While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski’s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating “rigorously” with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as “drawn formulas”, and formulas as “written (...) diagrams”, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski’s diagrammatic methods in number theory prompted Hilbert’s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert’s assessment of Minkowski’s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics. (shrink)

This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding mathematical (...) understanding prevail over mere preemptive reductionism to whole numbers. (shrink)

Of Euclid’s lost manuscripts, few have elicited as much scholarly attention as the Porisms, of which a couple of brief summaries by late-Antiquity commentators are extant. Despite the lack of textual sources, attempts at restoring the content of this absent volume became numerous in early-modern Europe, following the diffusion of ancient mathematical manuscripts preserved in the Arabic world. Later, one similar attempt was that of French geometer Michel Chasles (1793–1880). This paper investigates the historiographical tenets and practices involved in Chasles’ (...) restoration of the porisms, as well as the philosophical and mathematical claims tentatively buttressed therewith. Echoes of the Quarrel of the Ancients and the Moderns, and of a long-standing debate on the authority and usefulness of the past, are shown to have decisively shaped Chasles’ enterprise—and, with it, his integration of mathematical and historical research. (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)

This essay explores the research practice of French geometer Michel Chasles, from his 1837 Aperçu historique up to the preparation of his courses on ‘higher geometry’ between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have played (...) a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the ‘material history’ of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing. (shrink)

RésuméDans cet article, nous analysons en détail les deux mémoires de Legendre, Sur les intégrations par arcs d’ellipse (1786a, 1786b), en comparant les deux démonstrations qu’il donne de la rectification de l’hyperbole au moyen de deux arcs d’ellipse. Entre le premier et le second mémoire, Legendre prend connaissance d’un théorème analogue de Landen, qu’il cherche à incorporer à son projet de constitution d’un calcul des arcs d’ellipse et dont il propose une interprétation originale en le déduisant de ses propres principes. (...) En retraçant la préhistoire de la transformation de Landen, des premières méthodes d’intégration par arcs de sections coniques chez Maclaurin, d’Alembert et Landen jusqu’à l’élaboration par Legendre des échelles de modules, nous cherchons à mettre en lumière les conditions de son identification. Nous montrons comment Legendre fut amené à discerner dans le théorème de Landen ce qui entre ses mains allait devenir l’instrument privilégié de la mise en ordre systématique des transcendantes elliptiques, grâce à une lecture orientée par les exigences spécifiques liées à une méthode d’approximation fondée sur la variation continue des paramètres et à la constitution de tables numériques d’arcs d’ellipse. (shrink)

Avec A. A. Robb, A. S. Eddington et quelques autres, B. Russell est l’un de ceux qui, dès les années vingt, ont mis en avant la nécessité de distinguer la structure topologique, la structure métrique et la structure causale de la variété espace-temps de la théorie de la relativité. Je me propose dans cet article de montrer comment les thèses traditionnellement imputées à Russell, à savoir l’indépendance de la structure topologique par rapport à...

Résumé — L’objet de cet article est de chercher à déterminer quel est le statut des possibles en mathématiques en montrant comment, à partir de l’analyse proposée par Kripke du mécanisme des illusions modales, il serait possible, conformément aux intuitions initiales de Putnam, de concilier réalisme et modalités. Si l’enjeu du réalisme mathématique est en effet de rendre compte de l’objectivité des mathématiques et non de l’existence de prétendus objets, nous pouvons concevoir une forme de réalisme qui ne sacrifierait pas (...) une réelle ouverture des possibles, pour peu que l’on apprécie dans toute sa complexité la double spécificité sémantique que constituent, en mathématiques, le recours aux descriptions imparfaites et la concurrence des langages.— Are there Kripkean metaphysical possibilities in mathematics ? Or should we stick to epistemic possibilities ? Generalizing the Kripkean account of the mechanism generating modal illusions, this paper investigates how we could make mathematical realism and modal knowledge compatible. Following Putnam’s insight according to which realism should deal with objectivity of mathematics rather than with mathematical objects, I’ll try to sketch a variety of modal knowledge rooted in a twofold semantic fact, peculiar to mathematics, viz. the use of imperfect descriptions and the stacking up of alternative language levels. (shrink)

Le réalisme est-il mort? Il y a plus de vingt ans, le philosophe des sciences Arthur Fine partait du constat abrupt de ce qui lui apparaissait comme une déroute du réalisme scientifique et engageait à dépasser les affrontements doctrinaux opposant réalistes et antiréalistes. Il prônait à l’égard de la science une voie médiane débarrassée des présupposés généraux et des interprétations globales : une « attitude ontologique naturelle », plus ouve...

Une fois achevée la rédaction des Principia Mathematica, Russell se tourne vers la théorie physique et s’attaque au problème philosophique que pose notre connaissance du monde extérieur. Pendant plus d’une décennie, il élabore une « philosophie de la physique » à partir de l’analyse des avancées décisives de cette période marquée tant par l’avènement du « règne de la relativité » que par la transformation profonde des cadres conceptuels...

This paper sets out to show how Eddington's early twenties case for variational derivatives significantly bears witness to a steady and consistent shift in focus from a resolute striving for objectivity towards “selective subjectivism” and structuralism. While framing his so-called “Hamiltonian derivatives” along the lines of previously available variational methods allowing to derive gravitational field equations from an action principle, Eddington assigned them a theoretical function of his own devising in The Mathematical Theory of Relativity (1923). I make clear that (...) two stages should be marked out in Eddington's train of thought if the meaning of such variational derivatives is to be adequately assessed. As far as they were originally intended to embody the mind's collusion with nature by linking atomicity of matter with atomicity of action, variational derivatives were at first assigned a dual role requiring of them not only to express mind's craving for permanence but also to tune up mind's privileged pattern to “Nature's own idea”. Whereas at a later stage, as affine field theory would provide a framework for world-building, such “Hamiltonian differentiation” would grow out of tune through gauge-invariance and, by disregarding how mathematical theory might precisely come into contact with actual world, would be turned into a mere heuristic device for structural knowledge. (shrink)