Symmetries of Nature: A Handbook for Philosophy of Nature and Science 9783110886931, 9783110129908

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Klaus Mainzer Symmetries of Nature

Klaus Mainzer

Symmetries of Nature A Handbook for Philosophy of Nature and Science

W G DE

Walter de Gruyter · Berlin · New York 1996

The publishers wish to thank Inter Nationes, Bonn, who subsidized the translation of this book. Title of the Original German Language Edition: Symmetrien der Natur Copyright © 1988 by Walter de Gruyter & Co., Publishers, Berlin. Translated by Barbara H. Möhr, Boston USA, and Thomas J. Clark, The Language Center, Pittsburgh, Pa 15222 USA. ® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Mainzer, Klaus. [Symmetrien der Natur, English] Symmetries of nature : a handbook for philosophy of nature and science / Klaus Mainzer. Includes bibliographical references and index. ISBN 3-11-012990-6 (acid free paper) 1. Symmetry. I. Title, Q172.5.S95M3613 1996 500-dc20 95-50834 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication Data Mainzer, Klaus: Symmetries of nature : a handbook for philosophy of nature and science / Klaus Mainzer. [Transi, by Barbara H. Möhr and Thomas J. Clark], - Berlin ; New York : de Gruyter, 1996 Einheitssacht.: Symmetrien der Natur ISBN 3-11-012990-6

© Copyright 1996 by Walter de Gruyter & Co., D-10785 Berlin All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Converted by: Knipp Medien + Kommunikation OHG, Dortmund Printing: Arthur Collignon GmbH, Berlin Binding: Lüderitz & Bauer, Berlin Printed in Germany.

το άντίξουν συμφέρον έκ των διαφερόντων καλλίστη αρμονία.* Heraclitus

* "What opposes unites, and that the finest attunement stems from things bearing in opposite directions."

Preface of the English Edition After seven years my book on 'Symmetries of Nature' is offered to the English speaking readers. Symmetries are no longer only fundamental structures in the natural sciences. Symmetries have become a broad topic of interdisciplinary research in the humanities, too. Meanwhile the techniques of symmetry breaking have been applied to chaos theory and the theory of nonlinear complex systems, in order to explain the emergence of order in nature. These procedures which I started to study in this book (Chapter 4.3-4.4) are described in my English book 'Thinking in Complexity. The Complex Dynamics of Matter, Mind, and Mankind' which was published in 1994 (Berlin/Heidelberg/New York/Tokyo). At first I would like to express my gratitude to the translators for their careful work - Dr. Barbara H. Möhr (Boston) with Chapters 1-2.1, 5, and Thomas J. Clark (The Language Center/Pittsburgh) with Chapters 2.2-4. An interdisciplinary book with Greek, Latin, and French quotations, mathematical, physical, chemical, and philosophical terminologies needs special efforts of translation. The translation was financially supported by Inter Nationes (German Foreign Office). Thus I would like to thank Inter Nationes and the publisher of my book, De Gruyter, for efficient support. Last but not least, I thank my friends and colleagues Jürgen Mittelstraß (Konstanz) and Abner Shimony (Boston) for several arrangements which helped to realize the project. Augsburg, July 1995

