Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The (...) book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene. (shrink)

Although the introduction of the modern axiomatic method in physics is attributed to Hilbert, it is only recently that physicists and mathematicians have applied it significantly, i.e. on a basis extensive enough to promise fruitful results. Carnap, for one, stresses the importance of the axiomatic method, yet he considers its application in physics as a task for the future.

Le problème de la formalisation de la logique hégélienne a fait l'objet récemment d'études d'inspiration et d'importance diverses. II y a d'abord le travail d'envergure de Gotthard Guenther sur le projet d'une logique non-aristotélicienne, le long article de Michael Kosok et la note de F. G. Asenjo.

Dans cet essai de cosmologie sauvage, je ne me suis pas adressé aux marquises et aux abbesses, comme le fit Fontenelle, mon lointain prédécesseur, en son temps. Mon public est profane et il refuse d'emblée la docte ignorance (docta ignorantia) des métaphysiciens et des mystiques, femmes savantes ou esprits crédules. Il se méfie aussi bien des mystifications dont se parent parfois les savants et les scientifiques, aussi bien que des fabulations des auteurs de science-fiction, mais il tend l'oreille souvent aux (...) poètes, écrivains et philosophes inquiets. La cosmologie ou théorie de l'univers est-elle possible et n'est-elle pas limitée par un horizon, l'horizon du visible ou de l'observable? Imaginer un au-delà de l'horizon est la tâche de la cosmologie spéculative qui a remplacé la métaphysique et qui prend alors la relève de la mythologie pour inventer la scène primitive d'une mer originelle. Cette eau originaire, que l'on retrouve chez Thalès dans la philosophie grecque présocratique, n'est pas si éloignée des fluctuations du vide quantique et le mythe de l'océan primitif peut être perçu comme la matrice de toutes les cosmogonies. Mais qui n'abordera jamais les rives de l'origine? La limite antérieure de l'origine rejoint la limite de l'horizon : toutes deux sont récessives et l'on ne peut concevoir l'univers que comme une sphère, ce qu'un autre penseur présocratique, Parménide, avait bien vu. C'est dans cette sphère que l'observateur local se tient et tient ensemble l'origine et l'horizon pour mesurer l'empan de l'univers, avec la Terre en son centre. (shrink)

Dans cet article, je compare les vues de Lautman et Herbrand sur la théorie des nombres et la philosophie de l’arithmétique. Je montre que, bien que Lautman eût avoué avoir été marqué par l’influence de Herbrand, les postures fondationnelles des deux amis divergent considérablement. Alors que Lautman versait dans un réalisme platonicien, Herbrand est resté fidèle au finitisme hilbertien. Il est vrai que Lautman était philosophe et que Herbrand était avant tout arithméticien et logicien, mais il demeure que l’oeuvre de (...) Herbrand a une portée philosophique mieux accordée à la logique et aux mathématiques contemporaines.In this paper, I am contrasting Lautman’s and Herbrand’s views on number theory and philosophy of arithmetic. It is argued that despite the fact that Lautman had acknowledged Herbrand’s major influence on his own work, their foundational stances diverge profoundly. Lautman defended a variety of Platonism and Herbrand advocated a personal version of Hilbertian finitism. Of course, Lautman was a philosopher while Herbrand dealt mainly with number theory and logic. It remains though that Herbrand’s work is more in tune with contemporary logic and mathematics from a philosophical perspective. (shrink)

SummaryFinite, or Fermat arithmetic, as we call it, differs from Peano arithmetic in that it does not involve the existence of an infinite set or Peano's induction postulate. Fermat's method of infinite descent takes the place of bound induction, and we show that a con‐structivist interpretation of logical connectives and quantifiers can account for the predicative finitary nature of Fermat's arithmetic. A non‐set‐theoretic arithemetical logic thus seems best suited to a constructivist‐inspired number theory.

