Philosophy of Mathematics - Number Theory

Off in another part of my thoughts, which have been on hold for a while, I have been trying to work out some ideas on the philosophy of mathematics.

I have two theses. One is about the relationship of abstract mathematical ideas to various types of measurement or geometric properties. If you want to know how the various derivatives on curved spaces arise from the simple issues of co-ordinate changes, it's all there. The other is a methodological thesis, that the purpose of mathematics is to provide tools and techniques to solve problems that arise from modelling physical and other processes, and to understand the scope and limits of those techniques. Creating and solving the equations of the mathematical models is what's usually called "applied mathematics", while understanding the scope and limits of the techniques is a lot of what's called "pure mathematics".

And then there's Number Theory. Which is about numbers. Not mathematical models.

You know that Langlands thing that all the Kool Kids are working on?

Yep. Number theory. Finite field number theory at that. Geometric Langlands is even more abstruse.

It takes genius-level insight and technique to understand the more recent developments in Langlands. That's the point: if the specialists can barely follow it, how is it going to be any use to some poor post-grad working on differential geometry at the University of Ennui-sur-Blase?

The social purpose of mathematicians is to teach other people - physicists, statisticians, epidemiologists, computer scientists and programmers for example - how to use the problem-solving techniques mathematics offers. What mathematicians do in their spare time is their business: they need a decent laptop, a whiteboard and some paper and pens: math is cheap compared to fundamental physics.

The Langlands guys can do what they want in their spare time. But it's a rabbit-hole. Maybe it's a big, well-lit rabbit-hole with all the health and safety gear and plenty of mechanical digging tools, but it's still a rabbit-hole. Unlike some of the rabbit-holes mathematicians have buried themselves into (functional analysis, for instance), Langlands is not going to produce anything useful to regular working stiffs (for instance, functional analysis produced the theory of weak solutions to differential equations, which is very useful). I feel confident saying that because Langlands is about structures the rest of mathematics just doesn't use.

(Rabbit-holes are as opposed to specialisms, which are very specific subjects that have useful applications in the real world or other parts of maths with real world applications. Like research in PDEs.)

Maybe "rabbit-hole" should be a term of art in methodology. It's a line of research that has no obvious application to any existing problems or in other branches of maths. The scientific version would be a research programme that was making theoretical progress but no empirical progress (was not making new predictions). A rabbit-hole may branch up to the surface every now and then, as applications to problems in other branches of maths are found, but generally once dug, the researchers dig away happily underground.

In this case I would be saying that Number Theory was a mathematician's pastime, and that other very abstruse, or very off-beat, programmes, are for all the sophistication, esoterica for the aficionados. Which doesn't sound too dramatic.