This is an interesting, important, ambitious, and infuriating book, one that deserves both attention and response from the mathematics community. It has many good things to say, and it has the ambition to reshape the debate on the philosophy of mathematics.
There was a time, early in the twentieth century, in which mathematicians were passionately interested in the philosophy of mathematics. People were deeply concerned about what mathematics is, what sort of existence mathematical objects have, and their opinions on these questions actually influenced the mathematics they did.
There were three major points of view in the debate about the nature of mathematics. The formalists argued (roughly: the short summaries that follow are really caricatures) that mathematics was really simply the formal manipulation of symbols based on arbitrarily-chosen axioms. The Platonists saw mathematics as almost an experimental science, studying objects that really exist (in some sense), though they clearly don't exist in a physical or material sense. The intuitionists had the most radical point of view; essentially, they saw all mathematics as a human creation and therefore as essentially finite. The intuitionists refused to have any dealings with completed infinite sets, rejected the "law of the excluded middle" (i.e., the claim that a mathematical statement is always either true or false), and were willing to give up large tracts of classical analysis that didn't fit this point of view.
The fact that the debate never really got resolved, together with the complicating factor of Gödel's incompleteness theorems, seem to have caused most mathematicians to lose interest. In recent years, most mathematicians seem to have been content with an attitude best described by Jean Dieudonné. In everyday life, we speak as Platonists, treating the objects of our study as real things that exist independently of human thought. If challenged on this, however, we retreat to some sort of formalism, arguing that in fact we are just pushing symbols around without making any metaphysical claims. Most of all, however, we want to do mathematics rather than argue about what it actually is. We're content to leave that to the philosophers.
Reuben Hersh wants to change this. His book is an attempt to get us all involved in the debate about the nature of mathematics. To this end, he does a number of things. First, he argues that most writing on the foundations of mathematics is woefully ignorant of actual mathematical practice. Second, he tries to break the three-way tie by making a new proposal as to what mathematics really is. Third, he runs through the history of the philosophy of mathematics to argue that (a) his position is not really new, but has a distinguished pedigree, and (b) that all the other positions are clearly wrong. Finally, he connects philosophical positions on the nature of mathematics to broader philosophical and political issues.
The foundational debates of the early twentieth century turned on the issue of certainty. Everyone agreed that mathematical statements were true in an absolute sense, that one could be certain that they were true. The issue was to find a philosophy of mathematics that guaranteed that certainty. Hersh's position is that the desire for certainty is simply a mistake. In fact, he argues, regardless of our ideals, mathematics is done by fallible people, and so the traditional philosophies cannot really guarantee certainty. So let's give it up: mathematics is a human endeavor, and mathematical truths are uncertain like any other truths.
But if we locate mathematics as a human construction, we need to account for the very strong feeling that mathematical objects have some sort of independent existence. The number pi (or the number two) is not just something in my head! Hersh agrees, and proposes that mathematical "objects" are really socio-cultural constructs. As such, they really do transcend individual minds even while remaining human creations. Hersh calls this view of mathematics humanism.
So far so good: it's an interesting proposal, and has much in its favor. It makes modest claims for mathematics which actually correspond to our human experience as mathematicians, and it takes seriously the fact that mathematics is learned and taught. (It is also, which Hersh does not observe, completely compatible with a Platonist understanding of mathematical objects; all that one needs to do is to delete the (implicit) "merely" in Hersh's claim that mathematical objects are socio-cultural constructs. Our mathematical concepts can certainly be socio-cultural constructs that attempt to grasp and understand Platonist mathematical objects...) The humanist view allows us to escape from metaphysics and to focus attention on things we can actually observe first-hand: the mathematical community and how it learns, teaches, and develops mathematics.
There are other persuasive things about "humanism." For example, it is deeply aware that mathematics has a history. Both formalism and Platonism often give the impression that they deal with mathematics as a completed product, when in fact mathematics is produced by people working in socio-cultural contexts. A look at the history of mathematics certainly seems to undermine a naive formalism (since "formal proof" is a relatively recent phenomenon). History also asks difficult questions for Platonism; does it really make sense to claim that, say, Galois groups "really exist", and that they existed even in Euclid's time?
