This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...) does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

Jean Nicod, born in 1893, died far too young in 1924. He is remembered today as one of the foreign disciples (among them Ludwig Wittgenstein and Norbert Wiener) attracted to Cambridge by Russell after the publication of the Principia. We publish here a translation of “Les tendances philosophiques de M. Bertrand Russell”, which appeared in the Revue de Métaphysique et de Morale in 1922. The article is testimony not only to Nicod’s philosophical talents, but also of how Russell’s philosophy could (...) be received in France at the beginning of the twentieth century. (shrink)

Jean Nicod (1893–1924) is a French philosopher and logician who worked with Russell during the First World War. His PhD, with a preface from Russell, was published under the title La géométrie dans le monde sensible in 1924, the year of his untimely death. The book did not have the impact he deserved. In this paper, I discuss the methodological aspect of Nicod’s approach. My aim is twofold. I would first like to show that Nicod’s definition of various notions of (...) equivalence between theories anticipates, in many respects, the (syntactic and semantic) model-theoretic notion of interpretation of a theory into another. I would secondly like to present the philosophical agenda that led Nicod to elaborate his logical framework: the defense of rationalism against Bergson’s attacks. (shrink)

This article has two purposes: first, to describe some archival discoveries about Nicod’s family background, academic development and political life; and second, to publish and comment on a newly discovered article by Nicod about Russell. This article acclaims not only Russell’s achievements in logic and philosophy—as one might expect from such a devoted protégé as Nicod—but also (albeit only in glimpses permitted by France’s wartime censorship) his anti-war politics and writings. As the reader will realize, the two objectives are connected: (...) what the archives reveal and what the article illustrates is a hitherto underappreciated level of political awareness and pacifist commitment in Nicod. (shrink)

Jean Nicod (1893–1924) is a French philosopher and logician who worked with Russell during the First World War. His PhD, with a preface from Russell, was published under the titleLa géométrie dans le monde sensiblein 1924, the year of his untimely death. The book did not have the impact he deserved. In this paper, I discuss the methodological aspect of Nicod’s approach. My aim is twofold. I would first like to show that Nicod’s definition of various notions of equivalence between (...) theories anticipates, in many respects, the (syntactic and semantic) model-theoretic notion of interpretation of a theory into another. I would secondly like to present the philosophical agenda that led Nicod to elaborate his logical framework: the defense of rationalism against Bergson’s attacks. (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)

The goal of this paper is to connect Rota's discussion of the Husserlian notion of Fundierung with Rota's project of giving combinatorics a foundation in his 1964 paper ‘On the foundations of combinatorial theory I’. Section 2 gives the basic tenets of this seminal paper. Sections 3 and 4 spell out the connections made there between Rota's philosophical writings and his mathematical achievements. Section 5 shows how these two developments fit into Rota's analysis of the place of combinatorics in mathematics.

In Philosophical Investigations 193–94, Wittgenstein draws a notorious analogy between the working of a machine and the application of a rule. According to the view of rule-following that Wittgenstein is criticizing, the future applications of a rule are completely determined by the rule itself, as the movements of the machine components are completely determined by the machine configuration. On what conception of the machine is such an analogy based? In this paper, I intend to show that Wittgenstein relied on quite (...) a specific scientific tradition very active at the beginning of the twentieth century: the kinematic or the general science of machines. To explain the fundamental tenets of this line of research and its links with Wittgenstein, I focus on Franz Reuleaux, whose works were known to Wittgenstein. The first payoff of this investigation is to help distance the functionalist framework from which this passage is often read: Wittgenstein’s machines are not computers. The second payoff is to explain why Wittgenstein talks about machines at this place in his discussion on rule-following: it is not the machine model in itself that is criticized in PI 193–94, but the “philosophical” temptation to generalize from it. (shrink)

Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was sustained (...) by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell's global reductionist standpoint with his local defence of the specificities of geometry? * This paper was first presented at the conference ‘Qu'est ce que la géométrie aux époques modernes et contemporaines?’, organized by the Universität Köln and the Archives Poincaré. I would like to thank Philippe Nabonnand for having enlightened me about the issues relative to projective geometry. I would like also to thank Nicholas Griffin, Brice Halimi, Bernard Linsky, Marco Panza, Ivahn Smadja for their helpful discussions. Many thanks also to the two anonymous referees for their useful suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

