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This Publication was financially supported by the Adam Mickiewicz University, Poznan.
ISSN 2191-1878
ISBN 978-3-631-80714-9 (Print)
E-ISBN 978-3-631-82018-6 (E-Book)
E-ISBN 978-3-631-82019-3 (EPUB)
E-ISBN 978-3-631-82020-9 (MOBI)
DOI 10.3726/b16869
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About the author
Roman Murawski, studied mathematics, philosophy and theology; MSc 1972, PhD 1979, Master of Theology 1980, licentiatus in sacra theologia 1985, Habilitation 1992, Full professor 2001; Professor of logic and philosophy of mathematics at Adam Mickiewicz University in Poznań (Poland).
About the book
The volume consists of thirteen papers devoted to various problems of the philosophy of logic and mathematics. They can be divided into two groups. The first group contains papers devoted to some general problems of the philosophy of mathematics whereas the second group – papers devoted to the history of logic in Poland and to the work of Polish logicians and mathematicians in the philosophy of mathematics and logic. Among considered problems are: meaning of reverse mathematics, proof in mathematics, the status of Church’s Thesis, phenomenology in the philosophy of mathematics, mathematics vs. theology, the problem of truth, philosophy of logic and mathematics in the interwar Poland.
Dedicated to
my wife Hania and daughter Zosia
Foreword
The volume consists of thirteen papers devoted to various problems of the philosophy of logic and mathematics. They can be divided into two groups. The first group contains papers devoted to some general problems of the philosophy of mathematics whereas the second group – papers devoted to the history of logic in Poland and to the work of Polish logicians and mathematicians in the philosophy of mathematics and logic.
The first group is opened by the chapter “On the philosophical meaning of reverse mathematics” in which philosophical consequences of this research project in the foundations of mathematics are discussed. In particular, we are interested in its implications for Hilbert’s program. The chapter “On the distinction proof–truth in mathematics” is devoted to some historical, philosophical and logical considerations connected with the distinction between proof and truth in mathematics. The crucial role of Gödel’s incompleteness theorems as well as of the undefinability of truth vs. definability of provability and the role of finitary vs. infinitary methods are stressed. The problem of the need of extending the available methods by new rules of inference and new axioms is also considered. The problem of a proof in mathematics is discussed also in the chapter “Some historical, philosophical and methodological remarks on proof in mathematics”. It is devoted to historical and philosophical as well as methodological considerations on the role and meaning of proof in mathematics. In particular, the following problems are discussed: the role of informal proofs in mathematical research practice, the concept of formal proof and its role, the problem of the distinction: (formal) provability vs. truth and relations between informal and formal proofs. The chapter “The status of Church’s Thesis” is devoted to the problem of the status of this famous hypothesis of the theory of computability. The following possibilities are considered and discussed: Church’s Thesis as an empirical hypothesis, as an axiom or theorem, as a definition and as an explication. Next two chapters closing the first part of the volume are devoted to particular conceptions in the philosophy of mathematics. The chapter “Between theology and mathematics” is devoted to philosophical and theological as well as mathematical ideas of Nicholas of Cusa (1401–1464).He was a mathematician but first of all a theologian. Connections between theology and philosophy on the one side and mathematics on the other were by him bilateral. In the chapter, it is shown how some theological ideas were used by him to answer the fundamental questions in the philosophy of mathematics. The chapter “Phenomenological ideas in the philosophy of mathematics” is devoted to phenomenological ideas in conceptions of modern philosophy of mathematics. Views of Husserl, Weyl, Becker and Gödel are discussed and analysed. The aim of this chapter is to show the influence ←00 | 6→of phenomenological ideas on the philosophical conceptions concerning mathematics.We 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, are briefly discussed. Lastly, the connections between Husserl’s ideas and the philosophy of mathematics of Kurt Gödel are studied.
The second group of chapters is opened by a historical chapter on mathematical logic and the foundations of mathematics in the reborn (after the First World War) Poland. The rise of Warsaw School of Logic and of Polish School of Mathematics are described; and the background of this process, the cultural and scientific, in particular philosophical atmosphere in which those processes took place, is presented. The next chapter in this group “Tarski and his Polish predecessors on truth” is devoted to the description and analysis of Alfred Tarski’s views concerning the concept of truth. Conceptions of his Polish predecessors: Twardowski, Łukasiewicz, Zawirski, Czeżowski and Kotarbiński are also discussed. The chapter “Benedykt Bornstein and his philosophy of logic and mathematics” presents philosophical views on logic and mathematics of this rather unknown and almost completely forgotten significant Polish philosopher. Though he was a Ph.D. student of Kazimierz Twardowski, he was not a member of the Lvov–Warsaw Philosophical School – mainly because of his metaphysical views. In some way, he was an individualist; his research did not follow the main trend. However, his views and conceptions were interesting. The following three chapters are devoted to three centres of logic and mathematics in the interwar Poland, namely: Warsaw (Warsaw School of Mathematical Logic), Cracow and Lvov (Lvov School of Mathematics) and to the presentation of philosophical views on logic and mathematics developed and proclaimed there. In particular, views of Tarski, Andrzej Mostowski (Warsaw), Jan Sleszyński, Stanisław Zaremba, Zygmunt Zawirski, Witold Wilkosz and Leon Chwistek (Cracow; in fact Chwistek was active both in Cracow and Lvov) as well as Hugo Steinhaus, Stefan Banach and Eustachy Żyliński (Lvov) are discussed. The volume is closed by a chapter devoted to the description and analysis of philosophical views on logic and mathematics