Klaus Mainzer

Preface This book traces the concept of symmetry as it emerges in an examination of basic questions in philosophy of nature and of science. It has the character of a handbook on this theme and is addressed equally to philosophers, mathematicians, natural scientists and historians of science. It was at the University of Constance, to a great degree, that I was able to establish the link between current research in mathematics and the natural sciences, on the one hand, and classical and modern philosophy of nature on the other. My thanks go, first of all, to Horst Sund, rector of the university and biochemist, for his valuable advice and for following my work insightfully during my professorship at Constance. My thanks go also to my colleague in physics Klaus Dransfeld, who repeatedly invited me to lecture on the theme of the book in the Department of Physics at Constance. Another valuable stimulus came from physicists, chemists, biologists, mathematicians and philosophers participating in the program of General Studies, both from the University of Constance and visiting professors. My thanks go also to my friends and colleagues Jürgen Mittelstraß and Friedrich Kambartel in the Department of Philosophy at Constance for their support. Large parts of the book were written earlier, during my stay at the Center for Philosophy of Science at the University of Pittsburgh in 1985. Conversations with Nicholas Rescher and Adolf Grünbaum were important to me. Further thanks go to mathematicians of TH Darmstadt, especially Rudolf Wille, who invited me to participate in the symmetry symposium at Darmstadt in 1986. Last but not least I thank Hans Primas and his colleagues at the physical chemistry laboratory at ΕΤΗ-Zürich for invitations, collégial discussions and valuable suggestions. In Zurich especially, the memory of Hermann Weyl is linked to the theme of symmetry. I thank Mrs. Margit Güttler for the careful production of the manuscript. Cornelia Liesenfeld, doctoral candidate in philosophy helped with the proofreading and the production of the indices of literature, persons and facts. I thank my colleague Heinz Wenzel and Susanne Rade heartily for their thoughtful support in the printing process. Constance, Spring, 1987

Klaus Mainzer

Table of Contents Introduction

1

1.

Early History of Symmetry

15

1.1

Symmetries in Early Cultures

15

1.2 1.21 1.22 1.23

Symmetries in Antique-Medieval Mathematics Geometry Arithmetic and the Doctrine of Harmony Astronomy

23 25 39 49

1.3 1.31 1.32 1.33 1.34

Symmetries in the Antique-Medieval Philosophy of Nature Pre-socratic Beginnings Mathematical Atomism Physics and Cosmology Chemistry and Alchemy

64 65 75 82 97

1.4 Symmetries in Early Art and Technology 1.41 Technology 1.42 Art and Architecture

106 107 117

2.

Symmetries in Modern Mathematics

133

2.1 2.11 2.12 2.13

Symmetries of Ornamental Patterns and Crystals Discrete Plane Groups Discrete Groups in Space Color Symmetries and Symmetries of Music

134 134 148 162

2.2 Symmetry and Equation Theory 2.21 Galois Theory 2.22 Sample Applications

173 174 179

2.3 2.31 2.32 2.33 2.34

183 184 198 203 213

Symmetries and the Invariance of Geometric Theories Klein's "Erlanger Program" Continuous Lie Groups Differential Geometry and Symmetrical Spaces Representation Theory and Hilbert Spaces

XII

Table of Contents

3.

Symmetries in Classical Physics and the Philosophy of Nature . 223

3.1 3.11 3.12 3.13 3.14

Symmetries of Space and Time Pre-scientific Space-Time Space-Time Symmetry according to Newton and Kant Space-Time Symmetry according to Leibniz and Huygens Space-Time Symmetry of Classical Mechanics

224 225 234 238 242

3.2 3.21 3.22 3.23 3.24 3.25 3.26

Symmetry and the Classical Physics of Forces Newton's Program of Forces Theory of Gravitation Electrostatics Magnetostatics Electrodynamics Symmetry and the Unity of Forces

248 248 252 259 264 270 280

3.3 3.31 3.32 3.33

Symmetry, Laws of Conservation and the Principles of Nature . Lagrange and Hamilton Formalism Laws of Conservation and Symmetry Extremal Principles and the Pre-established Harmony of Nature

287 287 292 300

3.4 Symmetry and Thermodynamics 3.41 Invariance of Time and Irreversible Processes 3.42 Maxwell's Demons and Darwin's Theory of Evolution

315 317 332

4.

Symmetries in Modern Physics and Natural Sciences

341

4.1 4.11 4.12 4.13 4.14

Symmetries in the Theory of Relativity Special Relativity Theory: Global Symmetry of Space-Time ... General Relativity Theory: Local Symmetry of Space-Time Symmetry and Relativistic Cosmology Symmetry and the Unity of Gravitation and Electrodynamics ..