Abstract:Russellian incomplete symbols are usually conceived as an analytical residue—as what remains of the would-be entities when properly analyzed. This article aims to reverse the approach in raising another question: what, if any, does the incomplete symbol contribute to the completely analyzed language? I will first show that, from a technical point of view, there is no difference between the way Russell defines his denoting phrases in “On Denoting” and the way Frege defines his second-order concepts. But I will secondly (...) support that the two notions have two widely different conceptual meanings: the same logical procedures which are used by Frege to increase the deductive power of his system allow Russell to logically relate quantificational notation to ordinary language. Focusing on Russell’s treatment of OD’s puzzles, I will thirdly argue that this shift constitutes the source of a deep transformation in the way logic and language are related. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:January 28, 2009 (12:22 pm) G:\WPData\TYPE2802\russell 28,2 051red.wpd russell: the Journal of Bertrand Russell Studies n.s. 28 (winter 2008–09): 171–90 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 eviews A RELATIONAL DISPUTE Sébastien Gandon iufz/zphier / U. Blaise Pascal 63000 Clermont-Ferrand, France [email protected] Stewart Candlish. The Russell/Bradley Dispute and Its SigniWcance for TwentiethCentury Philosophy. Basingstoke and New York: Palgrave Macmillan, 2007. Pp. xix, 235. (...) isbn 978-0-230-50685-5. £49.99; us$84.95 (hb). Candlish’s book aims at giving a “fair” description of the notorious Russell/ Bradley dispute, that is, a more balanced one than the version that is usually told. The standard account is the one given by Russell, and, as Russell was a party of the debate, it is no surprise that he seriously misconstrued Bradley’s arguments. Attempting to set the historical record straight, Candlish then opposes Russell’s description of the dispute. The author is quite clear, however, on the fact that he does not himself endorse the Idealistic point of view (p. xi). The two Wrst chapters of the book are introductory: the Wrst one is a presentation of the stereotypical picture that the author intends to demolish, while the second is an outline of Bradley’s metaphysics. Chapters 3 and 4 are devoted to a confrontation between Russell and Bradley’s theories of judgment and truth. Chapters 5 and 6 are dedicated to a rehabilitation of Bradley’s writings about relation. In these four chapters, Candlish attempts to show that Russell’s arguments against Bradley were not as devastating as they are supposed to be, and that Bradley’s insights were quite powerful—so much so that they have been rediscovered by some later philosophers, such as Wittgenstein, for instance. The concluding chapter is an attempt to explain the decline of monistic idealism after Bradley. Candlish does not only aim at oTering a “fair” account of the Russell/Bradley dispute. This goal is extended by a more general plea for the introduction of an historical perspective in analytic philosophy. According to Candlish, “our implicit acceptance of a distorted version of philosophy’s recent past has serious consequences ” (p. 19): one of them is that, by misunderstanding our own history, we “misunderstand ourselves and the nature of what we are about” (ibid.); another is that, by being deprived of any way to connect what we do in philosophy January 28, 2009 (12:22 pm) G:\WPData\TYPE2802\russell 28,2 051red.wpd 172 Reviews with what some of the philosophers of the past have done, “we may, as a result, fall under the impression that only we, and those obviously like us, are really doing philosophy, at least as it ought to be done” (ibid.). From this perspective, Russell’s confrontation with Bradley seems to represent a rather good topic: not only does it epitomize a turning-point in Russell’s intellectual evolution and in the emergence of the new tradition, but it also shows how easily the new and victorious analytic philosophy has misconstrued its own history. If I entirely share Candlish’s conviction about the necessity of introducing an historical perspective in analytical philosophy, I am not convinced by Candlish’s own reconstruction of the dispute. In a nutshell, what I Wnd very useful in the book is all that concerns Bradley, and some of the parts devoted to Russell (especially, the analysis of his theory of judgment). What I do not agree with is, Wrst, Candlish’s way of doing history of philosophy; second, his treatment of Russell’s works; and third, his general conception of the use of logic in philosophy. Candlish really succeeds in expounding to the modern eye both the coherence of Bradley’s thought, and the diUculty of grasping it. His analysis of the role Bradley gave to “intellectual satisfaction” (pp. 26–7, 33) and to the “principles of suUcient reason” (pp. 46–7) is very enlightening—it explains why Bradley, while endorsing a kind of methodological scepticism, denied that ultimate contingencies were given in experience (p. 48). Candlish also does a great job of presenting... (shrink)

