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)

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)

Zygmunt Ratajczyk was a deep and subtle mathematician who, with mastery, used sophisticated and technically complex methods, in particular combinatorial and proof-theoretic ones. Walking always along his own paths and being immune from actual trends and fashions he hesitated to publish his results, looking endlessly for their improvement.

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)

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 the presentation and analysis of the philosophical views concerning logic and mathematics of the leading members of Cracow Circle, i.e., of Jan Salamucha, Jan Franciszek Drewnowski and Józef Maria Bocheński. Their views on the problem of possible applicability of logical tools in metaphysical and theological researches is also discussed.

The aim of this book is to present and analyze philosophical conceptions concerning mathematics and logic as formulated by Polish logicians, mathematicians and philosophers in the 1920s and 1930s. It was a remarkable period in the history of Polish science, in particular in the history of Polish logic and mathematics. Therefore, it is justified to ask whether and to what extent the development of logic and mathematics was accompanied by a philosophical reflection. We try to answer those questions by analyzing (...) both works of Polish logicians and mathematicians who have a philosophical temperament as well as their research practice. Works and philosophical views of the following Polish scientists will be analyzed: Wacław Sierpiński, Zygmunt Janiszewski, Stefan Mazurkiewicz, Stefan Banach Hugo Steinhaus, Eustachy Żylińsk and Leon Chwistek, Jan Łukasiewicz, Zygmunt Zawirski, Stanisław Leśniewski, Tadeusz Kotarbiński, Kazimierz Ajdukiewicz, Alfred Tarski, Andrzej Mostowski and Henryk Mehlberg, Jan Sleszyński, Stanisław Zaremba and Witold Wilkosz. To indicate the background of scientists being active in the 1920s and 1930s we consider in Chapter 1 some predecessors, in particular: Jan Śniadecki, Józef Maria Hoene-Wroński, Samuel Dickstein and Edward Stamm. (shrink)

Two crucial concepts of the methodology and philosophy of mathematics are considered: proof and truth. We distinguish between informal proofs constructed by mathematicians in their research practice and formal proofs as defined in the foundations of mathematics (in metamathematics). Their role, features and interconnections are discussed. They are confronted with the concept of truth in mathematics. Relations between proofs and truth are analysed.

Lvov-Warsaw School in Philosophy – as the very name suggests – was connected mainly with two academic centers: universities in Lvov and Warsaw. However, it had a broader impact. The members of this school were active also at other universities, in particular in Cracow, Vilnius and Poznań. The aim of the paper is to present and analyze the connections of Lvov-Warsaw School with the University in Poznań.

Nicholas of Cusa was first of all a theologian but he was interested also in mathematic and natural sciences. In fact philosophico-theological and mathematical ideas were intertwined by him, theological and philosophical ideas influenced his mathematical considerations, in particular when he considered philosophical problems connected with mathematics and vice versa, mathematical ideas and examples were used by him to explain some ideas from theology. In this paper we attempt to indicate this mutual influence. We shall concentrate on the following problems: (...) the role and place of mathematics and mathematical knowledge in knowledge in general and in particular in theological knowledge, ontology of mathematical objects and their origin, in particular their relations to God and their meaning for the description of the world and physical reality, infinity in mathematics versus infinity in theology and their mutual relations and connections. It will be shown that—according to Nicholas—mathematics and mathematical thinking are tools of rationalization of theology and liberating it in a certain sense from the trap of apophatic theology. (shrink)

In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated.

The paper is devoted to the description of life and scientific achievements as well as the influence of Seweryna Łuszczewska-Romahnowa.

The volume contains twenty essays devoted to the philosophy of mathematics and the history of logic. They have been divided into four parts: general philosophical problems of mathematics, Hilbert's program vs. the incompleteness phenomenon, philosophy of mathematics in Poland, mathematical logic in Poland. Among considered problems are: epistemology of mathematics, the meaning of the axiomatic method, existence of mathematical objects, distinction between proof and truth, undefinability of truth, Goedel's theorems and computer science, philosophy of mathematics in Polish mathematical and logical (...) schools, beginnings of mathematical logic in Poland, contribution of Polish logicians to recursion theory. (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.

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.

The paper is devoted to the presentation of Chwistek’s philosophical ideas concerning logic and mathematics. The main feature of his philosophy was nominalism, which found full expression in his philosophy of mathematics. He claimed that the object of the deductive sciences, hence in particular of mathematics, is the expression being constructed in them according to accepted rules of construction. He treated geometry, arithmetic, mathematical analysis and other mathematical theories as experimental disciplines, and obtained in this way a nominalistic interpretation of (...) them. The fate of Chwistek’s philosophical conceptions was similar to the fate of his logical conceptions. The system of rational meta-mathematics was not developed by him in detail. He worked on his own ideas without any collaboration with other logicians, mathematicians or philosophers. His investigations were not in the mainstream of the development of logic and philosophy of mathematics. (shrink)

In the paper the views and tendencies in the philosophical reflection on mathematics in Poland between the wars are analyzed. Views of most outstanding representatives of Lvov–Warsaw Philosophical School and of Polish Mathematical School are presented. Their influence on logical and mathematical researches is considered.