343 344 351 359 366

4.2 4.21 4.22 4.23 4.24 4.25

Symmetries in Quantum Mechanics Symmetries of Previous Atomic Models Symmetry of Quantum Systems Symmetry and EPR Holism Symmetry and Superselection Rules Symmetry and Complementarity

374 375 380 394 402 409

4.3 Symmetries in Elementary Particle Physics 4.31 Quantum Electrodynamics: Symmetry of Electromagnetic Forces 4.32 Symmetry and the Unity of Weak and Electromagnetic Forces . 4.33 Quantum Chromodynamics: Symmetry of the Strong Forces ...

414 414 433 449

Table of Contents

XIII

4.34 Supersymmetry and the Unity of Natural Forces

464

4.4 4.41 4.42 4.43 4.44

Symmetries in Chemistry, Biology and the Theory of Evolution Molecular Symmetry and Stereochemistry Symmetries of Biochemistry Symmetries of Organisms Symmetry, Chaos and Evolution

477 479 501 516 528

5.

Symmetry and Philosophy

561

5.1

Symmetries in Intuition and Perception

562

5.2 Symmetry as a Category of Cognition 5.21 Symmetry and the Category of Substance 5.22 Symmetry and the Categories of Causality and Interaction Effect 5.3 5.31 5.32 5.33

Symmetry Symmetry Symmetry Symmetry

568 569 575

in Philosophy of Science and Philosophy of Nature . 578 and the Methodology of Scientific Research 579 and the Structures of Scientific Theories 589 and the Dialectics of Nature 608

5.4 Symmetry in Modern and Post-Modern Art 5.41 Symmetry in the Art and Architecture of Modernism 5.42 Symmetry and Symmetry Breakings in Postmodernism

621 623 632

References

637

Author Index

661

Subject Index

667

Introduction Symmetries are a theme of current interest in the natural sciences. In the last few years, Nobel prizes in physics have been awarded for research into the symmetry of the elementary particles and of the universe. Questions of symmetry are discussed in chemistry and biology as well. H. Weyl's conjecture that the various basic laws of physics can be derived from unitary structures of symmetry seems to have been confirmed.1 Thus the theme of symmetry is closely tied to the demand for unity of the natural sciences, which in the modern era is at risk because of increasing specialization. But, long before any science, man was fascinated by symmetry. Symmetrical forms and symbols are to be found in art and architecture as well as in everyday useful objects and in the mythologies of the religions. Symmetry, therefore, is a theme that spans the human life world, technology, culture and nature, and thereby searches out a unity of the natural and human sciences, which seems to have been long since abandoned following C.P. Snow's thesis of the separated cultures of "science" and "the humanities." 2 The current references to the ecological balance of nature do, thereby, address a symmetry that is endangered by one-sided interests which also threaten human life supports. Only those who know about these connections, and who understand that research is a part of human history and culture, are capable of technically responsible action, in harmony with the natural world that has brought them forth by evolution. This demonstrates the necessity for an education in the natural sciences that will complete specialization and single-discipline education. It is from this perspective that the guiding theme of symmetries in the

1

2

"Kepler, Galileo and Bruno share with the Pythagoreans of Antiquity the belief in a cosmos ordered by the highest and most perfectly rational, mathematical laws, in divine reason as the origin of nature's rationality and in the relatedness, to it, of human reason. In the course of the long experience of the following centuries this belief increasingly found surprising partial fulfillments in physics (the best one, perhaps, in Maxwell's amazingly harmonious theory of the electromagnetic processes in the ether). But again and again nature proved to be superior to the human mind, and forced it to break with a premature conclusion in favor of a deeper harmony." H. Weyl, Philosophy of Mathematics and Natural Science, in: A. Baeumler/M. Schröter (Publisher), Handbook of Philosophy, Vol. 2, Munich 1927, 118. Weyl's Handbook article appears as a monograph in the 5th edition, Munich, Vienna, 1982. C.P. Snow, The Two Cultures: And A Second Look, London 1963.

