Theory without a concrete example.

Question

I heard in class that there is a branch of mathematics that has been studied for some decades, but still has no "concrete example" of the theory. My professor refused to speak out the name of this "theory" for some reason. What kind of math might it be?

Answer 1

One interesting example of such a theory is the field with one element. Of course, there is no field with one element in algebra, but this name refers to a hypothetical mathematical object, denoted $\mathbf{F}_1$ which, in some sense, would behave like a field. There is as yet no concrete (or even abstract) description of this object.

I let you read the expected properties of $\mathbf{F}_1$ here. The problem is open since over 60 years and has generated a number of high-level research articles.

Answer 2

Depending on what you mean by "concrete example" there could be many instances of such theories. For example, if by "concrete" you mean "not relying on the axiom of choice", then an example of a non-concrete theory would be the theory of $\sigma$-additive Lebesgue measures on $\mathbb R$; see this 2017 publication in Real Analysis Exchange for details.