I am slowly working on an article for Skeptical Inquirer about the ways in which religious apologists use mathematical arguments in their rhetoric. Among these arguments are the familiar creationist claims about probability and information theory, but there is also a family of arguments based on the effectiveness of mathematics itself. The basic argument is that mathematics is so useful for describing the world solely because God, in his benevolence, designed the world to be describable in that way. They will often cite Eugene Wigner’s 1960 article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” as supportive of their basic view.
In pondering what to say in reply to this dubious argument, I’ve been reading some philosophy of mathematics lately. Inevitably, the question of whether mathematical objects exist has arisen. So, in the spirit of my ongoing discussion about things that frustrate me about philosophy, let’s consider that question now.
Here’s mathematician Timothy Gowers, a Field’s Medalist, from his book Mathematics: A Very Short Introduction:
A few years ago, a review in the Time Literary Supplement opened with the following paragraph:
Given that 0 x 0 = 0 and 1 x 1 = 1, it follows that there are numbers that are their own squares. But then it follows in turn that there are numbers. In a single step of artless simplicity, we seem to have advanced from a piece of elementary arithmetic to a startling and highly controversial philosophical conclusion: that numbers exist. You would have thought that it should have been more difficult.
This argument can be criticized in many ways, and it is unlikely that anybody takes it seriously, including the reviewer. However, there certainly are philosophers who take seriously the question of whether numbers exist, and this distinguishes them from mathematicians, who either find it obvious that numbers exist or do not understand what is being asked. The main purpose of this chapter is to explain why it is that mathematicians can, and even should, happily ignore this seemingly fundamental question.
The absurdity of the `artlessly simple’ argument for the existence of numbers becomes very clear if one looks at a parallel argument about the game of chess. Given that the black king, in chess, is sometimes allowed to move diagonally by one square, it follows that there are chess pieces that are sometimes allowed to move diagonally by one square. But then it follows in turn that there are chess pieces. Of course, I do not mean by this the mundane statement that people sometimes build chess sets — after all, it is possible to play the game without them — but the far more `startling’ philosophical conclusion that chess pieces exist independently of their physical manifestations.
What is the black king in chess? That is a strange question, and the most satisfactory way to deal with it seems to be to sidestep it slightly. What more can one do than point to a chessboard and explain the rules of the game, perhaps paying particular attention to the black king as one does so? What matters about the black king is not its existence, or its intrinsic nature, but the role that it plays in the game.
This seems exactly right to me. Mathematical objects have precisely the same kind of existence as the pieces in a game of chess. The king has a role to play in the game of chess, and it has certain attributes bequeathed to it by the rules of the game. In that sense, it exists. But if you then say, “No, no. I want to know if the black king exists independently of anyone’s ideas about how to play chess,” then I no longer understand what you are talking about.
Another way of making the point is that questions about mathematical objects are essentially identical to questions about fictional characters. If someone asks, “Did Sherlock Holmes smoke a pipe?” then in one sense the answer is just no. Sherlock Holmes never existed in the real world, so obviously he didn’t smoke a pipe. But if we understand that question to be referring specifically to the fictional world created by Arthur Conan Doyle, then the question is meaningful. Moreover, within that context it makes sense to answer that yes, he did indeed smoke a pipe.
Likewise for mathematical questions. Mathematical objects exist in a fictional world created by mathematicians. For example, in the context of modern abstract algebra I know what it means to ask if finite, simple groups exist. But if you ask whether finite, simple groups existed prior to the establishment of modern abstract algebra, than I don’t know what you mean. It’s like asking if Sherlock Holmes existed prior to the writings of Arthur Conan Doyle.
There are obvious parallels here with our earlier discussion of realism vs. anti-realism in the philosophy of science. Perhaps there are some analogies we can find to provide guidance here. A scientific realist would say that electrons, say, existed before scientists ever thought to formulate the concept. Maybe we can say the same about prime numbers; they existed prior to the time when mathematicians thought to formulate the concept.
But this doesn’t help. The premise of scientific realism is that there is a physical world that exists independently of anyone’s ideas about it. While I would not want to have to write down a rigorous definition, it seems clear what we mean when we talk about the existence of physical objects. Not so with mathematical objects. Whatever sort of existence they have, they certainly are not part of the physical world. Mathematical objects just are whatever mathematicians say they are.
What about the usefulness of mathematics? The main reason for thinking that electrons really exist out there in the world is their incredible usefulness in the context of scientific explanations. It would be a miracle if they were so useful without actually existing. Likewise for mathematical objects, we could argue. What a miracle it would be if math were so useful, even indispensable, in the context of scientific explanations, if they did not actually exist.
But this argument also does not seem very persuasive. There is no more mystery in the usefulness of mathematics than there is in the usefulness of maps in navigating unfamiliar areas. The map is not the territory, but the pattern of dots and squiggly lines on the paper is inspired by the territory. Likewise for the particular abstractions mathematicians choose to study. A perfect circle has no existence outside of a geometry textbook, but our abstract notion of a perfect circle captures something important about certain physical objects. The abstract models of mathematics are like the abstract version of the territory presented on a map. They help you see reality more clearly without themselves being part of that reality.
The general view of mathematics I am defending is known as fictionalism. It’s main rival is Platonism, which holds that mathematical objects exist independently of anyone’s ideas about them. It might be objected that some of what I have said in this post is question begging. For example, I said mathematical objects just are whatever mathematicians say they are. But that’s precisely the point at issue.
It’s not that I am specifically trying to beg the question. It’s just that I don’t know how else to talk about mathematics except in the context of fictionalism. My reply to someone who insists that mathematical objects exist in some non-spatio-temporal realm that we come to understand through mathematical research is just to stare at him blankly. I don’t understand what he’s trying to convince me of. The existence of physical objects makes sense to me. The existence of abstract objects that cannot possibly be reduced to some physical phenomenon does not make sense to me. The concept of existence does not seem helpful here.
Hence my frustration. Mathematical Platonism has some passionate defenders, but the position makes no sense to me. More importantly, though, my objection to this argument is the same as my objection to the argument between realists and anti-realists in science. I don’t see how anything at all is riding on this, and I don’t believe that anyone is thinking more clearly about mathematics as a result of the point and counterpoint in this discussion.
Of course, this is far more high-falutin than anything the religious demagogues have in mind. So, in the interest of ending on a lighter note, here’s a quotation from Katherine Loop, from her book Revealing Arithmetic: Math Concepts From A Biblical Worldview:
For example, if we use counting to explore our hands, we find we have four fingers and one thumb on each hand. Counting helps us see the design God placed within hands! At the same time, though, if we were to count the fingers on every person’s hands, every once in a while we would find a person with fewer/more fingers than usual, evidence we live in a fallen universe.
See my earlier remarks re staring blankly.