What Numbers Could not Be
Author: Paul Benacerraf Publication: The Philosophical Review, 1965 Link: The Philosophical Review This is a compelling article about the nature of mathematical objects (here focusing the exposition on the natural numbers). Essentially, the paper argues that numbers cannot be any of their possible particular definitions (e.g., as particular sets, Church numbers, etc). Instead, when we talk of numbers, we speak of the abstract structure that relates them. So 2 is neither $(s (s 0))$ nor ${\varnothing, {\varnothing}, {\varnothing, {\varnothing}}}$ – not these particular objects, but the relation that 2 has to 1 and 0 and 3 and so on, in whatever definition you want to give to the whole system. ...