Inspired by Where Mathematics Comes From by Lakoff and Nunez, reviewed below, I am setting forth my own philosophy of mathematics. It is probably not original; in fact it seems like common sense. It also seems a bit like Kant’s philosophy of ontology, if I understand Kant. Before presenting my own ideas, I’ll summarize the major philosophies of mathematics prevalent today.
Lakoff and Nunez discuss the three major philosophies of mathematics prevalent today and put forth their own, called the philosophy of embodied mathematics. To me, these philosophies are rather like the proverbial blind men trying to describe an elephant. The man touching the trunk says an elephant is like a fire hose. The man touching the tail says the elephant is like a rope. The one touching a leg says an elephant is like a tree. Each blind man correctly describes a part of the whole.
The philosophy of formalism states that mathematics is the production of valid formulas that follow from a small number of axioms. A proponent, Bertrand Russell, defined mathematics as “the subject where we never know what we are talking about, nor whether what we are saying is true.” For formalists, mathematics exists independent of meaning. This seems preposterous on the face of it, and I doubt that many working mathematicians hold it to be true. However, it contains a grain of truth in that I (or any mathematician that I can think of) would be disinclined to accept as true something that could not, in principle, be given a formal proof. That Godel has shown that there are propositions about arithmetic that cannot be proven true or false does not invalidate this.
The Platonic philosophy of mathematics, or Platonism, states that mathematics is concerned with the discovery of truths about a realm of abstract mathematical ideas, and that this realm has an objective reality outside of any human minds. This is the philosophy that most working mathematicians intuitively have. The deeper one enters into a mathematical subject matter, the more it seems to have an objective reality. However, this view has been pretty well torn apart by the philosophers. Many of the arguments against Platonism are well summarized in Where Mathematics Comes From and elsewhere, and the gist of them seems to be the impossibility of a physical being knowing a non-physical reality. My philosophy of mathematics, given below, contains a modified form of Platonism, which I think meets the objections.
The post-modernist explanation of mathematics states that mathematics, like all systems of ideas, is purely a product of the culture in which it arises. In its extreme form, this philosophy is not only wrong, it is pernicious. If there is no objective reality and truth is what society wishes it to be, then we are in the world of Orwell’s 1984, where Winston Smith’s reeducation by the state is complete when he is willing to believe that two plus two is five. Extreme postmodernism is the philosophy of totalitarianism, and it seems to me that a professor who believes this theory is in the wrong line of business. That said, a moderate form of postmodernism does illuminate mathematical thought. Mathematicians, like everyone else, are products of their culture. The areas of mathematics that are deemed important and the methods of proof that are accepted are determined, in large part, by culture.
The philosophy of embodied mathematics has its own problems, and also supplies valid insights. This philosophy states that mathematics does not have an objective reality but is not totally culturally determined either. Instead, “Mathematics is the product of human beings. It uses the very limited and constrained resources of human biology and is shaped by the nature of our brains, our bodies, or conceptual systems, and the concerns of human societies and cultures.” Lakoff and Nunez take special aim at Platonism, which is what they see at the center of an elitist “romance of mathematics”.
I think that mathematics is more than a product of human beings. It can only be understood as a blend of internal and external reality. Let me offer an analogy. Imagine we are in a forest, and I (pointing) say “this is a tree” and “that is a tree”. Most people would agree that the trees are actually there, but I am saying much more. I am saying that both objects I am pointing to are instances of the same kind of entity, namely “tree”. So the concept of tree is rooted both in the external entities (which are apparent to any sentient beings, not just humans) and internal concepts and abilities, including language and the idea of a category of “tree”. In the same way, mathematics is implicit in the regularities of the universe but must be made explicit by human thought. It is both a feature of the universe and a product of our minds.
There is no way to prove this, but it would be hard to imagine that an alien that was intelligent in a way that would enable it to talk with us would not believe that 2 + 2 = 4. On the other hand, imagine that this alien, like Saint-Exupery’s Little Prince, lived on a very small spherical planet. Such an alien might have no idea of Euclidean geometry, because that would be irrelevant to its environment. The ratio of the circumference to the diameter of a circle (C/d) would not be pi, but rather it would be variable. A circle drawn around the equator would have C/d = 2, and progressively smaller concentric circles in the northern hemisphere would have ratios closer and closer to pi. Whether the alien would regard pi as an important number or not is unclear. Given the ubiquity of pi in our mathematics, it is hard to believe that it would not appear in alien mathematics as well. Our ideas about mathematics are not just about us, and not just about the external world, they are about the complex and interpenetrating interaction between the two.