Afterword

I have been largely negative as far as naturalism is concerned. But there is much to be learned from it and it is now time to say a few words along that line. We are certainly the products of evolution, and not just any evolutionary process, but a Darwinian one. Though I do not for a moment think that a simple Darwinian account of mathematics could be right, the truth, whatever it is, must be compatible with our biological origins. In this regard, I feel a bit like opponents of sociobiology who likewise accept Darwin gladly but deny that every aspect of our behaviour is the direct result of evolutionary adaptations. It may be, for example, that evolution has provided us with plastic minds and we then learn our various behaviours as a result of interacting with the environment. This, at least, is what I suppose to be true.

The onus seems to fall on those of us who resist letting Darwin tell the whole story of mathematics. This is a fair demand. Platonists will certainly allow that a sense of elementary arithmetic will have survival value, but we deny the fact that 5 + 7 = 12 reflects nothing more than a well-adapted brain. We should investigate how the human mind is able to grasp abstract entities, given that we claim this actually happens. As I mentioned earlier, naturalists often claim to have solved many more problems than they actually have. They, too, have difficulties with the mind–body problem. But that's no reason for gung-ho Platonists to ignore the matter. We should happily acknowledge that there is much work to be done in the realm of cognition, and we have the Darwinian naturalists to thank for making the urgency of this task manifest.

I see intellectual life in general as a history of rival research programmes. The basic idea is common in the philosophy of science where Kuhn's paradigms, Lakatos's research programmes, and many other similar ideas prevail. The same idea, at least in outline, is at work in all fields, including literary criticism, economics, musicology, and, of course, philosophy. Naturalism is a perfect example. It is a thriving research programme and it holds lots of promise. Platonism is another and, I think, just as promising, indeed more so. Each has its own aims and methods and each is ever ready to point out the problems with the others. Intellectual life is healthy when many such programmes are flourishing and competing. I have tried to point out some of the problems I see with mathematical naturalism and some of the advantages of Platonism. However, no argument, for or against, should be considered decisive. When it comes to research programmes, it is the weight of evidence over the long run that matters. The most that I can hope for is to make manifest a few new troubles for naturalists to consider and perhaps to generate some interest in the Platonic alternative.

It's not just a case of contrasts. There is much to be learned from naturalists, even if one does not share the outlook. One thing I particularly like is the detailed case studies that many naturalists provide. This is certainly the case with Maddy. It is also the case with some philosophers of mathematics, such as Mark Colyvn, Mary Leng, and David Corfield, who are making their mark with extensive examinations of mathematics in the making. This is extremely interesting and valuable work.

There are lots of topics I haven't covered but could have. David Bloor and other sociologists of knowledge hold self-avowed naturalist and highly anti-Platonist views. Mathematics, like natural science, reflects our interests, says Bloor. It is not a body of objective truths that we discover; it is, he claims, something we create to serve our various social interests. There are also accounts coming from those involved in neurophysiology. Butterworth and Dehaene claim that mathematical thought is simply encoded in our brains, the product of Darwinian evolution. It has survival value, but does not reflect any sort of objective truth. As well as avoiding sociologists and neurophysiologists, I have also ignored Wittgenstein. Some have suggested reading his “forms of life” in a naturalistic way. Wittgenstein's problem about “going on in the same way” has been likened to Hume's problem of induction, and the solution in each case is simply to say it's a brute fact about the way we act. This is Kripke's famous reading of Wittgenstein and it could quite legitimately be seen as another version of naturalism. Finally, Hartry Field's nominalism is motivated by naturalism, as he aims to defuse Platonism by showing that mathematics is not essential to science.

Each of these is worth a chapter in its own right, but I have resisted for two reasons. In the case of Field and Wittgenstein, there is already a huge literature on each. And what makes them interesting are for the most part issues that are only weakly associated with naturalism. Social constructivist and neurophysiological accounts of mathematics are also obviously naturalistic, but philosophers tend not to be all that interested in them. They should be, however, since these views have more in common with the typical naturalist views of philosophers than most philosophers realize. Nevertheless, I must draw a line somewhere. Working on sociological and neurophysiological accounts of mathematics will have to wait for another day.

