Mind and Nature: Selected Writings on Philosophy, Mathematics, And
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Computer Oral History Collection, 1969-1973, 1977
Computer Oral History Collection, 1969-1973, 1977 INTERVIEWEES: John Todd & Olga Taussky Todd INTERVIEWER: Henry S. Tropp DATE OF INTERVIEW: July 12, 1973 Tropp: This is a discussion with Doctor and Mrs. Todd in their apartment at the University of Michigan on July 2nd, l973. This question that I asked you earlier, Mrs. Todd, about your early meetings with Von Neumann, I think are just worth recording for when you first met him and when you first saw him. Olga Tauskky Todd: Well, I first met him and saw him at that time. I actually met him at that location, he was lecturing in the apartment of Menger to a private little set. Tropp: This was Karl Menger's apartment in Vienna? Olga Tauskky Todd: In Vienna, and he was on his honeymoon. And he lectured--I've forgotten what it was about, I am ashamed to say. It would come back, you know. It would come back, but I cannot recall it at this moment. It had nothing to do with game theory. I don't know, something in.... John Todd: She has a very good memory. It will come back. Tropp: Right. Approximately when was this? Before l930? Olga Tauskky Todd: For additional information, contact the Archives Center at 202.633.3270 or [email protected] Computer Oral History Collection, 1969-1973, 1977 No. I think it may have been in 1932 or something like that. Tropp: In '32. Then you said you saw him again at Goettingen, after the-- Olga Tauskky Todd: I saw him at Goettingen. -
The Scientific Content of the Letters Is Remarkably Rich, Touching on The
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Reviews / Historia Mathematica 33 (2006) 491–508 497 The scientific content of the letters is remarkably rich, touching on the difference between Borel and Lebesgue measures, Baire’s classes of functions, the Borel–Lebesgue lemma, the Weierstrass approximation theorem, set theory and the axiom of choice, extensions of the Cauchy–Goursat theorem for complex functions, de Geöcze’s work on surface area, the Stieltjes integral, invariance of dimension, the Dirichlet problem, and Borel’s integration theory. The correspondence also discusses at length the genesis of Lebesgue’s volumes Leçons sur l’intégration et la recherche des fonctions primitives (1904) and Leçons sur les séries trigonométriques (1906), published in Borel’s Collection de monographies sur la théorie des fonctions. Choquet’s preface is a gem describing Lebesgue’s personality, research style, mistakes, creativity, and priority quarrel with Borel. This invaluable addition to Bru and Dugac’s original publication mitigates the regrets of not finding, in the present book, all 232 letters included in the original edition, and all the annotations (some of which have been shortened). The book contains few illustrations, some of which are surprising: the front and second page of a catalog of the editor Gauthier–Villars (pp. 53–54), and the front and second page of Marie Curie’s Ph.D. thesis (pp. 113–114)! Other images, including photographic portraits of Lebesgue and Borel, facsimiles of Lebesgue’s letters, and various important academic buildings in Paris, are more appropriate. -
Council for Innovative Research Peer Review Research Publishing System
ISSN 2347-3487 Einstein's gravitation is Einstein-Grossmann's equations Alfonso Leon Guillen Gomez Independent scientific researcher, Bogota, Colombia E-mail: [email protected] Abstract While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say, Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993) [1]. Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996) [2]. -
Arxiv:1601.07125V1 [Math.HO]
CHALLENGES TO SOME PHILOSOPHICAL CLAIMS ABOUT MATHEMATICS ELIAHU LEVY Abstract. In this note some philosophical thoughts and observations about mathematics are ex- pressed, arranged as challenges to some common claims. For many of the “claims” and ideas in the “challenges” see the sources listed in the references. .1. Claim. The Antinomies in Set Theory, such as the Russell Paradox, just show that people did not have a right concept about sets. Having the right concept, we get rid of any contradictions. Challenge. It seems that this cannot be honestly said, when often in “axiomatic” set theory the same reasoning that leads to the Antinomies (say to the Russell Paradox) is used to prove theorems – one does not get to the contradiction, but halts before the “catastrophe” to get a theorem. As if the reasoning that led to the Antinomies was not “illegitimate”, a result of misunderstanding, but we really have a contradiction (antinomy) which we, somewhat artificially, “cut”, by way of the axioms, to save our consistency. One may say that the phenomena described in the famous G¨odel’s Incompleteness Theorem are a reflection of the Antinomies and the resulting inevitability of an axiomatics not entirely parallel to intuition. Indeed, G¨odel’s theorem forces us to be presented with a statement (say, the consistency of Arithmetics or of Set Theory) which we know we cannot prove, while intuition puts a “proof” on the tip of our tongue, so to speak (that’s how we “know” that the statement is true!), but which our axiomatics, forced to deviate from intuition to be consistent, cannot recognize. -
