[A dialogue between Kant and Bolzano on mathematical knowledge.].

This paper surveys Bolzano's Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a better-grounded presentation of mathematics) on the 200th anniversary of its publication. The first and only published issue presents a definition of mathematics, a classification of its subdisciplines, and an essay on mathematical method, or logic. Though underdeveloped in some areas (including,somewhat surprisingly, in logic), it is nonetheless a radically innovative work, where Bolzano presents a remarkably modern account of axiomatics and the epistemology of the formal sciences. (...) We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics. Here we find the beginnings of his theory of collections, for him the most fundamental of the mathematical disciplines. Though not exactly the same as the later Cantorian set theory, Bolzano's theory of collections was used in very similar ways in mathematics, notably in analysis. In retrospect, Bolzano's debut in philosophy was a remarkably successful one, though its fruits would only become generally known much later. (shrink)

Ce dialogue confronte deux conceptions qui dominent jusqu’à nos jours la philosophie des mathématiques : d’un côté la conception kantienne qui souligne l’irréductible apport de l’intuition dans la formulation des axiomes, ainsi que l’effectivité des procédés de construction ; de l’autre côté la conception bolzanienne qui s’efforce d’éliminer toute intervention de l’intuition au profit des démonstrations et des procédés purement conceptuels.In this dialogue, two opposed conceptions, which dominate the philosophy of mathematics till today, are confronted. Kant’s account of mathematics is (...) based upon the activity of constructing mathematical objects in pure intuition . In yielding objects for mathematics, our intuition contributes in an essential way to the formulation of mathematical truths. Against Kant, Bolzano argues that intuition has place neither in arithmetic nor in geometry and that mathematical existence consists in the possibility of the defined objects. i.e. in non-contradiction. For Bolzano, the central idea of mathematics is that of rigorous proof. (shrink)

Bernard Bolzano was born in Prague in 1781 and died in 1848 in the same city1. His major work Wissenschaftslehre was published in 1837 in Sulzbach, Germany.Since the publication of Jan Berg’s Bolza...

Review of S. Lapointe: Bolzano’s Theoretical Philosophy, An Introduction.

Jean-Paul Coujou a traduit – première en langue française –, commenté et annoté des ouvrages choisis dans l’œuvre monumentale de Francisco Suarez. Avant les trois présents volumes, il avait publié les trois premières Disputes et la XXXIe . Depuis trois..

Analytical philosophy begins with the first mathematical and philosophical works of Bolzano published between 1804 and 1817. There, Bolzano set out a project for the global reform of mathematics by means of the axiomatic method. Having completed the Wissenschaftslehre, Bolzano wrote a summary of his logic for the Größenlehre, which he sent to Exner in 1833. The correspondence between Bolzano and Exner covered some of the main subjects treated by analytical philosophy: the status of abstract objects (propositions and objective ideas), (...) intuitions, objectless ideas, the concept of object and many others. While Bolzano argued in favor of abstract entities independent of mind and of language, Exner considered them as abstractions obtained from the subjective judgments and representations. During the XlXth century, Bolzano's philosophy spread over Bohemia and Austria through manuscripts and through the first edition of Zimmermann's textbook of philosophy. The most important Brentanians, Kerry, Twardowski, Meinong and Husserl, discussed his doctrines which may also have influenced Wittgenstein and the Polish school. (shrink)

Analytical philosophy begins with the first mathematical and philosophical works of Bolzano published between 1804 and 1817. There, Bolzano set out a project for the global reform of mathematics by means of the axiomatic method. Having completed the Wissenschaftslehre, Bolzano wrote a summary of his logic for the Größenlehre, which he sent to Exner in 1833. The correspondence between Bolzano and Exner covered some of the main subjects treated by analytical philosophy: the status of abstract objects (propositions and objective ideas), (...) intuitions, objectless ideas, the concept of object and many others. While Bolzano argued in favor of abstract entities independent of mind and of language, Exner considered them as abstractions obtained from the subjective judgments and representations. During the XlXth century, Bolzano's philosophy spread over Bohemia and Austria through manuscripts and through the first edition of Zimmermann's textbook of philosophy. The most important Brentanians, Kerry, Twardowski, Meinong and Husserl, discussed his doctrines which may also have influenced Wittgenstein and the Polish school. (shrink)

« Pour la première fois peut-être la science n’est plus considérée comme simple intermédiaire entre l’esprit humain et l’être en soi, dépendant autant de l’un que de l’autre et n’ayant pas de réalité propre, mais comme un objet sui generis, original dans son essence, autonome dans son mouvement ». En écoutant les différentes contributions à ce colloque, je me suis retrouvé sur un terrain familier. Presque tous les noms des auteurs des xviie et xviiie siècles m’étaient bien connus : ceux (...) de La... (shrink)

