I work primarily in the philosophy and history of mathematics which has naturally led to discussion about the philosophy of mathematics with people outside my field, and outside of philosophy. In this post I will try to address, in a loose and (mostly) non-technical way, some of the questions that often arise in such discussions. What follows is not meant to be complete or precise, and is certainly biased, but I hope that it will provide a starting point for more in depth discussions about the nature of mathematics.
1. What is `philosophy of mathematics’?
To start, what it is not is mathematics simpliciter. Philosophers of mathematics, for the most part, are not engaged in solving equations, and we’re probably not any better than the proverbial man on the Clapham omnibus at calculating an 18% tip on a $47 bar tab. When we do do mathematics, it tends to be mathematical logic –proofs, derivations and model-building in formal systems that have some philosophical relevance. But I’ve still not answered the original question.
When asked the above question my response is usually along the lines of: the investigation whether mathematical objects (like sets, natural numbers, and functions) exist; in what way they exist; what their relationship is to the physical world, to us, and to each other; and how it is that we come to know about them. To that I really should add questions like why is mathematics (seemingly) universally applicable? How should we interpret the practices of actual mathematicians? And How do children acquire mathematical concepts? There is obviously much more that could be said here, but this should give some idea of the sorts of things philosophers of mathematics do.
2. Isn’t mathematics just a construction to help us reason about the world?
This is a very common response to my initial attempts to explain what philosophy of mathematics is, and my answer is: maybe, but I don’t think so. Such a view raises as many difficult questions as it answers. To my mind, the most pressing is how to explain the universality of mathematics. There are a few related issues here, but the general thought is, and our experiences seem to suggest, that numbers behave in the same way whether we were raised in Canada, Cambodia, or at the other end of the universe, and whether we are counting sheep, predicting economic growth, or solving Einstein’s equations. Most of us would want to say that if we made contact with another intelligent species that had had no previous contact with humans, their mathematics would at least be inter-translatable with ours. Such a though would be hard to reconcile with a psychologistic (mind-dependent) theory of mathematics.
My second worry is that we still haven’t said what mathematics is fundamentally about. What are we referring to when we say 2+3=5, or arguably even more problematic, that there are the same number of points on a straight line as on the surface of a sphere? If, on the one hand, we say that we are referring to mind-dependent, universal objects, we have just pushed the question back to a question about what those sorts of objects are like. If we say that mathematics is fundamentally just symbol manipulation, we run into the universality problem again, but also questions about whether, and how mathematical statements can be said to be true.
All of this is not to say that a psychologistic approach is impossible, but a proponent of such a view has some difficult metaphysical and semantic questions to answer.
3. What do you mean by the `foundations of mathematics’?
Very often you will hear philosophers of mathematics talking about the foundations of mathematics (or at least you will if you happen to find some of us). `Foundations’ has traditionally been seen as the core of the subject, but the term is a bit ambiguous. Working mathematicians, mathematical logicians, and more technically minded philosophers most often discuss what I call formal foundations –the (usually axiomatic) formal (read `formulated in logical notation’) system(s) that is most basic, or that the rest of mathematics be formulated in, or has the most promise for the unification of mathematics. Zermelo-Fraenkel set theory with the axiom of choice (ZFC, what philosophers and mathematicians generally mean when they talk about set theory) is the most well-known contender, but there are others.
Formal foundations don’t necessarily give us all we are looking for as philosophers however. To formal foundations I would add metaphysical foundations and epistemic foundation –fundamental theories about what mathematics is about, and how we come to know about and why we are justified in employing mathematics. This is not to say that there is a sharp line between the three kinds of foundation, indeed, there is usually overlap, but when you hear `foundations of mathematics’ it could mean one or more of the types of foundation I mentioned.
4. What do you mean by…?
There are many terms that get used in any discipline whose meanings are unclear, or mean something different than what we would first think. Here are a few from the philosophy of mathematics:
platonism: This is platonism with a small `p,’ ostensibly to disambiguate from a direct connection to Plato’s philosophy, though the term comes from the fact that there is some connection. To be a platonist about mathematics is to hold the view that mathematical objects have a robust existence, mind-independently, acausally, and outside of space and time.
Intuitionism: This refers to followers of the philosophy of mathematics of Brouwer and Heyting at the beginning of the 20th century. Brouwer indeed thought that mathematics should be based on intuition. These days, intuitionistic logic or mathematics usually refers to the denial that the law of the excluded middle holds universally. The law of the excluded middle says that any statement of the form `P or not P’ is true. Inuitionists also deny that P and not not P are equivalent.
Nominalism: The thesis that abstract objects, like numbers, but also propositions and possible worlds don’t exist.
Naive set theory: The view often erroneously attributed to Georg Cantor, that any collection of objects forms a set. This view leads directly to contradiction via Russell’s paradox. Consider the set of all sets that don’t contain themselves, it must both contain itself and not contain itself, contradiction.