Karl Menger Selecta Mathematica Volume 2

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3 Karl Menger Selecta Mathematica Volume 2 Bert Schweizer, Abe Sklar, Karl Sigmund, Peter Gruber, Edmund Hlawka, Ludwig Reich, Leopold Schmetterer (eds.) Springer-Verlag Wien GmbH

4 Prof. Dr. Bert Schweizer Department of Mathematics and Statistics University of Massachusetts, Lederle Graduate Research Center Amherst, MA, USA Prof. Dr. Abe Sklar Department of Mathematics, Illinois Institute of Technology Chicago, IL, USA O. Univ.-Prof. Dr. Karl Sigmund em. O. Univ.-Prof. Dr. Leopold Schmetterer Institute of Mathematics, University of Vienna, Austria O. Univ.-Prof. Dr. DDr. h.c. Peter Gruber em. O. Univ.-Prof. Dr. DDr. h.c. Edmund Hlawka Institute of Analysis and Technical Mathematics Technical University, Vienna, Austria O. Univ.-Prof. Dr. Ludwig Reich Institute of Mathematics, University of Graz, Austria This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks 2003 Springer-Verlag Wien Ursprünglich erschienen bei Springer-Verlag/Wien 2003 Softcover reprint ofthe hardcover 1 st edition 2003 Product Liability: The publisher can give no guarantee for all the information contained in this book Printed on acid-free and chlorine-free bleached paper SPIN: With 11 Figures CIP data applied for ISBN ISBN (ebook) DOI /

5 Table of Contents Menger's Work on Logic and Foundations... 1 Van Dalen, D.: Commentary on Menger and Intuitionism Bemerkungen zu Grundlagenfragen I: Über Verzweigungsmengen, Jahresbericht der Deutschen Mathematiker-Vereinigung 37 (1928), 2l Selected Papers on Analysis Reich, L.: Commentary on Karl Menger's Contributions to Analysis On Cauchy's Integral Theorem in the Real Plane, Proc. Nat. Acad. Sei. 25 (1939), On Green's Formula, Proc. Nat. Acad. Sei. 26 (1940), The Behavior of a Complex Function at Infinity, Proc. Nat. Acad. Sei. 41 (1955), 512-5l A Characterization of Weierstrass Analytic Functions, Proc. Nat. Acad. Sei. 54 (1965), Analytische Funktionen, Wissenschaftliche Abhandlungen der Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Vol. 33, Köln (1965), Une caracterisation des fonetions analytiques, Comptes Rendus Acad. Paris 261 (1965), Weierstrass Analytic Functions, Mathematische Annalen 167 (1966), v-

6 Table of Contents Selected Papers on Length Senechal, L.: Commentary on Menger's Path Length Papers On Shortest Polygonal Approximations to a Curve, Reports of Math. Colloq. Notre Dame, Indiana 2 (1940), Definition intrinseque de la notion de chemin, Comptes Rendus Acad. Paris 221 (1945), Stieltjes Integrals Considered as Lengths, Annales Societe Polonaise Math. 21 (1948), What Paths Have Length?, Fundamenta Mathematicae 36 (1949), A Topological Characterization of the Length of Paths, Proc. Nat. Acad. Sei. 38 (1952), Selected Papers on Algebra of Analysis 109 Sklar, A.: Commentary on the Algebra of Analysis and Algebra of Functions Kaiser, H.: The Inftuence of Menger's Ideas on the Work of Nöbauer and His School Algebra of Analysis, Report Notre Dame Mathematical Lectures, Vol. 3, 1944 (pp. I-lI, 1-44) Tri-Operational Algebra, Reports of Math. Colloq. Notre Dame, Indiana 5/6 (1944), General Algebra of Analysis, Reports of Math. Colloq. Notre Dame, Indiana 7 (1946), An Axiomatic Theory of Functions and Fluents, in: The Axiomatic Method (eds. L. Henkin, P. Suppes and A. Tarski) North Holland, 1959, The Algebra of Functions: Past, Present, Future, Rendiconti di Matematica 20 (1961), Vl-