To illustrate the view that a speaker can have a partial understanding of a concept, Burge uses the example of Leibniz’s and Newton’s understanding of the concept of derivative. In a recent article, Sheldon Smith criticizes this example and maintains that Newton’s and Leibniz’s use of their derivative symbols does not univocally determine their references. The present article aims at challenging Smith’s analysis. It first shows that Smith misconstrues Burge’s position. It second suggests that the philosophical lessons one should draw (...) from the practice of the historians of philosophy are more ambivalent than what Smith thinks. (shrink)

© The Authors [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] Grosholz is interested in the growth of knowledge: what happens when reasoning not only orders what we already know, but adds to what we know? In her previous works, especially in her [2007], Grosholz insisted on the fact that working scientists and mathematicians, when they add to what we know, often combine different ‘modes of representation’, taking advantage of the ambiguity that arises when (...) features of one sort of discourse are grafted onto another. In her new book she goes on along the same path but brings an important addition to the pattern: ‘ampliative’ thinking takes place ‘when discourse whose main intent is to establish and clarify reference is yoked with discourse whose main intent is analysis’. According to Grosholz, the basis of productive thinking is the difference between... (shrink)

James Levine has recently argued that there is a tension between Russell’s Moorean semantical framework and Russell’s Peano-inspired analytical practice. According to Levine, this discrepancy runs deep in Russell’s thought from 1900 to 1918, and underlies many of the doctrinal changes occurring during this period. In this paper, I suggest that, contrary to what Levine claims, there is no incompatibility between Moore’s theory of meaning and the idea of informative conceptual analysis. I show this by relating Moore’s view of meaning (...) to his Sidgwick-inspired criticism of the so-called naturalistic fallacy. I maintain that Moore’s semantical framework has a methodological intent: following Sidgwick, Moore wants to block any attempt to justify ethical principles through setting ad hoc conditions on the meaning of the terms involved. Thus, far from grounding philosophical knowledge on subjective intuitions, as Levine suggests, Moore’s framework would provide us with the means to make room for a discursive and dialectic conception of philosophical inquiry. (shrink)

In Die Zahlen (1888), Dedekind defines an infinite set as a set that is isomorphic with one of its proper parts. In The World and the Individual (1900), the American philosopher Josiah Royce relates Dedekind’s notion to Fichte’s and Hegel’s concept of Self defined as an entity that reflects itself into itself. The first aim of this article is to explain Royce’s analysis and to put it in its proper context, that of a critique of Bradley’s mystical idealism. The second (...) aim is to urge a shift in focus in Dedekind’s scholarship: instead of addressing the question of the relationship between mathematics and philosophy in Dedekind’s work through the supposed intentions of its author, it is more fruitful to analyze the reception that philosophers have made of his texts. (shrink)

La théorie russellienne des relations est ordinairement conçue comme le résultat d'une réflexion logique et ontologique sur l'ordre et l'asymétrie. Le présent article vise à présenter une autre généalogie, centrée sur les concepts de grandeur et de vecteur. Nous montrons en premier lieu que la thèse de l'irréductibilité des relations est avancée pour la première fois en 1897, à l'occasion d'une reformulation de la dialectique hégélienne de la quantité. Nous soulignons, en second lieu, que la notion de grandeur fait, autour (...) des années 1898-1899, l'objet d'une formalisation mathématique, complètement indépendante de la théorie des relations et fondée exclusivement sur les algèbres vectorielles. Nous montrons enfin que Russell, en possession de sa nouvelle doctrine logique, développe dès 1901 une théorie relationnelle des grandeurs vectorielles. Tout se passe donc comme si un des intérêts de la nouvelle théorie était, pour le philosophe, de permettre la substitution des relations aux grandeurs. (shrink)

Les conditions d’identité jouent un rôle fondamental dans le débat sur le statut épistémologique des diagrammes. À cet égard, les détracteurs comme les partisans de leur utilisation en tant que moyens de preuve partagent un même présupposé, qui consiste à étendre aux diagrammes la distinction entre type et token, normalement appliquée aux mots. Dans cet article, nous entendons discuter ce présupposé. Dans la première partie, nous montrerons que la distinction entre type et token, en tant que critère d’identité des mots, (...) ne fait pas consensus en philosophie du langage. Nous verrons, en particulier, comment David Kaplan a rejeté ce critère et proposé une approche alternative, inspirée de la conception de la monnaie courante développée par Kripke. Dans la deuxième partie, nous étudierons la possibilité d’étendre aux diagrammes l’approche kaplanienne, par moyen de quelques considérations de caractère historique concernant la transmission des textes mathématiques grecs. (shrink)

