Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike Connotea Del.icio.us What's this? (shrink)

RÉSUMÉ: Pasch est généralement considéré comme le premier à avoir proposé une axiomatisation de la géométrie. Mais ses Vorlesungen über neure Geometrie (1882) contiennent plusieurs éléments étrangers au paradigme hilbertien. Pasch soutient ainsi que la « géométrie élémentaire », dont il propose une axiomatisation complète, est une théorie empiriquement vraie. Les commentateurs considèrent généralement les différences entre la méthode de Pasch et celle qui deviendra standard après Hilbert comme autant de défauts affectant une pensée encore inaboutie. Notre but consiste au (...) contraire à reconstruire la cohérence et l’originalité de l’approche de Pasch. Nous donnerons d’abord un aperçu du contexte historique dans lequel les Vorlesungen s’inscrivent, pour tenter ensuite de réconcilier l’effort d’axiomatisation et l’empirisme de Pasch. Nous insisterons notamment sur les remarquables procédures logiques que Pasch met sur pied afin d’adapter sa pratique mathématique aux contraintes dictées par sa réflexion philosophique.ABSTRACT: Pasch is usually credited with having presented the first axiomatization of a geometrical theory, but the Vorlesungen über neuere Geometrie (1882) contains many features which do not fit the Hilbertian paradigm. Thus Pasch, while axiomatizing his “elementary geometry,” claims that it is an empirically true theory. Scholars usually regard the discrepancies between Pasch and the post-Hilbertian standard method as mere inconsistencies of Pasch’s theory. On the contrary, this article aims at reconstructing the coherence and originality of the Vorlesungen. We will first display the historical background of this work and then try to reconcile Pasch’s logical axiomatic claim with his empiricist stance. More importantly, we will insist on the remarkable logical procedures worked out by Pasch in order to adapt his mathematical development to the strictures of his broad philosophical position. (shrink)

Relativity is the most important scientific idea of the twentieth century. Albert Einstein is the unquestioned founder of modern physics. His Special and General theories of Relativity introduced the idea to the world. In this classic short book he explains clearly, using the minimum amount of mathematical terms, the basic ideas and principles of his theory of Relativity. Unsurpassed by any subsequent books on Relativity, this remains the most popular and useful exposition of Einstein's immense contribution to human knowledge.

A theory of ersatz impossible worlds is developed to deal with the problem of counterpossible conditionals. Using only tools standardly in the toolbox of possible worlds theorists, it is shown that we can construct a model for counterpossibles. This model is a natural extension of Lewis's semantics for counterfactuals, but instead of using classical logic as its base, it uses the logic LP.

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)

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)