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Chapter 4 The Semantic Challenge to Realism

4 .1 Introduction

The semantic challenge to realism comes from another of Benacerraf' s " "l papers, What Numbers Could Not Be. In that paper, he ([1965] 1983) argues that there is no principled way of deciding which of the settheoretic models of Peano arithmetic is the numbers , that there is no principled way of deciding what system of objects is the numbers , and hence that we cannot make sense of the idea that numbers are determinate objects. These conclusions are intended to motivate his ([ 1965] 1983, 291) structuralist view that Arithmetic is . . . the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects - the numbers. ' Concluding with a remark that is reminiscent of Quine s (1960, 76) " " claim that there are no language neutral meanings, Benacerraf ([ 1965] 1983, 294) says, " if the truth be known , there are no such things as numbers." 2 ' s (1996) 1. " What NumbersCould Not Be" doesnot representBenacerraf presentthinking on the issuesin question, but I shall focus on it becauseit has becomethe classic statementof skepticismaboutthe determinacyof referenceto the numbers, and, assuch, it long ago took on a philosophicallife of its own. Moreover, from a realistperspective , Benacerrafhasn't changedhis basicview all that much. In this connection , I will consider someof the things he (1996) now saysabout the issues. ' s statementat facevalue 2. In Katz (l996b) I expresseduneaseabout taking Benacerraf " becauseof his ([1965] 1983, 294) final remark, which is not to say that thereare not at leasttwo prime numbersbetween15and 20." Benacerraf(1996, 51 n. 10) saysthat this remark was meantto indicatethat his statement" should not be taken as contradicting establishedmathematicalresults." Recognizingthat this cannotbe an absoluterequirement , he gives a higher priority to hermeneuticsthan to revisionism. Then he (1996, 51- 52) saysthat " the reduction[in Benacerraf([1965] 1983)] doesnot answerthe fundamental ] true," adding regardingwhat makesthe true [mathematicalstatements " On thequestion assessment of WNCNB, they wouldn' t be true." This explanationremovesthe

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There are two responses to Benacerraf' s argument that should be dismissed at the outset. One is that , although it is a challenge to the mathematical realist' s claim that numbers are determinate objects, it is not , strictly speaking, a challenge to general realism, which could be true even if mathematical realism were false. Nothing should be made of this . On the one hand , mathematics is so much the central case in the philosophical controversy over the existence of abstract objects that indeterminacy of reference to numbers will translate immediately into doubts about the tenability of general realism. On the other hand , Benacerraf' s argument , as we shall see, is close enough to indeterminacy arguments against logical realism and linguistic realism that , were it to work in the case of mathematical realism, we could expect that them. something quite similar would work against The other response is that Benacerraf' s structuralism seems to fall prey to his own empiricist scruples about knowledge of abstract objects. Since the structure that " all progressions have in common merely in virtue of being progressions" is, on his own admission , an " abstract structure ," there must also be a problem about how spatiotemporal creatures like ourselves can have knowledge of that structure . Here, too, I think we should make nothing of the point . The problem that Benacerraf is raising about the determinacy of our referenceto numbers is independent of the question of whether his structuralism can be reconciled with his position on mathematical knowledge in " Mathematical Truth ." Benacerraf' s troubles would be cold comfort if we were forced to concede that reference to numbers is indeterminate . Contrary to general wisdom , I think Benacerraf-style arguments for the indeterminacy of reference to numbers and other mathematical objects are irredeemably flawed . I think that the flaw has escapednotice for two related reasons. One is that philosophers have not recognized that those arguments are special cases of indeterminacy arguments generally ; as a consequence, indeterminacy arguments in the philoso phy of mathematics have not been viewed from an abstract enough perspective to make the flaw in them apparent . The other reason is that 's unease , but replacesit with confusionabout Benacerraf priorities. One would have that an such assessment contradicts establishedmathematicalresultsand hence thought . Thedeeperissuebetween representsat leasta significantdowngradingof hermeneutics us is whether hermeneuticscan be separatedfrom ontological and epistemological . The view in chapter 1 is that questionsin the foundations of the formal sciences hermeneuticsis a featureof controversieslike that amongrealism, conceptualism , and nominalism, and that betweenempiricism and rationalism. Hermeneuticsservesas a criterion for assessingsuch positions. Roughly, the best philosophicalposition, other things being equal, is the one that preservesthe broadestrangeof establishedmathematical resultsand philosophicalintuitions.

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the flaw in indeterminacy arguments in the philosophy of language, the paradigm cases of indeterminacy arguments, has itself not been widely recognized. I will try to show that , once both these conditions are corrected, the flaw in Benacerraf-style indeterminacy arguments in the philosophy of mathematics becomes evident , and also that linking indeterminacy arguments in the philosophy of mathematics and the philosophy of language provides a deeper understanding of the nature of such arguments and of the forms of skepticism based on them. My approach will be as follows . First , I will present analyses of the two principal indeterminacy arguments in the philosophy of language that reveal their flaw. On the basis of those analyses, I will develop a conception of the structure of indeterminacy arguments in general. Next , I will explain how knowledge of the flaw in question can be used to develop a general strategy for resisting indeterminacy arguments . Finally , I will show that Benacerraf-style arguments fall under our general conception of indeterminacy arguments, and then show how our strategy can be applied to block such arguments . 4.2 IndeterminacyArguments in the Philosophyof Language Indeterminacy arguments are skeptical arguments . They claim that we lack the means to distinguish among the things we have to distinguish among in order to legitimize our belief that our talk about certain objects is talk about determinate objects. One characteristic of such skeptical arguments is that they are based on an allegedly unbreakable symmetry between the intended interpretation of such talk and certain deviant interpretations . The skeptic challenges us to break the symmetry . Indeterminacy arguments differ from one another with respect to the kind of knowledge in question, e.g ., about meanings, numbers , and so on, and also in the considerations they offer for the allegedly unbreakable symmetry . Nonetheless, they form a distinct class of skeptical arguments exhibiting a common pattern of philosophical reasoning. The class of indeterminacy arguments includes several celebrated arguments about meaning and reference in the philosophy of language. The most celebrated of them is, of course, Quine ' s (1960) argument for the indeterminacy of meaning and translation (and the derivative ' argument he [ 1969b] gives for the inscrutability of reference). Quine s indeterminacy argument has served as a model for philosophers in constructing other indeterminacy arguments . I will treat it as the paradigm of the class of indeterminacy arguments . ' Wittgenstein s (1953) well -known argument about rule following is sometimes taken as an indeterminacy argument , but , strictly speaking,

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it isn ' t one. Wittgenstein is not a philosophical skeptic putting forth an ' argument designed to show that we don t know what we are talking about. Given his metaphilosophical views , Wittgenstein would be the last philosopher to be in the business of creating new philosophical problems or trying to prove that no sensecan be made of our ordinary talk . As Wittgenstein (1953, sec. 133) wrote , " . . . the clarity that we are aiming at is indeed completeclarity. But this simply" means that the should completelydisappear. philosophical problems ' Wittgenstein s rule- following argument is rather an attack on certain philosophical theories that he thinks confuse us about the use of language, in particular those according to which private mental states have the normative force to fix meaning . Wittgenstein ' s (1953, sec. 201) famous paradox is the culmination of an argument - beginning , in effect, at sec. 139- designed to refute such philosophical theories and to locate the normative force to fix meaning in the rules of our public ' linguistic practice. For these reasons, Wittgenstein s rule- following argument will not be one of my examples of indeterminacy arguments in the philosophy of language, though I will consider it at the very end of the chapter in connection with Putnam ' s Skolemite skepticism . In contrast, Kripke ' s (1982) rule- following argument is put forth as a general skeptical argument . His paradox is intended as a philosophi cal problem for everyone. Moreover, since Kripke ' s argument is a highly influential indeterminacy argument and also since I wish to call attention to a feature of it in which it contrasts with Quine ' s indeterminacy ' argument , I will discuss Kripke s argument too.

