In my last post, I outlined the view of mathematical Platonism taken by Roger Penrose and other mathematicians. Briefly, in the words of Joel Spencer, “Mathematics is there. It’s beautiful. It’s this jewel we uncover” (quoted in The Man Who Loved Only Numbers, p 27).
However, some modern mathematicians feel that the time for such ideas is past. In the words of Brian Davies, “It is about time that we … ditched the last remnant of this ancient religion” (in his article “Let Platonism die,” European Mathematical Society Newsletter, June 2007).
One alternative commonly presented is Empiricism. Santa Claus and the Tooth Fairy do not exist, and neither do infinite decimals, or perfect circles, or the set of all natural numbers. Only the physical universe exists. In the words of astronomer Carl Sagan, “The Cosmos is all that is or ever was or ever will be” (the opening sentence of his book Cosmos). Our only truly certain knowledge is physics, and mathematics is in fact a branch of physics. When we say that 2 + 2 = 4, we are not talking about a relationship between Platonic number-objects. Instead, what we mean – and all that we mean – is the empirical truth that two atoms plus two atoms gives four atoms, and likewise for stars, rocks, or people. “Four” is not a noun, it’s an adjective.
The empiricist point of view seems to solve the mystery of the “unreasonable effectiveness of mathematics.” Mathematics is just part of physics, and it isn’t surprising that different branches of physics agree with each other. Empiricism also avoids the need to postulate a “soul” or some other mechanism for peering into an ethereal Platonic world.
There are two problems with the empiricist point of view, however. First, it isn’t true to the history of mathematics. Galileo used parabolas to describe the motion of falling objects, but the ancient Greeks had originally described parabolas in a quite different context, that of conic sections. Similarly, imaginary numbers were originally discussed without the slightest idea that centuries later they would become a fundamental part of quantum theory. There’s still a mystery there: why should mathematics from one context work so well in another?
The second problem is that scientific truths are contingent. We could, for example, live in a world where plants were purple. The fundamental forces of physics could be different from the way they are (although that could imply a lifeless universe). We can even imagine a universe containing no matter at all, only empty space. The laws of mathematics, however, could not have been different – they are necessary. It is impossible to imagine – at least, to imagine consistently – a universe in which 2 + 2 = 5. Even God cannot make 2 + 2 = 5. This indicates that there must be more to mathematical truth than just physics. Statements about mathematics, such as “2 + 2 = 4” are in a different category to statements about the universe, such as “light travels at 299,792,458 metres per second.” But if that is the case, then mathematical truth must in some sense lie outside the universe – which brings us back to Platonism.