L’idéalisme britannique est un mouvement qui a dominé les universités britanniques pendant une cinquantaine d’années à la fin du xixe siècle et au début du xxe siècle, mais qui est passé presque totalement inaperçu dans le monde francophone. Rejetés en bloc par les philosophes analytiques, ces auteurs ont aussi été ignorés pendant longtemps dans leur pays, mais certains d’entre eux, notamment Bradley et Collingwood, jouissent d’un regain d’intérêt à la faveur d’un renouveau des études sur les origines de la philosophie (...) analytique. Ce texte est une introduction à l’idéalisme britannique, qui retrace, dans une première partie plus historique, les grandes lignes de sa genèse, son développement et son déclin. Dans une deuxième partie, nous donnons quelques arguments en faveur d’une étude plus approfondie de ce mouvement.British Idealism is a philosophical movement that dominated British universities , for fifty years around the turn from the XIXth to the XXth century, but it went largely unnoticed in the French-speaking world. Condemned by analytic philosophers, these authors were also ignored in their own country, but some of them, notably Bradley and Collingwood, are now enjoying a newly found popularity within the larger trend towards a study of the origins of analytic philosophy. This text is an introduction to British Idealism that plots, in an historical first part, the outlines of its rise, development and decline. In the second part, we provide reasons for further studies of this movement. (shrink)

L’article vise à établir un lien entre les travaux de Russell sur les fondements de la géométrie et ses célèbres textes des années 1911-1914 sur la perception, la matière et les sense-data. Nous insistons d’abord sur le fait que la notion russellienne de donnée sensorielle n’est pas phénoménaliste : les sense-data des Problèmes de Philosophie sont des objets extérieurs aussi peu mentaux que les corps matériels. L’accent est ensuite mis sur l’article On Matter : Russell y introduit pour la première (...) fois l’idée que les sense-data n’ont pas une, mais deux positions — une position où ils sont vus et une position d’où ils sont vus. Cette manœuvre permet à Russell de considérer la donnée sensorielle comme le point d’intersection neutre de deux ensembles fondamentalement différents : une série « physique », matérielle, et une série de « perspectives », mentales. L’idée qu’une même chose peut être classée sous différentes catégories, donc que le mode de classification aristotélicien en espèce et en genre n’est pas le seul concevable, est à la fois la clé de voûte de la théorie des relations entre « sense-data » et « physical things » de 1914 et le nerf du ralliement ultérieur au « monisme neutre ». Nous défendons, dans une troisième partie, que ce canevas n’est lui-même qu’un lointain écho de la conception de la géométrie projective définie comme théorie générale des relations d’incidence développée par Russell dans The Principles of Mathematics (1903). (shrink)

L’article vise à établir un lien entre les travaux de Russell sur les fondements de la géométrie et ses célèbres textes des années 1911-1914 sur la perception, la matière et les sense-data. Nous insistons d’abord sur le fait que la notion russellienne de donnée sensorielle n’est pas phénoménaliste : les sense-data des Problèmes de Philosophie sont des objets extérieurs aussi peu mentaux que les corps matériels. L’accent est ensuite mis sur l’article On Matter : Russell y introduit pour la première (...) fois l’idée que les sense-data n’ont pas une, mais deux positions — une position où ils sont vus et une position d’où ils sont vus. Cette manœuvre permet à Russell de considérer la donnée sensorielle comme le point d’intersection neutre de deux ensembles fondamentalement différents : une série « physique », matérielle, et une série de « perspectives », mentales. L’idée qu’une même chose peut être classée sous différentes catégories, donc que le mode de classification aristotélicien en espèce et en genre n’est pas le seul concevable, est à la fois la clé de voûte de la théorie des relations entre « sense-data » et « physical things » de 1914 et le nerf du ralliement ultérieur au « monisme neutre ». Nous défendons, dans une troisième partie, que ce canevas n’est lui-même qu’un lointain écho de la conception de la géométrie projective définie comme théorie générale des relations d’incidence développée par Russell dans The Principles of Mathematics (1903). (shrink)

