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 (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.
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!)
> 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
I wonder if something similar could be done for probability distributions. It seems hard.
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.
Here's a longer article by Thurston on the same ideas https://arxiv.org/abs/math/9404236. Highly recommend irrespective of how involved you are with mathematics.
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.
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.
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.
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.
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.
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.
> 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.)
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?
|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.
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|.
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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Yeah, that's StackOverflow for ya (or MathOverflow I guess).
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.