DOCSLIB.ORG

Introduction: Hilbert's Axiomatics As 'Symbolic Form'?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction: Hilbert's Axiomatics As 'Symbolic Form'? Introduction: Hilbert’s Axiomatics as ‘Symbolic Form’? Rossella Lupacchini University of Bologna According to Ernst Cassirer, symbolic forms “are not different modes in which an independent reality manifests itself to the human spirit but roads which the spirit proceeds towards its objectivization, i.e., its self- revelation” (1923, p. 78). Can Hilbert’s axiomatics be viewed as a symbolic form? The aim of this introductory essay is to frame the question in a ‘map’ and to provide a few signs to orient it. Each of the essays collected in this vol- ume may contribute a different set of instructions to reach an answer. Both Hilbert’s axiomatics and Cassirer’s philosophy of symbolic forms have their roots in Leibniz’s idea of a ‘universal characteristic,’ and grow on Hertz’s ‘principles of mechanics,’ and Dedekind’s ‘foundations of arith- metic’. As Cassirer recalls in the introduction to his Philosophy of Symbolic Forms, it was the discovery of the analysis of inªnity that led Leibniz to fo- cus on “the universal problem inherent in the function of symbolism, and to raise his ‘universal characteristic’ to a truly philosophical plane.” In Leibniz’s view, the logic of ‘things’ cannot be separated from the logic of ‘signs,’ as “the sign is no mere accidental cloak of the idea, but its neces- sary and essential organ.” Every ‘law’ of nature assumes for our thinking the form of a univer- sal ‘formula’—and a formula can be expressed only by a combina- tion of universal and speciªc signs. [. .] The content of the spirit is disclosed only in its manifestations; the ideal form is known only by and in the aggregate of the sensible signs which it uses for its expression. (Cassirer 1923, p. 86) I am grateful to Wilfried Sieg for his detailed remarks on the ªrst draft of this essay. Perspectives on Science 2014, vol. 22, no. 1 ©2014 by The Massachusetts Institute of Technology doi:10.1162/POSC_e_00116 1 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 2 Hilbert’s Axiomatics as ‘Symbolic Form’? It was by means of a “grammar of the symbolic function” that Leibniz was able to broaden the traditional and historical concept of idealism. “Ideal- ism,” Cassirer observes, “has always aimed at juxtaposing to the mundus sensibilis another cosmos, the mundus intelligibilis, and at deªning the boundary between these two worlds” (Cassirer 1923, p. 87). But for the ‘universal characteristic’ this opposition is no longer irreconcilable and ex- clusive: “the senses and the spirit are now joined in a new form of reciproc- ity and correlation.” The question at issue, in the language of Kantian philosophy, is “the share which thought, on the one hand, and experience, on the other, have in our knowledge.” Hilbert addressed it at Königsberg, in 1930, in order to highlight the core of his scientiªc carrier. He entrusted his answer to the axiomatic method, “a general method for the theoretical treatment of questions in the natural sciences,” and even more: already in daily life methods and concept-formations are used that demand a great measure of abstraction and that can be understood only by the unconscious application of axiomatic methods. For ex- ample, the general process of negation, and in particular the con- cept “inªnite.” (1930, p. 1159) Precisely in the concept of ‘inªnite,’ Hilbert identiªed “an important par- allelism between nature and thought, a fundamental agreement between experience and theory.” The inªnite is not an ‘observable’ of nature; it is nowhere realized, and it is not to be admitted in our thought “without special precautions”. As far as the concept ‘inªnite’ is concerned, we must be clear to ourselves that ‘inªnite’ has no intuitive meaning [anschauliche Bedeutung] and that without more detailed investigation it has abso- lutely no sense. For everywhere there are only ªnite things. [. .] There is absolutely nothing continuous that can be divided inªn- itely often. Even light has atomic structure [. .] inªnity, because it is the negation of a condition that prevails everywhere, is a gigantic abstraction—practicable only through the conscious or unconscious application of the axiomatic method. (1930, p. 1159) It is mathematics that mediates between theory and practice, between thought and observation; it is mathematics that builds the connecting bridges. Hilbert also noticed that even though “the laws of the world can be acquired in no other way than through experience,” conic sections were studied “long before one suspected that our planets or even electrons move in such a course.” This is a very old and eloquent example of “an occur- rence which is virtually an embodiment and realization of mathematical Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 3 thoughts.” Mathematics, however, does not merge with nature, nor with thought. As Hilbert explained: We can understand this agreement between nature and thought, between experience