Review: [Untitled] Reviewed Work(s): Mathematical Knowledge by Mark Steiner W. D. Hart The Journal of Philosophy, Vol. 74, No. 2. (Feb., 1977), pp. 118-129. Stable URL: http://links.jstor.org/sici?sici=0022-362X%28197702%2974%3A2%3C118%3AMK%3E2.0.CO%3B2-6 The Journal of Philosophy is currently published by Journal of Philosophy, Inc..
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Mackie carefully reconstructs Hume's position, contending that Hume confuses two senses of necessity: (a) the distinguishing feature of causal sequences, and (b) the warrant for an a priori inference from cause to effect. Hence, his failure to find (a) rests illegitimately upon his failure to find (b). For, once distinguished, it is clear that the impossibility of finding (b) should not affect an independent search for (a). A third idea of necessity, to wit, the warrant for some, not necessarily a priori, inference from cause to effect, appears in Hume too, and, were it not for his doubts about induction, regularity would serve admirably for this sense. Mackie is highly critical of Kantian views on causation, arguing that neither Kant nor the partial reconstructions of the Kantian philosophy by Strawson or Bennett provide us with a deeper understanding of causation or an a priori justification of the use of the causal principle. BERNARD BEROFSKY
Columbia University
Mathematical Knowledge. MARK versity Press, 1975. 164 p. $9.50.
STEINER.
Ithaca, N.Y.: Cornell Uni-
During the decline of sense data and analyticity, epistemology a p peared to have lost that pride of place characteristic of post-critical philosophy and perhaps of modern philosophy in general. With the concomitant rise of semantics and the resurgence of ontology, epistemology seemed all but eclipsed. Frege's subversion triumphed; one almost felt more at home reading some ancients than most moderns. But even if epistemology deserved to be taken down a peg or two, it must remain a first-class citizen in the philosophical republic. The reason is evident. Some of the deepest problems of philosophy consist in reconciling natural but incompatible epistemologies and ontologies. Thus, for example, it is no accident that there is both a problem of other minds and a mind-body problem. But nowhere are such conflicts older than in the philosophy of mathematics. I t should be plain to a sympathetic reader of the Meno, the Symposium, and the middle books of the Republic that Plato strove heroically to find a plausible epistemology for his theory of forms. Platonism seems obvious when you are thinking about mathematical truth, but inlpossible when you are thinking
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about mathematical knowledge. And of course epistemology did not die in our century; it just changed. Causality, holism, and epistemology naturalized are supplanting sense data and analyticity. I t is thus both natural and welcome that the basic issues in the epistemology of mathematics should be rethought. Progress not just in mathematical logic but also in epistenlology itself makes this an intellectual duty. Mark Steiner's Mathematical Knowledge is aimed at doing that duty, and its aim is as timely as it is noble. The book falls into two parts. The first two chapters are a 69page discussion of logicism. T h e last two chapters are a 44-page discussion of the necessity of proof for mathematical knowledge, of platonism and causal theories of knowledge, and of mathematical intuition. There is also a fifteen-page appendix to chapter I which initiates a defense of Hilbert against one of Poincarb's criticisms. The thrust of chapter I is a conditional defense of logicist epistemology. Steiner writes, "There is no circularity in the reduction of arithmetic to set theory; if set theory is solidly based, a mathematician can in principle use it, to the exclusion of any other mathematical theory, in coming to know the rest of mathematics, including arithmetic." Steiner defends this thesis by attempting to rebut objections he credits to Poincark and to Wittgenstein. I n chapter 11 Steiner denies the antecedent of his conditional epistemic logicism. Set theory is not as solidly based as number theory; number theory is an epistemically autonymous science of objects. (Here Steiner denies Benacerraf's thesis that numbers are not objects.) The burden of chapter 1x1 is that proof is in no way a necessary condition of mathematical knowledge; so much for the phenomenology of doing ordinary mathematics. Chapter IV falls into two parts. The thesis of the first part is that "the most plausible version of the causal theory of knowledge admits Platonism, and the version most antagonistic to Platonism is implausible." I n the second part we are told that there exists a faculty of intuiting mathematical structures, and we are given some speculations as to the nature of this faculty. Whitehead said that all philosophy is a footnote to Plato; Steiner leads us back into the cave. T h e logical structure of chapter I has a byzantine complexity. (Steiner seems to prefer obiter dicta and dialectic for its own sake to disciplined organization. Then too, much as Wittgenstein's evasive style once had a pernicious influence on philosophical composition, Quine's truncated aphorisms seem to have exerted a deleterious effect on Steiner's prose.) The first principal objection to conditional epistemic logicism treated in chapter I is that the