7 Table of Contents On Substitutive Algebra and Its Syntax, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 10 (1964), Superassociative Systems and Logical Functors, Mathematische Annalen 157 (1964), Two Theorems on the Generation of Systems of Functions (with H. I. Whitlock), Fundamenta Mathematicae 58 (1966), Selected Papers on Didactics, Variables, and Fluents 301 Schweizer, B., Sklar, A.: Commentary on Didactics, Variables, Fluents Senechal, L., Schweizer, B.: A Mengerian Tour Along Caratheodory's Royal Road Ca1culus - A Modem Approach, Introduction and Chapter 1, Ginn and Co., Boston, 1955, xi-xvii and The Mathematics of Elementary Thermodynamics, American Journal of Physics 18 (1950), Random Variables from the Point of View of a General Theory of Variables, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability ( ), eds. L. M. Le Cam and O. Neyman, University of California Press, Berkeley, 1956, Vol. 2, What are x and y? The Mathematical Gazette 40 (1956), Rates of Change and Derivatives, Fundamenta Mathematicae 46 (1958), Selected Papers on Probabilistic Geometry Schweizer, B.: Commentary on Probabilistic Geometry Statistical Metrics, Proc. Nat. Acad. Sci. 28 (1942), Vll-

8 Table of Contents Probabilistic Theories of Relations, Proc. Nat. Acad. Sci. 37 (1951), Probabilistic Geometry, Proc. Nat. Acad. Sci. 37 (1951), Ensembles fious et fonctions aleatoires, Comptes Rendus Acad. Paris 232 (1951), Selected Papers on Group Theory and Algebra Lausch, H.: Commentary on Menger's Work on Algebra Beiträge zur Gruppentheorie I. Über eine Gruppenmetrik, Mathematische Zeitschrift 33 (1931), Une theorie axiomatique generale des determinants, Comptes Rendus Acad. Paris 234 (1952), Selected Papers on Sociology Leonard, R.: Commentary on Menger's Work on Sociology An Exact Theory of Social Groups and Relations, American Journal of Sociology 43 (1938), On Social Groups and Relations, Mathematical Social Sciences 6 (1983), Menger's Work on Economics Sigmund, K.: Commentary on Menger's 'Austrian Marginalism and Mathematical Economics' Austrian Marginalism and Mathematical Economics, in: Carl Menger and the Austrian School of Economics, eds. R. O. Hicks and W. Weber, Oxford, 1973, Miscellaneous Editors Comments Otto Schreier, Nachruf, Monatshefte für Mathematik 37 (1930), Hans Hahn t, Fundamenta Mathematicae 24 (1935), viii-

9 Table of Contents Memories of Moritz Schlick, in: Rationality and Science, ed. by E. T. Gadol, Springer, Wien-New York, 1982, Introduction to: Ernst Mach: The Science of Mechanics, 6 th American Edition, Open Court, La SaUe, Illinois 1960, v-xxi Introduction to: Hans Hahn: Empiricism, Logic and Mathematics, Vienna Circle CoUection Vol. 13, Reidel, Dordrecht, 1980, ix-xviii You Will Like Geometry, A Guide Book for the Illinois Institute of Technology Geometry Exhibition at the Museum of Science and Industry, Chicago, 1952, Table of Contents - Vol. I Table of Contents of Karl Menger: "Selected Papers in Logic and Foundations, Didactics, Economics" 659 List of Publications - Karl Menger IX-

10 List of Authors Prof. Dr. Dirk van Dalen, Department of Philosophy, Universiteit Utrecht, Heidelberglaan 8, NL-3584 CS Utrecht, The Netherlands Prof. Dr. Hans Kaiser, Institut für Algebra und Computermathematik, Technische Universität Wien, Wiedner Hauptstraße 8-10/118, A-1040 Wien, Austria Prof. Dr. Hans Lausch, School of Mathematical Sciences, Monash University, Victoria 3800, Australia Prof. Dr. Robert Leonard, Departement des Sciences Economiques, Universite du Quebec a Montreal, Succursale Centre-Ville, Montreal, Quebec H3C 3P8, Canada Prof. Dr. Ludwig Reich, Institut für Mathematik, Universität Graz, Heinrichstraße 36, A Graz, Austria Prof. Dr. Bert Schweizer, Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA , USA Prof. Dr. Lester Senechal, Mount Holyoke, South Hadley, MA 01075, USA Prof. Dr. Kar! Sigmund, Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria Prof. Dr. Abe Sklar, 5044 Marine Drive, Chicago, IL 60640, USA -x-

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