and theory, only if we take into account both the formal element and the mechanism that is connected with it; and we must do this both for nature and for our understanding. (1930, p. 1160; my emphasis) 1. The ‘Mechanism’ of Physical Knowledge The axiomatic design, brought about by mathematical constructions, also reveals an underlying connecting structure between various disciplines.1 This helps clarify why “the most important bearers of mathematical thought have always...cultivated the relations to the neighbouring sciences, especially to the great empires of physics and epistemology” (Hilbert 1918, p. 1107). This helps clarify in what way the most rigorous treatment of a problem, speciªc to one science, can extract beneªts from results and concepts that originate from neighbouring sciences. In Hilbert’s vision, rigor goes hand in hand with open-mindedness. While insisting on rigour as a requirement for a perfect solution of a problem, I should like [. .] to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are suscepti- ble of a fully rigorous treatment [. .] Such a one-sided interpreta- tion of the requirement of rigour would soon lead to the ignoring of all concepts arising from geometry, mechanics, and physics, to a stoppage of the ºow of new material from the outside world, and ªnally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an impor- tant nerve, vital to mathematical science, would be cut by the extir- pation of geometry and mathematical physics! (Hilbert 1900b, p. 1100) But unlike Hilbert, who saw in mathematics the key to understanding na- ture and considered the link between mathematics, geometry, and physics as vital, important bearers of physical thought at the end of the 19th cen- tury concerned themselves instead with keeping physics separate from mathematics. In the 1880s, the ‘atomist’ Boltzmann was accused of substituting mathematical deduction for physical thought, of reducing the physical science to an empty game of abstract symbols and monstrous algorithms. 1. Along this lines, Marletto and Rasetti’s essay in this volume is an eloquent develop- ment of Hilbert’s view. Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 4 Hilbert’s Axiomatics as ‘Symbolic Form’? The accuser was Peter G. Tait, co-author with Lord Kelvin of the Treatise on Natural Philosophy (1867) one of the most inºuential works of the 19th century. Located in the tradition of Newtonian empiricism, the Trea- tise supports energetics in the ‘atomistics vs. energetics’ dispute, which fu- eled the epistemological debate. The ‘crisis’ of atomism was claimed at the 1895 Lübeck meeting of the German Association of Scientists and Physicians,2 and corroborated by Ernst Mach’s phenomenalism. According to Mach, the variety of natural processes and phenomena is already an ar- gument against the primacy of mechanistic atomism. He admits that our efforts to mirror the world in thought would be futile if we were not able to grasp something stable in the ºow of experience. Yet, when we separate an object from its changeable environment, what we really do is isolate a group of sensations to which we ascribe a greater stability than to the oth- ers. On this basic group of sensations, we ªx our thoughts. Since the ob- ject does not lose its identity, while single elements are extracted from the group, we are led to consider that, after removing all the elements, some- thing still remains. Here comes the illusion of a ‘substance’ distinct from its attributes, a Ding-an-sich, and the useless attempt to reduce all phe- nomena to the motion of atoms. The physical model, proposed as an alter- native to classical mechanistic philosophy, is that which results from the collection of our sensory representations of physical data, mathematically organized according to a “principle of economy.” All of the conceptual in- struments involved in the cognitive process are regarded as devices for the economy of thought. But the advantage of logical-scientiªc thought, notices Husserl con- temporaneously, lies in the capacity of going beyond the limits of what is given in intuition. Logic ought to come before any economy of thought; hence founding the former on the latter would be nonsense (1900, Prole- gomena to Pure Logic IX). Husserl underlines a fundamental difference be- tween natural and logical theories: what makes a theory “logical” is an ideal nexus of necessity which prevails within it, whereas any properly theoretical component remains foreign to a “natural” theory. Thus, in the frame of phenomenalism, a physical theory is viewed as a natural theory. It was for mathematical reasons that Boltzmann intervened in the dis- pute between the physics of atoms and the physics of energy, or the phys- ics of the continuum.
Load more

Recommended publications
  • David Hilbert's Contributions to Logical Theory
    David Hilbert’s contributions to logical theory CURTIS FRANKS 1. A mathematician’s cast of mind Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Godel,¨ Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques.
