First published in 1903, _Principles of Mathematics_ was Bertrand Russell’s first major work in print. It was this title which saw him begin his ascent towards eminence. In this groundbreaking and important work, Bertrand Russell argues that mathematics and logic are, in fact, identical and what is commonly called mathematics is simply later deductions from logical premises. Highly influential and engaging, this important work led to Russell’s dominance of analytical logic on western philosophy in the twentieth century.

Kant's question 'How are synthetic judgments a priori possible?' pre- cipitated the Critique of Pure Reason. Question and answer notwith- standing, Mill and others persisted in doubting that such judgments were possible at all. At length some of Kant's own clearest purported.

In the early years of this century, Poincaré and Russell engaged in a debate concerning the nature of mathematical reasoning. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical in character. Russell urged the contrary view, maintaining that (i) the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that (ii) with the aid of recent developments in logic, it is possible to demonstrate its falsity. This (...) refutation of Kant's views consists in showing that every known theorem of mathematics can be proven by purely logical means from a basic set of axioms. In our view, Russell's alleged refutation of Poincaré's Kantian viewpoint is mistaken. Poincaré's aim (as Kant's before him) was not to deny the possibility of finding a logical ‘proof’ for each theorem. Rather, it was to point out that such purely logical derivations fail to preserve certain of the important and distinctive features of mathematical proof. Against such a view, programs such as Russell's, whose main aim was to demonstrate the existence of a logical counterpart for each mathematical proof, can have but little force. For what is at issue is not whether each mathematical theorem can be fitted with a logical ‘proof’, but rather whether the latter has the epistemic features that a genuine mathematical proof has. (shrink)

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of (...) inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type. (shrink)

Le principe d’induction est lié à la définition des nombres entiers d’une façon à la fois essentielle et sujette à controverse. Fonde-t-il ces nombres, ou bien trouve-t-il en eux son fondement ? Son statut lui-même peut être conçu de diverses manières. Est-il donné par l’expérience, par l’intuition, par la logique, par convention ? Ces questions furent l’objet d’une âpre discussion, autour des années 1905-1906, dans le cadre plus large d’un débat sur les fondements des mathématiques qui opposa Poincaré aux logicistes (...) et aux « axiomatistes » . Nous proposons dans cet article de faire le point sur les termes et les enjeux de cette discussion. Nous montrerons en outre qu’elle fut l’occasion d’une redéfinition de l’analytique et, par suite, des frontières de la logique.The principle of induction is linked essentially to the definition of “integers”, but at the same time has been the subject of controversy : Is the principle founded on integers or, on the contrary, are integers founded on it ? The principle itself can be understood in different ways : as a result of experiment, intuition, logic or convention. These topics were hotly debated in the years 1905-06, in the context of a larger debate on the foundations of mathematics, in which Poincaré opposed the logicists and the “axiomatists” . This article reconsiders what is at stake by looking at the debate in detail and also shows that the debate provided an opportunity for redefining analyticity, and in doing so, redefining the domain of logic. (shrink)