The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems (...) connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the side of mathematics. The knowledge of mathematics needed to understand the text is elementary. Reports on historical conceptions. Thinking about today‘s mathematical doing and thinking. Recent developments. Based on the third, revised German edition. For mathematicians - students, teachers, researchers and lecturers - and readersinterested in mathematics and philosophy. Contents On the way to the reals On the history of the philosophy of mathematics On fundamental questions of the philosophy of mathematics Sets and set theories Axiomatic approach and logic Thinking and calculating infinitesimally – First nonstandard steps Retrospection. (shrink)

The paper is devoted to phenomenological ideas in conceptions of modern philosophy of mathematics. Views of Husserl, Weyl, Becker andGödel will be discussed and analysed. The aim of the paper is to show the influence of phenomenological ideas on the philosophical conceptions concerning mathematics. We shall start by indicating the attachment of Edmund Husserl to mathematics and by presenting the main points of his philosophy of mathematics. Next, works of two philosophers who attempted to apply Husserl’s phenomenological ideas to the (...) philosophy of mathematics, namely Hermann Weyl and Oskar Becker, will be briefly discussed. Lastly, the connections between Husserl’s ideas and the philosophy of mathematics of Kurt Gödel will be studied. (shrink)

Dieses Werk gibt eine überwiegend elementare Einführung in philosophische Probleme und Hintergründe des mathematischen Denkens und Sprechens, Lehrens und Lernens. Sie wendet sich an Lehrende und Studierende der Mathematik und der Philosophie. Ausgangspunkt und immer wieder Bezugspunkt sind die reellen Zahlen. In pointierter Weise werden mathematische und philosophische Probleme und Fragen vermerkt, die sich auf dem Weg zu ihnen stellen. Ein umfangreicher Abriss von Auffassungen aus der Geschichte der Mathematik und der Philosophie bis hin zu aktuellen Strömungen bildet den Hintergrund (...) für ihre eingehende Diskussion. Kapitel über Mengenlehre, Logik und Axiomatik, über ungelöste und unlösbare Probleme und fundamentale Ergebnisse schließen den Text ab. (shrink)

Dieses Werk gibt eine überwiegend elementare Einführung in philosophische Probleme und Hintergründe des mathematischen Denkens und Sprechens, Lehrens und Lernens. Sie wendet sich an Lehrende und Studierende der Mathematik und der Philosophie. Ausgangspunkt und immer wieder Bezugspunkt sind die reellen Zahlen. In pointierter Weise werden mathematische und philosophische Probleme und Fragen vermerkt, die sich auf dem Weg zu ihnen stellen. Ein umfangreicher Abriss von Auffassungen aus der Geschichte der Mathematik und der Philosophie bis hin zu aktuellen Strömungen bildet den Hintergrund (...) für ihre eingehende Diskussion. Kapitel über Mengenlehre, Logik und Axiomatik, über ungelöste und unlösbare Probleme und fundamentale Ergebnisse schließen den Text ab. (shrink)

Dieses Werk gibt eine überwiegend elementare Einführung in philosophische Probleme und Hintergründe des mathematischen Denkens und Sprechens, Lehrens und Lernens. Sie wendet sich an Lehrende und Studierende der Mathematik und der Philosophie. Ausgangspunkt und immer wieder Bezugspunkt sind die reellen Zahlen. In pointierter Weise werden mathematische und philosophische Probleme und Fragen vermerkt, die sich auf dem Weg zu ihnen stellen. Ein umfangreicher Abriss von Auffassungen aus der Geschichte der Mathematik und der Philosophie bis hin zu aktuellen Strömungen bildet den Hintergrund (...) für ihre eingehende Diskussion. Kapitel über Mengenlehre, Logik und Axiomatik, über ungelöste und unlösbare Probleme und fundamentale Ergebnisse schließen den Text ab. (shrink)

Justification logics are epistemic logics that explicitly include justifications for the agents' knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting's semantics for the Logic of Proofs LP. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya's minimal bimodal (...) explicit evidence logic, which is a two-agent version of LP. We discuss the relationship of our logic to the multi-agent modal logic S4 with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic. (shrink)

The paper is devoted to the discussion of some philosophical and historical problems connected with the theorem on the undefinability of the notion of truth. In particular the problem of the priority of proving this theorem will be considered. It is claimed that Tarski obtained this theorem independently though he made clear his indebtedness to Gödel’s methods. On the other hand, Gödel was aware of the formal undefinability of truth in 1931, but he did not publish this result. Reasons for (...) that are also considered. (shrink)

This volume is written jointly by Witold Marciszewski, who contributed the introductory and the three subsequent chapters, and Roman Murawski who is the author of the next ones - those concerned with the 19th century and the modern inquiries into formalization, algebraization and mechanization of reasonings. Besides the authors there are other persons, as well as institutions, to whom the book owes its coming into being. The study which resulted in this volume was carried out in the Historical (...) Section of the research project _Logical Systems and Algorithms for Automatic Testing of Reasoning,_ 1986-1990, in which participated nine Polish universities; the project was coordinated by the Department of Logic, Methodology and Philosophy of Science of the Bia??l??ystok Branch of the University of Warsaw, and supported by the Ministry of Education 1987). The major part of the project was focussed on the software for computer-aided theorem proving called Mizar MSE ) due to Dr. Andrzej Trybulec. He and other colleagues deserve a grateful mention for a hands-on experience and theoretical stimulants owed to their collaboration. (shrink)

Roman Murawski; V*—The Development of Symbolism in Logic and its Philosophical Background, Proceedings of the Aristotelian Society, Volume 89, Issue 1, 1 June 1.