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logicist cannot carry out his reduction of mathematics to set theory without presupposing induction, and thus cannot reduce number theory without circularity. Steiner examines a version of this objection considered by Charles Parsons.1 Parsons points out that, on the Frege-Russell strategy, we define each formal numeral explicitly i n set theory and we also define explicitly a predicate for "is a natural number." The question is how we know that the predicate is true of all and only the denotations of the numerals; Parsons points out that the natural answer is that we know it by induction "in the metalanguage," as the jargon has it. Steiner makes a few replies. First he assumes for no apparent reason that the mathematician "does not speak the metalanguage." He can articulate neither the question nor its answer, so he lacks no knowledge he needs to deduce number-theoretic results from set theory. This move reminds me of D. C. Williams' burlesque of Nelson Goodman: Goodman says that there are no sets because there are too many sets. Steiner says there is no missing knowledge because the missing knowledge is ineffable. Next Steiner denies that there is in set theory a concept of the infinite set whose members are 0, 1, etc., while admitting in the next paragraph that one must know that the sequence 0, 1, etc. never leads out of the natural numbers (and not just that the successor of every natural is a natural). Somehow object-language inductions on iterations of the successor function are to resolve this apparent contradiction. Then too Steiner dismisses Parsons' subtle thoughts on the difficulties implicit in quantifying over all sets with the observation that it is enough to have the notion of set and to grasp the logical operator "all." This logical-atomist oddity aside, Steiner does seem vaguely aware of the deeper problems; for he says a few pages later that "the use of universal quantification does not necessarily presuppose reference to a completed totality of items in the range of the bound variable." This will be astounding news for those familiar with conventional semantics, model theory, and truth theory. T o bear out the glad tidings, Steiner refers us to his appendix. There we are presented with an elusive distinction between two sorts of induction, a weak form in which the conclusion does not "presuppose a completed totality of numbers" and a strong form in which presumably it does. This seems more like mere repetition to me than like justification, except that Steiner's appendix is not about set theory at all; so it says absolutely nothing about how to quantify 1"Frege's Theory of Nu~nber," in Max Black, ed., Philosophy in America (Ithaca, N.Y.: Cornell, 1965), pp. 197-203.
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over all sets without assuming a set of all sets for the bound variables to range over. Finally Steiner cites Parsons' insight on Poincark to the effect that if we do have a priori assurance that there will be no conflict between construction in individual cases and inference from general propositions proved by induction, then this assurance is not founded on logic or set theory. Steiner is not sure what Parsons means by "conflict." If conflict is contradiction, Steiner's answer is that the mathematician is assumed to know his axioms, so since they are therefore true, no contradiction is possible. But knowing the axioms does not guarantee that the mathematician knows that his system is consistent; if he is to argue from knowing his axioms to consistency, the mathematician should know that he knows the axioms; for this sort of assurance, he seems to need something uncomfortably close to the sort of semantic knowledge Steiner thinks is unnecessary. If by "conflict" Parsons means that there might fail to exist a proof by finitely many uses of modus ponens (from the basis case) of an instance which follows by universal instantiation from a universally quantified theorem proved by induction, Steiner's answer is that one proof is enough. This reply is stupefying. Had I evidence of the discrepancy envisaged, I would be suspicious, not complacent. I might first suspect that my number theory was "about" a nonstandard model; that is, that an instance I can deduce by U I from a theorem proved by induction (but which I cannot prove by "finitely" many uses of modus ponens) contains a numeral that can only denote a nonstandard integer. Suppose I actually had an inscription of such an instance before me with a numeral written into it. If somehow I could not count off from this numeral the required number of uses of modus ponens (and then prove the instance that way), then perhaps I might suspect that there was a nonstandard number of occurrences of the successor sign in this numeral (assuming somehow that I use only standard informal arabic numerals to count). Surely Parsons is profoundly right that the assurance that none of this will happen is not founded on (first-order) logic or set theory. Any consistent number theory in standard formalization has nonstandard models; the assurance that nonstandard numbers from them are not used to specify the formalism's syntax (e.g., which sequences of signs are finite) is not given by the theory alone; for that we need a prior conception of which numbers are standard numbers. (Other suspicions might also be sensible; for example, for some reason Steiner does not consider the discrepancy which reverses that discussed above.) Steiner makes other points against Parsons no less impotent
