We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in crypto. Some people say that the same applies to all of mathematics.
My question is : do you buy this ? More precisely, and to make this into a question that I hope fits the requirements of this website :
Do you have an example of a mathematical theory (i.e. not an isolated theorem, but a coherent set of mathematical concepts and theorems) that you believe will be of no use, ever, to let's say engineers or physicists or non-mathematician scientists in general ?
By of use I mean that it is a mathematical object so relevant to a field or a model that non-mathematicians have to think in terms of this mathematical theory, OR that it is a crucial ingredient to the mathematical proof of some other useful mathematical result. Giving one proof among other, more simple ones is not sufficient. And a theory which language can be used to describe certain models but gives no significant insight or power of prediction doesn't count.
A few examples :
- hamiltonian systems are a good way to model most mecanical systems : useful
- Fourier transforms are a useful tool in numerous calculations : useful
- not being an expert in mathematical physics I can't give a precise example, but I would classify homology theory in useful because it is such a powerful mathematical tool that I'm sure it is a necessary ingredient to something with real-life applications, or will be
- tilings : applications in chemistry, including recently one with Penrose's aperiodic tilings : useful
- p-adic analysis : to my knowledge not useful.
Let me stress the "to my knowledge", as I know next to nothing about p-adic analysis and what may or may not be its applications.
Best Answer
I think "all math is eventually useful" must be false... There is certainly math that gets abandoned when better things come along, as with cylindrical algebras (as I understand their history). There are things like first-order mereotopology which have seen little development outside philosophy departments (and not much in them either). Arguably (and I say this with much sadness), NF will never catch on as a serious set theory so much as a source of odd model theory; probably likewise with NFU (which is even less deserving of such a fate).
Not that I think uselessness is a bad thing, of course.