Gerhard Heinzmann

Poincaré’s position is often presented as semi-intuitionist, especially in relation to his position in arithmetic. This article examines Poincaré’s relationship to an intuitionist position. In the first part, we examine the relationship between the Kantian tradition and intuitionism, and the reception of Poincaré’s approach by Brouwer and Heyting. In the second part, we present a detailed analysis of Poincaré’s conception of induction as an a priori synthetic judgment based on pure intuition and in relation to the foundations of geometry. For the latter, Poincaré presupposes a power of the mind to form groups and continua, of which we possess an intuitive knowledge. We conclude that although we are accustomed to some mathematical conceptions of Poincaré and Brouwer are rather closely related, Poincaré and Brouwer differ fundametally in their approaches from a conceptual point of view. Contrary to the neo-intuitionists of the Brouwerian tradition, for Poincaré the use of language is not only an efficient tool for memorizing or communicating, but also, and above all, an essential part of the genesis of mathematical conceptualization.

The main thesis defended in this paper is that, interpreted in the light of reflections of Peirce and Poincaré, one can found in mathematical reasoning a non-logical symptom that may be aesthetic in Goodman’s sense. This symptom is called exemplification and serves to distinguish between only logically correct and even explanatory proofs. It broadens the scope of aesthetics to include all activities involving symbolic systems and blurs the boundaries between logic and aesthetics in mathematics. It gives a better understanding of Poincaré’s thesis that to affect aesthetic value to certain properties is not simply an added value, a bonus that somehow rewards the mathematician’s mechanical labor, but on the contrary, taking the aesthetic value into account can be helpful to mathematical practice. As an example, three proofs of the irrationality of √2 are compared for their aesthetic functioning.

Cet article tente de donner une vue d’ensemble du travail de Beth à une large communauté de philosophes des sciences. L’accent sera moins mis sur ses contributions importantes et largement discutées à la logique mathématique – où résident ses mérites les plus créatifs – que sur le contexte philosophique et culturel. J’adopte le motif que Beth a lui-même poursuivi pour la composition de son ouvrage majeur The Foundations of Mathematics, à savoir renforcer les liens entre la recherche fondamentale et la philosophie générale, en rendant les problèmes qui l’intéressaient accessibles à partir de leurs racines théoriques et biographiques. This article attempts to provide an overview of Beth’s work for a broad community of philosophers of science. Emphasis will be placed more on the philosophical and cultural context than on his important and widely discussed contributions to mathematical logic—where his most creative merits lie. I adopt the motive Beth himself pursued in writing his major work The Foundations of Mathematics, namely, to strengthen the links between fundamental research and general philosophy, making the problems he was interested in accessible from their theoretical and biographical roots.

There is no substantive agreement about the philosophical views of analytic philosophers. Nevertheless, for much of them logical analysis is widely recognized to be important. It is true that Poincare used no logical analysis but refused nevertheless the old metaphysics. Indeed, the analytic tradition of philosophy of science is perhaps better characterized by several overlapping similarities, which are the clincher for my main thesis: Analytic philosophy of science has one of its origins in the philosophical network in France around 1900 and, especially, in Poincare.

The goal of this paper is to answer the following question: Does it make sense to speak of thought experiments not only in physics, but also in mathematics, to refer to an authentic type of activity? One may hesitate because mathematics as such is the exercise of reasoning par excellence, an activity where experience does not seem to play an important role. After reviewing some results of the research on thought experiments in the natural sciences, we turn our attention to experiments and thought experiments in mathematics, especially in fundamental mathematics, and investigate what thought experiments can teach us there. If we accept the principle that mathematical practice can sometimes be best described by a pragmatist approach where the concept of experience in mathematics is considered from a new perspective, thought experiments can in some cases be a useful instrument different from both ‘mathematical experiments’ and ‘formal’ proofs: they are mathematical experiments using deviant methods.

This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen’s systematical ideas in today’s debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen’s work on lattice-groups and divisibility theory, and modern set theory and Lorenzen’s critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen’s consistency proof and Hilbert’s larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz.

L'histoire des fondements des mathématiques du xxe siècle montre qu'il nous faut réviser la signification des notions philosophiques traditionnelles comme « évidence », « existence », « expérience » ou « rationalité ». On expose comment le logicien Paul Bernays, familier des conceptions de la philosophie de Jacob Friedrich Fries et de Léonard Nelson, donne aux résultats techniques une interprétation philosophique dont il s'inspire de plus en plus - à partir du milieu du siècle - de la « philosophie ouverte » de Ferdinand Gonseth. Bien avant Thomas Kuhn, Bernays conçoit la révision envisagée non pas simplement comme une question de vérité ou de fausseté, mais comme un problème exigeant l'introduction d'un nouveau système conceptuel. Cependant, l'affaiblissement du concept de vérité est pour lui une conséquence épistémologique provenant de la théorie de la connaissance adoptée et non une conséquence méthodologique issue d'un changement de paradigme et donc d'une incommensurabilité.

Pascal Engel makes an overall distinction between two tendencies to conceive of the concept of truth. Some defend a substantialist theory according to which truth expresses a “real characteristic” while others maintain that truth is a deflationary rather than an authentic property of truth bearers. Taking realism as a default option, Engel argues for a stable median position which states that truth is a more substantial concept than deflationism claims and less substantial than realistic theories say it is. In this article we develop an alternative median solution to the position proposed by Engel. It is based on an anti-realistic and pragmatic (but not utilitarian) option. More precisely, we show the extent to which the dialogical conception of truth advanced by Kuno Lorenz is opposed to the theses defended by Engel while pursuing the same goal.

"Dialogisches Handeln" ist eine Sammlung von philosophischen BeitrAgen zum dialogischen Konstruktivismus im Sinne von Kuno Lorenz, der Philosophie als eine Kommunikation mit Selbstreflexion versteht. Dabei umfaAt der Dialog in diesem Buch verschiedene Disziplinen der Philosophie - von der Anthropologie A1/4ber die Grundlagen von Logik und Sprachphilosophie bis hin zur Wissenschafts- und Erkenntnistheorie. Die Reflexion auf die kommunikativen Grundlagen wissenschaftlichen Austauschs schlieAt dabei Denkperspektiven verschiedener Wissenschaften - von der Psychologie bis zu Physik und Chemie - ebenso ein wie die philosophischen Disziplinen.