There are a number of interesting problems about the nature of mathematical knowledge that this book tackles, not the least of which is that mathematicians don’t seem to care about what the philosophy of mathematics might say — a fact that upsets many philosophers of mathematics. The usual mathematician’s perspective is that mathematics is progressing very well and philosophical problems about the kind of knowledge mathematicians produce are disjoint from and irrelevant to honest mathematics. Mathematicians allow that certain parts of mathematical logic may help them, but debates about formalism, Platonism, structuralism, Quinean holism, realism and naturalism leave them cold. While this book does not go as far as some in admitting that mathematicians might be right and that there are better questions to ask in the philosophy of mathematics than has been the case for some time, it does profitably move in that direction.
A start is made in Mary Leng’s introductory essay. Taking as her opening question ‘How can we know about numbers?’ she notes the impact of Benacerraf’s famous paper of 1973 in which he more-or-less argued that such knowledge was impossible, because we have no epistemic access to the abstract objects that mathematicians say are numbers. This, as she remarks, sits ill with the high degree of certainty with which we say such things as ‘There are infinitely many numbers’. This clash is, of course, one of the reasons mathematicians have walked away from philosophy of mathematics, but it remains an interesting philosophical question nonetheless. Leng herself wishes to give up talk about abstract mathematical objects in favour of what is called fictionalism. This philosophy seeks to show that mathematics is consistent and that while mathematical objects no more exist that do Hamlet and Ophelia we can talk reliably about them in, as it were, the story of mathematics. Leng’s fictionalism, which she sets out in a later chapter, is focussed tightly on the question of how fictionalists can be assured of the necessary consistency of their stories. Michael Potter, in the essay that follows the introduction, argues oppositely that talk about mathematical objects can be defended, and by this he means mathematics and not merely logic. The failure of logicism and the continuing vitality of mathematics he sees as a challenge to philosophers to explain what mathematical knowledge is in precisely those areas (set theory, arithmetic, and analysis). where it is not reducible either to logic or to other branches of mathematics itself.
As Leng notes, there are other attempts to explain, or explain away, mathematical objects. Reuben Hersh has argued that mathematics is only true by convention, a philosophy disliked by philosophers and mathematicians, one suspects. Putman and Quine’s indispensability argument argued that if science offers the highest standards we have when dealing with ontological questions, and if mathematics is so closely tied into science as to be indispensible, it would be indefensible to grant existence to the strange objects of modern physics and withhold existence from the mathematical abstractions needed to make physical theory work. Why swallow a camel of quarks and strain at a gnat of numbers, so to speak?
Potter’s argument is that some philosophies of mathematics push ontological questions about mathematics back to some collection (of axioms, stories, what ever they might be) which are said to be good enough to bear the weight. Typically, the collection is to be consistent — but nothing in the mathematics can guarantee that, and nor, it would seem, can anything else. This suggests that some form or realism or naturalism might have to be tried instead, drawing on arguments for the existence of scientific objects. His interesting suggestion is that the nature of mistakes might be fruitful to contemplate. After all, it is our ability to be wrong in our thinking about the external world that makes us feel strongly that it is external to us and not purely a product of our minds. Could it be that there is a middle way between naïve Platonism (Gödel’s view that we just do grasp abstract mathematical objects, which is why we can say true things about them) and the view that we grasp abstract concepts (which we do poorly as many a confusion attests)?
Quinean empiricism is defended here by Mark Colyvan, who observes that there is a lot to be said for the view that mathematics is justified by the success of science, in which it plays so great a part. One question then is: what philosophical questions are at least partly answered by that success, and which remain untouched? But before we get there, Colyvan picks up another much-discussed topic: Quine’s own opinion that mathematics uninvolved in science lacks that justification. Quine, like many a philosopher of mathematics, had in mind the esoterica of higher cardinals, but I suppose mathematicians might wonder if modern algebraic geometry falls in or out of the Quniean pale. Colyvan’s tentative answer is that a chain of necessities can be produced to legitimate even the unapplied parts of mathematics. Then he turns to his ongoing debate with Leng. She has argued elsewhere that mathematics only appears in scientific theories in its modelling capacity, and that is all that is then supported is the adequacy of the model not its ontological presuppositions. To this he replies that mathematics plays an ineradicable role in explaining some matters of science, and this returns mathematics to the status of indispensible for science.
