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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

#11

Post by renewiltord »

benrbray wrote: post #4
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…

I do wonder where all the mathematicians / academics hang out online. Now that I'm out of (compsci) grad school, I find myself without like-minded peers to study math with. Currently working my way through Riehl's category theory book, but it's a bit tough without anyone to talk to! I know there's MathOverflow, and some people in the FP community hang out on IRC. What's the academic equivalent of HN? GitHub? I'm gues…

An ex-girlfriend of mine is a mathematician. Mostly conferences and symposia. It’s a very meatspace activity. I haven’t kept in touch but the pandemic might have upended that.

I’d speculate that it’s because it’s hard to understand much of mathematics without the back and forth of a lecture to help. But that could just be because I truly suffered through my Mathematics graduate degree and am glad a Masters is all I managed.

Re: Some stuff I found interesting about number theory research

#12

Post by gameshot911 »

shuckles wrote: post #9
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…

During an undergraduate research program, I discovered a gap in a theory in PDEs which I’m assuming nobody else did because I was the only one who tried to follow each step (just getting that far was the entire purpose of the effort!). My advisor said it would be best to fix the gap before publishing about it because multiple students entering the job market depended on the result being true. That was an eye-opening…

So were you able to fix it??

Re: Some stuff I found interesting about number theory research

#13

Post by laichzeit0 »

renewiltord wrote: post #10

I wonder if they have the equivalent the engineers who bemoan the fact that some React developers make functioning websites without “knowing anything about how CPUs or memory models work”. Bourbaki was a thing so probably. And we know the joke about Bourbaki and Lang. Though the reference is diminished by the latter’s relationship with the former.

It’s higher than that. People build software calling .search() and .sort() after a 4 month nanodegree and don’t know how to implement either. I’d say they are even using http requests without knowing what a TCP socket connection is, maybe even what an http request header is. The barrier to entry in the dev world is extremely low.

To relate it to maths: It would be equivalent to someone doing math research without knowing how to do proof by induction (in analogy to not knowing how to implement .sort())

Re: Some stuff I found interesting about number theory research

#15

Post by RajuVarghese »

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 fields. Comments?

Re: Some stuff I found interesting about number theory research

#16

Post by thaumasiotes »

> (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.)

I might suggest that these are broadly the set of people who are able to understand the proofs, and that their higher ability also causes them to produce higher-quality work, so the fame and the understanding are separate effects of a root cause. (With understanding then feeding into fame.)

I don't buy that spending time understanding the material you're working with takes time away from producing results. They are complementary efforts, not opposed ones.

Re: Some stuff I found interesting about number theory research

#17

Post by Someone »

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?

https://sciendo.com/journal/FORMA gets close, I think (I don’t think it only publishes formally verified results. It likely also accepts more philosophical texts about formal verification). There’s also http://mizar.org/JFM/

I don’t think such journals can be the full solution, though.

Problem is that to entice mathematicians who don’t like formal verification to publish in it.

Re: Some stuff I found interesting about number theory research

#18

Post by throwaway81523 »

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…

There is mathoverflow.net and its meta site as math researcher hangouts, plus ncatlab.org for category theory, and various sites, blogs, mailing lists etc. for other fields.

You might also like Bill Thurston's famous article "On Proof and Progress in Mathematics", which says not so much that math is social, but that human mathematicians want personal understanding of mathematical truths. So black box computer-checked but human-incomprehensible proofs aren't of that much appeal.

https://arxiv.org/abs/math/9404236

Re: Some stuff I found interesting about number theory research

#19

Post by onion2k »

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…

That sounds like something that'd be true for every field, not just math. The nature of anything that's cutting edge means very few people will have looked at it yet let alone actually understood it.

Re: Some stuff I found interesting about number theory research

#20

Post by redtexture »

shuckles wrote: post #9
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…

During an undergraduate research program, I discovered a gap in a theory in PDEs which I’m assuming nobody else did because I was the only one who tried to follow each step (just getting that far was the entire purpose of the effort!). My advisor said it would be best to fix the gap before publishing about it because multiple students entering the job market depended on the result being true. That was an eye-opening…

Partial Differential Equations for the rest of us.
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