4.2.1 Quine's Argument ' Quine s (1960,27) indeterminacythesiscomesfrom reflectionon radical translation. Quine (1960, 78- 79) saysthat the discontinuity of radical translation tries our meanings: really setsthem over againsttheir verbal embodiments, or, more typically , finds nothing there. He (1960, 53- 54) explains: We could equatea native expression[" gavagai" ] with any of the terms " rabbit," " rabbit stage," " undetached disparate English " rabbit part, etc., and still, by compensatorilyjuggling the translation of numerical identity and associatedparticles, preserve . conformity to stimulus meaningsof occasionsentences The consequence is indeterminacyof translation. Given two analytical about the translation of " gavagai," such as " rabbit" and hypotheses " rabbit " stage, Quine (1960, 72) claims:

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Both could doubtless be accommodated by compensatory variations in analytical hypotheses concerning other locutions , so as to conform equally to all independently discoverable translations of whole sentences and indeed all speech dispositions of all speakers concerned. And yet countless native sentencesadmitting of no independent check . . . may be expected to receive radically unlike and incompatible English renderings under the two systems. , Quine (1990) has again stressedthat his indeterminacy doubts "Recently come from reflecting on radical translation ." This point is also stressed by prominent Quine scholars', e.g ., Gibson (1994). The problem with Quine s argument is not that there is some difficulty in getting indeterminacy from radical translation . The problem is that it is far too easy. It is as easy getting the cards one has stacked the deck to get. In the game of radical translation , the deck containing the semantic evidence from which the linguist can draw in trying to choose among the available translations contains only extensional cards. Since the intensional cards have been left out entirely, it is hardly surprising that Quine ' s field linguist can find no semantic evidence to distinguish between extensionally equivalent 'translations like those pairing " gavagai" with " rabbit ," " rabbit stage," and " undetached rabbit part ." " Quine (1960, 28- 29) presents radical translation as translation of a " language of a hitherto untouched people, that is" , as a"situation that is nothing more than actual translation purged of hints from previous translations and shared culture . Quine (1960, 28, n. 2) even suggests that such situations are sometimes encountered by real linguists who have developed some " techniques" for working in them. Thus, radical translation is presented to readers of Word and Object as if it were an unobjectionable idealization of actual translation , one that merely abstracts " away" from biasing complications like historical or cultural connections between languages. We are encouraged to think that the deck is a new one, containing no cards with nicked edges or bend marks that might serve to mark them. But, in addition to the idealization ' s being set up to exclude genuinely biasing factors, it is set up to exclude all evidence about sense properties and relations of expressions. Such evidence seems prima facie precisely what is required to break the symmetry between referentially equivalent translations . It is, after all , a fact of some sort that speakers of a natural language make judgments about the meaningful ness, ambiguity , synonymy, and other senseproperties and relations of in their language. Mightn ' t a bilingual speaker tell us that expressions " " is " " " " gavagai synonymous with rabbit but not with rabbit stage and

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" " undetached rabbit part ? Since the question at issue is whether the intension of a sentence can in principle be objectively determined for actual translation , in excluding the most promising evidence for objectively determining intensions from the radical translation situation , Quine has set things up so that translation cannot help but come out indeterminate . I am not suggesting that there was any deliberate deck stacking . What I think is that Quine constructed the radical translation situation to reflect the situation that Carnap (1956b) uses in describing his method for determining intensions in " Meaning and Synonymy in Natural Languages." (Quine almost always had Carnap in mind when thinking about semantics.) Carnap (1956b, 236- 40) presents a method for choosing between translations that he thought established the superiority of intensionalism over extensionalism. Carnap imagines a situation in which a field linguist asks his German-speaking informant Karl questions designed to provide evidence that will choose between " horse" and " horse or unicorn " as translation of the German term " " Pferd. Carnap thinks that the linguist can only obtain the necessary evidence by switching from queries about Karl ' s application of " Pferd" to actual animals to queries about his application of the term to possible ones. Carnap thinks that Karl ' s responses to queries in German like whether he would apply " Pferd" to something that looks like a horse but has a horn in the middle of its forehead will settle the issue of translation . Now , Carnap (1956b, 239) thinks of the intension of a predicate extensionally, as the objects (in possible worlds ) to which the predicate applies, and, consequently, the situation of the linguist that Carnap assumes in his method involves no evidence about sense properties and relations. As a reconstruction of this Carnapian situation , Quine ' s radical translation situation assumes that the linguist ' s method involves no such evidence. Quine ' s insight was to see that Carnap ' s method does not work when the extensional evidence cannot discriminate between the competing translations if their application is the same in all possible cases. Because" rabbit ," " rabbit stage," and " undetached rabbit part " are such competing translations , they refute Carnap ' s method . ' ' Quine s error was to present what is essentially Carnap s translation situation as the general situation of translation . As a general argument ' against the determinacy of translation , Quine s argument fails because it rules out evidence about senseproperties and relations by fiat . It begs the question against the intensionalist who does not understand intension extensionally as Carnap does and accordingly takes a different view of intensional evidence. (See Katz 1990a, 1994b.)

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If actual translation is not seen through the lens of radical translation , then at least initially it is an open question , on the one hand , whether there are " linguistically neutral meanings," and , on the other hand , whether linguists can obtain evidence about them that , together with standard scientific methodology, will determine a best choice among competing translations . Since there is no a priori reason not to accord the judgments speakers make about the meaningfulness, ambiguity , synonymy , and other sense properties and relations of expressions the same benefit of the doubt that is accorded to prima facie evidence in other infant sciences, the only non - question-begging way to decide the issue of whether the intensions of sentences can be objectively determined is to try . Collect evidence about sense properties and relations , try to construct a theory that systematizes them , and see what comes of it . The hypothesis of determinate meanings must end up like the phlogiston hypothesis, but it also might end up like the gene hypothesis . Here' s how a non-question-begging attempt might go . Let the field linguist investigate not only the referential properties and relations of expressions of the language under study but also their senseproperties and relations. Let the field linguist collect the judgments informants would make about the sense properties of " gavagai" and the relations its sensebears to the sensesof other expressions. Such expressions are not restricted to the vocabulary of the informants language, but , with " " bilingual informants , include expressions like gavagai in the inform ant' s language and expressions like " rabbit ," " rabbit stage," and " undetached rabbit part " in the language into which the field linguist intends to translate the former expressions. Now , our method , as ' opposed to Carnap s, is for the field linguists to elicit a wide range of judgments about the sense properties and relations of the expression to be translated and its possible translations and use these judgments as evidence on which to decide which of the translations is best. Translation is samenessof sense (synonymy ) between expressions of different languages, and hence the best translation is the one with the same sense properties and relations. To elicit judgments , the linguist can ask an informant questions about the sense of " gavagai," starting with the direct question " With which , if any, of the English expressions ' rabbit : ' rabbit stage: or ' undetached rabbit part' is the term ' gavagai' synonymous ?" A choice of one of these " expressions is certainly evidence for the translation of gavagai" as that expression. If , for some reason, the informant is unable to choose, there are other informants and less direct questions. For example, the linguist can ask " Is ' gavagai' closer in sense to ' infancy,' ' adolescence: and ' adulthood ' ' ' ' , or to infant , adolescent: and ' adult ' ?" or " Does the

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sense of ' gavagai' bear the same part -whole relation to the sense of ' ' some word in the informants language that the sense of finger ' ' " bears to the sense of hand in English ? If some informants can make some such judgments , there is evidence relevant to distinguishing like " rabbit ," " rabbit stage," and " undetached rabbit among translations " part . Once such evidence is acknowledged , we have no reason to doubt that , in combination with methodological principles such as simplicity, such evidence can in principle enable us to choose a best translation . As noted , our method involves the use of a bilingual informant . How things could be otherwise when the method is supposed to deal with translation is hard to imagine . Quine is on record as objecting to bilinguals . He (1960, 74) thinks that , once we introduce bilingual informants with a common linguistic competence, we are already assuming determinate translation : Quine raises the possibility that different competences, and so correlate different bilinguals have different" linguistic with gavagai" . This is, of course, a real possibility . expressions ' Quine s objection would go through if we assumed that there actually are bilingual informants with a common linguistic competence, since that would mean that translation is determinate . But our method only assumes the possibility of such bilingual informants . Since that does not imply that translation is d ~terminate , Quine ' s objection does not go through . The possibility of a homogeneous community of bilingual informants is on a par with the possibility of a homogeneous community of monolingual speakers, which is something customarily assumed when linguists investigate the grammar of a single language. Thus, to entertain the possibility of such a community of bilingual informants is only to suppose it in the spirit in which an investigation in any science supposes that there are laws to be found . Such suppositions get investigations going and are tested in the course of them. The supposition of laws of translation , like the supposition of any laws , will be confirmed or disconfirmed depending on whether the investigation succeeds or fails in providing us with suitable confirmed statements of them. Since the issue is whether there are intensions of sentencesand whether they can be objectively determined , no question is begged in assuming natural languages have intensional structure for the sake of determining whether or not the assumption is true . Contrast this hypothetical no , which acceptance of an intensionalist framework ' s insistence thatbegs radical with the extensionalist , Quine question against translation is the right model for actual translation , which , because it requires categorical acceptance of an extensionalist framework , begs the question against the intensionalist .

