
Bill Thurston's answer to “What's a mathematician to do?” (2010) - adenadel
https://mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213#44213
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moultano
I don't have any post graduate education. I'm ok at math, but not exceptional.
I managed to publish a paper a few years ago that is mostly a math paper that
I'm really proud of.
[https://arxiv.org/abs/1809.04052](https://arxiv.org/abs/1809.04052) (It's a
useful algorithm for big-data work, but it's also a bit of math about how to
align probability distributions so they collide as much as possible. I tell my
kids that daddy discovered something new about how triangles fit together.)

There are lots of useful corners of math out there, lots of things that are
worth thinking about that no one has thought about just because there are so
many things to think. There are plenty of things worth poking at that aren't
The Big Problems.

~~~
JadeNB
You like math? You do math? Guess what? You're a mathematician, regardless of
post-graduate education or self-assessment of your ability!

(I love to see 'amateur' mathematics, not in the derisive sense of the word
but in the formal sense of "not done by a professional mathematician". Good on
you!)

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19f191ty
Here's a longer article by Thurston on the same ideas
[https://arxiv.org/abs/math/9404236](https://arxiv.org/abs/math/9404236).
Highly recommend irrespective of how involved you are with mathematics.

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roenxi
Thurston's answer is obviously well meaning and trying to breath inspiration
into people who are pursuing maths as a discipline, but it should be observed
anyone taking him at his word without context will be led astray. It isn't
useful guidance for a young aspiring mathematician because it is generic,
contains nothing actionable and not even a clue on how to judge a
successful/failed state as they move through life.

Even "your name will go down in history if you consistently find patterns in
daily life that other people can't see" would probably be more useful advice
to the next Euler or Gauss.

> The product of mathematics is clarity and understanding.

Mathematics is clarity and understanding of _mathematical objects_ which is a
very small subset of the things that most people seek when they go looking for
clarity and understanding. Anyone looking for clarity and understanding in the
abstract is better off starting their search in the Psychology or possibly the
Philosophy departments.

I don't think he was being arrogant with that quote, but I do think that it is
the perspective of someone who has spent so much time looking at maths they
might have lost track of all the social manoeuvring that is what satisfies
most humans. In my case I'd rather have a deep understanding of what someone
is saying to me than of Fermat's Last Theorem - communication abilities tends
to be more of a bottleneck to satisfaction than abstraction abilities. Even in
Thurston's answer, he is alluding to the fact that communicating with other
mathematicians is as important to him as understanding abstract concepts.

> follow your heart and your passion. Bare reason is likely to lead you astray

This is lousy advice. Following your passions only works for people lucky
enough to have productive passions. A lot of people are passionate about
eating good food - if they want to be productive they will need a plan other
than following their passions.

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auggierose
I think Thurston's answer is absolutely to the point.

If you are passionate enough about good food, you probably have a great shot
at becoming a famous chef. I agree though that it is very dangerous advice:
most people are just not passionate enough about something, but mistake
fondness for passion. I'd say that applies to your "good food" example. But on
the other hand, for truly passionate people it is very dangerous NOT to follow
this advice.

> The product of mathematics is clarity and understanding

Somewhere else Thurston qualifies this in a recursive definition of
mathematics that is bootstrapped with numbers and geometrical objects. I'd say
in the age of the computer this qualification becomes less and less necessary:
there are other things than numbers and geometrical objects that are of
interest (for example distributed file systems). So more and more things are
becoming amenable to clarity and understanding, if we try hard enough. I think
a lot of things in computing could use a good helping of clarity and
understanding.

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jgwil2
Love the phrase, "mistake fondness for passion." Thanks!

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ksd482
RIP Bill Thurston! Thank you for sharing!

His mini-essay made a lot of sense. I think he nailed it in the beginning by
saying that the world collectively benefits from Mathematics as a whole. Or
rather, benefits are a "side effect" of people's Mathematical achievements.

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P-NP
I like the Feynman quote in the comments: "You keep on learning and learning,
and pretty soon you learn something no one has learned before."

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jeffreyrogers
I like that quote and I like Feynman in general, but I think it's worth
keeping in mind that Feynman was much more gifted than most people, even other
physicists. Some colleague of Feynman once said he thought Feynman harmed some
students' development because Feynman didn't realize how intelligent he
(Feynman) was and that he would suggest students approach problems in ways
that they just didn't have the ability for.

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ISL
That's beautiful -- thank you for sharing the post. I don't get to think about
pure math very often any more, but this was a wonderful reminder of clarity in
a world that's a little more disordered than usual.

One of the little joys of math, well known, but always makes me smile: Though
there are infinitely many rational numbers, they are countable.

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JadeNB
> One of the little joys of math, well known, but always makes me smile:
> Though there are infinitely many rational numbers, they are countable.

I think 'infinite but countable' isn't so surprising—even without a formal
definition, I think most people would expect the counting numbers to be
countable—but maybe the fact that there are infinitely many _more_ rational
numbers than counting numbers, and yet there are exactly as _many_ rational
numbers as counting numbers?

(This is one of many ways of phrasing it, but it perhaps understates how much
bigger the rationals seem to be than the counting numbers; the description
I've given would apply as well to the set of all integers, whose countability
is still perhaps surprising, but not _as_ surprising.)

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User23
There's one for each of 'em, to use an idiom. A bijection exists between the Z
and Q. Both sets are exactly the same size. Cantor's argument is certainly
quite clever, but "there are infinitely many more rational numbers than
counting numbers, and yet there are exactly as many rational numbers as
counting numbers" is nonsensical to me. Am I missing something?

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verma7
|Q-Z| is countably infinite, but |Q| = |Z|. There are countably infinite
rational numbers that are not counting numbers, yet there are exactly as many
rational numbers as counting numbers.

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082349872349872
Russell once wrote "Work is of two kinds: first, altering the position of
matter at or near the earth's surface relatively to other such matter; second,
telling other people to do so." Maths is a constructive proof that his case
analysis was not exhaustive.

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Illniyar
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Yeah, that's StackOverflow for ya (or MathOverflow I guess).

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hyperpallium
closed as interesting