On the other hand, the humanist view reduces our feeling that mathematical truths are really true to a social consensus, something we learn. This opens the possibility that Little Green People from Mars, if they exist, have a mathematics that is not only different from, but actually contradictory to ours. This is a position that many mathematicians find extremely hard to take.
One should emphasize that the evidence for some sort of "certainty" in mathematical truths goes beyond our intuitive feelings about them. The history of mathematics also points in this direction. We may no longer accept some of Euclid's arguments as rigorous, but we do think every one of his theorems is still true! What other science can make such a claim? Or consider Fermat's Last Theorem: is there any other field of human endeavor in which a question posed in 1636 can still make sense, in exactly the original terms, 350 years later?
In arguing for his humanist philosophy of mathematics, Hersh has some very good points to make, but he seems to spend much more time attacking the rival positions of formalism and Platonism. Against formalism, he follows and develops the criticisms of Imre Lakatos: he argues that no one actually writes formal proofs of anything, and that the view that mathematics is simply meaningless symbol-pushing is impossible to believe. Against Platonism, he argues that believing in eternal mathematical objects existing independently of human thought is only possible if one believes that God exists, which, he says, no one does anymore.
While there is something to both arguments, neither of them really takes the opposing philosophical position seriously. Formalism is more solid than Hersh makes it seem, Platonism has been the position of serious philosophers who were not theists, and, after all, many philosophers do believe that God exists. So do many mathematicians. This points up the basic problem: all too often, Hersh is willing to dismiss the opinions of eminent thinkers after only a very shallow interaction with their thought.
As a result, Hersh's historical survey of the philosophy of mathematics is, to my mind, the weakest part of the book. The quick sketches of the thought of various philosophers really do justice to no one. (They remind me of John Rist's description of Bertrand Russell's History of Western Philosophy: "sophistic and simplistic misinterpretations of most Western ethics and metaphysics.") An example is his treatment of George Berkeley's very complex philosophy of mathematics, which gets reduced to "using the deficiency of mathematics to support religion." He then adds "His attack on mathematicians is unique since St. Augustine," which is hard to understand, since in his section on St. Augustine Hersh explains that Augustine's "attack on mathematicians" is really nothing of the sort!
Even less convincing is Hersh's attempt, in the last chapter, to correlate philosophers' positions on mathematics with their political stances, in order to reach the conclusion that "the Platonist view of number is associated with political conservatism, and the humanist view of number with democratic politics." Aside from the fact that he takes as axiomatic that being associated with the latter is better, the whole argument is based on a painfully arbitrary distinction of left versus right. (David Hume as a leftist?!).
The book concludes with an invocation of the story of the blind men and the elephant "as a metaphor for the philosophy of mathematics, with its Wise Men groping at the wondrous beast, Mathematics." All except, it seems, the humanist, who can see the whole elephant and laugh at the fallibility of all the others. Hersh doesn't seem to realize how arrogant this attitude is.
What I think is missing from the book is a realization that the philosophy of mathematics is indeed philosophy, and not science, and that therefore it cannot ignore the overarching philosophical issues that relate to it. It is clear that one's beliefs on metaphysics and epistemology, and particularly one's stance with respect to the notion of truth, are going to have an enormous impact on one's positions on the nature of mathematical objects and mathematical truths.
However, despite the serious limitations of Hersh's treatment of other thinkers, and despite the fact that he does not argue for his "humanist" position as forcefully as he might have, this is still a book that deserves attention. It should be widely read, discussed, and argued with. I hope it stimulates many responses, and most of all that it manages to convince more mathematicians of the importance of the questions with which it deals.
Fernando Gouvêa (firstname.lastname@example.org) is chair of the Department of Mathematics and Computer Science at Colby College and editor of MAA Online.