Review: [Untitled] Reviewed Work(s): Mathematical Truth by Paul Benacerraf Ontology and Mathematical Truth by Michael Jubien The Plight of the Platonist by Philip Kitcher W. D. Hart The Journal of Symbolic Logic, Vol. 52, No. 2. (Jun., 1987), pp. 552-554. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28198706%2952%3A2%3C552%3AMT%3E2.0.CO%3B2-U The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.
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THEJOURNAL or SYMBOLIC LOGIC Volume 52, Number 2, June 1987
REVIEWS The JOURNAL reviews selected books and articles in the field of symbolic logic. The Reviews Section is edited by J. Michael Dunn, Herbert B. Enderton, William A. Howard, Akihiro Kanamori, and Michael Makkai. In the selection of publications for review they are assisted by the Consulting Editors. Authors and publishers are requested to send, for review, copies of books to The Journal of Symbolic Logic, U.C.L.A., Los Angeles, California 90024. In a review, a reference "XLIII 148," for example, refers either to the publication reviewed on page 148 of volume 43 of the JOURNAL, o r to the review itself (which contains full bibliographical information for the reviewed publication). "XLIII 154" refers to one of the reviews or one of the publications reviewed or listed on page 154 of volume 43, with reliance on the context to show which one is meant. The reference "XLI 701(6)" is to the sixth item on page 701 of volume 41, i.e., to Russell's On denoting, and "XLVII 210(4)" refers to the fourth item on page 210 of volume 47, i.e., to Montague's Pragmatics. References such as 4910 or 2478 are to entries so numbered in A bibliography of symbolic logic (this vol. 1, pp. 121-218). Similar referencescontaining the fraction4 or a decimal point (such as 7011 JOURNAL, vol. 3, o r 3827.1) are to Additions and corrections to A bibliography of symbolic logic ( t h ~ sJOURNAL, pp. 178-212). PAULBENACERRAF.Mathematical truth. Thejournal of philosophy, vol. 70 (1973), pp. 661-679. MICHAEL JUBIEN. Ontology and mathematical truth. Noris, vol. 11 (1977), pp. 133-150. PHILIPKITCHER. The plight of the Platonist. Ibid., vol. 12 (1978), pp. 119-136. Mathematics elicits philosophy. In Mathematical truth, Paul Benacerraf articulates a basic philosophical problem about mathematics. It would seem to be a datum, which a philosopher could deny only with peril to there being something against which to test his philosophy of mathematics, that mathematics includes some known truths. Benacerraf's problem is that what seems necessary for truth in mathematics also seems to make mathematical knowledge impossible. His articulation of this dilemma, or antinomy, considerably illuminates a great deal of philosophy, and not only philosophy of mathematics. Consider truth. Correspondence accounts, which generally speaking persuade, link truth and denotation. Let us take sentences as at least ersatz truth vehicles. It is the core of a correspondence account of truth that sentences can be true only if some expressions denote; otherwise, words do not make the sort of contact with the world that truth requires. It is a virtue of Tarski's work on truth in formalized languages that it emphasizes the central place of the bound variables of quantification here, but the basic idea appeals even without formal elaboration. There is no empirical evidence that the adjective "true" predicated of sentences (in, say, mathematics o r other sciences) is ambiguous; and it is hard to understand how mathematics can have been so spectacularly successful when applied in natural science unless at least that much mathematics is no less true than, and in the same sense as, the rest of theoretical science. Indeed, it is implausible that theoretical natural science can be factored into a purely mathematical component and a purely natural component. But if some mathematics is true, and if truth requires reference, then there are mathematical objects, such as numbers, to be, for instance, values of the bound variables of the calculus. So far these objects are mathematical only in that the truth of mathematics requires them. But Frege argued that there will never be more than finitely many inscriptions of numerals. So if mathematics really says what it seems to say, namely and among other things, that there are infinitely many numbers, then the physical candidates favoured by some for reasons yet to be mentioned as (or in lieu of) values of the variables of analysis are too few. Moreover, if, as tradition has it, numbers exist necessarily if they exist at O 1987, Association for Symbolic Logic 0022-481 2/87/5202-0020/$03.30
