When I was a math major in college and in some of the years later, I went through such questions and responses. If you like digging into the issues and work of

When I was a math major in college and in some of the years later, I went through such questions and responses. If you like digging into the issues and work of G. Cantor, B. Russell, J. von Neumann, K. Godel, P. Cohen, etc., fine. But here is a shorter response: Q. Is the square root of 2 rational? A. Rational means the ratio of two whole numbers. So, suppose p and q are whole numbers and we have (p/q)^2 = 2 then we have p^2 = 2(q^2) So that the number of factors of 2 on the left side is even while on the right side it is odd. But 2 is a prime number and we have a theorem the fundamental theorem of arithmetic, that for each positive whole number there is only way to factor that number as a product of primes. So, the square root of 2 is not rational. Done. This argument illustrates the philosophy of mathematics. Here, never mentioned an "ism". Opinions can vary, but my opinion from my background in pure/applied math is that this little argument about the square root of 2 is about all there is that is solid and needed about the philosophy of math. Opinions can vary!