Mathematical or scientific Platonism is discussed here by Alexander Paseau. He notes the influence of Quine on many present-day philosophers of mathematics, Colyvan among them, and he argues that their Platonism is a two-step process. First one argues that saying ‘2 is prime’ commits one to the existence of the number 2 — this is a realist view. Then one argues that objects in mathematics, such as ‘2’, are abstract: whence Platonism (Plato was a hardline realist). To this argument, Paseau says, you can reply pragmatically or with indifference. Pragmatically, one can say that mathematics is not so deeply implicated in science as to be confirmed by it. Or one can say, indifferently, that while mathematics is confirmed by science a Platonist philosophy of mathematics need not be. I would interject here a personal feeling that most mathematicians are Platonists (if indeed they are) simply because they have not considered any other philosophy of mathematics than Platonism. Paseau himself thinks that the pragmatic objection can be beaten back, but that the indifference argument is stronger.
These last two articles share a number of interesting concerns, among them probing the consensus that scientific practice vindicates Platonism. Another is the issue of mathematical explanation, on which there is a small literature, less than the studies of explanation in science. Another is mathematical practice, whether what mathematicians do offers material for the philosopher of mathematics. That it does is made abundantly clear by the essays in The Philosophy of Mathematical Practice (Mancosu et al, 2008). But the importance of mathematical practice is well illustrated here by a charming essay by the one mathematician present: Tim Gowers. Gowers discusses in some detail how mathematicians think about proofs and the kind of things that can happen when a proof is said to be a good one. As he notes, mathematicians rely on a number of concepts that a certain kind of philosopher of mathematics would view with alarm, such as ‘amusing’, beautiful’, ‘elegant’, ‘important’, ‘natural’, ‘technical’. He asks why this is the case, and his answer has to do with memory and with understanding. To select just one provocative suggestion of his, proofs can be measured by width (a term drawn from computer science), where width is an informal concept connected to the amount of storage needed in a proof. Small width proofs are easier to understand and, he suggests, therefore to be preferred to ones of large width.
The book contains a number of other articles that can only be mentioned here for reasons of space. Alan Baker (the philosopher, not the mathematician) writes on whether induction is a problem for mathematics and its philosophy. Marinella Cappelletti and Valeria Giardino jointly offer some thoughts on the cognitive basis of mathematical knowledge, a topic that will surely grow in importance. And the book ends with Crispin Wright’s article ‘On quantifying into predicate position’, a technical article with a strong anti-Quinean position in line with Wright’s important neo-logicist programme.
The book is a valuable introduction to current thinking in the philosophy of mathematics as it looks for a new direction. What needs to be added is an attention to the actual practice of mathematicians that matches the attention paid to the practice of scientists. For too long now philosophers of mathematics have tried to answer their fundamental questions about mathematics without looking with much interest at what mathematicians do, and have seemed oblivious of the many reflections mathematicians have made about what is involved in doing mathematics. If, as Quine suggested, philosophers should give up on looking for a first philosophy and instead appreciate the philosophical qualities of science, it might also be that philosophers could profitably fall in with mathematicians as well.
Mancosu, Paolo (ed.) 2008 The Philosophy of Mathematical Practice, Oxford University Press.
Jeremy Gray has taught at the Open University since 1974, where he is now a Professor of the history of mathematics. He is also an Honorary Professor at the University of Warwick, where he lectures on the history of mathematics. His most recent book, Plato's Ghost – The Modernist Transformation of Mathematics, was published by Princeton University Press in autumn 2008.