Jamie Tappenden

Geometry and generality in Frege's philosophy of arithmetic

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining FregesGrundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Freges early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes ofGrundlagen are developed: the relationship Frege envisions between arithmetic and geometry and the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer.

Negation, Denial and Language Change in Philosophical Logic

This paper uses the strengthened liar paradox as a springboard to illuminate two more general topics: i) the negation operator and the speech act of denial among speakers of English and ii) some ways the potential for acceptable language change is constrained by linguistic meaning. The general and special problems interact in reciprocally illuminating ways. The ultimate objective of the paper is, however, less to solve certain problems than to create others, by illustrating how the issues that form the topic of this paper are more intricate than previously realised, and that they are related in delicate and somewhat surprising ways.

Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments?

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege’s mathematical environment. This feeds into a currently active scholarly debate because Frege’s supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege’s attitude toward metatheory in general. I show that this thesis gains no support from Frege’s puzzling remarks about independence arguments.

A Primer on Ernst Abbe for Frege Readers

Setting out to understand Frege, the scholar confronts a roadblock at the outset: We just have little to go on. Much of the unpublished work and correspondence is lost, probably forever. even the most basic task of imagining Frege’s intellectual life is a challenge. The people he studied with and those he spent daily time with are little known to historians of philosophy and logic. To be sure, this makes it hard to answer broad questions like: ‘Who influenced Frege?’ But the information vacuum also creates local problems of textual interpretation. Say we encounter a sentence that may be read as alluding to a scientific dispute. Should it be read that way? To answer, we’d need to address prior questions. Is it reasonable to think Frege would be familiar with the issue? Deep or superficial familiarity? Would he expect his readers to catch the allusion? Can he be expected to anticipate certain objections? Can people he knows be expected to press those objections? A battery of such questions arise, demanding a richer understanding of Frege’s environment. ernst Abbe might have been the only intellect of the first rank that Frege spent regular time with, apart from Frege’s years of graduate study in Gottingen (about which time we know little).1 Frege’s