In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege's attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik Frege hardly could have construed Hume's Principle as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck's arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

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. (shrink)

In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege’s attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume’s Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck’s arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)