A word or two about religion seems in order. Naturalists are overwhelmingly atheists. So am I. The relation between religion and mathematical Platonism arises occasionally in print and frequently in conversation. My naturalist friends and colleagues enjoy teasing me (knowing I'm an atheist) along the lines that being a Platonist is really no different than believing in God. I'm kidded about being soft on superstition, a closet religionist, and so on. While I enjoy the kidding, I've actually never seen the slightest connection between religion and mathematical Platonism. But others have, including Lakoff and Núñez, who (as we saw in Chapter 4) talk about belief in God and belief in Platonism equally as matters of faith rather than evidence, and dismiss both for being so. The physicist turned theologian John Polkinhorne also sees a connection. In his recent Belief in God in an Age of Science (1998), Polkinhorne favourably cites a number of prominent mathematicians who are also Platonists (Gödel, Hardy, Connes) in support of the notion that mathematics is transcendent. God, of course, is transcendent, too, so, Polkinhorne seems to suggest, there is a kind of mutual support—Platonists should believe in God.

But aside from transcendence, there is really no connection between belief in God and belief in a Platonic realm. The best way to make this point is to appeal to Aristotle's taxonomy of causes and corresponding explanations. When a rock breaks a window, the event of hitting the window is the efficient cause of the event of the glass breaking. When George walks down the hall to get a drink of water, his desire to quench his thirst and his belief that there is a drinking fountain down the hall make up the teleological cause of his action. However, Plato's abstract entities are different than either of these. They are formal causes. Being a Euclidean triangle is the formal cause of having interior angles sum to 180 degrees. When it comes to causation and explanation, religion is intimately tied to teleology, just as window breaking is tied to efficient causation. We explain in terms of God's plans and purposes. This has nothing to do with formal causes and formal explanations; there is nothing teleological about them. There is no danger of sliding from Plato's realm into the abyss of religion, except through sloppy thinking.

Finally, a different kind of issue. Bombs were falling on Baghdad as I was writing some of this book. People were dying and a criminal gang in Washington was plotting its next move. How can anyone think about the nature of mathematics when war is being waged? It's a question academics in any field frequently face. There is an amusing anecdote stemming from the Great War. A woman approached an Oxbridge don of obvious military age and suggested that he should be fighting to defend civilization instead of shirking his duty like a coward. “Madam,” he replied, “I am the civilization they're fighting to defend.” Amusing, but not really to the point. The crucial question is: Can we justify devoting our energy to issues in the philosophy of mathematics that are remote from daily life when human suffering is so enormous and so immediate?

In the penultimate chapter I quoted Russell, who famously remarked: “Remote from human passions, [the mathematical realm is a place] where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world” (1919, 61). Platonism lends itself to this sentiment, and it does so to a greater extent than other accounts of mathematics, decidedly more than naturalism. Russell, who was a Platonist at the time he wrote the passages from which I'm quoting, recognized the difficulty. He balanced the positive image of the Platonic realm as a sanctuary with a question, raising the obvious problem:

In a world so full of evil and suffering, retirement into the cloister of contemplation, to the enjoyment of delights which, however noble, must always be for the few only, cannot but appear as a somewhat selfish refusal to share the burden imposed upon others by accidents in which justice plays no part. Have any of us the right, we ask, to withdraw from present evils, to leave our fellow-men unaided, while we live a life which, though arduous and austere, is yet plainly good in its own nature? (1919, 72)

Russell says yes, some of us do have the right—even the duty—to withdraw. “When these questions arise, the answer is, no doubt, that some must keep alive the sacred fire, some must preserve, in every generation, the haunting vision which shadows forth the goal of so much striving” (1919, 72). The answer may be right, but it also seems much too self-serving. I'd shrink in disgust were I to hear myself saying, “Gee, I'd like to end the killing, but I'm busy right now showing what's wrong with naturalism.” Like so many others, Russell also adds that what seems pure, remote, and useless today may prove to be of great utility tomorrow. This, too, may be right, but again, too convenient. “Gosh, my refutation of naturalism might someday cure cancer or make war obsolete.”

Are we faced with an exclusive choice: Pursue mathematics or pursue justice? I hope one can do both.¹ Russell wound up in jail for his opposition to the Great War, but while imprisoned he managed to write Introduction to Mathematical Philosophy. Marx often put the cudgels down so that he could take up the classic literature from Aeschylus to Shakespeare of which he was so fond. Einstein worked not only on the purest physics, but was also very active in the politics of social justice. Indeed, he was sufficiently active that the FBI amassed a great file on him in the hope of building a case for deportation. Of course, these are remarkable people. Perhaps the most remarkable is Plato himself. No one was more focused on the purest of pure knowledge. And yet, no one was more engaged in the politics of his day. Mathematical Platonism has been crucial throughout this book and I'm a keen supporter of Plato in this regard. But being socially involved is equally important, and an essential part of a well-lived life. I happily commend Plato to all and to commend him in both respects. Even if we fall well short of the standards set by these remarkable people, we can still do something in the same direction. None of us should ever tolerate poverty or war crimes under any circumstances.