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?

Necromathcy. :-D Or he might be thinking of various cases where we know a lot of characteristics that must be fulfilled by a particular kind of number, even though we don't know whether there are any such cases - like odd perfect numbers. — Joffan, Apr 17 '18 at 14:53
she referred to it as a "pathological math". I think she refused to tell the name because there are still some good portion of people working on this theory. So curious — Focus, Apr 17 '18 at 14:58
The theory of efficient markets. (I know, I know: this is economics, not math. I just can't help but get in a dig.) — Stephen, Apr 17 '18 at 14:59
Seriously, I doubt very much that such a "branch" of mathematics exists. Maybe there is an old, dry twig though. — Stephen, Apr 17 '18 at 15:00
There are branches of math in which examples are difficult to construct, but none I've come across in which it's impossible — Exit path, Apr 17 '18 at 16:31

score 2 · Accepted Answer · answered Apr 17 '18 at 16:27

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.

score 0 · Answer 2 · answered Apr 18 '18 at 13:08

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.

Theory without a concrete example.

2 Answers2