Frege: Definitions and Conceptual Analysis

Frege and Peano started in 1896 a debate where they contrasted the respective conceptions on the theory and practice of mathematical definitions. Which was (if any) the influence of the Frege-Peano debate on the conceptions by the two authors on the theme of defining in mathematics and which was the role played by this debate in the broader context of their scientific interaction?

This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century (...) geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer. (shrink)

This paper describes both an exegetical puzzle that lies at the heart of Frege’s writings—how to reconcile his logicism with his definitions and claims about his definitions—and two interpretations that try to resolve that puzzle, what I call the “explicative interpretation” and the “analysis interpretation.” This paper defends the explicative interpretation primarily by criticizing the most careful and sophisticated defenses of the analysis interpretation, those given my Michael Dummett and Patricia Blanchette. Specifically, I argue that Frege’s text either are inconsistent (...) with the analysis interpretation or do not support it. I also defend the explicative interpretation from the recent charge that it cannot make sense of Frege’s logicism. While I do not provide the explicative interpretation’s full solution to the puzzle, I show that its main competitor is seriously problematic. (shrink)

All contributions included in the present issue were originally presented at an ‘Author Meets Critics’ session organised by Richard Zach at the Pacific Meeting of the American Philosophical Association in San Diego in the Spring of 2014.

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...) independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

Husserl and Frege on the analysis of the concept of number and primitive logical concepts.

This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.