Dans l'Appendice au livre I de The World and the Individual (1898), le philosophe américain Josiah Royce développe, en se fondant sur Was sind und was sollen die Zahlen ? de Dedekind, une critique détaillée du livre de Bradley Appearance and Reality. Se concentrant sur le fameux § 66, Royce maintient que la théorie de Dedekind peut être vue comme l'accomplissement du mouvement de pensée inauguré par Fichte et Hegel : le 'Soi idéal ' est infini et l'arithmétique est la (...) théorie de sa structure formelle. Cet article analyse le contenu de ce texte, en le mettant en relation avec la pensée de Russell et de Lotze. (shrink)

English summary: This volume offers French readers important texts of modern philosophical mathematics, addressing in particular ontological questions and others on mathematical objects, how to demonstrate the need for mathematical truths, and how to explain that mathematics apply to the real world. French description: Le compagnonnage entre la philosophie et les mathematiques ne date pas d'hier. Mais l'emergence des nouvelles logiques, au debut du XXe siecle, a profondement modifie la forme des interactions entre les deux disciplines, suscitant de nouvelles interrogations (...) et modifiant la formulation des problemes herites de la tradition. Ce premier volume vise a rendre accessible aux lecteurs francophones des textes importants de la philosophie contemporaine des mathematiques. Il est plus particulierement consacre aux questions ontologiques et a celles liees aux fondements: Qu'est-ce qu'un objet mathematique? Comment rendre compte de la necessite des verites mathematiques? Comment expliquer que les mathematiques s'appliquent au monde reel? (shrink)

Penelope Maddy's latest book, devoted to Wittgenstein's early and late philosophy of logic, can be seen as an extension of Maddy 2007, in which the thoughts of Descartes, Kant (and Car...

The paper aims at comparing Wittgenstein’s discussion of color exclusion in his to Johnson’s doctrine of determinable and determinate expounded in his. I first summarize Wittgenstein’s developments about the incompatibility of elementary propositions and about the logic of color statements. In the second part, I present and discuss Johnson’s doctrine in relation to Wittgenstein’s development. In a third conclusive moment, drawing on a early work of Prior, I argue that the distinction made by Wittgenstein and Johnson between predication and determination (...) should be looked at from a long term historical perspective. (shrink)

Ce texte discute certaines conclusions d’un article récent de F. Mülhölzer et vise à montrer que le logicisme russellien a les moyens de résister à la critique que Wittgenstein lui adresse dans la partie III des Remarques sur les fondements desmathématiques.This paper discusses some conclusions of a recent article from F.Mülhölzer. It aims at showing that Russell’s logicism has the means to overcome the criticisms Wittgenstein expounded in Remarks on the Foundations of Mathematics, part III.

At the beginning of the 1890s, Schwarz and Peano (independently of each other) showed that Serret’s definition of the area of a surface was flawed. This paper first aims at describing the various methods that the mathematicians have used for correcting Serret’s reasoning; its second goal is to compare and to present more in detail two solutions: Lebesgue’s notorious construction and Peano’s definition.

The paper is a defense of a strict form of anti-realism, competing the "in principle" form defended by Michael Dummett. It proposes to ground anti-realism on the basis of two principles ("immanence" and "implicitness") and to develop the consequences of these principles in the light of sub-structural logics.

Standard epistemic logic actually formalizes folk-psychology. The problem of logical omniscience shows that this logic is not adequate in a cognitivist setting. In order to describe realistic information-processing believers, we need both an ontology of particulars and a hyper-intensional taxonomy of representations. From this point of view, sub-structural logics, which do not overlook the syntaxic details of the processing of information, are an interesting alternative to possible worlds semantics.

