Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it (...) applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5). (shrink)

Frege's definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege's ill-fated Basic Law V. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.

In this article, I first analyze and assess the epistemological and semantic status of canonical value-range equations in the formal language of Frege’s Grundgesetze der Arithmetik. I subsequently scrutinize the relation between (a) his informal, metalinguistic stipulation in Grundgesetze I, Section 3, and (b) its formal counterpart, which is Basic Law V. One point I argue for is that the stipulation in Section 3 was designed not only to fix the references of value-range names, but that it was probably also (...) intended to play a regulative role with an eye to Basic Law V. I further intend to shed new light on the status of Frege’s identification of the truth-values with their unit classes in Grundgesetze I, Section 10 and determine its place and importance in his overall logicist project. In a subsequent section, I first discuss the hypothetical extendability of the logical system of Grundgesetze. In what follows, I analyze the incompleteness of that system—were it not inconsistent and hence trivially complete—by focusing on the role of the definite description operator and the axiom governing it (= Basic Law VI) as well as on the role of the application operator. The latter is defined by means of the former. I further comment on the redundancy of an axiom that could be designed to supplement Basic Law VI by incorporating the second clause of Frege’s two-part elucidation of the description operator. I argue that although it seems likely that Basic Law VI could have been shown to be dispensable in Grundgesetze, Frege would probably have been reluctant to dispense with it, despite his commitment to axiomatic parsimony. In my view, the reason is that he considers it essential that every primitive function-name of his formal language be governed and grounded by a basic law that is tailored to the nature and the role of the primitive name occurring at a key point in the law. Toward the end of the article, I propose solutions to two problems with Frege’s use of “=” in the concept-script sentences that result from his definitions involving the replacement of the double stroke of definition with the judgment-stroke. I conclude the article with a summary of the main results that I have achieved. (shrink)