Share on mail 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". 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.
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Philosophers and their Critics 8. Cambridge, MA: Blackwell Publishers, The complete bibliography included in this volume shows only eleven articles, together with his doctoral thesis on logicism and the two versions of the introduction to the excellent anthology in the philosophy of mathematics that he edited with Hilary Putnam.
Most of the articles in this volume comment on one or both of these two famous essays. The first essay, WNCNB, argues that natural numbers cannot be sets, since incompatible versions of the number sequence can be defined in set theory such as the versions of Von Neumann and Zermelo. It goes on to argue that num- bers can be any collection of things that satisfy an appropriate set of conditions. Jerrold J. Katz contributes an extended article on this essay.
We are given something X that is allegedly picked out by some theory or practice. Then it is argued that some other thing Y fits the bill just as well. For such arguments to work, X and Y have to be manifestly different. The second essay, MT, is the subject of four of the essays in the volume un- der review.
The argument in MT is simple. Benacerraf argues that accounts of mathematical truth suffer from a basic problem of incompatibility.
Accounts of mathematical truth are motivated by the desire to have a homogeneous seman- tical theory in which the semantics of mathematical propositions parallel those for the rest of language, and also by the desire for a mathematical epistemology. Platonists could reasonably take the view that if there is a conflict between platonistic mathematics and epistemology based on the causal theory of knowledge, then it is the latter which should give way.
Consider an EPR-type set up. A source sends two photons as the result of a decay process in opposite directions to remote detectors. The measurement results are correlated: if one has spin- up, the other has spin-down.
Standing at one wing of the apparatus, and on measuring getting the result, say, spin-up, we can know that the other has spin-down. Is there any causal connection between us and the remote photon? What about a connection via the source? It might be thought that this line of reasoning assumes a crude version of the causal theory of knowledge. Not so. No matter how sophisticated the causal story becomes, so long as any causal connection is required, it will run into this problem.
The upshot is simple: We know that the distant photon has spin-down, but we are not causally connected to it. So, the causal theory of knowledge is simply false. Platonists need fear it not. The classical continuum is not clearly required by fluid mechanics since it is an idealization rather than a direct representation of the substances involved. Continuity of space-time does not seem to have a clear physical meaning.
Maddy concludes that Platonists like Steiner, Resnik and herself have lost their central argument for the truth of mathematics. Broadly, three types of arguments for Platonism have been prevalent. We have little sympathy with this line of argument ourselves — much preferring the first two considerations.
However, the objections that Maddy raises against the indispensability argument seem off target. Answer: 2. In fact, a discrete measuring rod is completely adequate for all science. Maddy and others take examples like this to show that much of the mathematics used involves idealization, not accuracy. But not all mathematics 2 attaches in so direct a fashion. This is fairly straight-forward.
But the mathematics of quantum mechanics hooks on to the world at a much more abstract level. Steven J. Unlike Maddy, Wagner is not much impressed with the indispensability argument.
His hopes for justifying Platonism rest on the semantic argument: systematic accounts of truth and validity must treat predicates and logical op- erators as denoting various abstract objects.
Robert Stalnaker is worried about the causal theory of knowledge as it ap- plies to possible worlds. David Lewis thinks that there are infinitely many parallel universes spatially and temporally disconnected from each other. But if that is so, how can we know about these alternative universes? Two essays on logic stand somewhat apart from the other contributions, since they are fairly formal.
Richard Grandy thinks that mathematics is a collection of stories that we are telling ourselves. The concept of truth in number theory is just part of the story about numbers that we are telling — there is no external thing with which we are comparing our mathematical statements.
Grandy avoids the perplexities of WNCNB by the principle: numbers only have the properties we construct them to have. How can Kripke be correct in claiming that there are sceptical arguments in Wittgenstein since Wittgenstein pooh-poohs scepticism? According to Steiner, Wittgenstein does 3 use sceptical arguments but just to demolish certain academic philosophies. So, both Kripke and his critics are right. Adam Morton is concerned with the process of understanding mathematical language.
Parts of his article are written in terms of bizarre jargon adapted from computer languages. The essay by John Earman and John D. Their entertain- ing article shows by a mixture of simple physics and conceptual analysis that the arguments purporting to show incoherence in the very notion of a supertask rest either on fallacious reasoning or indefensible assumptions. On the other hand, they conclude that the traditional super-tasks no longer throw light on concepts of the infinite and continuity.
However, they have higher hopes for a new variety of super-tasks. It is a pity that the editors did not emulate the Schilpp volumes by including some commentary by Benacerraf on each of the essays — this would have fulfilled the stated goals of the series Philosophers and their Critics to which this volume belongs.
Instead, the collection begins with an article by Benacerraf himself, written in a facetious style. The first part contains some entertaining reminiscences about the background to WNCNB, the second part makes the point that to draw philosophical conclusions from mathematical and logical theorems, some extra premisses are needed.
All in all, this is a rather disappointing collection. Most of the articles cover well-trodden ground without much in the way of novelty.
Though the essays are largely concerned with the philosophy of mathematics, only three or four of them contain anything mathematical. Perhaps the great days of the philosophy of mathematics, when logicians, mathematicians and philosophers contributed to the lively debates of the first half of this century, are gone for good. The scholastic debates of the present volume do not offer much hope of a revival.
Benacerraf and his Critics
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It all adds up to a powerful figure