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If reflection on radical translation were all there is to Quine ' s argument against intensionalism , we could dismiss the argument as ruling out intensionalism arbitrarily . But Quine had another argument in his (1961c) famous paper " Two Dogmas of Empiricism ." If this attack on the analytic / synthetic distinction had been successful, it would have ruled out intensionalism in an entirely principled way. As will be recalled, Quine argued that we cannot make objective sense of synonymy and analyticity in any of the ways we have for explaining legitimate logico- linguistic concepts. He then argued , quite convincingly, that lexical definition , explication , abbreviation , substitution procedures , and meaning postulates are inadequate to make objective sense of the analytic / synthetic distinction . This he took to be enough (1961c, 37) to conclude: " That there is such a distinction to be drawn at all is an unempirical dogma of empiricists , a metaphysical article of faith ." As 1 (1988; 1990b, 175- 202) explained elsewhere, the argument overlooks what is, in fact, the most promising type of explanation for notions like synonymy and analyticity . This type , which is characteristic of mathematics and logic - and which subsequently became the standard type of explanation in generative linguistics - is to define a family of concepts within an axiomatic theory of the structure of a system of 3 objects. I will not repeat the details of this criticism here. The essential is that , in taking substitution procedures to be the proper approach point to explaining concepts in linguistics , Quine was buying into the pre - Chomskyan taxonomic conception of the science. Quine makes it quite clear not only that he (1961b, 56- 57) is assuming that the explanation of concepts in linguistics , from phonology to semantics, is based on " substitution criteria , or conditions of interchangeability " but also that this assumption underlies his argument that attempts to explain " synonymy via" substitution procedures involv [e] something like a " vicious circle. Since this means that Two Dogmas of Empiricism " contains no criticism of an approach that defines intensions within an axiomatic theory, Quine ' s exclusion of an intensionalist approach to translation is arbitrary . One further point . Katz (1990b, 197- 98) and Gemes (1991) have noted a paradoxical aspect to the symmetry between competing translations in the radical translation situation . " Rabbit," " rabbit stage," and " undetached rabbit part " are supposed to be semantically equivalent translations of " gavagai" - as Quine (1960, 27) says, they are equally 3. As I (l990b, 184- 93) argued elsewhere, and apparently as Quine (1990, 199; see also Clark 1993) now agrees, the question of whether there can be a theory of meaning is entirely a matter for linguistic research to decide.

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" yet they are compatible with the" totality of speech dispositions . And " . How can they also supposed to be radically unlike and incompatible ? be equivalent and also incompatible Since the question of indeterminacy turns on whether there is a " general relation" of synonymy in the terminology of Two Dogmas of Empiricism , a relation of samenessof meaning that holds for variables Sand L- determinacy is a condition for synonymy within a language as much as it is a condition for translation between languages . Thus, the earlier question arises about a single language: how can there be competing hypotheses about the synonymy of expressions of the language that are " compatible with the totality of speech " dispositions of its speakers and yet the question of "the synonymy of the expressions be indeterminate because they are radically unlike " and incompatible ? The incompatibility seems to establish the existence of dispositions to judge the expressions as different in meaning , " " " " " as indeed the incompatibility of rabbit , rabbit stage, and undetached " rabbit part " does, but then the hypotheses cannot be compatible " with the totality of speech dispositions . The paradox seems to arise becausesensesare smuggled into the radical translation situation to keep the linguist ' s choice from degenerating into a choice among synonyms . As we shall see below, this paradoxical element is, in one form or another, found in indeterminacy arguments generally and will be important ' for our response to Benacerraf s indeterminacy argument .

"

' 4.2.2 Kripke s Argument finite mind grounds the claim Kripke (1982) asks what fact about " " my that I mean 125 in literal uses of sixty - eight plus fifty -seven. The fact must somehow justify my thinking that the referent of that expression is the third member of the triple in the infinite set of triples of numbers that is the plus function . It must not equally justify my thinking that the referent is the third member of the ' corresponding triple in an infinite set of triples such as Kripke s (1982, 8) quus " function , i .e., the triple . As Kripke puts it (1982, 54), the " " idea in my mind is a finite object so why can it not be interpreted as " the quus function , rather than the plus function ? The finitude of our ideas makes successful reference depend on our justifying a projection from finitely many known addition triples to the infinite plus function . But such a projection situation faces us with the Humean problem that a finite number of observed casesis compatible with any prediction about unobserved cases. Since an idea in my mind can only represent a finite set of triples , and since the finite set it represents is a proper subset of the infinite quus function as well as a

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" proper subset of" the infinite plus function , the claim that sixty - eight plus fifty seven means 125 can be no better supported than a claim that it means 5. ' Kripke s doubt about whether the available evidence determines a for our uses of " sixty - eight plus fifty -seven" is parallel unique referent ' to Quine s doubt about whether the available evidence determines a ' " " unique translation for the native s term gavagai. Furthermore , their reason for skepticism is the same. Quine and Kripke both think that the evidence is inadequate to justify the intended interpretation because it supports the intended and deviant interpretations equally well . Last, but by no means least, the evidence supports them equally well because it is confined to extensional facts about expressions, and , in ' ' Kripke s case as in Quine s, the restriction to extensional evidence is a product of the way in which the semantic situation has been constructed . Just as intensional evidence is unavailable in choosing a translation for " gavagai" becauseQuine constructed his radical translation situation to exclude facts about sense properties and relations , so intensional evidence is unavailable for fixing the reference of " sixty " eight plus fifty -seven because Kripke constructed his rule- following situation to exclude facts about them. In the former situation , there is no appeal to information about the sensesof " rabbit ," " rabbit stage," and so on, and, in the latter, there is no appeal to information about the sensesof " plus ," " quus ," and so on. If , like Quine ' s, Kripke ' s puzzle arises because senses have been ' painted out of the semantic picture , the solution to Kripke s puzzle , ' like the solution to Quine s, ought to emerge once we paint them in ' again . Moreover , unlike Quine s overall argument , which contains the criticism that we cannot make objective senseof senses, Kripke ' s overall argument does not contain any criticism of senses. Consequently, the resolution of Kripke ' s puzzle ought to consist in introducing the sensesof " plus ," " quus ," and so on, and describing their role in the reference of mathematical expressions. Sensesare the linguistic objects that speakers have knowledge of in virtue of their semantic competence. On linguistic realism, a sense is an abstract object, but it is nonetheless a finite object because it is Cornposition ally constructed from the finite sensesof the finitely many lexical items in a sentence. Kripke is certainly right that our minds , and hence our ideas, must be finite things , but when senses are in the picture , we not only have the finite idea of the extensionof the numerical " " expression sixty - eight plus fifty -seven, that is, the number one hundred twenty five , we also have the finite idea of the intension of the " " expression, that is, the sense of sixty - eight plus fifty -seven. More generally, as competent speakers of English , we know the sense of

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numerical expressions and the sense of I' m plus n; which we may suppose is given by (P). (P) If n is 0, the sum is m, and if n is greater than 0, the sum is the number reached from m by the processof taking its successor, then the successorof the successor, and so on n times. , knowledge of the senses of the numerical Knowledge of this sense terms which can replace " m" and " n," and knowledge of composition compositional sense of expressions ality " constitute knowledge of the like sixty - eight plus fifty -seven." This knowledge too is finite , just as are the sensesthemselves, since both our knowledge of sensesand the senses of which it is knowledge are the product of finitely many combinations of finitely many elementary units - mental entities in the former case, abstract entities in the latter. Recall that Kripke ' s puzzle is a Hume- style problem of projecting from a finite number of observed casesto an infinite number of unobserved cases. With the recognition of senses and our knowledge of them , the puzzle disappears because we are no longer faced with a problem of projecting from the finite to the infinite . On the one hand , both our knowledge of the senseof " plus " and the senseitself are finite objects, and , on the other hand , neither of them serves as a basis for of the triple using sixty - eight plus fifty with addition triples to rule out what we recall from our acquaintance . projections to the quus function Rather, we refer to 125 because, in virtue of our grammatical knowledge as English speakers, we grasp the finite sense of " sixty - eight plus fifty -seven," because we use that expression with the intention to refer to the referent of its sense, and because, in virtue of falling under the sense of " sixty - eight plus fifty seven," the number 125 is the referent of the expression.4 Hence, the fact about my finite mind that grounds my meaning 125 in literal uses of " sixty - eight plus fifty -seven" is the finite fact that my communicative intention reflects my grammatical knowledge of the compositional senseformed from the sensegiven by (P) and the sensesof " sixty -eight " and " fifty -seven." 5 Kripke (1982, 54) seems to have a reply prepared in advance: 4. On my new intensionalism , it is not generally in the nature of sensesto do this , but it doesn' t matter here whether 125is the referent of " sixty - eight plus fifty -seven" because, as Fregeans would say, it is the nature of the sense of that expression to determine 125 as its referent or because, as I would say, the relevant contextual factors are in place and 125 falls under the sense of that expression. " 5. I anticipate the question how we can know that the expressions " and so on and " sameness" rather than a sense with the " iterate" in P have a sense with the ( ) concept " " " " concept schmameness so that plus expresses the notion of doing the same up to the

