Sanford Shieh

The days when Frege was more footnoted than read are now long gone; still, until very recently he has been read rather selectively. No doubt many had an inkling that there’s more to Frege than the sense/reference distinction; but few, one suspects, thought that his philosophy of mathematics was as fertile and intriguing as the present collection demonstrates. Perhaps, as Paul Benacerraf’s essay in this collection suggests, logical positivism should be held partly responsible for the neglect of this aspect of Frege’s thought, since it contributed to the following common impression of Frege’s philosophy of mathematics: Frege’s project was to reduce arithmetic to logic, in order to provide an empiricist account of mathematical knowledge; Russell’s paradox showed that this was not possible, and that the best that can be done is to reduce arithmetic to set theory; so, Frege’s logicism is demonstrably a philosophical dead end.

The central premise of Michael Dummett's global argument for anti-realism is the thesis that a speaker's grasp of the meaning of a declarative, indexical-free sentence must be manifested in her uses of that sentence. This enigmatic thesis has been the subject of a great deal of discussion, and something of a consensus has emerged about its content and justification. The received view is that the manifestation thesis expresses a behaviorist and reductive theory of meaning, essentially in agreement with Quine's view of language, and motivated by worries about the epistemology of communication. In the present paper I begin by arguing that this standard interpretation of the manifestation thesis is neither particularly faithful to Dummett's writings nor philosophically compelling. I then continue by reconstructing, from Dummett's texts, an account of the manifestation thesis, and of its justification, that differ sharply from the received view. On my reading, the thesis is motivated not epistemologically, but conceptually. I argue that connections among our conceptions of meaning, assertion, and justification lead to a conclusion about the metaphysics of meaning: we cannot form a clearly coherent conception of how two speakers can attach different meanings to a sentence without at the same time differing in what they count as justifying assertions made with that sentence. I conclude with some suggestions about how Dummett's argument for global anti-realism should be understood, given my account of the manifestation thesis.

The most puzzling and intriguing aspect of intuitionism as a philosophy of mathematics is its claim that classical deductive reasoning in mathematics is illegitimate. The two most well-known proponents of this position are L. E. J. Brouwer and Michael Dummett. Both of their criticisms of the use of classical logic in mathematics have, by and large, been taken to depend on the thesis that the principle of bivalence does not apply to mathematical statements; and the difference between these criticisms is taken to consist in their grounds for this thesis. Brouwer bases this thesis on an account of the nature of mathematics: it is an activity of mental construction essentially independent of language. Dummett, in contrast, bases the thesis on his anti-realist view of the nature of meaning. ;My first principal argument in this essay shows that the conception of anti-realism just outlined is problematic as a reading of Dummett. To begin with, Dummett explicitly rejects, not only the evidentiary restrictions of this conception, but also, more seriously, the use of specific epistemological theses in a philosophical account of meaning. Moreover, I give an alternative account of anti-realism, consisting of alternative interpretations of the manifestation requirement and of the rejection of realist truth conditions, that do not rely on an epistemology of meaning, and hence, a fortiori, not on evidential restrictions. ;My second principal argument in this essay consists of drawing a consequence of this conception of anti-realism, namely, that the failure of bivalence cannot, by itself, support a cogent criticism of classical logic. The basic reason is that, given my view of anti-realism, the claim that the legitimacy of applying classical reasoning to a class of statements depends on whether bivalence applies to that class already implies that the system of deductive reasoning applying to these statements is non-classical. Hence an argument against classical logic based on this claim is question-begging. ;Finally, I conclude by showing that Dummett's criticism of classical reasoning, as reconstructed by my third argument, fails