Benacerraf's dilemma and natural realism for arithmetic.

Abstract

A natural realist approach to the philosophy of arithmetic is defended by way of considering and arguing against contemporary attempts to solve Paul Benacerraf's dilemma (1973). The first horn of the dilemma concerns the existence of abstract mathematical objects, which seems necessitated by a desire for a unified semantics. Benacerraf adopts an extensional semantics whereby the reference of terms for natural numbers must be abstract objects. The second horn concerns a desirable causal constraint on knowledge, according to which "for X to know that S is true requires some causal relation to obtain between X and the referents of the names, predicates, and quantifiers of S". Within the philosophical tradition, Benacerraf's dilemma crystallizes the tension between realists (roughly the first horn) and empiricists (roughly the second horn). It is shown that natural realism and naturalism meet. Both horns of the dilemma are amended. Abstract objects are conceived along conceptualist lines such that their existence is not mind-independent. The causal constraint is refined by drawing upon the work of Mark Steiner (1973). Natural realism is the notion that the truth-values of arithmetical statements are recognition-transcendent. Natural realism explains why one and only one arithmetic is applicable. It is in some sense discovered, which is captured by the idea that arithmetical statements have truth-values even if one is not able to adduce what they are. It is argued that with proper emendation, like that of Philip Kitcher (1984), empiricism as envisioned by J. S. Mill is defensible. Furthermore, the epistemology of arithmetical knowledge is divided into two tiers. The first principles of arithmetic are acquired by causal interaction with physical objects. As Kitcher has shown, empiricism functions at the first-tier. The a priori is revised such that arithmetical knowledge generated from the first-tier is considered non-empirical. It is argued that pragmatists' indispensability argument, once qualified, allows that arithmetic's applicability is one reason to consider natural realism for that domain (Putnam 1979a). Furthermore, this thesis provides a case where aspects of Hilary Putnam's early and later views are rendered consistent. By way of utilizing both Putnam's writing in favour of realism (1979a), and those that break with them (e.g., 1981), it is suggested that he should not have abandoned his earlier view. In a nutshell, the realization of knowledge can depend upon agents' values, methods, and so on, without forcing the abandonment of realism (provided one has the correct values and methods). Putnam's early view sets the criteria for when one can be a realist about a given domain.