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than the four considered in this paragraph; but it seems only merciful to award the laurels to Parsons without further ado. I n the second half of chapter I, Steiner considers a circularity objection against epistemic logicism culled from Wittgenstein. Part of Steiner's aim is to argue that Wittgenstein should be taken seriously. T o this end, Steiner reads Wittgenstein as arguing that we would need arithmetic to check a proof written in the primitive notation of Principia Mathernatica with lines twelve miles long. Steiner's basic reply is that we need not use only primitive notation. It seems that if this reply is adequate, Steiner has not shown that Wittgenstein deserves serious attention. In the first half of chapter 11, Steiner argues that, since "set theory is shakier than number theory," set theory cannot be used to justify number theory. There is an old argument for this view that Steiner ignores. Suppose that in expositing Principia, Bertrand Russell had "proved" that there are only finitely many primes. I t is claimed that this would show not that Euclid was wrong but that Russell's system was wrong. Thus Russell and Whitehead write in the preface to Principia MathematicaZ "the chief reason in favour of any theory on the principles of mathematics must always be inductive, i.e. it must lie in the fact that the theory in question enables us to deduce ordinary mathematics." This hypothetico-deductive justification of set theory by number theory is deeper than anything Steiner offers in defense of his view. I n the second half of chapter 11, Steiner criticizes Paul Benacerraf's article in T h e Philosophical Review for 1965.3 Before Benacerraf, Russell toyed with the idea that numbers are not objects but rather positions which any old objects might occupy in progressions satisfying the Dedekind-Peano postulates. (Russell thought that getting the applications of number theory right would decide in favor of some one progression,4 but Quine argues persuasively that this is not so.5 I have my doubts about "positionism" in general. It should require that the order type of a (countable) model for number theory determine (the rest of the structure of) the model. This requirement is in general false: Kemeny and Henkin proved that countable models for first-order number theory have just two order types, and Ehrenfeucht proved 2 Cambridge: University Press, 1910, p. v. a"What Numbers Could Not Be," Philosophical Review, LXXN, 1 Uanuary 1965): 47-73. 4Zntroduction to Mathematical Philosophy (London: Allen & Unwin, 1956), ch. r. 6 W. V. Quine, Ontological Relativity, and Other Essajs (New York: Columbia, 1969), p. 44.
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that there are continuum-many non-isomorphic countable models for number theory. (However, addition and multiplication are effective only in standard models,s and Benacerraf requires "recursive" progressions; so Benacerraf may be able to handle the present difficulty.) Steiner agrees with Benacerraf that "the numbers have no properties but those they have in relation to one another." Strictly speaking, this is an odd view; the number 9 is related by the numbering relation to the planets, by the solution relation to many equations, and by the reference relation to many people. Steiner denies Benacerraf's conclusion that numbers are not objects. These two views are an interesting metaphysical position; I wish Steiner had worked it out. (By the way, Steiner seems to identify positionism with if-then-ism 2 la Putnam and the Russell of the PrincipZes; this is probably a mistake. According to if-then-ism, each mathematical sentence S must be associated with a unique complete theory T. S is true if and only if it is true in all models for T. This thesis secures the truth of the axioms of ZF too cheaply. If, on this notion of truth, truth still implies syntactical consistency, then it follows by the completeness theorem that ZF has a model. That is, if-then-ism has some of the ontological commitments it was expressly designed to avoid. On the other hand, positionism does not seem to incorporate explicit theses about truth. Moreover, i t is clear how if-then-ism intends to avoid ontological commitments, namely, by including sentences which, asserted, would express such commitments only in conditionals. But insofar as a good platonist will take "positions" as objects, it is not clear how positionism is even intended to avoid ontological commitments.) The thesis of chapter 111 is that proof is not necessary for mathematical knowledge; here Steiner makes his case. The ganglion of Mathematical Knowledge is the first half of chapter rv. Steiner's thesis is that the most plausible version of the causal theory of knowledge actually requires platonism and that "the" version most hostile to platonism is implausible. T h e first conjunct is the shocker. Steiner's argument is that the most plausible version of the causal theory requires that a sentence that says that p occur within any causal explanation of how a person knows that p. Echoing Quine's indispensability thesis, Steiner says that all explanatory theories contain number theory and analysis as subtheories. But any sentence in any explanation is true, and the platonist theory of mathematical truth due to Tarski is the only 6Paul J. Cohen, Set Theory and the Continuufn Eiypothesis (Men10 Park, Calif.: W. A. Benjamin, 1966), pp. 48/9.
