*Early Work in Poland—Geometry and Teaching*

**Author**: Andrew McFarland

**Publisher:** Springer

**ISBN:**

**Category:** Mathematics

**Page:** 499

**View:** 451

Alfred Tarski (1901–1983) was a renowned Polish/American mathematician, a giant of the twentieth century, who helped establish the foundations of geometry, set theory, model theory, algebraic logic and universal algebra. Throughout his career, he taught mathematics and logic at universities and sometimes in secondary schools. Many of his writings before 1939 were in Polish and remained inaccessible to most mathematicians and historians until now. This self-contained book focuses on Tarski’s early contributions to geometry and mathematics education, including the famous Banach–Tarski paradoxical decomposition of a sphere as well as high-school mathematical topics and pedagogy. These themes are significant since Tarski’s later research on geometry and its foundations stemmed in part from his early employment as a high-school mathematics teacher and teacher-trainer. The book contains careful translations and much newly uncovered social background of these works written during Tarski’s years in Poland. Alfred Tarski: Early Work in Poland serves the mathematical, educational, philosophical and historical communities by publishing Tarski’s early writings in a broadly accessible form, providing background from archival work in Poland and updating Tarski’s bibliography.

Professor Peter Mittelstaedt is a physicist whose primary concern is the foundations of current physical theories. This concern has made him, through his prolonged, incisive and detailed examinations of the structures and overall characteristics of these theories, into a philosopher of physic- of contemporary physics, to be precise, of relativistic theories of space and time, and of the logic of quantum mechanics, in particular. The present book, which expounds his main ideas in these matters, has seen four editions (in German), each including newer results - as indeed does the present translation: see the author's 1975 preface to the English translation. Perhaps this is the place to repeat the author's chief problem and mention his own approach, even though they are expounded in his Intro duction. How close is Mittelstaedt to Kant's understanding of science? We are at liberty to choose a framework for thought - a logic and a method ology - prior to experience (in the classic sense, to think a priori); yet we choose a framework so as to fit our empirical findings. How is this done? How may it be understood and justified? This is obviously the question of all philosophies that evolve from, and are in reaction to, Kant's system.

The text is addressed to students and mathematicians who wish to learn the subject. It can also be used as a reference book and as a textbook for short courses on geometry.

The main item in the present volume was published in 1930 under the title Das Unendliche in der Mathematik und seine Ausschaltung. It was at that time the fullest systematic account from the standpoint of Husserl's phenomenology of what is known as 'finitism' (also as 'intuitionism' and 'constructivism') in mathematics. Since then, important changes have been required in philosophies of mathematics, in part because of Kurt Godel's epoch-making paper of 1931 which established the essential in completeness of arithmetic. In the light of that finding, a number of the claims made in the book (and in the accompanying articles) are demon strably mistaken. Nevertheless, as a whole it retains much of its original interest and value. It presents the issues in the foundations of mathematics that were under debate when it was written (and in some cases still are); , and it offers one alternative to the currently dominant set-theoretical definitions of the cardinal numbers and other arithmetical concepts. While still a student at the University of Vienna, Felix Kaufmann was greatly impressed by the early philosophical writings (especially by the Logische Untersuchungen) of Edmund Husser!' He was never an uncritical disciple of Husserl, and he integrated into his mature philosophy ideas from a wide assortment of intellectual sources. But he thought of himself as a phenomenologist, and made frequent use in all his major publications of many of Husserl's logical and epistemological theses.

Introduces the problem of the symbolic structure of physics, surveys the modern history of symbols, proceeds to an epistemological discussion of the role of symbols in our knowledge of nature, and addresses key issues related to the methodology of physics and the character of its symbolic structures.