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But ultimately the sceptical problem cannot be evaded , and it arises precisely in the question how the existence in my mind of I' " any mental entity or idea can constitute grasping any particular sense rather than another . The idea in my mind is a finite object : can it not be interpreted as determining a quus function , rather than a plus function ? This reply is not to our solution , but to some other (psychologistic ) one . The pronoun " itll in Kripke /s question refers to an idea in my mind , and hence in asking the question , Kripke is only pointing out that this idea can be interpreted under a projection to the quus function . Once senses are in the picture , it is clear that trying to apply Kripke ' s argument in this passage to our view would involve an illegitimate shift from talking about (grasping ) a sense to talking about lI [ t ] he idea in my mind .1I If the pronoun in Kripke ' s question were to keep its reference to an objective sense, the answer to Kripke ' s question would be that there is no interpreting of senses: senses simply determine one function to the exclusion of others .6 Let us review the main aspects of our solution to Kripke ' s puzzle about rule following . There are only two points at which it makes any sense to claim that the puzzle arises . One is the relation between my ideas and the sense of " plus " that they enable me to grasp . The other is the relation between the sense of IIplus " and its referent , the plus function . To repeat our explanation of why the problem cannot arise at the former point , Kripke ' s problem is a Hume - style problem of what justifies projecting from an observed finite number of addition triples to the infinite plus function , and hence , when what is I " grasped is the sense of Iplus , what is grasped is finite too , so that there is no longer a problem of projecting from finite sample to infinite 7 population . n-kth caseand doing somethingelsethereafter(e.g., taking the squareof the successor ). This questionis a repeatof the Quineanquestiondiscussedabove. Wethus explainhow we can know this in the sameway that we explainedhow we can know that the sense of " rabbit" involves the concept" object" rather than the concept" temporalslice of an " " " object or the concept undetachedpart of an object. Wecanverify that we aregrasping the senseswe think we are graspingin the caseof the word " plus" by comparingits senseproperties and relations with those of the expressions" and schmo on" and " schmiterate " in the samemanner in which we comparedthe sensepropertiesand relations of " rabbit" with those of " rabbit stage" and " undetachedrabbit part" to determinetheir nonsynonymy . Seemy (l990b, 166- 67) discussionof suchverificationin connectionwith one of Kripke' s examples . 6. It is thus mistaken to think that the problem must resurfaceas a problem about ; seealso Boghossian(1994) and my (l994b and 1997, 10- 11). graspingsenses 7. It is, moreover , hard to seehow thereis any other relevantproblemabout how our finite ideasenableus to grasp finite senses . No problem can comefrom doubts about theexistenceof ideasor senses , sinceneitherbehaviorismnor skepticismaboutintensions

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' The other point at which it might make sense to claim that Kripke s " " problem arises is in the relation between the sense of plus and the plus function that is its referent. Here there is a relation' between something finite and something infinite , but even so, Kripke s problem cannot arise becausethere is no projection once sensesenter the picture . " The relation of the senseof " sixty -eight plus fifty -seven to its referent the number 125 is simply that the plus function exemplifies the sense of " plus " . As Kripke (1982, 11) puts his problem , it is one the solution of which must be " an account of what fact it is (about my mental state) that constitutes my meaning plus ." That problem cannot arise, since determination of the infinite plus function is out of our hands - or , perhaps we should say, outside our heads. Determination of the infinite plus function is an abstract semantic relation between two abstract objects. Kripke (1982, 54) himself says that there is no problem in connection with the infinite sequence falling under a finite concept: The " sense" in turn determinesthe addition function as the referent of the " +" sign . There is no special problem . . . as to the relation between the sense and the referent it detennines. plus function , not the quus Kripke (1982, 54) explicitly allows "that the function , falls under the sense of plus " in saying that " [i ] t simply is " in the nature of a sense to detennine a referent. Thus, Kripke can we succeed in detenninately referring hardly object to the claim that " " to 125 in our literal uses of sixty - eight plus fifty -seven because 125, -seven, is the referent that the being the" sum of sixty - eight and fifty " sense of sixty -eight plus fifty -seven determines. ' ' Finally , like Quine s radical translation situation , Kripke' s ruleelement. In Quine s radical following situation contains a paradoxical " " " " " translation situation , the expressions rabbit , rabbit stage, and undetached " " rabbit part were supposed to be equivalent but incompatible ' ' " " " English renderings " of gavagai. In Kripke s" rule- following situation , the reference of sixty - eight plus fifty -seven is supposed to be indeterminate between the semantically indistinguishable infinite entersthe picture. Also, it is an uncontroversialfact that fluent speakersgrasp senses ; for example, they graspsenseswhen they recognizean ambiguity. Further, they can be of the grammatical presumedto graspsenseson the basisof somementalrepresentation . Thereare, to be sure, psychologicalproblems structuresthat relatesentencesto senses about syntacticand semanticcompetenceand philosophicalproblemsabout the mental and the physical, but neitheris relevanthere. Furthermore , as indicatedin chapter2, no relevantepistemologicalproblemcould arisewithout the assumptionthat an idea is an . idea of somethingin virtue of their causalconnection

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triples of numbers that are the plus function and those that are the " " " " quus function , but , as the terms plus and quus have to be nonsynonymous expressions that respectively denote those functions , paradoxically, the functions are semantically distinguishable and the reference of " sixty -eight plus fifty -seven" determinate . 4.3 The GeneralForm of IndeterminacyArguments The common pattern of reasoning that we have found in Quine ' s and ' Kripke s indeterminacy arguments is the general form of such arguments . They begin with an explicit or implicit account of our informal ' knowledge of some domain , typically embodied in the skeptic s representation of the situation we face in relation to theories of the domain . The skeptic assumesthe account to be a complete representation of that knowledge at least insofar as issues of indeterminacy are concerned, and, on the basis of this assumption , the skeptic argues for an unbreakable " symmetry between the intended interpretation and certain deviant interpretations of the theory which precludes our ruling out the deviant interpretations . At this point , indeterminacy arguments go their separate ways . Those that are put forth as skeptical arguments, such as Quine ' s, proceed from there being no way to break the symmetry to the conclusion that no interpretation (e.g ., translation ) is correct because there is no fact of the matter. Those that are not put forth as skeptical arguments ' , such as Kripke s, stop with the symmetry puzzle and simply ask what is to be done. Going for the robust skeptical conclusion , however, involves a dubious inferential step. As I (1988; 1990b, 183) argued against Quine , symmetry at best delivers the unknowability of language neutral meanings, so unknowable meanings, like Kant' s noumena, could still exist. If realism is right - and it hasn' t been ruled out - it is absurd to think that existence of objects in a realm that is independent of us depends in any way on our cognitive powers . The skeptic must do something to shore up the final step in the argument , but the available props themselves are in need of bracing .8 It is even more critical for the skeptic to support the assumption that the representation of our informal knowledge on which the symmetry puzzle rests is complete . As we noted , Quine sought to do this by trying 8. Benacerraf(1996, 26) recognizesthis weaknessin the argument, and he (1996 , 52- 53 n. 15) arguesthat Wright's (1985, 122) attempt to shoreup the argumentat this point doesnot work. I think the issuegoesdeeperbecauseit involvesthe verificationisttheory of meaning. That, as I seeit , is what Wright' s conceptof perfectunderstandingrestson, but I can't pursuethe questionhere.

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to show that beliefs involving putative symmetry -breaking senseproperties and relations do not qualify as legitimate knowledge of the semantic domain , and hence that those properties and relations can be excluded from a full account of our informal semantic knowledge . . This , I believe, is because he is not Kripke makes no such argument ' playing skeptic, but only devil s advocate. Genuine skeptics, however, cannot neglect this aspect of their argument , since, as we have seen in ' the caseof Quine ' s and Kripke s arguments , failure to exclude symmetry -breaking properties opens up the possibility of a determinate specification of the objects in the domain based on those very properties and relations. An argument to illegitimatize the symmetry -breaking properties and relations is essential for an effective indeterminacy argument for skepticism about our knowledge of the domain . The properties and relations that make the interpretations that enter into the symmetry puzzle incompatible are, for this very reason, potential properties . Indeed , it must be clear even on symmetry breaking the basis of the skeptic' s own account of the puzzle that the intended and deviant interpretations are incompatible , since otherwise nothing stops us from saying that the objects involved in the deviant interpretation are just the objects involved in the intended interpretation under a deviant description . If skepticism is not to be lost for lack of a response to the claim that the objects in the allegedly deviant model are just the meanings or the numbers under a deviant description , then it must be clear in what respects the deviant interpretations conflict with the intended one. But if this is clear, the skeptic' s indeterminacy argument is threatened on the grounds that those differences show that our knowledge of the objects in the domain is rich enough to distinguish the objects assigned to expressions on a deviant interpretation from the objects assigned on the intended interpretation . The symmetry is unstable becausethe interpretations have to be both equivalent and incompatible - equivalent in order to be symmetrical and incompatible in order to be rivals . Recognition of the incompatibility among the various interpretations in indeterminacy arguments reflects the fact that what is being challenged is something we in fact know or know how to do - such as distinguishing synonymous and 9 nonsynonymous expressions or referring determinately to numbers. 9. Theship of Theseusis not a symmetrypuzzlein an indeterminacyargument, because it is not a skepticalchallengeto somethingwe know or know how to do. Rather,it reflects time. Thereis a gap in our knowledgeof the conditionsfor something ' s identity over ' no issue about how to distinguish a ship we know to be Theseuss from a ship we know not to be. Nor is there an issueabout how to distinguish the two ships; they are distinguishablehistorically, spatially, and materially. The issue is which ship is ' s. Theseus

