Thread by @benskuhn

I had a fascinating conversation yesterday with a former roommate who's now a postdoc in number theory. I learned a lot about how math research works! Some stuff I found interesting:

Most big number theory results are apparently 50-100 page papers where deeply understanding them is ~as hard as a semester-long course.

Because of this, ~nobody has time to understand all the results they use—instead they "black-box" many of them without deeply understanding.

(Probably worth noting that mathematicians have an extremely high bar for "deeply understanding." It's not like they're being dumb, the impression that I got is that making progress in number theory is actually just close to the limit of humanity's collective cognitive ability)

For instance people often don't understand techniques deeply enough to e.g. do calculations on examples (this is quite hard tbc!) Roommate told a story of using a technique to calculate a number and having a high-powered prof go "wow, I didn't know you could actually do that"

A lot of number theory is figuring out how to stitch together many different such black boxes to get some new big result. Roommate described this as "flailing around" but also highly effective and endorsed my analogy to copy-pasting code from many different Stack Overflow answers

Apparently this has caused research to become much more collaborative than math research used to be and every big result now is coauthored, sometimes with up to 10 coauthors covering different techniques/subfields.

One interesting thing about this structure is that this makes error-correction quite hard since for any given result, probably <10 people in the world understand it well enough to check it thoroughly. Because of this, gaps in papers are actually fairly common.

Roommate submitted his thesis for publication and one reviewer told him "oh, you cited this result from ~30y ago but it actually has a gap in the proof that no one's figured out how to fix yet." (People learn this stuff via the number theory gossip grapevine apparently?)

He had to put a big disclaimer on that section of his thesis (fortunately it was just a corollary) saying it assumed this result that currently had a gap. (He also tried for a bit to fix the gap, but spent 2-3 weeks getting nowhere.)

When I asked, roommate spitballed that 25% of papers have no gaps, 30% have some notation/precision issue that is "easy" to correct, 40% have a gap requiring some sort of insight but patchable with under a month of work, and 5% had gaps that would take real work to resolve.

(Apparently it's frequently the most famous people who discover these gaps. Roommate said it was surprising the degree to which a lot of what famous people do is actually what everyone is "supposed to do," eg deeply understanding techniques, checking things for yourself...)

(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.)

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."

To reiterate, I (and roommate) don't think this situation is dumb or anything. I'm posting about it because it's interesting, not because it's terrible. Math is freaking hard and I'm not surprised if this is what happens when new results are this difficult to prove + understand.

I did think it was kind of funny how similar this is in some ways to programming, 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 the vibe where everything is built on a stack of stuff that, if you look at it closely, doesn't actually work, is unexpectedly similar!

(PS epistemic status: this is me relaying a conversation with one number theorist. I'm not an expert and may be misinterpreting things; corrections welcome! Roommate is an expert but also just one person with his own take, other researchers may feel differently!)

Agreed, first tweet should have said "how number theory research works"!

https://twitter.com/dchudz/status/1419282690688303108

Oh, to be more concrete about what roommate meant by "deeply understand," I just remembered that he gave the benchmark of "as deeply as an undergraduate understands linear algebra" (e.g. deeply enough to, say, invert a matrix).

(I assume he means understanding deeply enough that *you can re-derive an algorithm* for inverting a matrix, since I'm sure there are many people who have memorized it but don't understand why it works the way it does.)

Roommate has decided he's ok being non-anonymous! He doesn't have a Twitter or a Soundcloud, but he has an academic page and you should send him fan mail because he's the best: math.berkeley.edu/~dalal/

To be clear these are 100% Rahul's observations, the only thing I take credit for is telling him to keep talking when he was worried he was boring (lmao)

I asked his permission to repeat them but would have made him publish himself if I'd known people would be *this* interested!