Crispin WRIGHT: Wittgenstein on the Foundations of Mathematics, London: Duckworth 1980.

Michael Dummett1 has long been encouraging philosophers of language to take up the task of constructing for natural languages semantics alternative to that originating with Frege. A classical Fregean or 'realist' semantics is characterized by its fascination with truth; for a realist, the understanding of a statement need in some way be analyzed in terms of a knowledge of its truth conditions. Dum• mett envisions a likely alternative or 'anti-realist' semantics in the form of a generalization of the intuitionistic critique of classical mathematics. The nub of an antirealist semantics would be verification; the meaning of a statement is to be explained in terms of the conditions under which its assertion would be justi• fied. Dummett and, recently, Crispin Wrighe have argued that the meaning-as-use doctrine of Wittgenstein's Philosophical Investigations3 and Remarks on the Foundaticns of Mathematics4, provides the materials for a thoroughgoing anti• realism. DummettS believes that the species of anti-realism deriving from Witt• genstein is compatible with intuitionism in mathematics, while Wright sees in Wittgenstein an antirealism far surpassing the intuitionistic in its strictures. Wright's Wittgenstein would repudiate any philosophical semantics which pre• supposes that linguistic expressions have determinate sense or extension. The austerity of this position derives from an argument for what one might term "the nonobjectivity of sense", an argument which Wright takes as summarizing the conclusions Wittgenstein intended his reader to draw from a consideration of the procedure of following a rule.

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To hold that sense is nonobjective is to believe that a shared understanding of an expression in no way enjoins on speakers an adherence to a determinate pattern

1. Dummett, Michael: Truth and Other Enigmas, London: Duckworth 1978. 2. Wright, Crispin: Wittgenstein on the Foundations of Mathematics, London: Duck• worth 1980. References by chapter and section number. 3. Wittgenstein, Ludwig: Philosophicallnvestigations. G.E.M. Anscombe, trans; Oxford: Basil Blackwell1963. References are to Part I by 'PI' followed by paragraph number. 4. Wittgenstein, Ludwig: Remarks on the Foundations of Mathematics. G.H von Wright, R. Rhees and G.E.M. Anscombe, eds; G.E.M. Anscombe, trans; Oxford: Basil Blackwell 1978. References to 'RFM' followed by part and paragraph number. 5. Dummett, Michael: "Wittgenstein's Philosophy of Mathematics", reprinted in [1). 166 of use of the expression as correct. On this reading, Wittgenstein's remarks on following a rule are intended to bring one to the unsettling realization that a11 the means at our disposal for fixing the sense of a term are incapable of exerting any real compulsion on us to continue to employ the term in a way consonant with prior or present dealings with it. Putting it metaphysica11y, we might say that denying objectivity to sense is refusing to countenance the existence of meaning-dependent facts about language, or insisting that there is no real aspect of the world, other than the obvious general tendency of speakers to agree in judgments, against which linguistic moves might be counted correct or incorrect, true or false. If any real accord between sense and use is illusory, we can only employ language in ways which seem to us to cohere with our previous understanding. I believe that, in ascribing the "nonobjectivity of sense" thesis to Wittgenstein, Wright overlooks what is arguably the main point of the passages in which Witt• genstein's thoughts on rule-following received most detailed exposition. (pI 139- 242; RFM VI) In his survey of the phenomena of understanding, Wittgenstein brought his readers back again and again, from a multitude of directions, to this point. Indeed, there is a paragraph of the Remarks, in which Wittgenstein's reflections on this point seem to have lead him to anticipate the nonobjectivity thesis and to discard it: My question was: ",How can one keep to a rule?" And the picture that might occur to someone here is that of a short bit of handrail, by means of which I let myself be guided further than the rai! reaches. (But there is nothing there; but there isn't nothing there!) (RFM VII, 66) In querying the propriety of the "handrail" metaphor, Wittgenstein was ques• tioning the objectivity of sense. Since the training received in the mIes which guide language-use comes to an end at some stage, what determines the correct application of the mIes from the explanatory stage forward? In the initial reply, "But there is no thing there", Wittgenstein was trying on Wright's answer: that correct use is wholly indeterminate. The rejoinder, "There isn't nothing there" , serves to locate in this context the point to which I alluded above: that the metaphor neglects the special situation of the human mle-follower. It is the familiar style of human response which underwrites the determinateness of be• havior as informed by rules. It is a plain fact of our "natural history" that a determinate pattern in the use of an expression is guaranteed by features of the human reactions which Wittgenstein saw to be closely tied to meaning. Indeed, it is the character of human reactions that allows for the possibility of meaning and gives rise to meaning-dependent facts. One discovers a similar question and reply at PI 198 directing our attention in the same direction. Let me ask this: what has the expression of a rule - saya sign-post - got to do with my actions? What sort of connexion is there here? Weil, perhaps this one: I have been trained to react to this sign in a particular way and now I do so react to it. Wittgenstein believed that, in the series of reactions which constitute the prac-