8. Gram-Schmidt Orthogonalization
Chapter 8 Gram-Schmidt Orthogonalization (September 8, 2010) _______________________________________________________________________ Jørgen Pedersen Gram (June 27, 1850 – April 29, 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include On series expansions determined by the methods of least squares, and Investigations of the number of primes less than a given number. The process that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram ________________________________________________________________________ Erhard Schmidt (January 13, 1876 – December 6, 1959) was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia (now Estonia). His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905. His doctoral dissertation was entitled Entwickelung willkürlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations. Together with David Hilbert he made important contributions to functional analysis. http://en.wikipedia.org/wiki/File:Erhard_Schmidt.jpg ________________________________________________________________________ 8.1 Gram-Schmidt Procedure I Gram-Schmidt orthogonalization is a method that takes a non-orthogonal set of linearly independent function and literally constructs an orthogonal set over an arbitrary interval and with respect to an arbitrary weighting function. Here for convenience, all functions are assumed to be real. un(x) linearly independent non-orthogonal un-normalized functions Here we use the following notations. n un (x) x (n = 0, 1, 2, 3, …..). n (x) linearly independent orthogonal un-normalized functions n (x) linearly independent orthogonal normalized functions with b ( x) (x)w(x)dx i j i , j . -
K-Theory and Algebraic Geometry
http://dx.doi.org/10.1090/pspum/058.2 Recent Titles in This Series 58 Bill Jacob and Alex Rosenberg, editors, ^-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara) 57 Michael C. Cranston and Mark A. Pinsky, editors, Stochastic analysis (Cornell University, Ithaca) 56 William J. Haboush and Brian J. Parshall, editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. -
Lotze and the Early Cambridge Analytic Philosophy
LOTZE AND THE EARLY CAMBRIDGE ANALYTIC PHILOSOPHY ―This summer I‘ve read about a half of Lotze‘s Metaphysik. He is the most delectable, certainly, of all German writers—a pure genius.‖ William James, September 8, 1879.1 Summary Many historians of analytic philosophy consider the early philosophy of Moore, Russell and Wittgenstein as much more neo-Hegelian as once believed. At the same time, the authors who closely investigate Green, Bradley and Bosanquet find out that these have little in common with Hegel. The thesis advanced in this chapter is that what the British (ill-named) neo-Hegelians brought to the early analytic philosophers were, above all, some ideas of Lotze, not of Hegel. This is true regarding: (i) Lotze’s logical approach to practically all philosoph- ical problems; (ii) his treating of the concepts relation, structure (constructions) and order; (iii) the discussion of the concepts of states of affairs, multiple theory of judgment, general logical form; (iv) some common themes like panpsychism and contemplating the world sub specie aeternitatis. 1. LOTZE, NOT HEGEL, LIES AT THE BOTTOM OF CAMBRIDGE ANALYTIC PHILOSOPHY Conventional wisdom has it that the early philosophy of Moore and Russell was under the strong sway of the British ―neo-Hegelians‖. In the same time, however, those historians who investigate the British ―neo-Hegelians‖ of 1880–1920 in detail, turn attention to the fact that the latter are not necessarily connected with Hegel: William Sweet made this point in regard to 1 Perry (1935), ii, p. 16. Bosanquet,2 Geoffrey Thomas in regard to Green,3 and Peter Nicholson in regard to Bradley.4 Finally, Nicholas Griffin has shown that Russell from 1895–8, then an alleged neo-Hegelian, ―was very strongly influenced by Kant and hardly at all by Hegel‖.5 These facts are hardly surprising, if we keep in mind that the representatives of the school of T. -
Einstein, Nordström and the Early Demise of Scalar, Lorentz Covariant Theories of Gravitation
