Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint

We now know of a number of ways of developing Real Analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals by Abstraction", differs by placing additional emphasis upon what I here term Frege's Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is therefore a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of Analysis. I The basic formal prerequisite for a successful neo-Fregean—or as I shall sometimes say: abstractionist—foundation for a mathematical theory is to devise presumptively consistent abstraction principles strong enough to ensure the existence of a range of objects having the structure of the objects of the intended theory. In the case of Number Theory, for instance, the task is to devise presumptively consistent abstraction principles sufficient to ensure the existence of a series of objects having the structure of the natural numbers: a series of objects that constitute an ω−sequence. As is now familiar, second order logic, augmented by the single abstraction, Hume’s Principle, accomplishes this formal prerequisite.1 The outstanding question is therefore whether Hume's Principle, beyond being presumptively consistent, may be regardedion, Hume’s Principle, accomplishes this formal prerequisite.1 The outstanding question is therefore whether Hume's Principle, beyond being presumptively consistent, may be regarded 1 This result is now commonly known as Frege's Theorem. It is prefigured by Frege in [6] §§82-3 and reconstructed in detail by Wright in [15], §xix. Other detailed accounts of the proof are given in Boolos [1], in an appendix to Boolos [2] and in Boolos & Heck [4]