2

Introduction

natural sciences will be developed historically and systematically in what follows. The first chapter deals with the early history of symmetry up to the beginning of the modern natural sciences in the Renaissance. A first section directs the reader to the use of symmetry patterns in early, especially nonEuropean cultures. In the mythologies of the nature religions, symmetry symbols still serve to exorcise natural powers. These are replaced in the Greek natural philosophy by symmetry models for the rational explanation of nature, and finally in the contemporary technical conquest of the forces of nature (e.g., nuclear energy). The mathematical formulation of the concept of symmetry was a decisive prerequisite. In the Pythagorean quadrivium geometry, arithmetic, music and astronomy - the harmony and proportionality (συμμετρία) of nature became the central concern of a mathematical philosophy. Animated by technical, aesthetic or religious motives, symmetry remains a favorite theme of Antique-Medieval mathematics. In geometry, theorems about regular plane figures and regular solids of Euclidean space are proven. In arithmetic and music, laws of proportion and harmony are explored. In astronomy, spherical models are used to explain the phenomena of the heavens. But what would harmony be without dissonance and violation of symmetry? Presumably, without charm, and boring, since it would be the everyday, the normal. But in fact we are flooded by information from the outer world in which chaotic multiplicity and change are the probable and order and lasting symmetries seem to be the improbable. The centro-symmetric spherical models of Eudoxos and Aristotle soon come into conflict with discrepant observations of the sky, which right up to N. Copernicus require increasingly elaborate geometrical and kinematic assumptions in order to save the symmetry of the model. What remains are artificial and complicated models of planets which lose their credibility because - as Copernicus in his Platonic tradition still believes - only the simple can be real, the simple which underlies the multiplicity of phenomena. The history of Antique-Medieval astronomy offers a convincing case study for investigating the interplay of original assumptions of symmetry on the one hand, and the breaks of symmetry on the basis of new realizations on the other hand, as a basic pattern of the development of research, which will be repeated on into modern physics. In J. Kepler we encounter the real revolutionary of this development. In his youth he was still a zealous Piatonist {Mysterium cosmographicum). In his main work Astronomia nova he breaks with the belief in the harmony of the spheres and the Platonic solids, and in his late work Harmonice mundi he detects a new symmetry of the universe based on natural science.

Introduction

3

In Antiquity, natural philosophy and mathematics remain extensively separate, since, as Aristotle says, physics deals with motion and change; mathematics with the unchanging. Therefore only the eternally recurring circular motions of the heavens are formulated mathematically and are considered divine. The complexity and variability of nature on earth is collected and ordered and explained by means of various qualitative principles of the natural philosophy from the time of the Presocratics. It is noteworthy that the categories of the Aristotelian natural philosophy operate by no means in an abstract and artificial manner. They are taken from the familiar world of the people (of those times). In the foreground are the organic developments and the life of humans and animals from birth to death, or the growth and decay of plants, which are understood to be goal-directed processes like the methodical actions of humans. Nature is seen as a great organism whose processes are tuned to each other harmoniously, and in which human life is only a part. Inanimate nature - substances and minerals - as well, is explained in Antique-Medieval alchemy by the familiar organic models. Minerals are said to be "grown," "fertilized," and "bred." In early alchemy and medicine, success is promised only to those who take notice of nature's cycles. The Greek atomists form a contrast to the organic conception of nature, not even comprehending life as it is holistically given, but instead wanting to trace all existent things back to the aimless collision of the smallest indivisible building blocks (ατομος) in empty space. Plato's natural philosophy introduces a mathematical model of the microcosm for the first time, explaining the elements by means of the geometric symmetry of regular bodies. Exhibiting the efficacy of history, W. Heisenberg will conceive modern atomic physics in this Platonic tradition, which the Aristotelian natural philosophy had thrust into the background as speculative during Antiquity and in the Middle Ages. The chapter ends with applications of symmetry in art, architecture and technology that appear in Antiquity and the Middle Ages and especially in the Neoplatonism of the Renaissance. Since the beginning of the modern age, symmetry and the unity of nature are sought in mathematical laws of nature, and tested, confirmed or discarded by experiment. Symmetry assumptions in the astronomy and natural philosophy of Antiquity were founded on plane and solid figure symmetries of Euclidean geometry: the circle, the sphere, and regular solids. But in order to understand the laws of nature of modern physics as assumptions of symmetry, it is necessary first to investigate modern developments in mathematics. It was first of all algebra and group theory from the end of the 18th century that created the basis for the rigorous general mathematical definition of the concept of symmetry (the "automorphism group"), which found its first application in the crystallography and stereochemistry of the 19th century

