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New math book rescues landmark topology proof

quantamagazine.org

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Re: New math book rescues landmark topology proof

#61

Post by PartiallyTyped »

liversage wrote: post #58
leephillips wrote: post #2

If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.

It's amazing how math can require so much work for what appears to be a single result, but I spend an entire semester proving a single mathematical theorem and to do that I had to work through an entire textbook, so I don't think it's that unusual.

I am not a mathematician, though I do very applied math (ML), I took a course this semester that is intended for Pure Math MSc, called Advanced Vector spaces, having only done some linear algebra and calculus at the undergraduate level, some abstract algebra and some geometric algebra.

I am consistently in awe of how well mathematicians have stacked layers of abstraction one on top of the other, and how many different ideas end up being very related to one another. Maybe I am romanticised it and the fact that I regret not going for pure math, but there is beauty in all those abstractions that fit together so nicely.

Re: New math book rescues landmark topology proof

#62

Post by mindcrime »

leephillips wrote: post #2

If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.

I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Gödel's Incompleteness Theorem(s)[1]. I know there are at least a couple of complete books dealing exclusively with this result.

[1]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

Re: New math book rescues landmark topology proof

#63

Post by Pyramus »

curtisf wrote: post #55
doublerabbit wrote: post #48

I really wish I could get math to stick. I just finished my National 5 maths (rough equivalent of a US High School Diploma) night-class today and all I ever seem to understand is how, but not why. I'm the one asking "why is that that." And today was recapping on trinomial, simplifying fractions. Simple I expect to anyone with a mathematical mind, but to me, it's just an insane implosion which leaves me exhausted. I h…

High school math does not look much like research math. High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ... Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving…

Very good example re continuity!

I fully agree that (high) school math is mostly calculus and very different from university/research math. I would recommend parent to start exploring what math actually is, which fields there are and where they come from, what some famous open problems are and what beauty means to mathematicians.

Good popular maths books are e.g. The Music of the Primes by Marcus du Sautoy [1] or Fermat's Last Theorem by Simon Singh [2].

It's also important to note that there are no shortcuts - it takes at least 1-2 years of studying maths to start to understand the why.

[1] https://en.wikipedia.org/wiki/The_Music_of_the_Primes

[2] https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book)

Re: New math book rescues landmark topology proof

#64

Post by broken_symlink »

nerdponx wrote: post #16
lostmsu wrote: post #11

Math is the most formal science there is. Why don't serious math journals require programmatically checked proofs for all publications, and instead rely on what essentially is a code review for a very large change request with extremely convoluted code to be vetted bug free by only a handful of experts? P.S. Reviews are still needed to check novelty etc.

Is it possible to even check such a proof programmatically? Do we have that kind of technology? Do you also have to review the proof checking code to make sure you didn't make any errors in transcription from the original paper?

There is a language called lean which is popular among mathematicians. Here is an article: https://www.quantamagazine.org/building-the-mathematical-lib...

A more recent article as well: https://www.quantamagazine.org/lean-computer-program-confirm...

Re: New math book rescues landmark topology proof

#65

Post by dls2016 »

My buddy had “beat a Fields medalist at a foot race” on his CV for a while after completing the Pier to Peak half marathon faster than (a ~60 year old) Prof Freedman.

I’d run to Goleta beach playground with my kids and would see Freedman wander down from Station Q and bust out a few chin ups on the monkey bars.

I hope I’m in that kind of shape!

Re: New math book rescues landmark topology proof

#66

Post by kevinventullo »

mindcrime wrote: post #62
leephillips wrote: post #2

If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.

I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Gödel's Incompleteness Theorem(s)[1]. I know there are at least a couple of complete books dealing exclusively with this result. [1]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

I would say that most hallmark results in theory-heavy mathematics (number theory, algebraic geometry, topology, …) could stretch to at least a book length treatment if you wanted a mostly self-contained proof.

It sounds like the main difference with this theorem is that the writeup of the original proof was so complicated, terse, and error-prone, that very few experts even understood it.

Re: New math book rescues landmark topology proof

#67

Post by kevinventullo »

PartiallyTyped wrote: post #61
liversage wrote: post #58

Earlier quoted context omitted.

It's amazing how math can require so much work for what appears to be a single result, but I spend an entire semester proving a single mathematical theorem and to do that I had to work through an entire textbook, so I don't think it's that unusual.

I am not a mathematician, though I do very applied math (ML), I took a course this semester that is intended for Pure Math MSc, called Advanced Vector spaces, having only done some linear algebra and calculus at the undergraduate level, some abstract algebra and some geometric algebra. I am consistently in awe of how well mathematicians have stacked layers of abstraction one on top of the other, and how many differen…

As a former mathematician, I would say the stacked layers get a lot messier closer to the cutting edge.

Re: New math book rescues landmark topology proof

#68

Post by adgjlsfhk1 »

lostmsu wrote: post #41
klyrs wrote: post #38

Earlier quoted context omitted.

> Are proof checkers much more complicated than LaTeX? In terms of lines of code, no. In terms of the learning curve, yes -- latex does a very good job of getting out of the way and letting a mathematician just write in English, where proof assistants require rigid structure that doesn't remotely resemble how (most) mathematicians think. In terms of runtime, oh my god, get out of town.

> In terms of runtime, oh my god, get out of town. I think you're mixing two different things here. Runtime is large for proof assistants, e.g. programs that can actually generate pieces of proof for you. Specifically the generation part. Verification of a complete proof were all the steps are provided like you would do in a paper should not, AFAIU, take a long time.

The problem is that the way people write proofs, you don't go axiom by axiom. You might evaluate an integral in one step, or make an argument that some sequence is obviously asymptotically less than a function. Pretty much every step in a human proof requires a computer proof generation step to verify.

Re: New math book rescues landmark topology proof

#69

Post by karatinversion »

mindcrime wrote: post #62
leephillips wrote: post #2

If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.

I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Gödel's Incompleteness Theorem(s)[1]. I know there are at least a couple of complete books dealing exclusively with this result. [1]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

I don’t think the incompleteness theorems are nearly as wordy - this translation of Gödel’s original paper is under 40 pages after subtracting the introduction (not part of the original).

Re: New math book rescues landmark topology proof

#70

Post by Koshkin »

spekcular wrote: post #4

This has been a long time coming. There is an infamous MathOverflow thread about the proof (also linked in the article), with the following comment summarizing the state of affairs ~10 years ago: > There is no other evidence. In fact there is absolutely no evidence what so ever. I have never met a mathematician who could convince me that he or she understood Freedman`s proof. I attempted to read that monstrosity of a…

> what so ever

I've always thought it was connected.

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