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Some stuff I found interesting about number theory research

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Re: Some stuff I found interesting about number theory research

#91

Post by mhh__ »

pramodbiligiri wrote: post #65
mhh__ wrote: post #58

Earlier quoted context omitted.

I am kind of sceptical of these verified-mathematics projects in the sense that my litmus is whether I could easily verify my workings rather than the proofs I rely upon, but nonetheless I have enormous respect for those doing this work and really hope it comes to fruition. I really hope this work trickles down to our programming - state machine problems are basically "solved" (for my needs at least), but more comple…

When you say that state machine problems are basically solved, are you referring to any tools that you're using to reason about them?

By solved I really mean "in commercial use", there's probably a lot of research let to be done, but people are already verifying TLA+ specs all day everyday.

Re: Some stuff I found interesting about number theory research

#92

Post by mhh__ »

jkhdigital wrote: post #67
mhh__ wrote: post #58

Earlier quoted context omitted.

I am kind of sceptical of these verified-mathematics projects in the sense that my litmus is whether I could easily verify my workings rather than the proofs I rely upon, but nonetheless I have enormous respect for those doing this work and really hope it comes to fruition. I really hope this work trickles down to our programming - state machine problems are basically "solved" (for my needs at least), but more comple…

I'm a PhD student in computer science, and after spending some time on this subject I'm pretty well convinced that parts of mathematics and CS are converging towards a more unified foundational framework that will benefit both fields tremendously. In a nutshell, the key challenge is taking the core of complexity theory (which is easy to formalize in terms of state machines) and migrating it into the language(s) of ca…

I mostly hope you're right, good luck. Post your thesis here when the time comes.

I'm a bit of a philistine when it comes to cutting edge CS in this area so I'll leave you to it!

Re: Some stuff I found interesting about number theory research

#93

Post by Hermitian909 »

deadbeef57 wrote: post #79
Hermitian909 wrote: post #70

Earlier quoted context omitted.

I think the main thing missing from this analysis is how much of the mathematical corpus ends up being "uninteresting" over time. Take Algebraic Geometry, a field primarily concerned with the question "what are the zeroes of multivariate polynomial equations" e.g. x^2-y+z^3. Papers that could be considered part of the field were published as early as the 16th century but very little is worth reading from before the w…

I just want to say that in the case of the classification of finite simple groups, there's actually a bit of a problem. Exactly because "a lot of the deep expertise mathematicians" have left the field; but on the other hand "a small, curated subset of these techniques" is still missing. A small group of experts (all in their 70's or 80's) are currently writing a dozen volumes on this classification, and the rest of t…

Oh you're right! I hadn't been aware that the book series was incomplete, my face is a bit red and I share your concerns. Here's to hoping!

Re: Some stuff I found interesting about number theory research

#94

Post by Sniffnoy »

hannob wrote: post #56
williamstein wrote: post #6

I did a Ph.D. in number theory, published a few dozen research papers, and have programmed a lot and this post sounds about right to me. I did CS as an undergrad, before doing a math Ph.D., and remember being very surprised that math papers weren't a lot more wrong than they actually are (since computer software is so often full of bugs, and all it takes is one single bug to completely invalidate an entire paper). Wh…

Given that you seem to know this space, I have been wondering: Was there ever any major flaw in an important mathematical proof that was taken as a given, and subsequently plenty of papers building on that turned out to be wrong as well? Like I don't mean "this proof has a gap in its justification, but the thing claimed likely is still true". I mean "we thought someone had a proof for X, but actually later someone el…

This[0] may be a good place to start, but I'm not sure if it contains proper examples of what you say (of serious collapse). E.g., I expect a lot of people probably relied on the false result mentioned by Matheus, but most of them could probably be repaired with the corrected statement. So it seems like this probably hasn't happened?

OTOH, there was the Italian school of algebraic geometry, but that was more than one single flaw...

[0] https://mathoverflow.net/questions/35468/widely-accepted-mat...

Re: Some stuff I found interesting about number theory research

#95

Post by nullc »

> Anyway, despite the fact that error-correction is really hard, publishing actually false results was quite rare because "people's intuition about what's true is mysteriously really good."

I don't find that mysterious. One aspect of number theory is that results have arbitrarily deep implications and relations. If you are wrong about something you can often substitute it in somewhere else and produce a wrong answer or a contradiction.

Doing something wrong which still looks right after being tested from a few different perspectives is extremely tricky.