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For example, in the case of Quine ' s skeptical challenge, the incompatibility shows that we know that the expressions " rabbit ," " rabbit stage," " " and undetached rabbit part are nonsynon~ ous. The reason that there is a symmetry issue at all is that , at least tacitly, we have the knowledge required to distinguish the intended interpretation from deviant ones. Here we can go on the offensive against the skeptic . If our knowledge were as meager as the skeptic wishes us to think , the interpretations in s~ metry puzzles would not be incompatible . Thus, on the one hand , the skeptic has to underrate our knowledge to make us think that we are impotent to make such distinctions , and , on the other hand , the skeptic has to rate our knowledge properly in order to construct a symmetry between competing interpretations . Since skeptics can' t have it both ways , they face a dilemma : either the account of our knowledge on which their symmetry claim rests is complete and the interpretations are not rivals , or the interpretations are rivals and the account is not complete . In the former case, there is s~ metry, but only because the interpretations are equivalent , and hence not rivals . In the latter case, there is nonequivalence and rivalry, but only because the interpretations are nons~ metrical . In neither case is there a real symmetry to puzzle . 10. Simply in virtue of his descriptionof the " gavagai" puzzle in Wordand , we Object know that Englishcontainsthe sensepropertiesand relationsnecessaryto distinguish " "" amongthe putatively symmetricaltranslations rabbit, rabbit stage," and " undetached rabbit part." If these expressionsdid not have different senses , they could not be . It would be absurdfor Quine (1960, 27) to claim that competinghypotheses manualsfor translating. . . can be set up in divergentways, all compatiblewith the totality of speechdispositions,yet incompatible with oneanother. (Italicsmine) Moreover,our knowledgeof the nonsynonymyof thoseexpressions is knowledgeof the sensedifferencesin their compositionaland decompositionalstructure. For example, the " " different meaningsof stage and " part" that combinewith the meaningof " rabbit" to form meaningsfor the whole expressions" rabbit stage" and " undetachedrabbit part" exhibit sucha sensedifference.The meaningof " stage" involvesthe concept" temporal " sliceof someenduringobject, event, or process , whereasthe meaningof " part" involves the concept" oneof the divisionsof a whole." Oncewe recognizesuchsensedifferences , thepuzzleabouttranslationdisappears , sincethat recognitionreflectsspeechdispositions that confirm one of the divergenttranslationsand disconfirmthe others. In the caseof Kripke' s puzzle about rule following, our knowledgeof the senseof " "' plus something like that givenby (P) - and Kripke s (1982,7- 54) definitionsof " quus" " and quaddition" (expressedin termsof the senseof " plus" ) makesthe puzzle comprehensible , but, as explainedabove, that semanticknowledgealso enablesus to resolve the puzzle becauseit explainswhy there is no step of projectingnumericalreference . Similar points apply to Kripke' s (1982 19- 20) version of the argumentinvolving the , terms " table" and " tabair." Seemy (1990b , 163- 74) discussionof this and the other versionsof his argument.

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The dilemma shows the need for the skeptic to have an argument that illegitimatizes the symmetry -breaking properties and relations . To escape the dilemma , the skeptic must establish that our recognition of the incompatibility of the intended and the deviant interpretations does not show that we have knowledge of differences between them that breaks the symmetry . The skeptic has to argue, as Quine argued in " " Two Dogmas of Empiricism , that the properties and relations that are involved in the incompatibility are not , in the final analysis , legitimate (e.g ., they cannot be made objective sense of ). Skeptics have to show something like what Quine tried to show when he argued that our belief in synonymy and analyticity is due to our subscribing to the myth of the museum , that our recognition of incompatibility is not based on knowledge , but on false belief. If skeptics do not illegitimatize the features on which our recognition of incompatibility ' depends, either because the argument intended to do so doesn t work or because no argument is given , the symmetry is broken and the indeterminacy argument fails. A successful illegitimatizing argument is the only way skeptics can avoid having their account of our informal knowledge discredited for not doing that knowledge full justice . 4.4 The Strategyfor ResistingIndeterminacy Given these considerations, there is a straightforward strategy for resisting indeterminacy arguments . First identify the paradoxical aspect of the symmetry puzzle . Then focus on the properties and relations of the intended interpretation that make it and the deviant interpretations ) rivals , that is, those features of the intended interpretation but not the deviant one(s) that prevent us from treating the latter as just the former under a deviant description . On this basis, we take those features of the objects in the intended model to be ones those objects have, and hence we take them to be the features the skeptic has omitted in representing our informal knowledge of the domain . The next step, which mayor may not be necessary depending on the nature of the skeptic' s argument , is to counter the skeptic' s claim that the features are illegitimate . Since the claim cannot be backed up with an appeal to indeterminacy without begging the question , the skeptic must come up with an independent way of excluding symmetry -breaking properties , one that explains why we are mistaken to think that they carry the weight we have put on them. This requires the skeptic to invoke some a philosophical perspective that enables the skeptic to impose ' s casegeneral In and relations. , it was condition on legitimate properties Quine the taxonomic that made the naturalist / empiricist perspective theory

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of grammar with its operationalist methodology of " substitution criteria " or " conditions of " interchangeability seem to be the proper test of . Using the Quinean example as a guide to concepts in linguistics 's a analyzing skeptic attempt to come up with an independent way of excluding the symmetry -breaking properties , we expose the philo sophical perspective the skeptic has invoked to judge what are and are not legitimate properties and relations. The final step is to observe that the skeptic' s claim of illegitimacy is of illegitimacy from that philo sophical perspective but not from our philosophical perspective. This frees us to employ the properties and relations in question to rule out the skeptic ' s deviant interpretations ). This strategy can be illustrated with an indeterminacy argument based on Frege' s (1953, 68) famous Caesar question . In criticizing " Leibnizian definitions " of the individual numbers , Frege puts the Caesar problem in these terms: . . . we can never- to take a crude example - decide by means of our definitions whether any concept has the number JULIUS CAESAR it to or whether that same familiar of Gaul , belonging conqueror is a number or is not. Let us suppose that a skeptic challenges us to distinguish between an " " interpretation of Peano arithmetic on which seventeen refers to the number 17 and an interpretation on which it refers to Julius Caesar. Now the Caesar question is easily answered. We can reject identity statements like " 17 = Caesar" on the grounds that Caesar has the property of being a concrete object, while the number 17 has the property of being an abstract object, or that Caesar was a sentient creature, a human being , a Roman, and so on, while the number 17 is none of these thing S.l1 11. Wedon' t haveto go to all the troubleof developingstructuralismto solveproblems like the Caesarproblem. Of course, once we develop it , we can say, as Shapiro(1996) does, that since17 is an elementin a structureand Caesaris not, their identificationis a categorymistake. But, from our perspective , developingstructuralismfor this purpose is overkill. Further, from our perspective , a structuralist solution to such symmetry . problemsperpetuatesthe myth that we have no choicebut to cave in to Benacerraf Moreover, we cavein without really getting out of the woods. Frege(1953, 68) pointed out that a solution would provide us with " authority to pick out [numbers] as selfsubsistent " ' , objectsthat canberecognizedasthesameagain. Aren t the integerseventeen the rational number seventeen and the natural number , the real number seventeen , seventeen" the sameagain" ? DespiteWittgensteinianphilosophersof mathematicswho would deny it, it is pretty plausibleto say that they are. But evenif one thinks that they are not all the same, the structuralistought at least to say that they have a structural " . This, however,asKastin(1996)observes , is preciselywhat is ruled propertyin common ' " out by Shapiros polystructuralism.