Among the main theorems obtained in mathematical logic in this century are the so called limitation theorems, i.e., the Löwenheim-Skolem theorem on the cardinality of models of first-order theories, Gödel’s incompleteness theorems and Tarski’s theorem on the undefinability of truth. Problems connected with the latter are the subject of this paper. In Section 1 we shall consider Tarski’s theorem. In particular the original formulation of it as well as some specifications will be provided. Next various meanings of the notion of (...) a satisfaction predicate will be studied. In Section 2 the problem of definability of the notion of truth, in particular of the notion of truth for the language of Peano arithmetic PA, will be discussed. It will be explicitly shown that the notion of satisfaction for the language of PA can be defined in a certain weak fragment of the second order arithmetic. Finally the axiomatic characterization of satisfaction and truth as well and its mathematical and philosophical meaning will be discussed. (shrink)

Contents: Preface. SCIENTIFIC WORKS OF MARIA STEFFEN-BATÓG AND TADEUSZ BATÓG. List of Publications of Maria Steffen-Batóg. List of Publications of Tadeusz Batóg. Jerzy POGONOWSKI: On the Scientific Works of Maria Steffen-Batóg. Jerzy POGONOWSKI: On the Scientific Works of Tadeusz Batóg. W??l??odzimierz LAPIS: How Should Sounds Be Phonemicized? Pawe??l?? NOWAKOWSKI: On Applications of Algorithms for Phonetic Transcription in Linguistic Research. Jerzy POGONOWSKI: Tadeusz Batóg's Phonological Systems. MATHEMATICAL LOGIC. Wojciech BUSZKOWSKI: Incomplete Information Systems and Kleene 3-valued Logic. Maciej KANDULSKI: Categorial Grammars with (...) Structural Rules. Miros??l??awa KO??L??OWSKA-GAWIEJNOWICZ: Labelled Deductive Systems for the Lambek Calculus. Roman MURAWSKI: Satisfaction Classes - a Survey. Kazimierz _WIRYDOWICZ: A New Approach to Dyadic Deontic Logic and the Normative Consequence Relation. Wojciech ZIELONKA: More about the Axiomatics of the Lambek Calculus. THEORETICAL LINGUISTICS. Jacek Juliusz JADACKI: Troubles with Categorial Interpretation of Natural Language. Maciej KARPI??N??SKI: Conversational Devices in Human-Computer Communication Using WIMP UI. Witold MACIEJEWSKI: Qualitative Orientation and Grammatical Categories. Zygmunt VETULANI: A System of Computer Understanding of Texts. Andrzej WÓJCIK: The Formal Development of van Sandt's Presupposition Theory. W??l??adys??l??aw ZABROCKI: Psychologism in Noam Chomsky's Theory . Ryszard ZUBER: Defining Presupposition without Negation. PHILOSOPHY OF LANGUAGE AND METHODOLOGY OF SCIENCES. Jerzy KMITA: Philosophical Antifundamentalism. Anna LUCHOWSKA: Peirce and Quine: Two Views on Meaning. Stefan WIERTLEWSKI: Method According to Feyerabend. Jan WOLE??N??SKI: Wittgenstein and Ordinary Language. Krystyna ZAMIARA: Context of Discovery - Context of Justification and the Problem of Psychologism. (shrink)

In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.

Praca poświęcona jest prezentacji i krytycznej ocenie poglądów filozoficznych Jana Franciszka Drewnowskiego na matematykę i logikę. Podstawą rozważań są cztery źródła. Podstawowym źródłem jest Drewnowskiego "Zarys programu filozoficznego", dalej dwa artykuły, a mianowicie "Stosowanie logiki symbolicznej w filozofii" oraz "Uwagi o stosowaniu logiki symbolicznej", w końcu fragmenty dziennika. Pokazuje się w niej, jak Drewnowski rozumiał matematykę i teorie matematyczne, jak widział logikę i jej rolę w nauce oraz na ile był zaznajomiony ze współczesnymi osiągnięciami w zakresie logiki matematycznej i podstaw (...) matematyki oraz z ich konsekwencjami dla filozofii matematyki. (shrink)

The paper is devoted to two Polish philosophers and logicians Henryk Struve and Władysław Biegański. They represented the traditional pre-mathematical approach to logic and stood on the threshold of the new paradigm of logic as formal mathematical logic. Their views concerning logic and its philosophy as well as its relations to philosophy and mathematics will be analyzed. Those views will be compared with views of their contemporaries and with views of Jan Łukasiewicz—one of the main leaders of Warsaw School of (...) Logic who represented the new paradigm. (shrink)

The aim of this paper is to present and analyse the views on logic of the members of the so-called Cracow Circle, namely the Dominican Father Józef M. Bocheński, Rev. Jan Salamucha, and Jan Franciszek Drewnowski. They tried to apply the methods of modern formal/mathematical logic to philosophical and theological problems. In particular, they attempted to modernise contemporary Thomism by employing logical tools. The influence of Jan Łukasiewicz, the co-founder of the Warsaw School of Logic will be also discussed. --------------- (...) Received: 16/01/2021. Reviewed: 24/03/2021. Accepted: 13/05/2021. (shrink)