Bernard Bolzano is increasingly recognized as one of the greatest nineteenth-century philosophers. A philosopher and mathematician of rare talent, he made ground-breaking contributions to logic, the foundations and philosophy of mathematics, metaphysics, and the philosophy of religion. Many of the larger features of later analytic philosophy first appear in his work: for example, the separation of logic from psychology, his sophisticated understanding of mathematical proof, his definition of logical consequence, his work on the semantics of natural kind terms, or his (...) anticipations of Cantor's set theory, to name but a few. To his contemporaries, however, he was best known as an intelligent and determined advocate for reform of Church and State. Based in large part on a carefully argued utilitarian practical philosophy, he developed a program for the non-violent reform of the authoritarian institutions of the Hapsburg Empire, a program which he himself helped to set in motion through his teaching and other activities. Rarely has a philosopher had such a great impact on the political culture of his homeland. Persecuted in his lifetime by secular and ecclesiastical authorities, long ignored or misunderstood by philosophers, Bolzano's reputation has nevertheless steadily increased over the past century and a half. Much discussed and respected in Central Europe for over a century, he is finally beginning to receive the recognition he deserves in the English-speaking world. This book provides a comprehensive and detailed critical introduction to Bolzano, covering both his life and works. (shrink)

Les origines grecques de la conception d'harmonie (H. Dufourt) - Du monde au matériau : le devenir-musical du timbre (A. Soulez) - La musique est-elle un langage? (R. Casati et J. Dokic) - Le devenir du matériau musical au XXe siècle (M. Solomos) - Hauteur, timbre, harmonie, synthèse (J.C. Risset) et d'autres articles...

The relevance of metamathematical researches for philosophy of math- ematics is an indubitable matter. In the paper I shall speak about impli- cations of metamathematics for general philosophy, especially for classical epistemological problems. Let us start with a historical observation con- cerning Hilbert's programme, the rst research programme in metamathe- matics as a separate study of formal systems. This programme was strongly in uence by epistemological considerations. In fact, Hilbert wanted to se- cure all classical mathematics against inconsistencies and this (...) aim had be achieved with the help of nitary consistency proof. Hilbert's claim is like the Cartesian project of reduction of all concepts to clarae et distinctae ideae. The epistemological commitment of original Hilbert's programme may be treated as a prelim- inary heuristic motivation for looking for epistemological implications of metamathematical results. It seems reasonable to examine in this respect three so-called limitative theorems: the rst Godel's incompleteness theo- rem , the second Godel's incom- pleteness theorem consistency of S, cannot be proved in S, providing that S is consistent) and Tarski's undenability theorem ; \S" stands for formal system containing elementary number theory. From the above-mentioned theorems we obtain an important, from philosophical point of view, conclusion: semantics of S cannot be expressed in S. Now, I shall present three applications of limitative theorems to the analysis of classical epistemological problems. (shrink)

Stephenson (2022) has argued that Kant’s thesis that all transcendental truths are transcendentally a priori knowable leads to omniscience of all transcendental truths. His arguments depend on luminosity principles and closure principles for transcendental knowability. We will argue that one pair of a luminosity and a closure principle should not be used, because the closure principle is too strong, while the other pair of a luminosity and a closure principle should not be used, because the luminosity principle is too strong. (...) Stephenson’s argument also depends on a factivity principle for transcendental knowability, which we will argue to be false. (shrink)

1. Die erste Hälfte des Kommentars, mit Ausnahme der Kapitel über die Weltseele.

'What is real?' has been one of the key questions of philosophy since its beginning in antiquity. But it is not just a question that philosophers ask. This Very Short Introduction discusses what reality is by looking at a variety of arguments, theories, and thought-experiments from philosophy, physics, and cognitive science.

The Natural Philosophy of Jan Bayer The main focus of this study is a reconstruction of the natural philosophy of the early modern Prešov's scholar Jan (Johannes) Bayer (1630–1674), with special regard to its Mosaic profile. After a critical reading of the research done on Bayer up to this point, the author concludes that Bayer’s natural-philosophical work, as such, has not yet been satisfactorily analyzed, nor has its connection to its supposedly two most important sources, Francis Bacon and Jan Amos (...) Comenius, been sufficiently explored. The study therefore concentrates on a critical analysis of Bayer’s interpretation of creation and his natural history, which are then both situated in the philosophical context of their time. The author then notes that although Bayer’s project of erecting an original and on the (especially Aristotelian) tradition independent Mosaic natural philosophy ended in failure, the thinker’s work can be assessed as a valuable attempt to synthesize Baconian and Comenian motifs while at the same time reflecting broader intellectual contexts of the period. (shrink)