    [Show full text]
  • Einstein and Hilbert: the Creation of General Relativity
    EINSTEIN AND HILBERT: THE CREATION OF GENERAL RELATIVITY ∗ Ivan T. Todorov Institut f¨ur Theoretische Physik, Universit¨at G¨ottingen, Friedrich-Hund-Platz 1 D-37077 G¨ottingen, Germany; e-mail: [email protected] and Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria;∗∗e-mail: [email protected] ABSTRACT It took eight years after Einstein announced the basic physical ideas behind the relativistic gravity theory before the proper mathematical formulation of general relativity was mastered. The efforts of the greatest physicist and of the greatest mathematician of the time were involved and reached a breathtaking concentration during the last month of the work. Recent controversy, raised by a much publicized 1997 reading of Hilbert’s proof- sheets of his article of November 1915, is also discussed. arXiv:physics/0504179v1 [physics.hist-ph] 25 Apr 2005 ∗ Expanded version of a Colloquium lecture held at the International Centre for Theoretical Physics, Trieste, 9 December 1992 and (updated) at the International University Bremen, 15 March 2005. ∗∗ Permanent address. Introduction Since the supergravity fashion and especially since the birth of superstrings a new science emerged which may be called “high energy mathematical physics”. One fad changes the other each going further away from accessible experiments and into mathe- matical models, ending up, at best, with the solution of an interesting problem in pure mathematics. The realization of the grand original design seems to be, decades later, nowhere in sight. For quite some time, though, the temptation for mathematical physi- cists (including leading mathematicians) was hard to resist.
    [Show full text]
  • Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg
    Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg By “elementary” plane geometry I mean the geometry of lines and circles—straight- edge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a “missing link,” Bolyai’s discovery—proved and generalized by William Jagy—of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incom- pleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called “the purity of methods of proof,” exemplified in his and his early twentieth century successors’ works on foundations of geometry. 1. AXIOMATIC DEVELOPMENT 1.0. Viewpoint. Euclid’s Elements was the first axiomatic presentation of mathemat- ics, based on his five postulates plus his “common notions.” It wasn’t until the end of the nineteenth century that rigorous revisions of Euclid’s axiomatics were presented, filling in the many gaps in his definitions and proofs. The revision with the great- est influence was that by David Hilbert starting in 1899, which will be discussed below. Hilbert not only made Euclid’s geometry rigorous, he investigated the min- imal assumptions needed to prove Euclid’s results, he showed the independence of some of his own axioms from the others, he presented unusual models to show certain statements unprovable from others, and in subsequent editions he explored in his ap- pendices many other interesting topics, including his foundation for plane hyperbolic geometry without bringing in real numbers.
    [Show full text]
  • Foundations of Geometry
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.
    [Show full text]
  • Mechanism, Mentalism, and Metamathematics Synthese Library
    MECHANISM, MENTALISM, AND METAMATHEMATICS SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Florida State University Editors: ROBER T S. COHEN, Boston University DONALD DAVIDSON, University o/Chicago GABRIEL NUCHELMANS, University 0/ Leyden WESLEY C. SALMON, University 0/ Arizona VOLUME 137 JUDSON CHAMBERS WEBB Boston University. Dept. 0/ Philosophy. Boston. Mass .• U.S.A. MECHANISM, MENT ALISM, AND MET AMA THEMA TICS An Essay on Finitism i Springer-Science+Business Media, B.V. Library of Congress Cataloging in Publication Data Webb, Judson Chambers, 1936- CII:J Mechanism, mentalism, and metamathematics. (Synthese library; v. 137) Bibliography: p. Includes indexes. 1. Metamathematics. I. Title. QA9.8.w4 510: 1 79-27819 ISBN 978-90-481-8357-9 ISBN 978-94-015-7653-6 (eBook) DOl 10.1007/978-94-015-7653-6 All Rights Reserved Copyright © 1980 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1980. Softcover reprint of the hardcover 1st edition 1980 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS PREFACE vii INTRODUCTION ix CHAPTER I / MECHANISM: SOME HISTORICAL NOTES I. Machines and Demons 2. Machines and Men 17 3. Machines, Arithmetic, and Logic 22 CHAPTER II / MIND, NUMBER, AND THE INFINITE 33 I. The Obligations of Infinity 33 2. Mind and Philosophy of Number 40 3. Dedekind's Theory of Arithmetic 46 4.
