What is … the philosophy of mathematics?
During calendar year 2005, I attended two weddings and a handful of other events where I met strangers (gasp!), many of whom were not just unfamiliar with philosophy (gasp gasp!) but didn’t even have much in the way of higher education (gasp gasp gasp!). As is both natural and expected on these sorts of occasions, I was asked what I did. And also quite naturally and expected, I was at a lost for how to explain the work of a philosopher of mathematics to a man who owns and runs a convenience store and got an associate’s in business management twenty-five years ago.
Even once we’re safe and sound back in the ivory tower, it’s still a useful exercise to try to figure out how to explain just what it is we do in the department of philosophy. Quite frankly, most of our fellow intellectuals won’t even have a good idea of what we’re up to: my friend Annie—a psych major who took a class or two on the history of ethics and is about to start her second year of med school—thinks that philosophy is basically just history. Natural scientists, mathematicians, and engineers have plenty of jokes about those crazy philosophers, and the things that literary theorists call `philosophy’ just make my soul hurt.
So let’s imagine you’re the latest junior hire at the University of Somplace, Department of Philosophy, and you meet the young dean of Arts and Sciences at the new faculty mixer. The dean is a psychologist, and doesn’t know much about philosophy beyond `Freud was wrong and Lacan is crazy, but Skinner had some interesting things to say’. How are you going to explain what you, a philosopher of math, write about in your research?
Since I fancy myself more of an epistemologist than a metaphysician, let’s start by talking about mathematical knowledge. While coming up with a solid definition of knowledge is a non-trivial project, let’s work with the Theaetetus definition: knowledge is justified true belief. We’ll work backwards through these three things.
For the most part, we seem to have beliefs about things: I believe that Bush should never have been elected, that the Moon is not made of green cheese, that the sum of the angles of a Euclidean triangle is π radians, and so on. Though Kierkegaard would disagree, it seems weird to have beliefs that aren’t about some thing or another. And we have mathematical beliefs, so we can immediately ask the question: What are mathematical beliefs about?
Let’s take the triangle-sum theorem (about the sum of the angles of a Euclidean triangle) as an example. One group, mathematical realists, will say that there are real things, somewhere and somehow, called triangles, and this belief is about a certain property that these real things really have. Another group, the formalists, will say that the theorem is just a statement in some language-game, with no real meaning or reference to the terms. A third major group, the structuralists, will say that the theorem is talking about the relationships between ordinary things, such as the one these three wine glasses are standing in.
Next up is truth. In what sense are mathematical theorems true? Realists and structuralists will say that mathematical theorems are objective reports of the properties and relations of some real things (though, of course, they’ll disagree over what things). Formalists are more likely to say that these truths are very different from truths about, say, the colour of the tablecloth; the strongest formalists are even liable to say that mathematics is more about convention than truth in any very strong sense. Intuitionists may want to say there’s a certain subjective yet not exactly conventional component that cannot be neglected.
Finally we have justification. Mathematicians justify theorems by writing proofs, where we start with a given assumption or definition or two and infer one claim from the next. (It would probably be helpful to actually run through a proof here as an example.) Kant was famous for calling mathematical proofs arguments that were `synthetic’ and `a priori’. `A priori’ means we don’t need to appeal to experience—run experiments, the way most scientists do—to arrive at these absolutely certain facts. Yet `synthetic’ means that these are substantive truths about the world. How can mathematics be both of these at once?
Logicists claim that mathematical proofs are really just logical proofs. Formalists say that proofs are the manipulation of symbols according to rules, like in a game of chess. Kant had his own theory, that mathematical cognition involves representations of the world using both concepts or terms and `pure intuition’.