By Feynman's argument, you can prove that square numbers almost certainly keep on existing.

Roughly, it goes as such:

1) the probability of N being a perfect square is proportional to 1/sqrt(N).

2) For any N_0 arbitrarily high, if you integrate from N_0 to infinity the expression (1/sqrt(N) dN), you get infinity.

3) The expression in 2) is the "Feynman equivalent" of the expected number of square numbers above N_0.

So Feynman's nonproof actually turns out to be true, despite it not being a proof in this case as well.