Don't know how many people are interested in discussing the philosophy of math, but I didn't want to hijack the Absolute Truth thread with this related, yet sufficiently different, topic. So...
There are two main competing philosophies of math, Platonism and formalism. Platonism charges that the various mathematical objects that are dealt with in the pursuit of math (numbers, groups, fields, etc...) are immutable, eternal and transcendent. They are likened to Plato's Forms (hence the name Platonism), in that they exist independent of human thought and have properties which are immune to our triflings. Formalism maintains that math is purely a human construct, a logical system whose rules we choose and thus is purely dependent on our triflings. An essay I read a few years back claims that most mathematicians are Platonists at heart, but when having to defend their work they will go running to formalism as a justification for the various theorems and whatnot they produce.
The discussion that was started in the Absolute Truth thread hinged on whether mathematical truths can be considered a priori or a posteriori. This is not completely analogous to the two schools of thought mentioned above, though there is another school of thought, call it "historical empiricism" say, which claims that mathematics is an empirical science, as absurd as the notion sounds. There is some evidence for this, however. Throughout the ages, various mathematical "truths" have been found to be untrue. Indeed, this is partly responsible for the birth of formalism as mathematicians of the 19th century grew tired of resting all of their great work on wishy-washy definitions and assumptions. Also, much of the progress of math has been driven by roundly practical needs (though that's not to say most mathematicians care for applying math to the real world). The study of linear equations was prodded along by the need for solutions to distributing sheaths of wheat (or some other such problem faced by administrators of yore). The study of trigonometry was strongly tied to the pursuit of astronomy. Calculus was invented for its practical use in studying physics, so on and so on.
At any rate, what are other's thoughts on this? Is math just a human construct, or are our endeavors merely models of a purer, absolute Truth? Can we say all of our math is completely a priori, uninfluenced by our experiences in the physical world, or is it completely a posteriori, driven by simple observations of our common perceptions? Or is there a mixture of any combination of these, or something else I missed? Discuss at your leisure.