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Benacerraf rejects identity statements like " 17 = Caesar" on the " ' " stronger grounds that they are senselessor unsemantical' because the terms in them do not belong to a common superordinate category. He ([ 1965] 1983, 286- 87) writes : If an expression of the form " x = y " is to have a sense, it can only be in contexts where it is clear that both x and y are of some kind or category C, and that it is the conditions which individuate things as the sameC which are operative and determine its truth value. There is no such C in the case of 17 and Caesar. The basic onto logical category of the former is Abstract and the latter Concrete, and these categories are disjoint . Hence, Benacerraf recognizes that we can rule out all interpretations that identify numbers with Julius Caesar, Marcus Brotus , Lassie, the moon , or other concrete objects. None of those things could be a nurnnberbecause each of them has or has had spatial or temporal location , whereas numbers can have neither spatial not temporallocation . It perhaps goes without saying that putative identities in indeterminacy " " " " arguments do not have to be shown senseless or unsemantical for us to have grounds for rejecting them. It is enough if they are shown false. In the question of whether numbers are sets, unlike the Caesar question, all of the objects in the intended and deviant interpretations belong to the category Abstract . Nonetheless, we can reject the identification of nurnnberswith sets on the grounds that setsbut not nurnnbers have members, have subsets, contain the null set, and so on, while nurnnbersbut not sets can be prime or composite, even or odd , perfect or imperfect , and so on. There are, of course, many mathematical systems of abstract objects with the same structure as the nurnnbers . Thus, we have questions like whether the natural nurnnbersare distinguishable from the even numbers . Again , the identification of the natural nurnnberswith a deviant system of nurnnbersis refutable on the basis of the features of the system that make it another collection of objects. In this example, a symmetry claim identifying the even nurnnberswith the natural nurnnbersis refuted by the fact that none of the former but every other one of the latter is odd , the fact that the latter contains an odd prime but the former doesn' t , and so on. The parallel question in a similar but nonmathematical case is whether nurnnbersare English sentences, supposing that the sentences of English form a recursive sequence. The sentences of English are types, so they too belong to the category Abstract . In this example too , the putative identification of the numbers with the deviant model is

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neither " senseless" nor " unsemantical." Still , we can reject the identification of numbers with sentences as false, since sentencesbut not numbers are interrogatives and declaratives, have subjects and direct objects, have morphemes as constituents, and so on, while numbers but not sentencesare odd or even, prime or composite, and so on. Cases like the one in which the deviant interpretation is the system of even numbers are trivial , since the application of our strategy does not require going beyond the properties in Peano arithmetic . In cases like those in which the deviant interpretation consists of Roman emperors , sets, and sentences, the application of our strategy requires going beyond the properties and relations that appear in the mathematical theory. This might seem to be something we ought not to do if we share the assumptions behind Frege' s criticism of " Leibnizian definitions " as improper . On the basis of Frege' s semantical view that sense is the determiner of reference, a proper definition of a mathematical term determines its referent, and hence a proper definition of number decides inter alia that Julius Caesar is not a number. The assumption that definitions of number terms do what a Fregean definition is supposed to do pushes the philosopher of mathematics into seeing the relation between the Dedekind / Peano definition of arithmetic and the numbers in terms of the Fregean relation between senseand reference. The question whether numbers are objects thus comes to depend on the question whether Peano arithmetic uniquely determines them as its model . Since Peano arithmetic does not do what a Fregean definition is supposed to do , but nonetheless expresses the mathematician ' s concept of arithmetic , structuralism follows as the most plausible explanation of number theory in the absence of determinate numbers. This is how, as I see it , Frege' s semantics misleads philosophers of mathematics. If they didn ' t accept Frege' s notion of definition , ' they wouldn t think that the properties that enter into the mathematical definition of arithmetic bear the entire weight of determining the ' numbers, or, to put it the other way around , they wouldn t think , as " Benacerraf ([ 1965] 1983, 291) does, that the properties of numbers which do not stem from [their arithmetic structure ] are of no consequence whatsoever." ' Given that we are not committed to Frege s semantics, our strategy can presuppose that Peano arithmetic , since it is silent on whether numbers are abstract, have members, have grammatical properties and so on, is an incomplete account of our informal knowledge of the numbers that is involved in determinate reference to them. If there is an argument , independent of Fregean semantics, that such nonnumber -theoretic properties are illegitimate , as we shall see there is in Benacerraf' s indeterminacy argument , we have to analyze it to find out

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what philosophical perspective it invokes. But here that is not necessary . We can simply argue that , in providing features that establish the incompatibility of those interpretations , the skeptic has provided us with the features required to break the symmetry .12 4.5 Benacerraf ' s Argument At one point in his argument , Benacerraf ([ 1965] 1983) assumes that , if we want to say that one set-theoretic account of the numbers is correct, we must be able to justify saying it . One might balk at this assumption in the same spirit in which , as I suggested in section 4.3 above, one ' might balk at Quine s indeterminacy argument : granting everything he says about radical translation , language neutral meanings could exist unknow ably. So why couldn ' t numbers exist unknow ably ? Realists, after all , hold that numbers are abstract objects that exist independently of us and our epistemic capacities, so, of all philosophers , realists are in the best position to insist that numbers can be unknowable things in -themselves. In reflecting on his argument , Benacerraf (1996, 26) acknowledges that " the need to refer to [the assumption ] is a genuine weakness of the argument ." Nonetheless, I don ' t want to make anything out of this weakness. I agree with Benacerraf' s ([ 1965] 1983, 284) earlier claim that " the " position that this is an unknowable truth is hardly tenable. Dependency on the principle that there cannot be a correct account without some way to show that it is correct is a weakness of his indeterminacy argument , to be sure, but , by the same token, it would be a weakness of realism if it had to say that there is a correct account but concede that it has no way to show that it is. It would be theoretically lame for realists to take this position , since it would raise questions about realism just as Kant' s things -in -themselves raised questions about his transcendental idealism . Benacerraf' s argument moves from a preliminary stage in which he concludes that numbers could not be sets to a final stage in which he concludes that numbers could not be any other objects either. If the 12. Acceptingour strategydoesnot requireone to deny that thereare casesof theories of which two or more membersof a setof competingmodelsareequallygood interpretations , whereit is indetenninatewhich of the modelsis the " right" interpretationof the " . , the indeterminacy theory But it requiresoneto saythat, in suchcases " is a consequence of the incompleteness of the theory. When an argumentfor the indetenninacyof the choicebetweenone and anothermodel for a theory turns on an incompleteness in the specificationof the theory, we havean argumentfor completingthe specificationrather than an argumentfor skepticismabout the choice. The completionof a theory resolves the choiceof an interpretationfor the theory in favor of one of the models.

TheSemantic to Realism 107 Challenge final stage of his argument were just a generalization of the preliminary ' stage, we could apply our strategy to Benacerraf s overall argument and conclude that it is self- defeating ( because, e.g ., descriptions of Ernie' s and Johnny' s set-theoretic constructions enable us to distinguish their interpretations of Peano arithmetic from the intended interpretation , the numbers themselves). This would be the end of it . Benacerraf ([ 1965] 1983, 290) would have been right to claim that "' ' " objects do not do the job of numbers singly, but wrong to take that " as grounds for claiming that numbers could not be objects at all ." Singly, they and the objects of a deviant interpretation that structurally masquerade as numbers exhibit conflicting properties that our strategy can use to distinguish one from the other. But the final stage of the argument is more than just a generalization of the preliminary stage. In the final stage, Benacerraf attempts to illegitimatize properties that could be used to rule out deviant interpretations . Benacerraf ([1965] 1983, 291) is aware that there are such I' properties : . . . it would be only [the properties of numbers that do not stem from the relations they bear to one another in virtue of their fonning a progression] that would single out a number as this object or that." And he takes specific steps to illegitimatize them. In this ' ' ' respect, Benacerraf s argument is like Quine s rather than Kripke s. ' Thus, the central question concerning Benacerraf s argument is whether the argument provides a compelling reason for thinking that properties of numbers that do not arise from the relations they bear to one another in virtue of their fonning a progression are illegitimate for us to use in distinguishing the numbers from other systems of objects that also form a progression . Benacerraf ([ 1965] 1983, 291) claims that " [t ]he search for which numbers really independently identifiable particular objects the are (sets? Julius Caesars?) is a misguided one." It is " misguided " because, as he puts the driving principle of his ([ 1965] 1983, 290) argument , If one theory can be modeled in another . . . then further questions about whether the individuals of one theory are really those of the second just do not arise. It is in explaining why such questions do not arise that Benacerraf makes his Quine-like attempt to illegitimatize properties that rule out deviant interpretations . Benacerraf ([1965] 1983, 290) says that such because " the mathematician ' s interest stops at questions do not arise " the level of structure , and hence it is " mistaken " for philosophers " not ' " [to be] satisfied with [the mathematician s] limited view of things and /Ito want to know more."