4

Introduction

and then in almost all parts of modern natural science. The second chapter treats the discrete symmetries of the ornaments and crystals (including the symmetries of color and music) from the point of view of group theory. But historically the group concept was first applied in the algebraic theory of equations ("Galois theory"), which can also be used to solve construction problems of symmetries from Antiquity. The continuous groups of differential geometry ("Lie groups") became important for modern physics. Finally, in the 2nd chapter, the mathematical presuppositions of the concept of symmetry both of relativity theory and quantum mechanics are examined. In the 3rd chapter the symmetries of classical physics are examined. Symmetry is understood to be invariance of natural laws or physical theories with respect to continuous transformation groups. This establishes epistemologically that a natural law is objectively valid - independently of changes in the position or the point in time of its examination by an experimenter or observer. Symmetries correspond to the freedom to choose the system of coordinates of the observer. All natural laws are invariant with respect to translation, rotation and reflection of the system of coordinates. The symmetries are global in the sense that natural laws are invariant with respect to equivalent transformations for all points in space. Historically this principle of relativity of classical physics appears first with G. Galilei, I. Beeckman, R. Descartes, E. Torricelli, and C. Huygens. First to be examined is the time-space-symmetry of Newton's absolute space and absolute time, as well as Kant's reference to chirality (left-right symmetry) in his pre-critical writings. This is to be distinguished from the space-time-symmetry according to Leibniz and Huygens, which does establish the kinematic group of classical physics, but cannot explain any dynamic effects such as Newton's centrifugal forces ("absolute movement"). But Newton's space-time-symmetry proves to be inadequate also with its assumptions that cannot be achieved empirically, e.g., the assumption of absolute rest. It was L. Lange's concept of the inertial system that first contributed the defintion of the space-time manifold of classical physics and made possible the establishment of the classical laws of nature by Galilean invariance. In the 18th century Kant took up the right-left symmetry ("parity") in nature and created a theory of matter that is determined by the polarity forces that attract and repel each other. But while Kant remains committed to the mathematical principles of Newtonian physics, at the beginning of the 19th century a philosophy of nature emerges that attempts a speculative total grasp of nature, which nevertheless falls into opposition to the mathematical-experimental method of modern natural science. Hegel's and Schelling's Romantic natural philosophy, which seeks to transcend polarity in a unity of mind and nature, arises against a background of the new doc-

