Paul Benacerraf is not the most prolific of contemporary philosophers. In a publishing career spanning 35 years, he has published not a single book and only a handful of articles. If he were working in a British university now, it would scarcely be enough to have him classified as "research active". But Adam Morton and Stephen Stich are not exaggerating (or, anyway, only a bit) when they claim Benacerraf's work on the philosophy of mathematics has "influenced every subsequent writer on the subject". Apart from the (not inconsiderable) influence he has had on the discipline through his teaching at Princeton, Benacerraf's huge reputation rests chiefly on two seminal articles - "What numbers could not be" (1965) and "Mathematical truth" (1975) - and a somewhat lesser known, though still widely discussed, paper he delivered to the Aristotelian Society in 1985 called "Skolem and the skeptic".
"What numbers could not be" posed a powerful challenge to Platonists - those who believe in the objective and external reality of mathematical entities - by presenting a devastatingly straightforward and apparently compelling argument that numbers could not possibly be sets (as Frege and Russell had held) or indeed any other kind of "object". The argument proceeds by showing that, faced with two different set-theoretic definitions of, for example, the number 3, nothing could compel us to choose one rather than the other - provided, that is, that under both definitions 3 was larger than 2, smaller than 4 and so on. In other words, as supposed "objects", numbers are entirely indeterminate; what matters about numbers is not what they are, but the relations in which they stand to one another.
In "Mathematical truth" the challenge to Platonists is renewed, only this time combined with an equally powerful challenge to anti-Platonists (Constructivists, Intuitionists and so on). In a taxonomy that has since become standard, Benacerraf distinguished two central problems in the philosophy of mathematics, problems that, he suggested, pulled the philosopher in two opposite directions. The first concerns the nature of mathematical truth, the second the nature of mathematical knowledge. Broadly speaking, Benacerraf argued that Platonists had a rather good answer to the question of what makes a mathematical proposition true (it corresponds to a fact in Platonic mathematical reality), but only a rather feeble answer to the question of how we can ever know it to be true (we "see" it). Conversely, anti-Platonists have no problem with the question of how we acquire mathematical knowledge (we are taught it, we make it up, and so on), but are at a loss to explain what a true mathematical proposition is true of.
"Skolem and the skeptic" is more technical than the other two, and apparently on a more narrowly focused topic. In it, Benacerraf pours cold water on those philosophers who have attempted to establish far-reaching metaphysical claims on the basis of a result in mathematical logic called the Lowenheim-Skolem theorem. This says that any finite, formal theory that has a model at all (ie, that is consistent) has a denumerable model, that is, one in which the elements of its domain can be put in a one-to-one correspondence with the natural numbers. This appears to some to be paradoxical because it seems to run counter to Cantor's proof that the set of real numbers is nondenumerable. And, according to the philosophers Benacerraf is concerned to challenge, this means that we have, after all, no clear concept of non-denumerable. Not so, says Benacerraf, for who is to say that the test of a concept's clarity is its ability to be "captured" in a formal theory?
For the most part, the articles in this excellent new collection present attempts to rise to the challenges set by these three papers. Together they show what difficult and far-reaching problems Benacerraf has raised, and what sophisticated thinking about the most fundamental questions in philosophy - questions of meaning, truth and knowledge - is needed even to begin to solve them. The degree of technical mastery required varies enormously. The papers by George Boolos and Richard Jeffrey - each intended, in different ways, to breathe new life into the question of the Frege/Russell definitions of numbers - are impenetrable to all but advanced students of mathematical logic. The more general papers on Platonism by Penelope Maddy and Steven Wagner, on the other hand, can be read with profit by those with only a smattering of logic, as can those on the language of mathematics by Richard Grandy and Adam Morton. Somewhere in between is "Skepticism about numbers and indeterminacy arguments" by Jerrold J. Katz, which offers a subtle analysis of Benacerraf's argument in "What numbers could not be", linking it to the issues raised by Quine's famous arguments for the "indeterminacy of translation".
A virtue of many of the papers is their attention to actual mathematical practice, though the lessons they draw from this vary widely, from Maddy's view that mathematical methodology provides some support for Platonism, to Morton's conclusion (supported by some instructive discussions of the kind of reasoning used in mathematics) that mathematical knowledge is, essentially, not knowledge that such-and-such is the case, but rather knowledge of how to do things: "We count, measure and subdivide, and mathematics tells us how to do this." More extreme than Morton are Richard Grandy's view that mathematics is "the epitome of story-telling" and Mark Steiner's defence of Wittgenstein's refusal to construct any theory of mathematical truth.
The book begins with a reappraisal by Benacerraf himself of his previous arguments. Called "What mathematical truth could not be", Benacerraf's contribution contains an engaging and enlightening "bit of amateur history" describing the philosophical microclimate in Princeton during the 1950s and 1960s, in which his views about mathematics were cultivated. In this, he emphasises the contrast between the "scientism" of American analytical philosophers and the more Wittgensteinian view characteristic of their British counterparts that regarded, for example, formal mathematical results as philosophically irrelevant. Within this contrast, however, Benacerraf himself seems now prepared to meet his British colleagues half-way. For the central theme of his essay is that formal results do not on their own have any philosophical implications: "The philosophical juice can be validly extracted only if a substantial portion of the yield is simultaneously supplied through the back door." Benacerraf then exposes the back-door supply of philosophical juice that lies hidden in some of the recent arguments for the philosophical implications of Godel's theorem and the Lowenheim-Skolem theorem. It is a characteristically brilliant piece of work and a fitting opening to a celebration of one of the most penetrating minds in contemporary philosophy.
Ray Monk is senior lecturer in philosophy, University of Southampton.
Benacerraf and his Critics
Editor - Adam Morton and Stephen P. Stich
ISBN - 0 631 19268 9
Publisher - Blackwell
Price - £40.00
Pages - 1
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