Why Worry about Weierstrass? Frege on the Paradox of Analysis

This paper investigates Frege's treatment of the paradox of analysis, focusing on the discussion in his posthumously published essay, “Logic in Mathematics”. Against an interpretation offered by Michael Beaney, it is argued that Frege is committed to the idea that analysis should faithfully capture already expressed senses (Sinne). Furthermore, and in line with a reading offered by Tyler Burge, it is maintained that such Fregean senses are not to be identified with conventional linguistic meanings, if such conventional meanings are thought of as fixed by actual linguistic practice – given that “actual linguistic practice” is understood in the narrow, empiricist sense that characterizes much analytic philosophy from logical positivism onward. Like Burge, the paper also argues that Fregean senses require a thicker, normative or teleological element: an aiming toward the logical perfection of the practice. However, the paper rejects Burge's conception of this normative or teleological element as an add-on to actual practice – an add-on which involves a systematic theoretical endeavor that goes beyond the competence and efforts of even the best real-life mathematicians. Instead, it is argued that Frege conceives the aiming toward logical perfection as inherent in the practice. This reading is justified by a detailed investigation of the polemical centerpiece of Frege's essay, namely, his attack on Weierstrass's conception of number.