Introduction

5

trine of electricity and magnetism and influenced heuristically many of the natural scientists of that time such as J.W. Ritter, H.C. Oersted, and perhaps also M. Faraday. The Romantic natural philosophers' tendency to speculation, however, contradicts a positivistic basic attitude of many natural scientists, and their deficient mathematical precision elicits only scorn from Gauss. Goethe's organic conception of nature, with its harmonious metamorphoses and its comprehensive vision, had come into conflict with natural science. And thus, especially in Germany, the cleavage between natural philosophy in the Aristotelian tradition and mathematical natural science grew ever wider. The joining of electricity, magnetism and optics succeeds mathematically in electrodynamics, which can be formulated invariantly to the Lorentz group. J.C. Maxwell had already predicted that light could be reduced to the electromagnetic field. In fact, the wave equations for the phase velocity of light were derived from Maxwell's equations and confirmed experimentally by H. Hertz. This brought about, for the first time, a unification of the phenomena of nature in mathematical physics. Until that time there had been only speculation about them in various approaches in natural philosophy. Further, Maxwell's electrodynamics constituted, for the first time, a physical theory for which the modern physical concept of symmetry could be expressed with precision. The electromagnetic field has both "global" symmetry in accordance with the Lorentz invariance - in which all spacetime coordinates can be altered - and also "local" symmetry in the sense of a gauge field. Even in 1923 H. Weyl will characterize the theory of the electromagnetic field as "the most perfect piece of physics that we know today..." The application of physical symmetry concepts is closely connected with mathematical developments in algebra and geometry in the 19th century. In 1872, F. Klein, in his well-known "Erlanger Program", had characterized and classified various geometric theories by means of continuous transformation groups. Under the direct influence of Klein, E. Noether in 1918 expanded this program for physics and showed how physical conservation principles can be characterized by means of transformation groups and traced back to space-time symmetries. Historically, there are already rudiments of these in lectures by the mathematician C.G. Jacobi in 1866. The mathematical variational and extremal principles are of central significance for the physical concept of symmetry. Historically they arise out of the background of Leibniz' natural philosophy of pre-established harmony and are determined by the search for a coherent basic principle of nature. Voltaire's critical debate with Leibniz' Theodicee in the 18th century proclaims - in a way that is parallel to modern intellectual history - a secularization of natural law, which now is no longer understood as having an

6

Introduction

ontological and theological basis. In conclusion, the third chapter discusses how well the time symmetry of classical physics agrees with thermodynamics. How can the invariance of natural laws to time translation be brought together with the irreversible natural processes of thermodynamics and Darwin's evolutionary theory of life? The problem of the asymmetrical arrow of time is examined in the classical discussions from L. Boltzmann, E. Zermelo et al., up to the work of J. Monod et al. The 4th chapter deals with the symmetry concepts in modern physics and natural science. First to be discussed is the Lorentz invariance of the forcefree four-dimensional Minkowski space of the special relativity theory, in which two observers have constant velocity relative to each other. It is a matter of global symmetry, since the transformations refer to all space-time coordinates. The local Lorentz invariance of the general theory of relativity has to fulfill much stricter requirements. Now the physical laws have to keep the same form even when every single point is transformed independently of all the others. This mathematical sharpening has the same significance as the physical postulate that two observers may also increase their speed relative to each other, that is, that gravitational forces come into play. A key concept of modern physics is to describe the introduction of fundamental forces mathematically by means of the transition from a global symmetry to a local symmetry. Relativistic cosmology applies the differential-geometric theory of symmetrical spaces, which E. Cartan had developed in the twenties from the theory of spaces with constant curvature (Riemann, Lie, Helmholtz et al). The solutions to Einstein's gravitation equation allow for varying symmetrical models, e.g., that the spatially homogeneous universe expands isotropically, or collapses, or oscillates. The natural-philosophical discussions about eternal matter in the sense of materialism, the finiteness of the world in the sense of Christianity, or the eternal recurrence of the same in Nietzsche's sense can serve heuristically for mathematical theories and verifiable hypotheses. The Platonic belief in a macroscopically symmetrical cosmos in the large is once more urgent, even if mathematically more complicated and no longer in the form of the ancient harmony of the spheres. In this connection it is noteworthy that D. Hilbert derived the relativistic equations together with Mies' electrodynamic equations from a variation principle independently of Einstein. This was the first attempt at a unification of the fundamental forces in modern physics, which, however, succeeded only later, under the conditions of quantum mechanics. Next to the relativity theory, quantum mechanics is the framework theory of modern physics. Recall first, the spherically symmetrical characteristics of the early atom models by which N. Bohr explained the discontinuous spectral lines of the chemical elements. According to these models the electrons move on fixed paths around the nucleus, like the ancient planets. By

