This soft question is, I think, borderline mathematical philosophy.
One of my former professors used to say that induction does not prove that a statement is true, but it merely eliminates the possibility of existence of any finite counterexample. For instance, when we prove the typical example that the sum of the first $n$ positive integers is $n(n+1)/2$, we are not "proving" the formula is true for all positive $n$, but rather we assert that it is impossible to find a finite $n$ for which the formula claimed fails to work.
One may say that they are exactly the same thing, but I've always felt that there is a subtle difference between showing something is true and showing something is not false. I just am curious if this is a somewhat of a shared belief in the mathematical society, or an isolated idiosyncrasy of mine. Any resources, insights, or opinions are all welcome.