heh. this is going to be *so* weird to people who think math is real...
It's also a total mess as an encyclopedia article.
This article should describe the various different philosophies of mathematics.
Also, I doubt it should be "weird" to anyone. Anyone who works with math pretty soon hits some philosophical issues.
A lot of scientists (particularly if you get into the social sciences) don't consider mathematics neutral.
- It differs from the philosophy of science by not taking mathematics as a neutral point of view - rather, investigating such subjects as the willingness to accept mathematical proofs, the validity of induction or analogy, what combination of metaphors constitutes an isomorphism, mathematicians' social capital and the meaning of the well-known collaboration graph.
- A lot of 'hard scientists' don't consider 'social science' to be science -- they point to the lack of falsifiability in many of their theories, and assert that abandoning objective standards of mathematical and logical proof removes the basis for falsifiability, leaving only opinion, fashion, and popularity contests.
- There are physicists who do think that. But that's a relatively small minority opinion (something like 10% or so). My experience with "math-fetishists" is that they actually tend to be social scientists. Very few physicists (or even mathematicians) seem to believe that an idea that can't be expressed in mathematics isn't an idea at all, however I've known/read a number of social scientists that seem to think that. The problem is that regarding mathematics in social science *greatly* limits the hypothesis that you can form and the theories that you can test. Also, pretty much anyone who does qualitative research would disagree with non-mathematical means non-falsifible, and I'm pretty sure that most physicists don't think this (even though I know of a few who do).