5 ms·

Bill Thurston's answer to “What's a mathematician to do?” (2010)

104 points by adenadel 6y ago

ksd482 6y ago
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.
hyperpallium 6y ago
closed as interesting
- deleted 6y ago
  [deleted]
ISL 6y ago
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.
- JadeNB 6y ago
  > 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.)
  - User23 6y ago
    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?
    - roenxi 6y ago
      Posting in agreement with you, I think it is more intuitively satisfying to say something like "|X| is the smallest well-defined mathematical object that is >= the size of X. |Q| = |Z|". It isn't intuitively satisfying to say that there are "as many" rationals as integers because that is obviously not true; there are multiple rationals between any two integers. An argument to ignore that pattern as we scale up to infinity is rather flimsy. But if we talk about infinite sets we are forced to admit the existence of well defined things that are plainly larger than |Z|, and no such well defined objects that exist between |Z| and |Q|.
    - verma7 6y ago
      |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.
deleted 6y ago
[deleted]
moultano 6y ago
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 6y ago
  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!)
- voz_ 6y ago
  Your paper is lovely and your work is valuable. Thank you for pushing the envelope just a little bit. Our modern world is built by as much of things like this - that push out science but by bit, as it is by advanced measured in monumental leaps and bounds.
- thomasahle 6y ago
  I love this quote from a follow up paper: > Interestingly, hash algorithms with collision probabilities equal to JP have already been unintentionally presented before JP was actually discovered and thoroughly analyzed in [8]. In [7] a data structure called HistoSketch was proposed to calculate signatures for JN ... after some simplifications and thanks to a nonequivalent transformation that eliminated the scale dependence, the final HistoSketch algorithm had a collision probability equal to JP instead of the originally desired JN. It's none the less a very interesting measure. Thanks for sharing! I recently worked on a project trying to determine "the best" locality sensitive hashing amoung all measures of similarity for sets: https://arxiv.org/abs/1904.04045 https://arxiv.org/abs/1904.04045 I wonder if something similar could be done for probability distributions. It seems hard.
19f191ty 6y ago
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.
roenxi 6y ago
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.
- orbifold 6y ago
  There is a very real sense in which working in mathematics can be suffering. I feel like any research mathematician sooner or later has that feeling. Because you will be confronted with your limitations and weaknesses eventually. So I think this a shared experience that is the implicitly understood background of his answer. What I took away from Thurstons answer was a way to deal with that suffering. I eventually decided that I did not really want to continue in a research mathematics direction, but for a while his answer really inspired me.
- 6y ago
P-NP 6y ago
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."
- andi999 6y ago
  Well. For him 'pretty soon' for me 'after a long time'
- jeffreyrogers 6y ago
  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.
082349872349872 6y ago
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.
Illniyar 6y ago
"This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help " Yeah, that's StackOverflow for ya (or MathOverflow I guess).

104 points by adenadel 6y ago

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.

hyperpallium 6y ago

closed as interesting

deleted 6y ago
[deleted]

ISL 6y ago

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.

JadeNB 6y ago
> 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.)
- User23 6y ago
  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?
  - roenxi 6y ago
    Posting in agreement with you, I think it is more intuitively satisfying to say something like "|X| is the smallest well-defined mathematical object that is >= the size of X. |Q| = |Z|". It isn't intuitively satisfying to say that there are "as many" rationals as integers because that is obviously not true; there are multiple rationals between any two integers. An argument to ignore that pattern as we scale up to infinity is rather flimsy. But if we talk about infinite sets we are forced to admit the existence of well defined things that are plainly larger than |Z|, and no such well defined objects that exist between |Z| and |Q|.
  - verma7 6y ago
    |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.

deleted 6y ago

[deleted]

moultano 6y ago

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 6y ago
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!)
voz_ 6y ago
Your paper is lovely and your work is valuable. Thank you for pushing the envelope just a little bit. Our modern world is built by as much of things like this - that push out science but by bit, as it is by advanced measured in monumental leaps and bounds.
thomasahle 6y ago
I love this quote from a follow up paper: > Interestingly, hash algorithms with collision probabilities equal to JP have already been unintentionally presented before JP was actually discovered and thoroughly analyzed in [8]. In [7] a data structure called HistoSketch was proposed to calculate signatures for JN ... after some simplifications and thanks to a nonequivalent transformation that eliminated the scale dependence, the final HistoSketch algorithm had a collision probability equal to JP instead of the originally desired JN. It's none the less a very interesting measure. Thanks for sharing! I recently worked on a project trying to determine "the best" locality sensitive hashing amoung all measures of similarity for sets: https://arxiv.org/abs/1904.04045 https://arxiv.org/abs/1904.04045 I wonder if something similar could be done for probability distributions. It seems hard.

19f191ty 6y ago

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.

roenxi 6y ago

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.

orbifold 6y ago
There is a very real sense in which working in mathematics can be suffering. I feel like any research mathematician sooner or later has that feeling. Because you will be confronted with your limitations and weaknesses eventually. So I think this a shared experience that is the implicitly understood background of his answer. What I took away from Thurstons answer was a way to deal with that suffering. I eventually decided that I did not really want to continue in a research mathematics direction, but for a while his answer really inspired me.
6y ago

P-NP 6y ago

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."

andi999 6y ago
Well. For him 'pretty soon' for me 'after a long time'
jeffreyrogers 6y ago
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.

082349872349872 6y ago

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.

Illniyar 6y ago

"This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help " Yeah, that's StackOverflow for ya (or MathOverflow I guess).