Why using number theory as an example of useful pure maths is not compelling
This is a short summary of ideas lazily taken from this thread.
The usual argument for funding pure maths goes:
Much maths is practically important today. The maths I am working on is not practically important today, but maybe it will be the maths that is practically important tomorrow. How can we predict what will be useful? It seems like pushing maths generally forward is the best response to this uncertainty.
And usually number-theory pops up as an example of above phenomena and trumps all the objections. Here’s some thoughts of a mathematician working in the field:
I’m a number theorist and I don’t find this argument compelling. Much of the number theory that is used in crypto is not deep. There’s nothing about Diffie-Hellman or RSA that requires the hundreds of years of number theory research that has gone into it. One could explain the algorithms for such procedures to a mathematician in the early 1800s with little effort (although the idea of having very efficient methods of arithmetic might strike them as very odd). Moreover, while other parts of number theory have turned out to be relevant it is still a very tiny fraction of all number theory.
Other objections to the original argument are relevant too:
You could just as well wait until the application appears and then do the maths.
And concerns about the lack of proper cost-benefit analysis pop up:
There is a fact that a lot of pure math does not (or havent yet anyways) lead to applications. It pays to put effort only into math that is immediately practically useful.
We need proper counterfactuals here, cases where a practical use of math counterfactually would not have been possible without previous development as pure math. And also, what-if the pure mathematicians have been directly working on practical math instead?
And the point of it all?
That a proper cost-benefit analysis of the value of research in pure maths is long overdue. Using number-theory as an example is just not convincing enough.