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satisfactory theory of truth. Hence, "the axioms of analysis, as interpreted by the Platonist, will indeed necessarily be used in whatever causal explanation can be given of our belief that the axioms, as interpreted by the Platonist, are true." Steiner takes a metalinguistic view of explanation. I suspect that he has Hempel's deductive-nomological model in mind: All explanations are deductions from true premises. Alonzo Church argues persuasively that all deductions are finite.7 But first-order number theory and analysis have infinitely many axioms. Therefore, no causal explanation of my belief that there do not exist four perfect squares in arithmetic progression can contain all axioms of number theory and analysis. (Elsewhere Steiner says, "For if N be an arithmetic truth, certainly it will appear in any explanation of any perception." On any view, there are infinitely many arithmetic truths,s even in second-order number theory; so Steiner's view requires that all explanations of all perceptions be infinitely large.) If we insist, as we should, that deductions be finite, then, although some axioms of number theory and analysis will probably occur in a correct and complete explanation of my believing that there do not exist four perfect squares in arithmetic progression, there is no a priori guarantee that this very theorem itself will occur in that explanation. Quine's indispensability thesis does not by itself trivialize causal theories of knowledge construed in terms of explanation. I think it would require heavy natural science to show that the law of quadratic reciprocity must be stated in the explanation of my belief in that law; doing this heavy science would be the opposite of trivializing the causal theory of knowledge, and would probably go a long way toward a deep and honorable reconciliation of platonism and epistemology naturalized. What after all is the problem? It has some recent roots. Edmund Gettier showed that justified true belief is not always sufficient for knowledge. Some thought, and for good reason, that a causal clause might handle Gettier counterexamples.9 But it is a metaphysical axiom that natural numbers are causally inert. So a causal condition on mathematical conditions seems unsatisfiable. This is the shape of the recent problem of platonism and the causal theory of knowledge. But there lies behind it a much older problem. Con7 Introduction to Mathematical Logic (Princeton, N.J.: University Press, 1956), pp. 51-54. a Thus, Kleene's result that any axiomatic theory in standard formalization with finitely many extra-logical primitives is finitely axiomatizable by adding new primitives is not relevant. a See, for example. Alvin I. Goldman, "A Causal Theory of Knowing," this JOURNAL,LXIV, 12 uune 22, 1967): 357-372.
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sider: Empiricism is the only real theory of knowledge; all the rest are nonstarters. Despite the failure of nerve by most classical empiricists, empiricism is the doctrine that all knowledge is a posteriori (analyticity and so forth were confused ideas). All a posteriori knowledge is justified ultimately by experience. As H. P. Grice argues,1° experience necessarily requires causal interaction with the objects experienced. But mathematical objects are necessarily causally inert. So platonism seems incompatible with empiricism. Yet platonism is the only adequate theory of mathematical truth. So platonism and empiricism also seem separately undeniable. This is the ancient form of the problem. I think Plato himself was aware of it, and I shall always be grateful to Paul Benacerraf for teaching it to me. I have tried to argue elsewhere that Quine's holism gives us a way into a reconciliation. But the problem abides. Quine's epistemology naturalized is the legitimate heir to empiricism: scientific knowledge (which includes mathematics) is a natural phenomenon that must be accounted for by science. Granted just conservation of energy, then, whatever your views on the mindbody problem, you must not deny that when you learn something about an object, there is a change in you. Granted conservation of energy, such a change can be accounted for only by some sort of transmission of energy from, ultimately, your environment to, at least proximately, your brain. And I d o not see how what you learned about that object can be about that object (rather than some other) unless at least part of the energy that changed your state came from that object. I t is all very well to point out that the best and (thus) true explanation of our state changes in learning probably requires the postulation of objects, like numbers, which cannot emit energy but about which we nevertheless have beliefs. For this still leaves unexplained how our beliefs could be about energetically inert objects. As Richard Grandy put it,ll i t is not clear that platonism is compatible even with de re mathematical belief. Platonists like me may need a "physics" for abstract objects, a metaphysics if you will. I agree with Steiner that number theory is the natural history of the natural numbers. T h e trouble is that we d o not understand what is natural about this history. And it is a crime against the intellect to try to mask the problem of naturalizing the epistemology of mathematics with philosophical razzle-dazzle. Superficial worries about the intellectual hygiene of 10 "The Causal Theory of Perception," in Robert J. Schwartz, ed., Perceiving, Sensing and Knowing (Garden City, N.Y.: Anchor, 1965), pp. 438-472. 11"Reference, Meaning, and Belief," this JOURNAL, LXX, 14 (Aug. 16, 1973): 489-452, p. 446.