Like finding a pseudoprime for some probabilistic primaility test-- it's not likely to also be a pseudoprime for an unrelated test. So pretty much the only way to pass a collection of independent probabilistic tests is to actually be prime. (Note the 'pretty much'-- a bunch of approximate proofs isn't a necessarily proof, but it at least explains why false things are seldom believed even when their headline proofs aren't solid)

I don't just mean through different reviewers -- when you publish a proof of something you've probably satisfied yourself that it was true through multiple avenues. The proof you published was just the one you thought was the most convincing. Unfortunately, making an error in it can be something that makes a proof more convincing (until the error is found), but fortunately having an incorrect proof of something doesn't make the underlying fact untrue.

Re: Some stuff I found interesting about number theory research

#96

Post by aaronlevin »

benrbray wrote: post #5

This is quite interesting, especially since one of the major objections to computer-aided proofs has that they are more difficult for humans to understand. Common wisdom has been that proof techniques matter more than the results, but if human-authored proofs are no longer being inspected, it seems like it's just a matter of time before computer-aided proofs take over. I wonder how long it will be before proof assist…

The main objection to computer-aided proofs is not that they are more difficult for humans to understand.

The main objection is that most proof-assistants use a different logical foundation than modern mathematics. Modern mathematics is built on ZFC[0] whereas most proof assistants, such as Coq, Isabelle, Agda, etc use different logical foundations, such as the Calculus of Constructions[1].

Many important results in modern mathematics are not easily stated or proven in systems such as CoC[1]. For example, Brouwer's _Fixed Point Theorem_[2], a pretty bog standard result in Topology that is useful to proof many things in Functional Analysis, has a clear statement in ZFC, but does not, to my knowledge, have an equivalent in CoC (and if it does it will be stated radically different).

[0]: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... [1]: https://en.wikipedia.org/wiki/Calculus_of_constructions [2]: https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem

Re: Some stuff I found interesting about number theory research

#97

Post by jaksmit »

If anyone is interested in number theory and would like to help some incarcerated people that are working on a number theory paper:

I'm a director of the Prison Math Project: http://prisonmathproject.org/

We have a few advanced mathematicians in jail who don't have access to computers. They need help with formatting their work in LaTeX and stuff like that to make it ready for publication (credit offered).

They have previously co-authored some papers from within jail eg

"Linear Fractional Transformations And Nonlinear Leaping Convergents Of Some Continued Fractions"

"Math And Art In Prison: A Collaborative Effort Across The Ocean"

please reach out if any interest in helping them.

Re: Some stuff I found interesting about number theory research

#98

Post by yellowcake0 »

daxfohl wrote: post #75
Hermitian909 wrote: post #70

Earlier quoted context omitted.

I think the main thing missing from this analysis is how much of the mathematical corpus ends up being "uninteresting" over time. Take Algebraic Geometry, a field primarily concerned with the question "what are the zeroes of multivariate polynomial equations" e.g. x^2-y+z^3. Papers that could be considered part of the field were published as early as the 16th century but very little is worth reading from before the w…

In a way that's what makes the question of AI math interesting. Probably AIs will be better than us at math in a couple decades. But that might not change anything because it's still up to humans to determine whether a result is interesting. And it may take us just as long to understand AI-created concepts as to build them up ourselves. Maybe math starts looking more like archaeology at that point. (Though it's argua…

There is no way that AI will be better than us at math in a couple of decades.

Re: Some stuff I found interesting about number theory research

#99

Post by smallnamespace »

ska wrote: post #90
nicoburns wrote: post #52

Earlier quoted context omitted.

> He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof, and it was unlikely that multiple very different reasons for something being true would all be wrong. This was the most frustrating thing for me when I was studying undergraduate maths. Often a result was just stated, with the reasoning behind how it was generated completely omitted. Goo…

There is a fundamental tension here because a) doing this stuff properly takes a long time and b) lots of undergraduate math teaching is "service" teaching, meant to get students up to a level where they can do the manipulations needed in other courses. This is why there is typically separate Calculus and Real Analysis streams in undergraduate - the former is needed for all the hard sciences and engineering, the latt…

Knowing and using a thing and proving it are (usually) different skills.

Re: Some stuff I found interesting about number theory research

#100

Post by daxfohl »

yellowcake0 wrote: post #98
daxfohl wrote: post #75

Earlier quoted context omitted.

In a way that's what makes the question of AI math interesting. Probably AIs will be better than us at math in a couple decades. But that might not change anything because it's still up to humans to determine whether a result is interesting. And it may take us just as long to understand AI-created concepts as to build them up ourselves. Maybe math starts looking more like archaeology at that point. (Though it's argua…

There is no way that AI will be better than us at math in a couple of decades.

Yeah I think I agree now. They'll probably be better at proving straightforward things that don't require any big new insights, but I don't think machines will ever be any good at determining what kinds of new concepts will be interesting to define and explore. Like a machine isn't going to develop calculus just for fun from base principles.
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