, by far, he was not the
most brilliant person in the room)

>" ..Since then I've had the chance, in the world of mathematics that bid me
welcome, to meet quite a number of people, both among my "elders" and among
young people in my general age group, who were much more brilliant, much more
"gifted" than I was.

I admired the facility with which they picked up, as if at play, new ideas,
juggling them as if familiar with them from the cradle—while for myself I felt
clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox
faced with an amorphous mountain of things that I had to learn (so I was
assured), things I felt incapable of understanding the essentials or following
through to the end.

Indeed, there was little about me that identified the kind of bright student
who wins at prestigious competitions or assimilates, almost by sleight of
hand, the most forbidding subjects. ..

In fact, most of these comrades who I gauged to be more brilliant than I have
gone on to become distinguished mathematicians.

Still, from the perspective of thirty or thirty-five years, I can state that
their imprint upon the mathematics of our time has not been very profound.

They've all done things, often beautiful things, in a context that was already
set out before them, which they had no inclination to disturb.

Without being aware of it, they've remained prisoners of those invisible and
despotic circles which delimit the universe of a certain milieu in a given
era.

To have broken these bounds they would have had to rediscover in themselves
that capability which was their birthright, as it was mine: the capacity to be
alone.”

..."

see also discussion on HN here

[https://news.ycombinator.com/item?id=8604814](https://news.ycombinator.com/item?id=8604814)

[https://www.goodreads.com/author/quotes/405977.Alexander_Gro...](https://www.goodreads.com/author/quotes/405977.Alexander_Grothendieck)

------
RickJWagner
"The awards and honors have only multiplied...."

I see what they did there.