    [Show full text]
  • Georg Cantor English Version
    GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle.
    [Show full text]
  • Definitions and Nondefinability in Geometry 475 2
    Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal.
    [Show full text]
  • John Von Neumann's “Impossibility Proof” in a Historical Perspective’, Physis 32 (1995), Pp
    CORE Metadata, citation and similar papers at core.ac.uk Provided by SAS-SPACE Published: Louis Caruana, ‘John von Neumann's “Impossibility Proof” in a Historical Perspective’, Physis 32 (1995), pp. 109-124. JOHN VON NEUMANN'S ‘IMPOSSIBILITY PROOF’ IN A HISTORICAL PERSPECTIVE ABSTRACT John von Neumann's proof that quantum mechanics is logically incompatible with hidden varibales has been the object of extensive study both by physicists and by historians. The latter have concentrated mainly on the way the proof was interpreted, accepted and rejected between 1932, when it was published, and 1966, when J.S. Bell published the first explicit identification of the mistake it involved. What is proposed in this paper is an investigation into the origins of the proof rather than the aftermath. In the first section, a brief overview of the his personal life and his proof is given to set the scene. There follows a discussion on the merits of using here the historical method employed elsewhere by Andrew Warwick. It will be argued that a study of the origins of von Neumann's proof shows how there is an interaction between the following factors: the broad issues within a specific culture, the learning process of the theoretical physicist concerned, and the conceptual techniques available. In our case, the ‘conceptual technology’ employed by von Neumann is identified as the method of axiomatisation. 1. INTRODUCTION A full biography of John von Neumann is not yet available. Moreover, it seems that there is a lack of extended historical work on the origin of his contributions to quantum mechanics.
    [Show full text]
  • The Foundations of Geometry
    The Foundations of Geometry Gerard A. Venema Department of Mathematics and Statistics Calvin College SUB Gottingen 7 219 059 926 2006 A 7409 PEARSON Prentice Hall Upper Saddle River, New Jersey 07458 Contents Preface ix Euclid's Elements 1 1.1 Geometry before Euclid 1 1.2 The logical structure of Euclid's Elements 2 1.3 The historical importance of Euclid's Elements 3 1.4 A look at Book I of the Elements 5 1.5 A critique of Euclid's Elements 8 1.6 Final observations about the Elements 11 Axiomatic Systems and Incidence Geometry 17 2.1 Undefined and defined terms 17 2.2 Axioms 18° 2.3 Theorems 18 2.4 Models 19 2.5 An example of an axiomatic system 19 2.6 The parallel postulates 25 2.7 Axiomatic systems and the real world 27 Theorems, Proofs, and Logic 31 3.1 The place of proof in mathematics 31 3.2 Mathematical language 32 3.3 Stating theorems 34 3.4 Writing proofs 37 3.5 Indirect proof \. 39 3.6 The theorems of incidence geometry ';, . 40 Set Notation and the Real Numbers 43 4.1 Some elementary set theory 43 4.2 Properties of the real numbers 45 4.3 Functions ,48 4.4 The foundations of mathematics 49 The Axioms of Plane Geometry 52 5.1 Systems of axioms for geometry 53 5.2 The undefined terms 56 5.3 Existence and incidence 56 5.4 Distance 57 5.5 Plane separation 63 5.6 Angle measure 67 vi Contents 5.7 Betweenness and the Crossbar Theorem 70 5.8 Side-Angle-Side 84 5.9 The parallel postulates 89 5.10 Models 90 6 Neutral Geometry 94 6.1 Geometry without the parallel postulate 94 6.2 Angle-Side-Angle and its consequences 95 6.3 The Exterior
    [Show full text]
  • Foundations of Mathematics
    Foundations of Mathematics November 27, 2017 ii Contents 1 Introduction 1 2 Foundations of Geometry 3 2.1 Introduction . .3 2.2 Axioms of plane geometry . .3 2.3 Non-Euclidean models . .4 2.4 Finite geometries . .5 2.5 Exercises . .6 3 Propositional Logic 9 3.1 The basic definitions . .9 3.2 Disjunctive Normal Form Theorem . 11 3.3 Proofs . 12 3.4 The Soundness Theorem . 19 3.5 The Completeness Theorem . 20 3.6 Completeness, Consistency and Independence . 22 4 Predicate Logic 25 4.1 The Language of Predicate Logic . 25 4.2 Models and Interpretations . 27 4.3 The Deductive Calculus . 29 4.4 Soundness Theorem for Predicate Logic . 33 5 Models for Predicate Logic 37 5.1 Models . 37 5.2 The Completeness Theorem for Predicate Logic . 37 5.3 Consequences of the completeness theorem . 