Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought (1879), by G. Frege.--Some metamathematical results on completeness and consistency; On formally undecidable propositions of Principia mathematica and related systems I; and On completeness and consistency (1930b, 1931, and 1931a), by K. Gödel.--Bibliography (p. [111]-116).

Textual and historical subtleties aside, let's call the idea that numbers are properties of equinumerous sets ‘the Fregean thesis.’ In a recent paper, Palle Yourgrau claims to have found a decisive refutation of this thesis. More surprising still, he claims in addition that the essence of this refutation is found in the Grundlagen itself – the very masterpiece in which Frege first proffered his thesis. My intention in this note is to evaluate these claims, and along the way to shed (...) some light on relevant passages of the Grundlagen. I will argue that Yourgrau does not make his case.The arguments with which we are concerned are found in the last three sections of Yourgrau's paper. A pervasive difficulty in these sections is that it is not clear exactly what Yourgrau is arguing against. The stated object of his attack is the Fregean thesis, a thesis about what numbers are; however, instead of a frontal assault, his strategy is to embark on a foray into the ill-defined issue of what it is that numbers number, where, roughly speaking, a number n numbers an object x just in case n can be legitimately assigned to x. The reason for this shift in emphasis appears to be rooted in a misconception. As we’ll see in more detail shortly, Yourgrau's argument against the Fregean thesis is based on an extension of a well known argument of Frege's found in §§22-3 of the Grundlagen, which Glenn Kessler has tagged the ‘relativity argument,’. According to Yourgrau, this is an argument ‘to the effect that what is literally numbered cannot simply be concrete objects’. This is incorrect. Frege himself clarifies the point of the argument in §21 with the following preface:In language, numbers (i.e., numerals] most commonly appear in adjectival form and attributive construction in the same sort of way as the words "hard" or "heavy" or "red," which have for their meanings properties of external things. It is natural to ask whether we must think of the individual numbers too as such properties, and whether, accordingly, the concept of number can be classed along with that, say, of color. (shrink)

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

Pity is an emotion which is intimately connected with virtue. If I were impervious to anger I could still be a paragon of rectitude. My emotional peculiarity might even be explained by moral saintliness. If I had a pitiless heart my entire life would surely be an abject moral failure. The imputation of an inability to pity strikes us as a damning moral criticism; it is one we are likely to make, for example, against those who commit acts of extreme (...) cruelty. Yet pity is hardly ever welcomed by its recipients, and for that reason it differs in a puzzling way from other emotions which are closely associated with virtue, such as gratitude or compassion. The prospect of becoming an object of pity is alarming, and not merely because we fear the misfortune that would evoke pity in others; it is alarming in part because we suspect that being on the receiving end of that emotion could seriously aggravate our plight. We also regard an aversion to being pitied as commendable, perhaps even morally commendable. There is something shameful in wanting to be pitied, just as there is in the indulgence of self-pity. The aged and the physically disabled do not want our pity, as a rule, and we think better of them for not wanting it. Finally, we know that those who give pity are frequently guilty of serious wrongdoing. A pitiless heart may be a terrible thing, but a fondness for dispensing pity is scarcely any better. (shrink)

This paper defends the view that Frege’s reduction of arithmetic to logic would, if successful, have shown that arithmetical knowledge is analytic in essentially Kant’s sense. It is argued, as against Paul Benacerraf, that Frege’s apparent acceptance of multiple reductions is compatible with this epistemological thesis. The importance of this defense is that (a) it clarifies the role of proof, definition, and analysis in Frege’s logicist works; and (b) it demonstrates that the Fregean style of reduction is a valuable tool (...) for those who would investigate the nature of arithmetical knowledge. (shrink)