Reals by Abstraction
Bob Hale · The Western Ontario series in philosophy of science · 2000
The neo‐Fregean argues that principles having a certain `abstractive’ form can play a distinctive foundational role in the philosophy of mathematics––the paradigm being the reduction of elementary arithmetic to the single second‐order sentence often referred to as ’Hume's Principle’ (HP). This essay considers an abstractionist account of another mathematical domain––real analysis. The proposal has two parts: (1) an abstraction principle (’ratio abstraction’) is given, which represents the real numbers as ratios of quantities. However, this procedure is effective only if we are already given a domain of objects with a particular kind of structure. (2) A series of abstractions (starting with HP, and including a Dedekind‐inspired ’cut abstraction’) gives a domain with the required kind of structure to underpin (1).