108 Chapter4 If Benacerraf were right , then the only properties available to us for discriminating between the individual of one theory and the individuals of another theory in which the first is modeled would be the structural properties represented in the theories (or in mathematics generally ?). Depending on how widely the notion of a structural property is taken, we might still be able to resist Ernie' s and Johnny' s identifications of numbers with sets, since properties of sets like having members and properties of numbers like being prime are mathematical 13 properties . But the Caesar question and others would be reopened, since abstractness, the various properties it entails (e.g ., causal inertness ), grammatical properties , and so on are not mathematical properties . Not only would questions like the Caesar question be reopened, but our general strategy for resisting indeterminacy arguments would be invalidated . If the only properties we can use for our strategy are those in which the mathematician takes an interest, the discovery of a paradoxical element would no longer guarantee that our strategy applies . We could not generalize from the way in which indeterminacy was resisted in the cases where Peano arithmetic is modeled in set theory to all caseswhere Peanoarithmetic is modeled in another theory. Such a generalization assumes that any properties of the deviant interpretation , including nonmathematical ones, can be used in applying our strategy. The real weakness of Benacerraf' s argument is his claim that it is " mistaken " for " [to be] satisfied with [ the mathemaphilosophers not ' tician s] limited view of things ." Benacerraf ([1965] 1983, 291) thinks that philosophers who " want to know more" are making a mistake because there is nothing more of consequence to know about the numbers than the properties of them that " stem from the relations they bear to one another in virtue of being arranged in a progression." Other " " properties of the numbers are of no consequencewhatsoever, Benacerraf ([ 1965] 1983, 291) argues, because . . . in giving the properties . . . of numbers you merely characterize an abstractstructure- and the distinction lies in the fact that the " elements" of the structure have no properties other than those relating them to other " elements" of the same structure . This begs the question . In the quotation that heads Benacerraf' s ([ 1965] 1983, 272) paper, R. M . Martin says that the philosopher who is " more

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sensitive [than the mathematician ] to matters of ontology " thinks there is more to be said about the properties of numbers than arithmetic structuralism allows . Benacerraf ([ 1965] 1983, 290) believes that Martin is " mistaken," but his reply can' t just be a reassertion of the claim that there is no more to be said about the properties of numbers than arithmetic structuralism allows . Benacerraf ([ 1965] 1983, 290) is surely right to say that a concern with the answer to philosophical questions about numbers " miss[es] the " point of what arithmetic , at least, is all about , but that' s hardly decisive in a philosophical controversy. He is surely wrong to think that a concern with the answer to philosophical questions about numbers misses the point of what the controversy concerning the indeterminacy of numbers is all about. That controversy is at least as much philo as it is mathematical . It is hard to see why philosophers who sophical " want to know more" are not simply philosophers wanting to do their own philosophical thing . Wetzel' s (1989b, 282) careful analysis of Benacerraf' s argument fingers premise (A ) as the real culprit , (A ) Structural properties (i .e., the properties of numbers that stem from the relations they bear to one another in virtue of being arranged in a progression) do not suffice to individuate [numbers ]. but , although I agree with her that (A ) is mistaken , I think that the real culprit is (B). (B) The essential properties of numbers are all structural . Numbers , she (1989b, 282) observes, are not the sorts of things that have members, but , essential as that negative property is to numberhood ' " " , it isn t a structural property (Zermelo numbers and von Neu " " mann numbers both have members). It is essential to seventeen that it has no spatiotemporallocation , that it is causally inert , that it is not mind - dependent, and so on, but these are not structural properties . (B) is mistaken because, although some essential properties of numbers perhaps even those that matter most for most purposes - derive from the relations they bear to one another in virtue of being arranged in a progression, others, at least as the realist sees it , derive from the relations that numbers bear to one another, to sets, to propositions , to sentences, and so on in virtue of all of them being abstract objects.14 14. Thepointheredoesnotdependonassuming . In providingnonmathematical realism absence of spatiotemporal location , causal propertiesof numberssuchas necessary

110 Chapter4 The good name of mathematics does not suffer because our knowledge of the numbers goes beyond mathematics. Arithmetic is not intended to be more than a theory of the mathematical structure of the numbers. Mathematics in the broadest sense has nothing to say about some essential properties of numbers because it has no metaphysical , of course, to criticize mathematics, but only to aspirations . This is not ' impugn Benacerraf s philosophicalassumption that number -theoretic properties exhaust those that are relevant to distinguishing numbers from other things . Hence, the fact that a property is one " in which the mathematician professes no interest" does not prevent it from playing a role in distinguishing the intended interpretation of arithmetic from a deviant interpretation . Benacerraf' s attempt to illegitimatize nonmathematical properties is the equivalent of Quine ' s attempt to illegitimatize nonreferential properties such as synonymy and analyticity . Like Quine , Benacerraf has to exclude the objectionable properties in order to establish his indeterminacy thesis, but , also like Quine, his grounds for exclusion are inadequate. The seriousnessof this inadequacy can be appreciated from the perspective of a general conception of indeterminacy arguments . The cost to Benacerraf of failing to illegitimatize nonmathematical properties is the failure of his argument for the indeterminacy of referenceto numbers, just as the cost to Quine of failing to illegitimatize intensional properties was the failure of his argument for the indeterminacy of translation . For Benacerraf' s ([ 1965] 1983, 291) claim that " The number words do not have " single referents follows only if number theory encompasses the full range of properties that can be used to exclude unintended models of arithmetic . Since it doesn' t , his indeterminacy argument cannot take number theory as a complete explication of our knowledge of the numbers, and the alleged symmetry on which the argument rests can be rejected on the same grounds on which we rejected the alleged symmetries on which Quine ' s and ' Kripke s arguments rest. They underestimate our informal knowledge of the domain . inertness , and so on , mathematical realism makes it easy to refute identity statements " " like 17 = Julius Caesar . But on mathematical conceptualism and mathematical nominalism too , we can make short work of those identities . The non - mathematical properties that function as counterexamples change , reflecting the respective psychological and or fictionalist conception of numbers . Conceptualists can refute " 17 = inscriptionalist Julius Caesar " by observing that Caesar , unlike 17, crossed the Rubicon . Inscriptionalist nominalists can refute it by observing that 17, unlike Caesar , is a construction out of orthographic symbols . Fictionalist nominalists can refute it by observing that 17, unlike Caesar , never existed in reality .

to Realism 111 TheSemantic Challenge 4.6 The Metaphysicsof Number-TheoreticSkepticism Reflection on the selectivenessof formalization and theory construction ought in itself to have made us suspicious about the claim that only arithmetic properties are legitimate to use in distinguishing numbers from other things . It is in the nature of formalization and theory construction to select those properties that have a distinctive role in the structure chosen for study. Moreover , selectiveness is essential in the formal sciencesbecause numbers and the other objects they study are not the private property of anyone discipline . Numbers belong to the ' domain of mathematics in virtue of the mathematician s concern with their arithmetic structure , but they also belong to the domain of phi ' losophy in virtue of the philosopher s concern with the most general facts about reality . The domains ought to be thought of not as disjoint but as overlapping . Numbers are, as it were, communal property . What is proprietary is only the special interest a discipline takes in them. The mathematician ' s special interest in numbers is with their arithmetic ' structure ; the philosopher s is with their ontology and epistemology. From the standpoint of the inherent selectivenessof formalization and ' theory construction , the assumption of Benacerraf s argument that we know nothing about the numbers except what is in number theory seems truly bizarre . Or, rather, it seems so today. Back in the sixties when Benacerraf " wrote " What Numbers Could Not Be, the assumption seemed anything but bizarre . At that time , philosophy was in the throes of a naturalistic revolt against traditional metaphysics. The idea of philoso phy as an a priori discipline with an independent subject matter consisting of the most general facts about reality was widely seen as a dangerous piece of speculative metaphysics. To some of those in the vanguard of the revolution , the idea was nonsense; to practically all , it was responsible for most of the ills of traditional philosophy . The new movements of the day- logical empiricism , Oxford philosophy, late Wittgensteinianism , and Quinean naturalism all took the matter of its own . Only no has naturalistic view that philosophy subject . of their own matters the sciences have subject Philosophy, properly conceived, is a second-order discipline , concerned with the semantic clarification of ordinary and / or scientific language. It is no wonder that , writing at such a time , Benacerraf regarded it as a mistake for philosophers to inquire into' philosophical facts about numbers that fall outside the mathematician s sphere of interest. There are none. The is, in metaphysical' philosopher who sets "out to discover such facts " It chimeras. is no Wittgenstein s (1953, sec. 94) words , in pursuit of

112 Chapter4 wonder , therefore, that the assumption that mathematicians tell us whatever there is to know about numbers did not seem bizarre in those halcyon days when it seemed to almost everyone that the metaphysical storms of the past had finally blown themselves OUt.t5 4.7 Putnam and Wittgenstein Putnam (1983, 423) says that , as well as showing that no formal system can capture our intuitive notion of a set, Skolemite considerations show that no " formalization of total science" (or even of our total belief system) can capture our intuitive notion of the world . Putnam (1983, 424) seesthis " paradox " as a serious problem for any philosopher or philosophically minded logician who wishes to view set theory as the description of a determinate independently existing reality . From the perspective of this chapter, this is a " serious problem " only for those who think that axiomatic set theory exhausts the information on the basis of which we determinately refer to sets, or even that it exhausts the information on the basis of which it is taken to be a description of the domain of sets. Putnam (1983, 424) is clearly aware that mathematical realists can the reasoning underlying the " paradox " on the grounds that challenge " it is natural to think that somethingelse- our ' understanding '" rather than axiomatic set theory captures the intuitive notion of a set. Since this , in its inchoate form , is one of the thoughts underlying our strategy, we need to look at why Putnam rejects the thought that our understanding is what captures the intuitive notion of a set. 15. In reflectingon " What NumbersCould Not Be," Benacerraf(1996,28) considersthree " waysthat realistsmight try to escapethe argumentin that paper. Oneof them, stick[ing] to [their] realist guns and refusing [. . .] to yield on the matter of the possibility of " taken establishingthe right answeras the right answer, is the approachthat we have in this chapter. Benacerraf(1996,28-29), however, finds this approachthe least" appealing " of the three possibleescaperoutesbecause Unlessa strong argumentcan be mounted for its realist under-pinnings that is itself groundedin a persuasiveanalysisof mathematicalpractice, . . . [this escape ." route] is " simply too theoretical , too metaphysical SinceBenacerrafsattack on the determinacyof referenceto numbersitself relieson a naturalist metaphysicsto support the claim that philosophy is a second-order discipline with no aspectof reality as its subjectmatter, his claim that the realist's escape route is " simply too metaphysical " is somethingof the pot calling the kettle black. Furthermore , given the epistemologicalexplanationin chapter2, the kettleis not soblack anymore.