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all, then since (as Hume argued) no physical object exists necessarily, no number is physical. Because mental things seem even more contingent, we seem forced to conclude that truth in mathematics requires that there be objects such as numbers, which are neither mental nor physical, and thus are abstract objects; mathematical truth issues into metaphysical Platonism. Mathematics itself is probably the best positive statement of the nature of abstract objects. But a negative axiom recommends itself: (very) abstract objects, such as numbers, are causally inert; they neither absorb nor emit energy. (The "very" is an attempt to skirt questions about the causal powers of geometrical objects.) That axiom is connected with the second, epistemological, horn of Benacerraf's antinomy. One pictures knowledge that p as an agreement between two explicably connected states, one of mind, the other of the world. A belief that p is knowledge only if the information expressed in that belief was obtained from whatever makes it true that p. The natural history of acquired information is its t'ransmission from the subject matter thereby known. (Since perception is the only familiar means of transmitting information that justifies belief, empiricism is everyone's favoured epistemology, even Plato's.) But the transmission of information (perhapsstructured energy)from subject matter to belief is a causal transaction between a person and (the rest of) the world. (All this is part of a very general project of naturalizing the mind by fitt~ngit into the causal nexus which is nature.) So if the subject matter of, say, analysis is very abstract, and thus causally inert, then no information can be transmitted from it to us, so we can acquire no knowledge of it. As promised, what seems required for mathematical truth seems also to make mathematical knowledge impossible. So long as both truth and knowledge are conceived as attaching to sentences pretty much one-by-one (though via orderings, one by complexity, and the other by justification), Benacerraf's dilemma remains acute; and even afterward, one worries whether a holisitic empiricism (via inference to the best explanation of what is observed even if the subject matter of the explanation cannot be experienced) coheres adequately with a molecular, inductive account of (satisfaction and) truth. Any serious pursuit is well served by an insightful articulation of its basic problems, and Benacerraf has unquestionably done exactly that for the philosophy of mathematics. Moreover, his antinomy makes sense of the otherwise odd array of philosophies of mathematics available; Platonists such as Frege and Giidel get a metaphysics that does justice to mathematical truth at considerably epistemic cost, while formalists, and intuitionists in their positive moods, achieve certain epistemic virtues while putting the infinite, the mathematician's paradise, at risk. Moreover, the pattern of Benacerraf's dilemma, an antinomy between a natural metaphysics and the natural epistemology, is hardly peculiar to the philosophy of mathematics. As only one example, belief that non-actual possible worlds are out there independent of us serves objective modal truth while making modal knowledge problematic. (The slogan that what matters is not objects, but objectivity, is a paradox; for only independently existing objects seem independent enough to make for objective truth.) In the first horn of his dilemma, Benacerraf argues that only abstract objects suffice for objective mathematical truth. Michael Jubien is not easy to understand, but he seems to see himself as, in part, blunting this horn by showing that not even (pure) abstract objects suffice to exposit mathematical truth. (He does not explain purity in general, but he counts among the pure the sets of the cumulative hierarchy without concrete urelements.) His argument seems to be that if we assume only that there are pure abstract objects, but not which, then neither acquaintance nor description suffice to isolate a pure model for, say, number theory. The argument is, in a way, plausible enough. After all, if we have only the predicate "is an object," how can we distinguish by acquaintance o r description objects of any one kind? As Wittgenstein asked, cannot the ostensive definition be misunderstood? But the upshot of such reflections might more plausibly be that we master notions such as those of number, abstract object, and mathematical truth together in a body, rather than not at all. As it were, why should one expect a description other than mathematics of the subject matter of mathematics? Of course, Jubien does not deny the data. Instead, he sketches a modal account of mathematical truth; various bits of mathematics are actually true if it is merely possible that there be various structures of concrete objects. However this sketch might be filled in, one worries about the analogue of Benacerraf's antinomy for modality. If mathematics is a matter of modality, and if mathematical truth is to be objective, then we seem pushed toward non-actual possibilia whose existence and nature is independent of anything we say o r do or nestle protectively in our warm little conceptual schemes. But non-actual possibilia are no less causally inert than actual abstracta. So by what sort of transaction could we acquire information about the structures of other possible worlds? O r if possible worlds are, say stipulated, then
