On the Philosophical Interest of Frege Arithmetic

Traditional “Fregean” logicism held that arithmetic could be shown free of any dependence on Kantian intuition if its basic laws were shown to follow from logic together with explicit definitions. It would then follow that our knowledge of arithmetic is knowledge of the same character as our knowledge of logic, since an extension of a theory (in this case the “theory” of secondorder logic) by mere definitions cannot have a different epistemic status from the theory of which it is an extension. If the original theory consists of analytic truths, so also must the extension; if our knowledge of the truths of the original theory is for this reason a priori, so also must be our knowledge of the truths of its definitional extension. The uncontroversial point for traditional formulations of the doctrine is that a reduction of this kind secures the sameness of the epistemic character of arithmetic and logic, while allowing for some flexibility as to the nature of that epistemic character. Thus, it is worth remembering that in Principles (p. 457), Russell concluded that a reduction of mathematics to logic would show, contrary to Kant, that logic is just as synthetic as mathematics.