But surely there is just such an organ? I know, with absolute certainty, that there is an infinite number of primes. But nobody has ever seen, heard, touched &c as much as one measly prime number. That's just a reprise of Socrates' argument in the Meno, but I find it very convincing.
Hence, (a) there is a place containing the prime numbers, and it can't be this finite universe since a finite container cannot hold an infinite set. And (b) we can reason about some of the things in this funny place with - at least - a degree of confidence that cannot be obtained from our five physical senses. And finally (c) the physical universe around us obeys, systematically and totally, abstract mathematical laws.
Which strongly suggests that this universe was designed, and it tells us something about the designer. If a watch ran for ever by itself, that would be strong evidence of a perfect - and hence divine - watchmaker. To me, the very regularity of the universe - the absence of miracles, to put it plainly - is evidence of a perfect Designer.
Re a: There doesn't have to be a place containing prime numbers. To believe so would be taking an extreme Platonist position. Neither does one need infinite space to discuss infinite things. One only makes finitely derived statements about infinite things. There is no "infinite list of primes" only a proof that given any prime, one can produce a larger prime. These methods are discussed in books about "foundations of mathematics" and similar topics.
Re b: We reason about mathematics with more certainty because the whole subject is man-made.
Re c: There's a book called something like "The Unreasonable Effeciveness of Mathematics." We do invent things in math to describe the real world. It's often surprising that math works so well. We make a mathematical system do describe one thing (electrons, for example) then extrapolate the math to other things (positrons) and ofter, the other things exist physically. Pythagoras suggested "everything is number."