41 5.4 Isomorphism and elementary equivalence . 43 5.5 Axioms and Theories . 47 5.6 Exercises . 48 6 Computability Theory 51 6.1 Introduction and Examples . 51 6.2 Finite State Automata . 52 6.3 Exercises . 55 6.4 Turing Machines . 56 6.5 Recursive Functions . 61 6.6 Exercises . 67 iii iv CONTENTS 7 Decidable and Undecidable Theories 69 7.1 Introduction . 69 7.1.1 Gödel numbering . 69 7.2 Decidable vs. Undecidable Logical Systems . 70 7.3 Decidable Theories . 71 7.4 Gödel’s Incompleteness Theorems . 74 7.5 Exercises . 81 8 Computable Mathematics 83 8.1 Computable Combinatorics . 83 8.2 Computable Analysis . 85 8.2.1 Computable Real Numbers . 85 8.2.2 Computable Real Functions .
    [Show full text]
  • Chapter 1 the Foundation of Euclidean Geometry
    Chapter 1 The Foundation of Euclidean Geometry \This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state." | Clifford 1. Introduction. Geometry, that branch of mathematics in which are treated the properties of figures in space, is of ancient origin. Much of its development has been the result of efforts made throughout many centuries to construct a body of logical doctrine for correlating the geometrical data obtained from observation and measurement. By the time of Euclid (about 300 B.C.) the science of geometry had reached a well-advanced stage. From the accumulated material Euclid compiled his Elements, the most remarkable textbook ever written, one which, despite a number of grave imperfections, has served as a model for scientific treatises for over two thousand years. Euclid and his predecessors recognized what every student of philosophy knows: that not everything can be proved. In building a logical structure, one or more of the propositions must be assumed, the others following by logical deduction. Any attempt to prove all of the propositions must lead inevitably to the completion of a vicious circle. In geometry these assumptions originally took the form of postulates suggested by experience and intuition. At best these were statements of what seemed from observation to be true or approximately true. A geometry carefully built upon such a foundation may be expected to correlate the data of observation very well, perhaps, but certainly not exactly. Indeed, it should be clear that the mere change of some more-or-less doubtful postulate of one geometry may lead to another geometry which, although radically different from the first, relates the same data quite as well.
    [Show full text]
  • David Hilbert and the Axiomatization of Physics (1898–1918) from Grundlagen Der Geometrie to Grundlagen Der Physik
    L. Corry David Hilbert and the Axiomatization of Physics (1898–1918) From Grundlagen der Geometrie to Grundlagen der Physik Series: Archimedes, Vol. 10 ▶ Presents a totally fresh and comprehensive picture of Hilbert’s intense, original, well-informed, and highly influential involvement with physics, that spanned his entire career and that constituted a truly main focus of interest in his scientific horizon David Hilbert (1862-1943) was the most influential mathematician of the early twentieth century and, together with Henri Poincaré, the last mathematical universalist. His main known areas of research and influence were in pure mathematics (algebra, number theory, geometry, integral equations and analysis, logic and foundations), but he was also known to have some interest in physical topics. The latter, however, was traditionally conceived as comprising only sporadic incursions into a scientific domain which was 2004, XVII, 513 p. 36 illus. essentially foreign to his mainstream of activity and in which he only made scattered, if important, contributions. Printed book Based on an extensive use of mainly unpublished archival sources, the present book Hardcover presents a totally fresh and comprehensive picture of Hilbert’s intense, original, well- ▶ 249,99 € | £219.99 | $299.99 informed, and highly influential involvement with physics, that spanned his entire career ▶ *267,49 € (D) | 274,99 € (A) | CHF 295.00 and that constituted a truly main focus of interest in his scientific horizon. His program for axiomatizing physical theories provides the connecting link with his research in more eBook purely mathematical fields, especially geometry, and a unifying point of view from which to understand his physical activities in general.
    [Show full text]