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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

#71

Post by madcaptenor »

webmaven wrote: post #63
linspace wrote: post #26

Earlier quoted context omitted.

A little offtopic: It's telling how you can tell the article was written by a mathematician, apart from the obvious fact that it's about mathematics. I'm talking about the structure. For example: "How do mathematicians prove theorems? This question introduces an interesting topic, but to start with it would be to project two hidden assumptions: (1) that there is uniform, objective and firmly established theory and pr…

There are variatons on this joke that are even more apropos to HN, for example: A man is flying in a hot air balloon and realizes that he is lost. He reduces height and spots a man down below. He lowers the balloon further and shouts, "Excuse me. Can you help me? I promised a friend I would meet him half an hour ago, but I don't know where I am." The man below says, "Yes. You are in a hot air balloon, hovering approx…

That answer isn't even technically correct unless the man on the ground is in a boat: https://www.google.com/maps/place/42%C2%B000'00.0%22N+60%C2%...

Re: Some stuff I found interesting about number theory research

#72

Post by tragomaskhalos »

Maths is, I believe, unique as a discipline in that it displays a distinct set of discrete difficulty steps the further into it you go. The first step is encountered at school: many students find they may be OK with basic arithmetic but anything conceptual like algebra is beyond them. A second step is encountered at college/university: someone who is perfectly good at school-level mathematics can find themselves totally at sea within literally week 1 of a degree course. And someone who copes fine with this level may in turn find that the more rarefied and demanding level demanded by a very top university is beyond them (source: this was my own experience - fine at degree level at an 'ordinary' university having flamed out spectacularly in an Oxbridge interview, demonstrating absolutely that I'd have been totally out of my depth there). Then, there are those few who push through even that level with enough ability to become productive and innovative academics.

When you start with a population for whom even arithmetic can be a challenge, it's not too surprising that only a miniscule priesthood emerge from the other side.

Re: Some stuff I found interesting about number theory research

#73

Post by [deleted] »

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…

[deleted]

Re: Some stuff I found interesting about number theory research

#74

Post by SAI_Peregrinus »

z77dj3kl wrote: post #42
etaioinshrdlu wrote: post #14

It would be nice if we had a mathematics-wide push for formal verification of proofs, not just done by the mathematicians who really like formal verification. Maybe there could be a journal of only formally-verfified results?

What do you mean? Mathematics is all about formal verification, i.e. proofs.

I think they mean computer-verified proofs. As you note, the existing proofs are already formally verified, so this might not add much.

Re: Some stuff I found interesting about number theory research

#75

Post by daxfohl »

Hermitian909 wrote: post #70
RajuVarghese wrote: post #15

In a book on mathematics by Alex Bellos he says that any contemporary cutting-edge research mathematics paper can be understood by no more than 100 mathematicians. The field is getting bigger and bigger and there are less and less mathematicians that have background knowledge sufficient to _understand_it much less critique it. The bar for the word 'understand' may be much higher in mathematics compared to other field…

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 arguably archaeology anyway -- the proofs are all "out there", we just have to find them).

Re: Some stuff I found interesting about number theory research

#76

Post by daxfohl »

plake wrote: post #53
benrbray wrote: post #33

Earlier quoted context omitted.

That's basically what I meant by "ergonomics" -- do you think we're still quite far from researchers being able to develop their own "tactics" to automate the sort of reasoning that would normally be "left as an exercise" to the reader of a research publication?

Yeah, "left as an exercise" can mean anything from "this is an undergraduate exercise" to "if you understand both the field and this paper, you could write a different, much longer paper, and we both know this, so let's assume I wrote that one." It's probably not impossible to build a proof language that makes that kind of thing doable, but I suspect that (a) it would be genuinely difficult to operate it skilfully, m…

I came here to say that maybe it makes things simpler though when everyone speaks the same language, and all proofs are written in the same format.

However then I remembered the new codebase I inherited and OMG it's such a mess because somehow just converting a UI enum to a DB enum is a 25-line function that's repeated over and over everywhere. (Needs to be version-safe for different clients that may have old enum values, needs to create error message in appropriate language, etc, and the callers are in different layers and all have different ways of getting the corresponding info).

So yeah maybe the high-level "I get it, you get it, we all get it, let's just state this an move on" is better.

Re: Some stuff I found interesting about number theory research

#78

Post by concreteblock »

caddemon wrote: post #59
hyperbovine wrote: post #41

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. I don't do pure math, but I write the occasional theory paper, and this resonates. So much ends up on the cutting room floor--usually you proved the key result three or four different ways before finding a proof that is actually incisive/aesthetically pleasing/whatever to justify signing…

As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere. Is it left out because it's not worth the effort to include compared to the number of people that would actually read it? Or is it something that runs the risk of negative perception by those established in the field?

I think it's fundamentally a question of incentives. A mathematician's career prospects and status depends heavily on being able to prove things that others can't. Why spend extra time helping your 'competitors'? Not saying this is ideal, but it seems to explain many behaviours in the community, and it seems unavoidable given the ever-increasing competition for jobs.

Re: Some stuff I found interesting about number theory research

#79

Post by deadbeef57 »

Hermitian909 wrote: post #70
RajuVarghese wrote: post #15

In a book on mathematics by Alex Bellos he says that any contemporary cutting-edge research mathematics paper can be understood by no more than 100 mathematicians. The field is getting bigger and bigger and there are less and less mathematicians that have background knowledge sufficient to _understand_it much less critique it. The bar for the word 'understand' may be much higher in mathematics compared to other field…

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 the community hopes that they get it done before they pass away.

This proof has a seriously low bus factor at the moment :scared:

Re: Some stuff I found interesting about number theory research

#80

Post by deadbeef57 »

jkhdigital wrote: post #47
etaioinshrdlu wrote: post #14

It would be nice if we had a mathematics-wide push for formal verification of proofs, not just done by the mathematicians who really like formal verification. Maybe there could be a journal of only formally-verfified results?

Homotopy type theory is just that: https://www.ias.edu/ideas/2014/voevodsky-origins

Homotopy type theory (HoTT) can mean several things: it's a new foundation of mathematics that was developed from the start with computer-formalization in mind. But you can also work on HoTT without ever touching a computer.

Conversely, there are many ways to do computer-formalization of mathematics, without every doing anything with HoTT.

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