Ostensibly, this begins a thread on the philosophy of mathematics, but the neat thing about mathematics is that, because its examples are so tackeddown and clearlydefined, it easily slides into other kinds of arguments. Arguments about truth and meaning, and what have you. Helpful, I think, for forging a subtler and more concrete approach to things.
I, for instance, am always the first and loudest in defense of the deep contingency of things; of self, society, and concepts. Most things that we humans can get our hands on are, in some significant sense, human creations. Mathematics seems, at the outset, to be the most resistant to this contingency; its results seem hard, fast, incorrigible, and inhuman. But I stick to my guns, and willingly teeth on the challenge. This places me somewhere in camp with the Constructivists, Intuitionists, Cognitive Scientists like Georges Lakoff, socalled Fictionalists like Hartry Field— the back of the bus, so to speak. But I've been having trouble placing myself exactly in the right club, and agreeing to premises of the disagreements... The ontological status of numbers, for instance.
The point of disagreement within many of these ornery schools revolves around ontological status; whether something like numbers or mathematical objects really exist or not. In many ways I'm on one extreme, according to my ontology, Kronecker did not go far enough when he said "God created the natural numbers; all the rest is the work of man." I think man is responsible for the natural numbers as well; that they do not exist in the sense most people think they do— out there, somehow. Notice I said "according to my ontology." This is because I think ontology is not easy to sort out— it is simultaneously analytic and synthetic, semantic and substantial, in ways that are not easy to disentangle.
In other words, many times when we are arguing over the existence of something— like numbers— we are simultaneously arguing over our definition of existence. Two stubborn mathematicians might even be imagining the same relationships and metaphors, but one counts that as existing; the other's not so charitable. They just cannot tell from argument. The one quality that's supereasy to tack down— "having physical mass"— does not really come into play with mathematics; so this puts us in a pinch. Because outside of that, we have nothing solid to compare and contrast our metaphors with— would this realm of mathematical objects be out there, in here, distinct from the physical world, implicit in it, latent in it, descriptive, mental, inherently mental, mental in such a way that it is inscribed in us so deeply that in fact it is the deepest part of reality? We have all these innumerable ways to imagine how and where things exist, in a general ontological way, that I have trouble seeing how we could ever get onto applying this definition to something like numbers, mathematical objects, or mathematical truths.
We would have to really get down on our hands and knees, and work out the details of latency, probably with a pencil, before we could say whether mathematical truths were latent— and I'm not sure 2300 years of Aristotelianism ever got that over and done with.
So rather than the reality— the ontological status— of mathematical creatures, I like speaking of Contingency, and demonstrating how utility at the very least shaped our mathematics, even if the utility and contingency is dependent on the historical quirks of human biology. I also love the created aspect of mathematics, in a subtle sense. But— man— I would love to have a Mathematical Platonist and Realist on board, on this thread; to help me rid myself on any facile notions I might have— to get a stronger grip on things in general....
