Semantical Rules and Misinterpretations: Reply to R. M. Martin Hilary Putnam Philosophy and Phenomenological Research, Vol. 42, No. 4. (Jun., 1982), pp. 604-609. Stable URL: http://links.jstor.org/sici?sici=0031-8205%28198206%2942%3A4%3C604%3ASRAMRT%3E2.0.CO%3B2-C Philosophy and Phenomenological Research is currently published by International Phenomenological Society.
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DISCUSSION
SEMANTICAL RULES AND MISINTERPRETATIONS:
REPLY T O R. M. MARTIN
Following Quine and (on this point) Wittgenstein, I take it that the notion of a "rule" cuts no philosophical ice when it comes to accounting for our understanding of our first language. It may have a use in talking about formalized languages; but then all the problems reappear when one takes a formalized language to be a model of a natural language or of unformalized scientific practice.' Professor Martin writes as if Quine and Wittgenstein had never written. He and I thus inhabit different philosophical worlds. It is not surprising that by his standards everything I say is wrong, u n ~ l e a rand , ~ unscholarly. Rather than trying to say it all again, I shall examine in detail Professor Martin's criticism of one of my articles ("Truth and Necessity in Mathematicsw-here abbreviated TNM), and just indicate what the response would be in the case of one or two of the remaining articles. In the first part of TNM I show that part of mathematics can actually be done in a nominalistic language. Hence reference to abstract entities is not a necessary property of mathematical statements. Professor Martin registers two objections: (1) The nominalistic statements I produce are "logically true," but there is no reason to call them "mathematics"; and (2) They do not have the same meaning as the sentences of ES and, thus, I was wrong to speak of them as "translations." Speaking to (2) first: the assertion that my nominalistic statements which are mathematically true are "translations" of the 'See my "The Analytic and the Synthetic," in Mirzd, Larzguage, arzd R e a l i t y , Philosophical Papers, vol. I1 (Cambridge: Cambridge University Press, 1975). 2Professor Martin's idea of what clarity requires is unusually demanding! I define a n a priori truth as one it could never be ratlorzal to give u p , or even to doubt. This, Professor Martin says, "is not up to sophisticated contemporary standards." (At the same time, it is a "straw m a n easily to be knocked down.") It is not clear whether it is the use of the notion of rational revisability or the use of the modal notion ("could") that puts me below "sophisticated contemporary standards."
theorems of ES occurs only in Professor Martin's imagination. I nowhere speak of them as "translations." My argument was that these sentences can be mathematically proved without using anything nonnominalistic (even the induction principle needed can be nominalistically stated), that they express what it is reasonable to regard as mathematical facts, and that the activity of proving them in the mathematical way described is a characteristically mathematic activity. Professor Martin disagrees. He would rather refer to these truths as logical truths. But their proof requires induction, and however fuzzy the line between logic and mathematics, if one is going to insist on a line at all, then what needs induction for its proof would seem clearly to be on the "mathematics" side of the line. The second part of TNM criticized Wittgensteinian views (I referred to them coyly as "a currently fashionable view"-TNM was written in 1963). Professor Martin was misled into thinking Carnap was my target, although I speak of "a late refinement" of the CarnapAyer view, which I briefly criticize. Let me now quote Professor Martin: Putnam writes that "according to a currently fashionable view, to accept a statement as mathematically true is in some never-quite-clearlyexplained way a matter of accepting a rule of language . . . " This view is ascribed to Carnap who is said to maintain that "mathematical statements are consequences of 'semantical rules'." Of course, if suitable semantical rules are available, the relevant language is obviously a n interpreted one. And if it is a language for mathematics, the semantical rules are such as to "interpret" the mathematical expressions occurring. Interpreted mathematical statements are not then logical consequences of the semantical rules alone, but only of them together with the specifically mathematical assumptions of the language. T h e Carnap view is thus said to consist of a n "uninteresting tautology," in effect, that "mathematics = language plus mathematics. " A "crushing rejoinder" to this view, Putnam says, is that is denies that "there is any such thing as Discovery in mathematics" other than in a purely psychological sense.
A glance at TNM will verify that what I referred to as a "crushing rejoinder" to the Carnap-Ayer position is that consequence, in Carnap's sense, is itself a mathematical notion. Thus the statement that "Gue mathematical statements are consequences of semantical rules" does reduce to the uninteresting tautology-uninteresting to an epistemologist, that is-that mathematics = language plus mathematics. I did remark that the Carnap-Ayer position goes along with the denial that there is any such thing as discovery in mathematics, other
than in a purely psychological sense. Carnap and Ayer redefine "information" so that mathematically equivalent propositions are said to give the same "information." What I say in TNM is that this is a Pickwickian sense of "information," and that when Carnap and Ayer said that the conclusion of a valid mathematical argument contains no new "information," they "only enunciated the uninteresting tautology that in a valid mathematical argument the conclusion is mathematically implied by the premises." Note that my central point, that the use of the mathematical notions of consequence, equivalence, etc., makes the Carnap-Ayer theory epistemologically empty-has been totally misrepresented by Martin. Professor Martin writes (in the paragraph just noted) as if Carnap distinguished between "semantical rules" and "mathematical assumptions of the language." This is part of Professor Martin's tendency to treat "semantical rule" as a philosophically noncontroversial, standardized notion. But Carnap in fact treated at least some of the mathematical assumptions of a language as semantical rules ("Meaning Postulates"). In "Semantics, Empiricism, and Ontology" Carnap treats the verification/refutation procedures (or the confirmation/infirmation procedures, in the case of empirical sentences) for the sentences of a language as part of what we specify when we specify the interpretation of the language. It is just this elastic quality of the notion of a "semantical rule" in the writings of those who wanted the notion to do epistemological work that I was referring to (of course, Quine pointed this out in "Truth by Convention" [I9361 and "Two Dogmas of Empiricism" [1951]). Professor Martin goes on to write that: However, Putnam never-quite-clearly-explains what Discovery in his sense is. T o be sure, he uts forward "the picture of the Mathematical Fact as Eternally T r u e [with capitals, notice] independent of H u m a n Nature a n d human mathematical activity" and comments that he does "not find this picture a t all misleading" and in fact has never "been told just h o w or why it is supposed to be misleading." But surely now, any "intelligent reader," to use Putnam's own words, will be "irritated" a t this contention, especially if no account is given of what Mathematical Facts are, regarded as Eternally T r u e independent of Human Nature. No one has ever, in the entire history of the subject, given a satisfactory account of these, and Putnam does not even attempt to d o so. It is not just that the view is never-quite-clearly-explained;it is not explained a t all, but rather referred to with capitals as though this raises the view above all suspicion and prevents it from being "misleading."