Hermann Weyl as a founding father of field theory in relativistic physics and quantum theory always stressed the internal logic of mathematical and physical theories. In line with his stance in the foundations of mathematics, Weyl advocated a constructivist approach in physics and geometry. An attempt is made here to present a unified picture of Weyl's conception of space-time theories from Riemann to Minkowski. The emphasis is on the mathematical foundations of physics and the foundational significance of a constructivist philosophical (...) point of view. I conclude with some remarks on Weyl's broader philosophical views. (shrink)

Léon Brunschvicg a été un pionnier de la philosophie des mathématiques et de l’épistémologie des sciences exactes en France. Gaston Bachelard peut être considéré comme un des héritiers les plus importants de l’épistémologie constructiviste de Brunschvicg. Les deux philosophes partagent la même conception d’une dialectique immanente de la pensée scientifique et l’idée que la philosophie consiste à construire et à analyser la logique interne du discours scientifique.

The author has endeavoured to define two main trends in the research on the foundations of mathematics, constructivism and structuralism. He gives many examples in axiomatic set theory, e.g. the continuum hypothesis, and in intuitionism, e.g. the notion of choice sequence, in order to show that the two approaches are complementary. The paper contains some original ideas, concerning the structure of the continuum and the constructive horizon, and is completed by an appendix. The paper is an attempt at the justification (...) of a constructivist philosophy in the making. (shrink)

Le discours fictif présente certaines anomalies qu'il est difficile d'évaluer dans un contexte logique. La sémantique formelle du discours ordinaire doit pouvoir souffrir certaines modifications pour rendre compte des incongruités du discours fictif. Woods, en s'attaquant à un probléme aussi épineux, n'a pas réussi à éviter toutes les épines, malgré un bel arsenal de tactiques.

The theme « Truth and Certainty » is reminiscent of Hegel’s dialectic of prominent in the Phänomenologie des Geistes, but I want to treat it from a different angle in the perspective of the constructivist stance in the foundations of logic and mathematics. Although constructivism stands in opposition to mathematical realism, it is not to be considered as an idealist alternative in the philosophy of mathematics. It is true that Brouwer’s intuitionism, as a variety of constructivism, (...) has idealistic overtones, but my main concern in this paper is located in the mathematical tradition of constructive mathematics from the Greeks to Fermat, Gauss and Kronecker, and from the logical side, in the finitist doctrine of Hilbert and his followers. (shrink)

In this article, I wish to discuss in an informal way the motivations and the motifs of the constructivist approach to logic and mathematics and by a natural extension to the general field of science, particularly theoretical physics. Foundational questions in those domains are not ruled by philosophical principles, but a critical philosophy of foundations could be the leitmotiv to the extent that it can be used as a criterion to decide between the theoretical options of scientific practices that are (...) often oblivious to their own doctrinal presuppositions. My objective is to provide the justificatory reasons for a constructivist option or posture in the field of scientific knowledge. (shrink)

The notion of transcendental observer (transcendentaler Zuschauer) is introduced in Husserl’sCartesianische Meditationen.The idea of the observer first appears in Kant’s Copernican revolution putting the emphasis on the observer as he announces in hisKritik der reinen Vernunft.Contemporary physics also has a notion of a local observer and it plays a central role in the foundations of quantum mechanics, general relativity and cosmology.

Hilbert's programme is shown to have been inspired in part by what we can call Kronecker's programme in the foundations of an arithmetic theory of algebraic quantities.While finitism stays within the bounds of intuitive finite arithmetic, metamathematics goes beyond in the hope of recovering classical logic. The leap into the transfinite proved to be hazardous, not only from the perspective of Gödel's results, but also from a Kroneckerian point of view.

What Gödel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist ideal which (...) is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent” . An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s general or polynomial arithmetic, thus returning into arithmetic without the detour of Cantor’s transfinite arithmetic of ideal elements. (shrink)

ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does not correspond to the infinite (...) descent of classical Fermatian arithmetic or number theory. However, from the point of view of classical logic, the principles of complete induction and transfinite induction, the least number principle and infinite descent are all equivalent. We find here a focal point for a foundational critique that aims for a a clarification of philosophical options in the foundations of logic and mathematics. (shrink)

Le réalisme se porte mal: des philosophes comme Putnam ou van Fraassen qui naguère encore défendaient des points de vue à tout le moins apparentés au réalisme proposent des thèses anti-réalistes ou qui sont aux antipodes du réalisme métaphysique. Et ceux qui tiennent encore au réalisme le voilent prudemment. Quant au réalisme critique de Popper et des poppériens, il est relégué aujourd'hui au rôle d'une thèse historique sur le développement rationnel de la science que contredit la doctrine kuhnienne. Reste le (...) réalisme naïf. (shrink)