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causal theories of knowledge are irrelevant to and misleading from this problem, for the problem is not so much about causality as about the very possibility of natural knowledge of abstract objects. Can we sketch even a fragment of a naturalized, epistemology lor mathematics? Since at least "Natural Kinds,"l2 Quine has urged that evolutionary ideas play a role in naturalizing epistemology.13 Can they play a role in naturalizing the epistemology of mathematics? Consider belief-forming animals. Fiist, a species whose members' beliefs are more likely to be true than another's has ceteris paribus a greater chance to survive; for action based on true belief is more likely ceteris paribus to achieve its ends than action based on false belief. Second, a species whose members' beliefs are typically simpler than another's has ceteris paribus a greater chance to survive. I t is easier ceteris paribzss to act on a simple belief than on a complex one; and action based on simple belief is more likely ceteris fiaribus to be economical and efficient in achieving its end? than action based on complex belief, and the more efficiently a creature acts, the better its chances ceteris paribus of surviving in the long run. Third, the better creatures can explain themselves and their environments, the more likely they are to survive. For being able to explain things (knowing how they work) makes possible manipulation of the environment (at least frequently), and a creature that can change its environment to suit its ends has ceteris paribus less chance of being done in by that environment and more chance of expanding its territory. Putting the three points together, since we are belief-fonning creatures and we have survived so far, it is natural that we should have simple, true explanations of how things about us work. This construction is confirmed by the existence of technology. Moreover, ceteris paribus, the survival advantage of simple, true explanation is proportional to the range and quantity of items explained (so long as giving explanations does not take u p so much time as to exclude other activities that promote survival) ; so it is natural that the drive to explain should tend to generalize. Now let us combine these evolutionary ideas with Quine's indispensability thesis: beyond a quite primitive level, simple true explanation (that is, science) is impossible without mathematics. I-Ience it is natural that we have mathematical knowledge. I t also stands to reason that the earliest branches of mathematical knowl12 In N. Keschcr, ed., Evays in Honor of Carl G. Hcmpel (New Yolk: Humanities, 1970). 1 3 See PP. 126 ff in Ontological Relativity and Other Essays.
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edge-geometry and arithmetic-are useful in solving immediate, day-to-day practical problems. Some intermediate branches of mathematical knowledge serve to reduce a generalized drive to explain physical phenomena not all or always of immediate or practical importance for survival, though it confirms my construction that many eventually did turn out to be of practical importance for survival; here I have in mind the development of the calculus for use in physics and astronomy. Now consider some abstract modern branches of mathematics like topology and algebra. Since it is natural for the drive to explain to tend to generalize, it is not surprising that it might generalize in a reflexive search for, as it were, second-level mathematical explanation (organization, unification, extension, simplification) of earlier first-level, more concrete and practical mathematics, by appeal to more abstract and theoretical mathematics. Here I have in mind the impact of algebra on number theory and geometry and the impact of ;apology on geometry and analysis. I n a curious way, if this construction is correct, it shows that Quine's indispensability thesis is crucial to one naturalistic explanation of mathematical knowledge, just as Steiner suggests. But it is equally important to note that in this construction Quine's thesis plays nothing like the role it played in Steiner's argument. In particular, nothing in this construction makes it plausible that the prime-number theorem must itself be stated in any correct naturalistic explanation of, say, Artin's belief in that theorem. For this reason I think that the present construction, even if it is correct, cannot be all there is to the natural science of mathematical belief. But perhaps it can inform further speculation. T h e second half of chapter IV is one of the few attempts to treat mathematical intuition seriously. I t is a difficult but important topic; Steiner deserves praise for his courage in taking u p a subject usually dismissed with ignorant contempt. I found his discussion hard to follow; here are, I think, some of its main points. First, Steiner denies that there is any naturalistic link between an intuition and its subject matter; he writes, "Intuition, of course, is not considered to be initiated from without but self-induced." I n the light of my remarks above, but not Steiner's explicit doctrine, one might thus expect intuition to have little to do with justification. Surprisingly enough, this seems to be Steiner's view. H e says that to speak of intuition is to offer a promissory note that a certain sort of explanation exists for a true mathematical belief; note that this is not to state such an explanation and thus perhaps justify
