Frege's Approach to the Foundations of Analysis (1874–1903)
The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ‘theory of quantity’ (‘Größenlehre’) in the second volume of his opus magnum Grundgesetze
der Arithmetik (Frege
1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege
1903. I begin with a few introductory remarks. In Section
2, I first analyze Frege's use of the term ‘source of knowledge’ (‘Erkenntnisquelle’) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of
knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege
1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of
real numbers in Die Grundlagen der Arithmetik (Frege
1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege
1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section
6, I return to Frege's own constructive treatment of analysis in Frege
1903 and succinctly describe what I take to be the quintessence of his account.
Document Type: Research Article
Affiliations: Munich Center for Mathematical Philosophy, Universität München, Ludwigstr. 31 80539, München, Germany
Publication date: 01 August 2013
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