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In Mathematics, It Often Takes a Good Map to Find Answers

quantamagazine.org

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Re: In Mathematics, It Often Takes a Good Map to Find Answers

#11

Post by tzs »

amirhirsch wrote: post #9

I liked this article. One notable point that felt like it was missing in the article is that the Prime Number Theorem, that the count of primes grow like (n / ln n) was provided such a map by Riemann in the letter in which he put forward his infamous eponymous hypothesis. That letter introduced the idea of using analysis to the Prime Number Theorem, extending the groundbreaking work of Riemann's friend Dirichlet who…

> and the proof of the Prime Number Theorem came only in the early 1900's

That's just a little too late. It was proved in 1896 independently by Hadamard and de la Vallée Poussin.

Hadamard, J. "Sur la distribution des zéros de la fonction zeta(s) et ses conséquences arithmétiques (')." Bull. Soc. math. France 24, 199-220, 1896

de la Vallée Poussin, C.-J. "Recherches analytiques la théorie des nombres premiers." Ann. Soc. scient. Bruxelles 20, 183-256, 1896

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#12

Post by ZenOfTheArt »

knzhou wrote: post #8
physicsgraph wrote: post #6

Earlier quoted context omitted.

I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible. The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.

Well, I recommend doing a concrete, nontrivial derivation from start to finish just to see how this approach scales. As a basic example that is typically covered in about half a page in books, try doing a full derivation of the wave equation for a wave on a string. I would bet that once you set up the 1000 nodes required to do this, you'll be completely exhausted, and moreover will have gotten no new insight! If you'…

A Fitch derivation of the existence of the intersection of all members of a nonempty set is a better place to start because it can be done in less than ten sheets of paper longhand. The ratio of triviality to pages consumed is quite shocking when you finally confront it. It is at that point that you realize intuition has no formal translation but is vital since the level of detail seems to blur and darken intuition when holding a proof to the standard of formal derivation rather than the ordinary informal standard. So far, I’ve seen relatively little interest in mathematical intuition or even honest appraisal of what it is or how mathematicians should develop it. Rather the trend seems to be pretending that mathematical intuition doesn’t exist and treating formalization as a no-op. I think this is due to an anti-intellectual atmosphere that views mathematics as a source of problems for the military as opposed to pastimes for civilians.

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#13

Post by jordigh »

mmhsieh wrote: post #7

The difficulty in coming up with a good map of mathematics is summarized by this quote by Banach: "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

Haha, it's like Maclane said: "I did not invent category theory to talk about functors. I invented it to talk about natural transformations."

You gotta go at least to the third level of abstraction to get the real meat.

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#14

Post by CatsAreCool »

I liked this article since it points out a problem in math where it can be hard to know what is currently known.

Perhaps a result can be proven using a little known proposition in a completely different area of math, but it is hard to find that result in the literature.

That is one reason I came up with https://mathlore.org. It is a place to collect mathematical info (with links to articles for a deeper look) so you or others can find it later when you need it.

It supports of public collection of math info as well as allowing you to build your own private collection so you can keep track of what you have learned.

The hope is it will be useful to others to help learn math and prove new theorems.

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#15

Post by sabas123 »

ssivark wrote: post #3

I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics https://arxiv.org/abs/math/9404236 Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.

It was a good read, thank you for sharing.

> "It was an interesting experience exchanging cultures. It became dramatically clear how much proofs depend on the audience. We prove things in a social context and address them to a certain audience. Parts of this proof I could communicate in two minutes to the topologists, but the analysts would need an hour lecture before they would begin to understand it. Similarly, there were some things that could be said in two minutes to the analysts that would take an hour before the topologists would begin to get it. And there were many other parts of the proof which should take two minutes in the abstract, but that none of the audience at the time had the mental infrastructure to get in less than an hour"

I wonder if we would ever get to a point where we would find an effective and desirable mental infrastructure such that this wouldn't happen.

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#16

Post by raincom »

ssivark wrote: post #3

I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics https://arxiv.org/abs/math/9404236 Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.

Just finished reading Thurston's paper. A great paper esp for those in "philosophy of mathematics".

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#17

Post by mturmon »

ssivark wrote: post #3

I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics https://arxiv.org/abs/math/9404236 Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.

It's a great read. See also: https://news.ycombinator.com/item?id=12280139

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#18

Post by ssivark »

sabas123 wrote: post #15
ssivark wrote: post #3

I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics https://arxiv.org/abs/math/9404236 Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.

It was a good read, thank you for sharing. > "It was an interesting experience exchanging cultures. It became dramatically clear how much proofs depend on the audience. We prove things in a social context and address them to a certain audience. Parts of this proof I could communicate in two minutes to the topologists, but the analysts would need an hour lecture before they would begin to understand it. Similarly, the…

Category theory is supposed to be one such tool, even though some find it very abstract. It's very much in the spirit of finding analogies among theories and analogies among analogies. (I swear I'm not trolling :P) I'm still working on my understanding of category theory, but somebody who has the mathematical fortitude might enjoy: http://groupoids.org.uk/pdffiles/Analogy-and-Comparison.pdf

In general better abstractions(similar ideas as in a recent discussion of Peter Naur's "Programming as theory building").

Re: In Mathematics, It Often Takes a Good Map to Find Answers

#19

Post by Koshkin »

knzhou wrote: post #8
physicsgraph wrote: post #6

Earlier quoted context omitted.

I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible. The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.

Well, I recommend doing a concrete, nontrivial derivation from start to finish just to see how this approach scales. As a basic example that is typically covered in about half a page in books, try doing a full derivation of the wave equation for a wave on a string. I would bet that once you set up the 1000 nodes required to do this, you'll be completely exhausted, and moreover will have gotten no new insight! If you'…

This has reminded of the Two Capacitor Paradox [0]. (The moral of the story is that you have to know the limits of your model.)

[0] https://en.wikipedia.org/wiki/Two_capacitor_paradox

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