The capitals that irritate Professor Martin are there for a reason.
I was mocking Wittgensteinian ways of mocking mathematical realism. If a mathematical fact (without capitals) is, e.g., the fact that a certain Turing Machine will halt if started in a certain state scanning a blank tape, then isn't this "independent of human nature"? Certainly it is causally independent of human nature! Isn't this "eternally true"? Certainly it's tenselessly true! What sort of "account" is it that we have never been given? A philosophical account. My suggestion was that the assumption that such an account is possible or needed is what we ought to r e e ~ a m i n e . ~ Professor Martin extends his criticism of TNM by complaining that: Putnam contends that the linguistic account of mathematics ascribed to Carnap leads to the "silly suggestion" that "once a mathematical assertion has been accepted by me, I will not allow anything to count against this assertion." But this is a far cry from anything Carnap ever believed or wrote. T h e decision to accept a given mathematical formalism is in part a pragmatical matter, according to him, and subject to continual revision.
There are three points I wish to make in response to this: (1) Professor Martin apparently does not allow rules to have the form 'S is to be held immune from revision'. But Wittgenstein's "rules of description" and Carnap's "Meaning Postulates" seem to have something like this very form. (2) Of course there is, on any "Rule of Language" account, a possibility of revising rules. This is what Carnap regarded as a "pragmatic matter." But then one is not talking about the same assertion, since the meaning of the sentences has been changed. (3) The distinction (which Martin ignores) between revising a rule and revising an assertion is crucial to the Rule of Language position. Once it is given up, one has accepted Quine's view that there is no clear sense in which mathematics is "analytic." Finally, Professor Martin writes: Putnam's penultimate paragraph must be read to be believed that it could have been seriouslv intended. T h e "Rule-of-Lanmarre" account " " is supposed by him never to allow revising a rule. He contends that 'any statement can be regarded as a 'rule of language' " on this view. Such a contention fails to take into account the very nature of semantical rules, which interrelate word and object in very special ways. Only statements of a quite unique kind are recognized as semantical rules, contrary to 3 B ~now t I think that one has to say more about the notion of truth-in and out of mathematics- than I thought we did in 1963. A more recent account of my view appears in "Analyticity and Apriority: Beyond Wittgenstein and Quine," Midwest Studies in Philosophy, 4.
Putnam's contention. This latter is so awry, in fact, as to lead us to think that one more straw m a n has been set u p in order to be knocked down with one more quixotic argument.
What I argued is not that rules cannot be revised, on the Rule of Language account, but that the distinction between revising a rule and revising an assertion is hopelessly unclear in natural language. If I was brief on this point, it was because I cover it at length in "The Analytic and the Synthetic."
Professor Martin's criticism of the modal-logical interpretation of mathematics depends on misinterpretation. (He ignores "Mathematics without Foundations," which gives a fuller explanation than "What is Mathematical Truth.") My proposal, briefly put, was to use a language with primitive modal operators to do the work of set theory. (Charles Parsons has since verified that this can be carried out.) It was not to quantify over possibilia, nor to refer to formal systems. The description that "the first item to observe is that here mathematics is in effect identified with the study of logical systems, no matter what their ontology, so long as one proves theorems about its objects" is (so far as I can understand it) wrong. I leave it to the interested reader to consult my papers and not Professor Martin's description of my view for details.
On quantum mechanics: In the 3-valued logic approach, "true" is not defined as "accepted" (any more than in classical physics). In quantum logic: I reject Professor Martin's view that an "interpretation" for the logical connectives must be provided before we know what quantum logic is. This disagreement goes back to his view that a language is understood through "semantical rules" and my view that a language is understood by learning how to employ it. IV. I am glad to be able to agree with Professor Martin on something. I agree that there was a tension in these papers between
my realism and my conceptualism. Subsequent writings4 have indicated how I would attempt to work out this tension.
HILARY PUTNAM. HARVARD UNIVERSITY.
'Especially "Realism and Reason," in my Meaning and the Moral Sciences (Routledge and Kegan Paul, 1978) and "Models and Reality," in the Journal of Symbolic Logic, vol. 4 5 , no. 3 (Sept. 1980). See also my Reason, Truth and History (Cambridge, 1981) and Realism and Reason; Philosophical Papers, vol. 3 (forthcoming, Cambridge, 1982).
