Though arithmetic is as real and concrete as anything, but that's where the reality of math ends for me.
To me math is not what runs the world, math is simply man's attempt to rationalize and categorize nature. It is like my attempt and essentially my motivation for reading about everything, to get rid of the feeling I have of something that is out there, just beyond the grasp of my mind. It is similar to a feeling of uncertainty. Math was basically man's attempt to get rid of uncertainty of the things that were happening around him, an answer for his need for control, which reached its climactic and ironic end with the introduction of the field of probability into math. :)
And so numbers are as real as "chair" "hair" or "space" in the English language. And math is simply a language that gets new words in its vocabulary derived out of the original set sometimes, sometimes incorporated anew to enable it to able to describe all of nature - a different way of looking at nature, in which we try to see what is unseen to the untrained eye, look a little deeper and try to find the patterns and the abstracts. And just like any other language, maths, its symbols and its grammar needs to mastered. The problem here is that there are so many rules derived out of the fundamentals, so many abstractions, you are forced to accept them as real just for simplicity's sake. Case in point would be calculus. Integral Calculus in my view is nothing but an abstraction of an extremely large addition, but the abstraction itself has so many rules associated with it, we are forced to look at it like it were in itself a fundamental mathematical action and so as we keep building on top of it, we lose track of the fact that all those symbols, are just a way to tell us something about reality rather than real things. Just like calling a four legged form with a back and a seat, a chair becomes natural for us and chair is very real, even though it consists of different things like a seat, a back and legs.
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