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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

#31
This reminds me of my favourite description of academic research, from To the Lighthouse:

"For if thought is like the keyboard of a piano, divided into so many notes, or like the alphabet is ranged in twenty-six letters all in order, then his splendid mind had no sort of difficulty in running over those letters one by one, firmly and accurately, until it had reached, say, the letter Q. He reached Q. Very few people in the whole of England ever reach Q. Here, stopping for one moment by the stone urn which held the geraniums, he saw, but now far, far away, like children picking up shells, divinely innocent and occupied with little trifles at their feet and somehow entirely defenceless against a doom which he perceived, his wife and son, together, in the window. They needed his protection; he gave it them. But after Q? What comes next? After Q there are a number of letters the last of which is scarcely visible to mortal eyes, but glimmers red in the distance. Z is only reached once by one man in a generation. Still, if he could reach R it would be something. Here at least was Q. He dug his heels in at Q. Q he was sure of. Q he could demonstrate. If Q then is Q—R—. Here he knocked his pipe out, with two or three resonant taps on the handle of the urn, and proceeded. "Then R..." He braced himself. He clenched himself.

Qualities that would have saved a ship's company exposed on a broiling sea with six biscuits and a flask of water—endurance and justice, foresight, devotion, skill, came to his help. R is then—what is R?

A shutter, like the leathern eyelid of a lizard, flickered over the intensity of his gaze and obscured the letter R. In that flash of darkness he heard people saying—he was a failure—that R was beyond him. He would never reach R. On to R, once more. R—

Qualities that in a desolate expedition across the icy solitudes of the Polar region would have made him the leader, the guide, the counsellor, whose temper, neither sanguine nor despondent, surveys with equanimity what is to be and faces it, came to his help again. R—

The lizard's eye flickered once more. The veins on his forehead bulged. The geranium in the urn became startlingly visible and, displayed among its leaves, he could see, without wishing it, that old, that obvious distinction between the two classes of men; on the one hand the steady goers of superhuman strength who, plodding and persevering, repeat the whole alphabet in order, twenty-six letters in all, from start to finish; on the other the gifted, the inspired who, miraculously, lump all the letters together in one flash—the way of genius. He had not genius; he laid no claim to that: but he had, or might have had, the power to repeat every letter of the alphabet from A to Z accurately in order. Meanwhile, he stuck at Q. On, then, on to R."

Re: Some stuff I found interesting about number theory research

#32
post #27

Earlier quoted context omitted.

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 kno…

These devs don't need to. As long as they get work done, create `value', their skills are sufficient. Admittedly, there are times when people who know better are needed for specific tasks and that's why we are paid better. Proof by induction is more akin to for-loop than optimized sort algorithms. It's often taught in the first mathematics course that includes proofs which can be Calculus II or even Linear Algebra. I…

It's funny, I'd say that the reason these devs don't need to understand the lower layers is because the layering abstraction was built well enough. That's a good thing!

Re: Some stuff I found interesting about number theory research

#33
post #24
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…

I think it will be quite some time yet. As a researcher, my objection to computer-aided proofs is not that they're hard to read -- you would of course write a human readable version to go with it -- but rather that they're extremely time-consuming to write. (And writing papers is hard enough already.) A typical research paper is written at a very high level; often steps in the argument will assume the reader is also…

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?

Re: Some stuff I found interesting about number theory research

#34

Earlier quoted context omitted.

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…

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…

A little further off-topic. I knew (well) a mathematician who most of the department thought a bit odd, and he was. If you asked him "how are you?", he would take longer than is usual to respond. I fairly quickly realised it was because he considered that nicety as an actual question and gave a considered response.

Re: Some stuff I found interesting about number theory research

#35
post #4

Earlier quoted context omitted.

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…

One answer to "What's the academic equivalent of HN?" is "going to week-long conferences and workshops", which often involves a bunch of presentations of papers (with Q&A) and discussions long into the night. Before the pandemic, an academic (such as me) could easily spend the entire summer just going from one conference to another. With the pandemic, there's now a lot of live video chat, and seminar series where eve…

> In math there's also a lot more "think deeply and send an email" that happens, rather than things like Hacker News.

That would have been my guess, too. I shared an office for a while with a math professor, and he spent around half of his time in the office writing emails to colleagues.

Re: Some stuff I found interesting about number theory research

#36
post #21

Earlier quoted context omitted.

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 kno…

Lack of gatekeeping and general accessibility of software development is the beauty of this profession.

Sorry, let me rephrase. I didn't mean it in a negative way. The last time I needed to actually implement a sort method was when I was doing very low level embedded work and there wasn't one available on that platform (at least not one with the memory/compute trade offs I needed). I'm sympathising with mathematicians who have to build stuff on black-box stuff they don't full understand. Things are too complex to know it down to the CPU level, even so complex that modern devs don't even know how search and sort is implemented, but still get shit done.

Re: Some stuff I found interesting about number theory research

#37
post #9

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…

Can you share what the gap was?

Re: Some stuff I found interesting about number theory research

#38
post #3

https://threadreaderapp.com/thread/1419281153983500290.html

I always wonder why would someone write a thousands-words article as a Twitter thread? Why not publish it on any other platform and link it?

Prob a few reasons. This was written in a very conversational tone.

If he wrote it in a blog, it would read weird or he’d spend a lot more time on it. I’ve started blog posts before and just kinda gave up when it didn’t seem like it came together well.

Re: Some stuff I found interesting about number theory research

#39

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…

I find the further off the "main path" you go, the easier it is to find incorrect papers.

In particular, I found a few papers which claim to provide algorithms which solve some problem. If there is no evidence the algorithm was ever implemented, there is a distribingly good chance it doesn't work -- I used to sometimes give such papers as projects for students, but now I always start on day 1 saying "Be prepared for this to be wrong".

Re: Some stuff I found interesting about number theory research

#40

Earlier quoted context omitted.

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…

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…

You can also tell that the article was written by a mathematician by noticing that the author is Bill Thurston, a very famous mathematician, who we sadly lost a couple of years ago. He was amazing. I guess it never would have occurred to me to identify the writing style as mathematician-like in its own right, since I was expecting it to be what it was.
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