Eckersley-Waites, T; (2012) Neo-logicism and a priori arithmetic. Masters thesis , UCL (University College London).

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Abstract

In this thesis I argue that the neo-logicist philosophy of arithmetic provides a possible response to a challenge that stems from Benacerraf's paper 'Mathematical Truth'. I discuss the precise nature of the challenge and argue that it deserves to be taken seriously. Frege's own solution to something like this challenge is to appeal to the context principle – that one should “never... ask for the meaning of a word in isolation, but only in the context of a proposition.” (Grundlagen der Arithmetik, Introduction). I evaluate this principle (interpreted as a principle governing both sense and reference) and argue that if it is accepted, the neo-logicist response becomes an epistemic possibility. I then assess Wright's version of neo-logicism as developed in his Frege's Conception of Numbers as Objects and subsequent papers. I highlight the way in which his argument for the analyticity of arithmetic appeals to the analyticity of Hume's Principle (HP). I focus on his argument for the claim that HP is analytic, and give a novel argument that it is not. I argue that HP is false in its most general formulation, and that any variant of HP that is true and from which the axioms of secondorder Dedekind-Peano arithmetic follow is not analytic on any reasonable conception of analyticity that can bear the requisite epistemological burden. However, it would be premature to give up on neo-logicism because of the inadequacy of HP. I emphasise the epistemological nature of the problem to which neo-logicism is supposed to be a solution, and hence assess whether some alternative, non-analytic but nonetheless a priori, principle can be used to salvage neo-logicism. Without coming to any firm conclusions, I argue that the most plausible version of this kind of neo-logicism is one that adopts Heck's 'Finite Hume's Principle' rather than HP itself.

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