In mathematics, there are many things that can be rigorously proven to exist, but at the same time rigorously proven that no concrete examples of it can be given. This is a crazy idea for non-mathematicians.
Have any philosophers studied things that certainly exist but can never be seen? Descartes, based the existence of many things on our mind. (I think, therefore I am.) However, in the above situation in mathematics, the mind can tell us that (1) something must exist, or else there would be a contradiction, and (2) that thing does not exist in my mind since neither examples of it nor method to produce it are existent, which are two contradicting messages.
Examples of things that exist but do not exist:
- Non-Lebesque measurable sets: the existence of this is established by the axiom of choice, so no concrete examples can be given.
- Brouwer fixed point theorem: this is proved by contradiction, giving no methods of finding such a fixed point.
Just tell me some theories on this topic. (I am not professional in philosophy.)