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Putnam (1983, 424) says: " But what can our " understanding come to, at least for a natu ralistically minded philosopher , which is more than the way we use our language? The Skolem argument can be extended . . . to " " show that the total useof the language. . . does not fix a unique " " intended interpretation . So the naturalistically minded philosopher is in the soup . The Platonistically minded philosopher , according to Putnam (1983, 424), has an immediate response to the Skolem argument , namely, . . . he will take this as evidence that the mind has mysterious " " faculties of " grasping concepts (or perceiving mathematical " which the naturalistically minded philosopher will objects ) never succeedin giving an account of. . . . the extreme positions Platonism and verificationism - seem to receive comfort from the " " wwenheim -Skolem Paradox; it is only the moderate position " " " (which tries to avoid mysterious perceptions of mathematical " objects while retaining a classical notion of truth ) which is in deep trouble . " " Putnam is wrong to think realists take comfort from the indeterminacy arguments, which purport to show that the intended interpretation cannot be uniquely fixed . I imagine that Putnam is thinking that realists might claim , as Benacerraf ([ 1965] 1983) notes, that they can hold that the intended interpretation is correct even though they are unable to justify it as the correct interpretation . But, as I argued in section 5, such a claim is not only theoretically lame but it raises doubts about realism parallel to the doubts raised by Kant' s claim that there are unknowable things -in -themselves. Hence, realists, no less than ' naturalists , need to explain why Putnam s extension of the Skolem ' ' " to show that the total use the of ' language. . . does not fix a argument ' " doesn ' intended t prevent them from holding interpretation unique that we can refer determinately to mathematical objects. If , as Putnam " (1983, 423) claims, a fonnalization of total science. . . could not rule out " " unintendedinterpretations ," the realist may not be in the deep trouble " that the naturalistically minded philosopher " is in , but the realist' s trouble is deep enough . But, in fact, realists are not in any trouble at all , providing they subscribe to the right semantics. Benacerraf (1985, 110) has put his ' finger on what is the nub of Putnam s skepticism : Of course, . . . no explanation [of how an unintended interpretation can be ruled out ] can be satisfactory. . . . The reason resides

114 Chapter4 in our logo centric predicament : Any explanation must consist of additional words . Words which themselves are going to be said to need interpretation . [Putnam ' s] strategy has a wondrous simplicity and directness. He will construe any account we offer as an uninterpreted extension of our already deinterpreted theory by explaining we merely produce a new theory which , if consistent , will be as subject to a plethora of (true ) interpretations as was the old . (" Any interpretation is itself susceptible to further " interpretation . ) ' s ruleWittgenstein following argument is the engine that drives Put' nam s Skolemite argument . This might seem at first to be bad news for the realist , but , on closer examination , it turns out that the realist, unlike ' Wittgenstein s conceptualist interlocutor in the PhilosophicalInvestigations , is in a good position to avoid Wittgenstein ' s paradox . ' ' ' Wittgenstein s (1953, sec. 141) criticisms of Frege s (and Russell s) approaches to semantics were supposed to show that his own usebased approach to meaning and his interlocutors mentalistic approach are the only ones left as serious contenders for our semantic allegiance. If those criticisms had succeeded, Wittgenstein ' s use-based approach would have emerged as the clear winner because his rule- following " " argument refutes the idea that what comes before the mind provides the normative basis for fixing meaning in the use of language. But if those criticisms do not succeed in eliminating all other approaches to meaning , a refutation of the mentalistic approach does not settle the issue of what provides normative force. In The Metaphysicsof Meaning, I (1990b, 21- 133) argue that Wittgen stein' s criticisms do not work against an approach to semantics that is realist rather than mentalistic and non -Fregean rather than Fregean. ' Wittgenstein s criticisms are too closely tailored to Fregean semantics and the Fregean conception of an ideal- logically perfect- language. I argue that the criticisms do not work against a theory in which senses are non -Fregean and reside in natural languages, understood as a system of sentencetypes systematically correlated with abstract senses. I then go on to argue that abstract sensessupply the normative element missing in mentalistic approaches, and non -Fregean intensionalism supplies a conception of how grammatical norms are applied in the use of language that avoids Wittgenstein ' s criticisms of the interlocu tor ' s Fregean conception of the application of language. On the interlocutors conception of meaning , what comes before the mind when we understand a word , being an uninterpreted symbol , " " requires a method of projection to interpret it , but the addition of such an interpretation is only the addition of another uninterpreted

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lI symbol . Thus, as Wittgenstein (1953, sec. 198) says, [it ] hangs in the air along with what it interprets , and cannot give it any support . II do not determine meaning . This is the by themselves Interpretations " I Ilogocentric predicament . Something more than symbols, words , and other specimens of orthography is required to determine meaning . Our linguistic realism provides that something more: a semantics of abstract senses. Abstract senseshave the normative semantic force that mental meanings lack because, rather than uninterpreted symbols, they are the semantic content of the sentence types that configurations of symbols express in the use of language. On my (1990b, 86- 111, 142 55) account in a speaker' s which it down affair of the use of language, is a top knowledge of the sentence-sense correlations of the language results from his or her ability to use configurations of spoken or written symbols as tokens of ' sentential types. The literal use of language consists in the speaker s realizing an intention to use a token of a word type with the sensethat the type has in the language. Since the abstract senseof a word type is not a symbol but a semantic universal , there is no appeal to interpretations that are only further symbols, and hence " there is no " logocentric predicament . Furthermore , since the sense of a word type is the norm for the literal use of its tokens, the sense has the force to determine their proper application . Since sensedetermines ' ' " what we call ' cases," obeying the rule and going against it' in aCtual ' this account of the use of language avoids Wittgenstein s (1953, sec. 201) paradox about rule following . Once sensesof the right sort enter the piCture, the predicament disappears. The logo centric predicament is a senselesspredicament . ' In form and resolution , Putnam ' s argument is close to Kripke s, ' which shouldn t be surprising in light of their common debt to Wittgen ' ' stem. But in one respect, Putnam s argument is closer to Quine s and ' Benacerraf s arguments : it contains, not a full fledged argument for illegitimacy , as their arguments do , but a reason for thinking about abstract objects - abstract senses, too , presumably - are not philosophi quotations show, is cally legitimate . The reason, as "the above Putnam " that naturalism precludes such mysterious objects. '" ' In characterizing realism as resorting to " mysterious perceptions , Putnam lines up with Gottlieb , Field , Chihara , and Dummett in unwarrantedly disparaging realism. Thus, the criticism of them in chapter 2, section 4 applies to Putnam: he confuses mystery with mysticism . Further, as shown in chapter 1, section 3.1, and in chapter 2, talk of ' " ' ' " faculties of ' grasping concepts (or perceiving mathematical objects ) does not necessarily imply a connection to abstract objects that puts such faculties beyond a naturalistic psychology of our cognitive proc-

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esses. There is, as far as I can see, no reason whatsoever to think that scientific psychology cannot in principle explain the rational faculties on which our account of mathematical knowledge is based, though , to be sure, current scientific psychology is far from telling us much about them. While it is true that some realists are guilty of obfuscation in talking about grasping facts about abstract objects in terms of causal connections, others - GOdel in particular - are entirely innocent. Hence, Putnam ' s implicit illegitimatizing argument fails too. 4.8 Conclusion If the argument in the present chapter is right , then the true significance of the fact that there are indefinitely many models that qualify structurally as well as the numbers themselves as interpretations of arithmetic is not , as Benacerraf ([ 1965] 1983) claimed , that we cannot refer determinately to the numbers, or, as Putnam (1983) claimed , that we cannot refer determinately to " the real world ," but rather that determinate reference to the numbers and to the real world depends as much upon our language and philosophy as it does upon our science.