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what is to prevent us from stipulating six impossible things before breakfast, thus putting objectivity at risk? Those who prefer the concrete should be no less wary of modality than of mathematics. In another justly famous essay (Thephilosophicalreview, vol. 74 (1965), pp. 47-73), Benacerraf argued that there is no good reason for claiming that the natural numbers really are, say, the sets in the domain of Zermelo's model for Dedekind's axiomatization of number theory, rather than, say, the sets in von Neumann's model, and that, consequently, numbers are not sets. Philip Kitcher develops this argument as an attack on metaphysical Platonism with respect to number theory; since it would be a sin against economy to believe in abstracta other than sets, Kitcher argues that numerals are not singular terms that denote abstract objects. (One does wonder whether practice might not be making it true that the natural numbers are the finite von Neumann ordinals.)The first two sections of Kitcher's essay defend the attack he extracts from Benacerraf against objections Kitcher develops from work by Mark Steiner, Nicholas White, Hartry Field, and others; the dialectic here is too complex to permit lucid summary in a brief review. In the final section, Kitcher sketches a non-Platonist account of mathematical truth; the axioms of number theory are actually true if it is merely possible for idealized versions of actions of collecting and correlating always to be performed. Kitcher, like Jubien, replaces actual abstracta with possible concreta. O r at any rate, Kitcher's possibilia are actions, and we are confident that mostly we know what we are actually doing, even if we are not sure how we know. Both Kitcher and Jubien drop abstracta in favour, first, of actualia to which we are confident of epistemic access; then the metaphysical pressure of the infinite prompts each to add many merely possible instances of those accessible actualia; but those possibilia, if independent of us enough to make for objective truth, seem no more epistemically accessible than the original abstracta. Kitcher's essay includes no reference to the paper by Benacerraf under review here, but thedates makeit seem fair to urge the modal analogue of Benacerraf's antinomy to Kitcher as well as to Jubien. W. D. HAKT EDWARI) L. KEENAN and L E ~ N A KM. I ) FALTZ. Boolean semantics for natural language. Synthese language library, vol. 23. D. Keidel Publishing Company, Dordrecht, Boston, and Lancaster, 1985, xii + 387 pp. Model-theoretic semantics for natural languages (such as English) uses the tools of mathematics and logic to model linguistic phenomena. The phenomena to be modeled are the judgments of entailment concerning (sets of) sentences in a natural language. For example, the fact that John wants an apple a n d a banana entails .lohn wants a banana is taken as something to be represented by a semantic theory. Like all modeling, when done successfully it leads to explanation and to the feeling that it is discovery and not merely invention. There are two principles which to one degree or another guide formal semantics. The first principle is that syntactic expressions are organized in categories which have some formal definition. The second is the Fregean principle of compositionality according to which the meaning of a complex expression is a function of the meanings of its subexpressions. Taken together, these principles lead to a view that corresponding to the syntactic categories there are semantic categories; each semantic category is a function space occurring in the type hierarchy arising from a set E of basic entities and the set of truth values. As Keenan and Faltr state, Boolean semantics for naturallanguage (BSNL) "situates itself squarely in the tradition of model theoretic semantics." In several ways, however, it differs from previous work, especially Montague grammar. The most important difference is that it isolates laws that should hold in the semantic spaces. The spaces are thus endowed with an algebraic structure. A secondary difference is that it changes theclass of ontological primitives from entities to properties. This shift is not essential with regard to the class of entailments modeled, so I will not dwell on it except to say that (extensional) properties are modeled by sets of basic entities. The structural property that the book isolates is the closure of many of the syntactic categories under Boolean combinations and the uniform interpretation of those combinations. For example, in the first paragraph above we find a noun phrase (NP) that is a Boolean combination of other NP's. The meaning of the whole is a function of the meanings of the conjuncts. For another example, consider John and Mary walked to the stadium hut not t o the beach. This sentence has a complex N P and a complex prepositional phrase (PP). By studying the inference patterns among such sentences, Keenan and Faltr are led to the conclusion that the semantic spaces for categories such as NP, PP, verb phrases (VP), and adjective phrases (AP) should be Boolean algebras. Other categories turn out to be interpreted by homomor-
