Some stuff I found interesting about number theory research

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 experts you'd think there would be a thriving online community out there! I only have an undergrad's (very limited) understanding of these topics but feel the ability to create a network for effective communication on these topics is 100% doable. (Isn't this the kindof thing the DAT and IPFS protocols were supposed to solve? P2P protocols/networks for bettering communication?)

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 guessing there must be quite a few hard-to-find Discords for specific research communities.

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). When I was a grad student at Berkeley, I once asked Serge Lang (who I think wrote the most advanced math textbooks of anybody who ever lived) why mathematics research papers don't have a lot more incorrect results in them. 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. Of course, there are serious mistakes and gaps in the literature, though what one person considers a gap, somebody else who knows more might not consider a gap. For example, I remember once that Ernst Kani was complaining to me about a big gap in a paper by Ribet; I didn't believe it, so I read the relevant section and was able to fill in the gap in a few hours sufficiently to satisfy him. In any case, I've spent decades both programming and doing number theory, and despite some similarities, they are very different things. Another related way these threads come together is with Kevin Buzzard's big project to formalize mathematics using LEAN; he's another number theorist, and he told me that a few years ago he started really worrying about the correctness of a lot of number theory papers, and that strongly motivated his interest in LEAN.

One of my takeaways: When I see "stitch together black boxes," to me that says that aspects of academic math are becoming similar to opportunistic software development.

When such an effect happens in group psychology it is typically part of the "grounding" effect happening. There is a high supply of theory without the energy to go into analyzing so many individual black boxes anymore, and a correspondingly high demand arises for integrative applications which bring the high-minded ideas down to more fundamental, widely-scoped, or practical effect. Down to earth, so to speak.

Some would think of this as an economic-benefit effect. The point is, it tends to bring all those stitched-together black boxes much closer to the end user space, where application is key, not so much theory.

So I wonder if in the coming years we will see a dramatic closing of the gap between direct access to mathematics inputs and daily living. It may be that the typical non-math-minded individual will soon readily and happily bootstrap themselves into the language of mathematics via the closing in of a perception of direct competitive or economic value.

To share an example, in 2003 I started to use a node-based editor to edit 3D textures for graphics projects. This node-based tool gave more control than standard sliders, color pickers, and checkboxes. But soon I discovered that in a lot of cases it was easier to enter a math formula in a node than to chain nodes together. Maybe a single checkbox designated whether the function was periodic, but other than that, enter math here, please. Your mathematical wish is our graphics-producing command.

So I wonder if we'll soon get to the point where formulaic systems control with direct mathematic input will be more desirable and common in unified system interfaces that are found in office and household work.

Let's say you want to write the perfect formula for a given air quality factor in your office...

Or your competitive game which allows you to program AIs is made much more efficient if you can tweak formulae by hand...

Or you need to design a new form of printed steak, but simple toggles and switches just won't do...

(Then...when the time is right for the black-box theory end to pick up a lot of momentum again, imagine how large a pool of new mathematicians will be ready...)

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.