Abstraction and Epistemic Entitlement: On the Epistemological Status of Hume's Principle

Abstract

The abstractionist programme of foundations for classical mathematical theories is, like its traditional logicist ancestors, first and foremost an epistemological project. Its official aim is to demonstrate the possibility of a certain uniform mode of a priori knowledge of the basic laws of arithmetic, real and complex analysis, and set theory (or as much set theory as anyone might soberly suppose to be indeed knowable at all.) It is a further issue whether a successful execution of the abstractionist project for a particular branch of mathematics would amount to a local vindication of logicism in some interesting sense of that term. Traditional logicism aimed to show that mathematical knowledge could be accomplished using only analytic definitions and theses of pure logic and hence is not merely a priori if logic is but is effectively a proper part of logic. Abstractionism, however, adds abstraction principles to the base materials employed in the traditional logicist project—principles that, at least in the central, interesting cases, are neither pure analytic definitions3 nor theses of pure logic as conventionally understood. Thus, whatever significance they may carry for the prospects for logicism, in one or another understanding of that doctrine, the epistemological significance of technically successful abstractionist projects must turn, one would suppose, on the epistemological status of the abstraction principles used, with any demonstration of a priority in particular being dependent on whether those principles can themselves rank as knowable a priori even if they are neither definitions, nor truths of logic, strictly understood. My primary focus here will be to critique this natural thought.