According to the realist, the meaning of a declarative, non-indexical sentence is the condition under which it is true and the truth-condition of an undecidable sentence can obtain or fail to obtain independently of our capacity, even in principle, to recognize that it obtains or that fails to do so.1 In a series of papers, beginning with “Truth” in 1959, Michael Dummett challenged the position that the classical notion of truth-condition occupied as the central notion of a theory of meaning, (...) and proposed that it should be replaced by the anti-realist (and intuitionistic) notion of assertability-condition. Taken together with normalization results obtained by Dag Prawitz, Dummett’s work truly opened up the anti-realist challenge at the level of proof-theoretical semantics.2 There has been since numerous rejoinders from partisans of classical logic, which were at times met with by attempts at watering down the anti-realist challenge, e.g., by arguing that anti-realism does not necessarily entail the adoption of intuitionistic logic. Only a few anti-realists, such as Crispin Wright and Neil Tennant, tried to look instead in the other direction, towards a more radical version of anti-realism which would entail deeper revisions of classical logic than those recommended by intuitionists.3 In this paper, which is largely programmatic, we shall also argue in favour of a radical anti-realism which would be a genuine alternative to the traditional anti-realism of Dummett and Prawitz. The debate about anti-realism has by now more or less run out of breath and we wish to provide it with a new lease on life, by taking into account the profound changes that took place in proof theory during the intervening years. We have in mind in particular the considerable development within Gentzen-style proof theory of non-classical, substructural logics other than intuitionistic logic, which seriously opens up the possibility that anti-realism, when properly understood, might end up justifying another logic, and the development of closer links between proof theory and computational complexity theory that has renewed interest in a radical form of anti-realism, namely strict finitism. (shrink)

Bolzano was the first to establish an explicit distinction between the deductive methods that allow us to recognise the certainty of a given truth and those that provide its objective ground. His conception of the relation between what we, in this paper, call "subjective consequence", i.e., the relation from epistemic reason to consequence and "objective consequence", i.e., grounding however allows for an interpretation according to which Bolzano advocates an "explicativist" conception of proof: proofs par excellence are those that reflect the (...) objective order of grounding. In this paper, we expose the problems involved by such a conception and argue in favour of a more rigorous demarcation between the ontological and the epistemological concern in the elaboration of a theory of demonstration. (shrink)

Logic and psychology overlap in judgment, inference and proof. The problems raised by this commonality are notoriously difficult, both from a historical and from a philosophical point of view. Sundholm has for a long time addressed these issues. His beautiful piece of work [A Century of Inference: 1837-1936] begins by summarizing the main difficulty in the usual provocative manner of the author: one can start, he says, by the act of knowledge to go to the object, as the Idealist does; (...) one can also start by the object to go to the act, in the Realist mood; never the two shall meet. He is himself inclined to accept the first perspective as the right one and he has eventually developed an original version of antirealism which starts, not from considerations about the publicity of meaning, in the manner of Dummett, but from an epistemic standpoint, trying to search in a non-Fregean tradition of analysis of judgement and cognate notions a way of founding constructivist semantics. The present paper ploughes the same field. We concentrate on the significance, for Sundholm’s program, of the perspective that has been opened by Twardowski in his important essay on acts and products (1912. (shrink)

The paper deals with the contributions of Jacques Herbrand to logic and philosophy in historical context.

This paper can be read as an attempt at providing philosophical foundations to linear logic. The only plausible form of philosophical antirealism deals with practical feasibility rather than with effectivity in principle. The very notion of recognizability is ambiguous, audit has to be considered from a stricter perspective than currently done. The intuitionistic assertability conditions are to be reinforced. This change requires a move towards a frame in which the circumstances of the application of a logical rule can be specified. (...) Gentzen's sequential calculi provide such a frame, when some structural rules have been removed or limited. (shrink)

Providing Single Sign-On (SSO) between service providers and enabling service providers to share user personal attributes are critical for both users to benefit from a seamless access to their services, and service providers to realize new business opportunities. Today, however, the users have several independent, partial identities spread over different service providers. Providing SSO and attribute sharing requires that links (federations) are established between (partial) identities. In SAML 2.0 (Maler et al. 2003), the links between identities are stored and managed (...) at the network side by the identity providers (network-side identity federation). This model prevents the service providers from mass-correlating the partial identities they have, but the users must fully trust the identity providers. In this paper, we propose a complementary approach where the users have a full control of the links between their partial identities. It is a client-side identity federation approach, which relies on the introduction of a new cryptographic tool, called invariable partially blind signature scheme, that may be of independent interest. (shrink)

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...) developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic. (shrink)

Philosophy of Probability provides a comprehensive introduction to theoretical issues that occupy a central position in disciplines ranging from philosophy of ...

According to foundationalism, mathematical propositions form a complex objective structure we have to discover and in which each proposition depends for its truth on the propositions that precede and ground it. Proofs that only establish the truth of their conclusions are contrasted with proofs that give also the objective reason of that truth. Foundationalism claims that the cognitive resources required by these proofs are universally available. This paper tries to undermine that claim and to show that foundationalism is a dead (...) end. (shrink)