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that belief. Earlier Steiner said that "the ultimate court of appeal" for mathematical knowledge is "holistic considerations," not intuition, and that holism grounds the indubitability of mathematics. But, he says, such considerations obviously play no role in the mind of the average mathematician, and they do not explain how human beings come to know mathematical truths, though they justify them ex post facto. I t seems to me that Steiner thus places intuition squarely in the context of discovery, not the context of justification. I am not sure he accepts this old distinction, nor am I sure it is a legitimate cousin of the illegitimate analytic/synthetic distinction; but it seems like an obvious way to try to organize Steiner's views. I shall let Steiner speak for himself on the nature of intuition. H e writes, "One imagines or looks at material bodies, and then diverts one's attention from their concrete spatial arrangement. One gathers u p in one's mind the objects into a manifold, and then has an intuition of their structure. This is how one might become familiar with the standard model of ZF set theory-by abstracting from dots on a blackboard arranged in a certain way." Later he continues, "Once one has abstracted from the geometric features of the array, the mental state is qualitatively different from simple imagining." He does not tell us how it is different. Perhaps you can judge for yourself whether Steiner achieves his declared objective; he says, "I have tried to dissolve some of the main confusions regarding mathematical intuition, so that its empirical study can become possible." One of the things that stluck me about Alathematical K7zowledg.e is its neglect of two central issues in the epistemology of mathematics. At first blush, it is astonishing that in a book so short no room was made for a direct discussion of either a priori knowledge of mathematical truth or knowledge of a mathematical truth that it is necessary. But on reflection it is clear that both omissions stem from Steiner's reluctance to take seriously the problem of naturalizing the epistemology of mathematics. Consider a priori knowledge first. Once analyticity and company drop out of mathematical epistemology, the claim that mathematical knowledge is a priori becomes the purely negative claim that mathematical knowledge is not justified through natural intercourse with objects. This makes Kant's question, How is such knowledge possible? quite acute. I think we have to face the music: there just is n o a priori knowledge (at least as traditionally characterized). Now consider the knowledge that a mathematical truth is necessarily true. (I am making the controversial assumption that we have such knowledge.)
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Kant had a good intuition here; he said that experience teaches us that a thing is so and so, but not that it cannot be otherwise. T h e point is that it is just as hard to see how to naturalize knowledge of necessity and essence as it is to see how to naturalize knowledge of abstract objects. I suspect that knowledge of necessity can be justified by appeal to the imagination. (This is a thesis about knowledge of necessity, not about the nature of necessity.) My suspicion is compatible with Kant's intuition, since imagining an object is not experiencing it; but it may also naturali~eknowledge of necessity, since imagining an object-especially de re imagining of it-should be a natural phenomenon linked naturally to it. Knowledge justified by the exercise of imagination fits a liberal epistemology naturalized; but, since imagination is not experience, it may also deserve a refurbished title to a priori knowledge. Of course, problems remain about knowledge of necessities of abstract objects, as in mathematics; but I shall try to develop my views elsewhere. For now, the point is that Steiner says next to nothing about a priori knowledge and knowledge of necessity and that he does so because he does not take seriously the problem of naturalizing epistemology. Afathematicnl K~zozuleclge takes u p many of the issues in the epistemology of mathematics which should be treated these days. Steiner's work has that honest creative courage which stimulates philosophical thinking arid which kindles hope for philosophical insight. I t would be a great shame if we were to be denied axess to future philosophical work by Adark Steiner. \V. D. HART
University College, London
Convention: A Philosophical Study. DAVID K. LEWIS. Cambridge, Mass.: Harvard Univer~ityPress, 1969. xii, 213 p. $9.00. I t has been a philosopher's platitude that language use is conventional." I n this book and in a recent paper which suggests some modifications, David Lewis draws on various concepts from fields as diverse as game theory and modal logic in order to give some
'
I am indebted to Richard Warner and David K. Lewis for helpful discussions of Convention and convention; Arthur Kuflik and Dale Jamieson pointed out errors in an earlier draft. 1"Languages and Language," in Minnesota Studies i n tire Philosophy of Science, University of Minnesota Press, forthcoming.
