Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts (...) that such constraints are ‘toothless’, showing they both assuage Frege’s original concerns and accommodate neo-logicist intents by dismissing ‘arrogant’ definitions. (shrink)

Dedekind’s methodology, in his classic booklet on the foundations of arithmetic, has been the topic of some debate. While some authors make it closely analogue to Hilbert’s early axiomatics, others emphasize its idiosyncratic features, most importantly the fact that no axioms are stated and its careful deductive structure apparently rests on definitions alone. In particular, the so-called Dedekind “axioms” of arithmetic are presented by him as “characteristic conditions” in the _definition_ of the complex concept of a _simply infinite_ system. Making (...) sense of Dedekind’s method may be dependent on an analysis of the classical model of deductive science, as presented by authors from the eighteenth and early nineteenth centuries. Studying the modern reconstructions of Euclidean geometry, we show that they did not presuppose deductive independence of the axioms from the definitions. Authors like Wolff elaborated a mathematics _based on definitions_, and the Wolffian model of deductive science shows significant coincidences with Dedekind’s method, despite the great differences in content and approach. Wolff had a conception of definitions as _genetic_, which bears some similarities with Kant’s idea of synthetic definitions: they are understood as positing the content of mathematical concepts and introducing thought objects (_Gedankendinge_) that are the objects of mathematics. The emphasis on the spontaneity of the understanding, which can be found in this philosophical tradition, can also be fruitfully related with Dedekind’s idea of the “free creation” of mathematical objects. (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 paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern propositional systems, and show that Frege does not have a separable propositional logic, definable in terms of primitives of Grundgesetze, that corresponds to modern formulations of the logic of “not”, “and”, “or”, and “if…then…”. Along the way we prove a number of novel results about the system of propositional logic found in Grundgesetze, and the broader system obtained by including identity. In particular, we show that (...) the propositional connectives that are definable in terms of Frege’s horizontal, negation, and conditional are exactly the connectives that fuse with the horizontal, and we show that the logical operators that are definable in terms of the horizontal, negation, the conditional, and identity are exactly the operators that are invariant with respect to permutations on the domain that leave the truth-values fixed. We conclude with some general observations regarding how Frege understood his logic, and how this understanding differs from modern views. (shrink)

Dans le deuxième volume, inédit, de La Philosophie de l’Algèbre, Jules Vuillemin fait une lecture inattendue et suggestive de l’œuvre de Richard Dedekind. Nous avons essayé de comprendre, en mobilisant les idées et outils de Vuillemin, les résultats de cette lecture. Ceux-ci nous semblent poser en particulier le problème des rapports entre histoire des sciences et philosophie des sciences. Notre article propose un diptyque pour présenter les questions que nous avons voulu poser au texte de Vuillemin. D’une part, nous analysons (...) de quelle manière Vuillemin continue et approfondit le travail de Jean Cavaillès. D’autre part, nous souhaitons accentuer la distance qu’établit Vuillemin entre l’histoire mathématique et son interprétation par les filiations conceptuelles qu’il propose comme essentiellement distinguées des relations historiques. (shrink)