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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

#81

Post by thehappypm »

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?

It's actually pretty tough to write out a proof, and going from a whiteboard where you're 99.9% sure it's right to something in LaTeX in a paper in your field is actually a long number of steps to take.

Re: Some stuff I found interesting about number theory research

#82

Post by rocqua »

webmaven wrote: post #61
rocqua wrote: post #28

Earlier quoted context omitted.

I think the reason for this black-boxing in software and mathematics is quite different. In top-level mathematics its that you have no choice but to treat results as black-boxes because the complexity otherwise is just too high. Its the only way forward, and mathematicians would prefer not to take this approach. In software development, black-boxing is a way of accelerating development. It allows people to abstract a…

But "accelerating development" and "lowering complexity" are, if not synonymous, then at least highly correlated.

The difference is in why. Mathematicians do it out of necessity because they are at the edge of complexity they can handle. Developers do it because it is convenient. Developers have a choice.

Re: Some stuff I found interesting about number theory research

#83

Post by defterGoose »

echopurity wrote: post #64

It's a huge failure of the (mis)education system that almost no one learns math.

I've been thinking a lot recently about math curriculum and how i think students would be well served by an earlier injection of set theoretical topics. At once more abstract and yet possibly more concrete for many of the "never-math" brains one encounters in a typical school setting.

Re: Some stuff I found interesting about number theory research

#84

Post by soheil »

> except that for programmers it's possible (and advantageous IMO) to deeply understand way more than they actually do, which seems less clearly true in number theory.

But that's not true for programmers either. How often do we look at the compiler or interpreter and check each opcode to make sure the right transistor gets turned on at the right time? The fact that number theory papers are 50-100 pages long may be an indication they're missing some useful level of abstraction, but then again if they're already operating at the most abstract level limited by the human mind then maybe that's why the papers are so long.

Re: Some stuff I found interesting about number theory research

#85

Post by rwallace »

> I suggested that perhaps this is because the famous people are the only ones who are fast enough to actually do everything you're "supposed to do" and still be productive.

Conjecture: it is instead or also because famous people are the only ones who are given enough slack to be able to take the time to actually do everything you're supposed to do.

Proposed test to distinguish between the conjectures: do famous mathematicians become famous by finding gaps, or find gaps after becoming famous?

Re: Some stuff I found interesting about number theory research

#86

Post by Y_Y »

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…

I like this idea. People often ask if maths is "created" or "discovered". Maybe we can say that it's "excavated" and that we know it's down there somewhere, we just need to put in a few years of shoveling.

Re: Some stuff I found interesting about number theory research

#87

Post by webmaven »

rocqua wrote: post #82
webmaven wrote: post #61

Earlier quoted context omitted.

But "accelerating development" and "lowering complexity" are, if not synonymous, then at least highly correlated.

The difference is in why. Mathematicians do it out of necessity because they are at the edge of complexity they can handle. Developers do it because it is convenient. Developers have a choice.

I think you may be underestimating how much "because they are at the edge of complexity they can handle" is a driver of mainstream SE. In particular, as a factor in the popularity of frameworks and libraries.

I'm not taking a stance like "if it was hard to write it should be hard to use", but quite a lot of software only seems to exist in order to hide complexity away in black boxes, often in trivial ways.

Re: Some stuff I found interesting about number theory research

#88

Post by dynrzk »

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…

Somewhat famously, symplectic geometry had its fair bit of foundational issues.

Re: Some stuff I found interesting about number theory research

#89

Post by d0mine »

z77dj3kl wrote: post #44
adam_ellsworth wrote: post #2

What really jazzes me here is this: > "(People learn this stuff via the number theory gossip grapevine apparently?)" With the panoply of dev-oriented social platforms I find it curious that – to my knowledge – nobody who's studied Number Theory, Category Theory, Set Theory, etc. have established a kind of social network for sharing ideas formally. Between LaTeX for Markdown and the limitless Compsci-leaning Maths exp…

The social networks are offline. Universities are great at open collaboration, it's just not a kind of online collaboration where anybody can walk in and partake. People spend a lot of time on a different type of communication: going to conferences, attending seminars, and meeting one-on-one. Why should it be online? Also keep in mind that a lot of mathematicians are also older, less tech-savvy. This might change in…

> Why should it be online?

Compare programming before and after Google or before and after StackOverflow.

There is https://mathoverflow.net/

Re: Some stuff I found interesting about number theory research

#90

Post by ska »

nicoburns wrote: post #52
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…

> 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 latter for math students and fellow travelers who really want to understand what is going on and be able to prove things properly.

Typically the analysis stream will start with a pretty in-depth look at what real numbers actually are (spoiler: weirder than you thought) and the implications of that. You get a much better understanding this way but you don't cover ground nearly as quickly.

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