This is actually a very good question to illustrate how everyday intuitions can lead you astray in thinking about philosophical problems -- or about any problems, really, that require finer and more sophisticated distinctions than needed in typical situations of everyday life. Everyday intuitions, embedded in the categories of our inherited natural languages, are insensitive to certain fine distinctions made in the more finely-tuned artificial languages we've invented. From an everyday point of view, there is no reason to think that your argument about pi being infinitely large is flawed at all. But in mathematics, we've had to deal with the problem that that argument works in some cases and not in other cases -- some infinite series "converge," as mathematicians say, while others "diverge." The sum of all the reciprocals of the natural numbers (i.e. one half plus one third plus one fourth plus one fifth, and so on) diverges, i.e. it's infinite, as in your intuition. But the sum of all the...

I have no problem at all with what Stephen says, but would add a couple of things. First, Stephen didn't address what might actually be the questioner's main concern, i.e. whether the fact that "all physical things seem to conform to it" makes mathematics metaphysical. What is "it" here? Mathematics keeps growing, and one of the main sources of growth is that new things keep coming along (such as new scientific findings) for which existing mathematics is no help. The formulation of general relativity, for instance, required new mathematics that had been developed to some degree (by Riemann and others) before 1915, but without any thought that it might someday actually apply to something in the world out there. The further development of differential geometry was largely in response to its employment in theoretical physics (though of course it then took on a life of its own, as mathematical ideas do). And these new developments invariably (perhaps inevitably) don't quite fit, in various ways,...