Neo-Fregeanism and Quantifier Variance

Sider argues that, of maximalism and quantifier variance, the latter promises to let us make better sense of neo-Fregeanism. I argue that neo-Fregeans should, and seemingly do, reject quantifier variance. If they must choose between these two options, they should choose maximalism. 1: Introduction Benacerraf’s problem lies at the heart of philosophy of mathematics: the truth of mathematical statements appears to require the existence of infinitely many abstract mathematical objects, yet we seem to know these truths without having perceptual access to abstracta. What grounds our knowledge of mathematics? Philosophical responses typically take one of two paths, attempting either to dispense with abstract objects, or else to show how we can have knowledge of them. Neo-Fregeans take the second path, arguing that our knowledge of logic together with our knowledge of a definitional fact about the word ‘number’ secures our knowledge of abstracta and arithmetic. The definitional fact is Hume’s Principle: ∀F∀G [the number of Fs = the number of Gs iff the Fs stand in one-one correspondence with the Gs] We know that there are concepts which stand in one-one correspondence with each other (e.g. the concept being non-self-identical stands in one-one correspondence with itself); via the biconditional, we can thus establish that there are numbers. Moreover, we can derive the axioms of Peano Arithmetic from this starting point using logic alone.