JOHN D. NORTON EINSTEIN, NORDSTRÖM AND THE EARLY DEMISE OF SCALAR, LORENTZ COVARIANT THEORIES OF GRAVITATION 1. INTRODUCTION The advent of the special theory of relativity in 1905 brought many problems for the physics community. One, it seemed, would not be a great source of trouble. It was the problem of reconciling Newtonian gravitation theory with the new theory of space and time. Indeed it seemed that Newtonian theory could be rendered compatible with special relativity by any number of small modifications, each of which would be unlikely to lead to any significant deviations from the empirically testable conse- quences of Newtonian theory.1 Einstein’s response to this problem is now legend. He decided almost immediately to abandon the search for a Lorentz covariant gravitation theory, for he had failed to construct such a theory that was compatible with the equality of inertial and gravitational mass. Positing what he later called the principle of equivalence, he decided that gravitation theory held the key to repairing what he perceived as the defect of the special theory of relativity—its relativity principle failed to apply to accelerated motion. He advanced a novel gravitation theory in which the gravitational potential was the now variable speed of light and in which special relativity held only as a limiting case. It is almost impossible for modern readers to view this story with their vision unclouded by the knowledge that Einstein’s fantastic 1907 speculations would lead to his greatest scientific success, the general theory of relativity. Yet, as we shall see, in 1 In the historical period under consideration, there was no single label for a gravitation theory compat- ible with special relativity. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Math 126 Lecture 4. Basic Facts in Representation Theory
Math 126 Lecture 4. Basic facts in representation theory. Notice. Definition of a representation of a group. The theory of group representations is the creation of Frobenius: Georg Frobenius lived from 1849 to 1917 Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups. Find out more at: http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Frobenius.html Matrix form of a representation. Equivalence of two representations. Invariant subspaces. Irreducible representations. One dimensional representations. Representations of cyclic groups. Direct sums. Tensor product. Unitary representations. Averaging over the group. Maschke’s theorem. Heinrich Maschke 1853 - 1908 Schur’s lemma. Issai Schur Biography of Schur. Issai Schur Born: 10 Jan 1875 in Mogilyov, Mogilyov province, Russian Empire (now Belarus) Died: 10 Jan 1941 in Tel Aviv, Palestine (now Israel) Although Issai Schur was born in Mogilyov on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja. In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius. -
ACTA UNIVERSITATIS UPSALIENSIS Uppsala Studies in History of Ideas 48
ACTA UNIVERSITATIS UPSALIENSIS Uppsala Studies in History of Ideas 48 ANDREAS RYDBERG Inner Experience An Analysis of Scientific Experience in Early Modern Germany Dissertation presented at Uppsala University to be publicly examined in Geijersalen (Eng6-1023), Engelska Parken, Humanistiskt centrum, Thunbergsvägen 3P, Uppsala, Friday, 9 June 2017 at 10:00 for the degree of Doctor of Philosophy. The examination will be conducted in Swedish. Faculty examiner: Kristiina Savin (Lunds universitet, Institutionen för kulturvetenskaper). Abstract Rydberg, A. 2017. Inner Experience. An Analysis of Scientific Experience in Early Modern Germany. Uppsala Studies in History of Ideas 48. 245 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9926-6. In the last decades a number of studies have shed light on early modern scientific experience. While some of these studies have focused on how new facts were forced out of nature in so-called experimental situations, others have charted long-term transformations. In this dissertation I explore a rather different facet of scientific experience by focusing on the case of the Prussian university town Halle in the period from the late seventeenth till the mid-eighteenth century. At this site philosophers, theologians and physicians were preoccupied with categories such as inner senses , inner experience, living experience, psychological experiments and psychometrics. In the study I argue that these hitherto almost completely overlooked categories take us away from observations of external things to the internal organisation of experience and to entirely internal objects of experience. Rather than seeing this internal side of scientific experience as mere theory and epistemology, I argue that it was an integral and central part of what has been referred to as the cultura animi tradition, that is, the philosophical and medical tradition of approaching the soul as something in need of cultivation, education, disciplination and cure. -
Relativistic Quantum Mechanics 1
Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4).