Introduction

7

analogy to the development of the Aristotelian planet models, Bohr's originally simple atom model must also be assimilated by means of certain artifices (in this case the quantum numbers) to the complicated relationships that reveal themselves for various elements in the laboratory. Thus the basic equation of quantum mechanics, the Schrödinger equation, exhibits two kinds of symmetry. It can be assumed, at least approximately, that the electrons have a spherically-symmetrical potential energy for which no direction is distinguished, so that the corresponding Hamiltonian function is invariant towards the symmetry operations of the sphere. Further, electrons are indistinguishable (Leibniz: "indiscernibiles") in the sense that it makes no difference for the Hamiltonian function whether the positions of the electrons are exchanged and permuted. This permutation symmetry is closely connected to Pauli's principle of exclusion, according to which two electrons do not have the same quantum number. With reference to Leibniz' principle of indistinguishability, H. Weyl also speaks of the Leibniz-Pauli principle. Mathematically, the states of quantum systems (atoms, electrons, etc.) can be represented by vectors of a Hilbert space. The symmetry (automorphism group) of the Hilbert-space formalism of Neumann's quantum mechanics has been investigated especially by H. Weyl, E.P. Wigner, et al. since the end of the twenties, and related to the unitary transformations of the Hilbert space. The space-time symmetries are determined by a subgroup which can be represented by the Galileo group of classical physics. But the decisive distinction from classical physics (and from relativity theory) is that quantities in (von Neumann's) quantum mechanics which can be used as measuring quantities ("observables"), are not exchangeable ("commutative"). This group-theoric characteristic of quantum systems comes to expression in terms of measurement technique in that it makes a difference in what sequence quantities are measured. There is an aggravating disadvantage in von Neumann's quantum mechanics in that no classical ("commutative") observables are allowable. But how can the demonstrable existence of commutative quantities in the quantum realm, e.g., spin or rest masses, be understood (the problem of "Superselection rules," respectively "violation of the principle of superposition")? How can the measuring process of quantum mechanics be described? It is an interaction between a classical macroscopic measuring instrument and a quantum system. How can the quantum theory be understood as a framework theory for the chemist, who, for example in his molecular investigations, deals with non-classical quantities (e.g., "position" and "momentum" of an electron) and classical quantities (e.g., "temperature" or "chemical potential" of a thermodynamic process)? Hence, generalized formalisms of quantum mechanics (e.g., C*- algebra, quantum logic) have recently been developed which also admit classical ob-

8

Introduction

servables and are determined by symmetry groups. Thereby, from the perspective of uniting natural-science theories, a framework is built in which classical systems, quantum systems (in von Neumann's sense), generalized quantum systems and thermodynamic systems can be examined. Philosophically the non-classical quantities have led to considerable difficulties of interpretation, since - in contrast to classical physics - their measurement has to depend on the context of measurement ("measuring instrument"); else, their values are not unequivocally determined. Is the quantum world basically different from the macroscopic world, or does physics project a merely fictitious and grossly idealized picture of reality? Must realism in the sense of Aristotle and of classical physics have to be abandoned altogether? Then what symmetries are at the basis of reality? It is, finally, a matter of the unification of the natural forces insofar as they are known today in elementary particle physics. The historical development of physical theories is defined by a step-by-step unification. Newton achieved the first great unification when he traced trajectories of free-falling or projected terrestial bodies to the same conformity with celestial bodies. Next came Maxwell who based electricity, magnetism and optics on electrodynamics. Newton's gravitation theory had to be replaced by Einstein's general relativity theory, and Maxwell's electrodynamics had to be enlarged by special relativity theory, and quantum mechanics by quantum field theories, especially quantum electrodynamics. The first step in that direction was made, already in 1928, by P.A.M. Dirac, when he predicted - with a relativistic quantum mechanical wave equation - an anti-particle to the electron ("positron"), which in fact was discovered in 1932. The breakthrough for the theory of the electromagnetic interaction of electrons, positrons and photons came at the end of the forties with the work of R.P. Feynman, J.S. Schwinger et al.. The group of (unitary) transformations, which leave the laws of this theory invariant, has the so-called U(l)-symmetry. Physically these transformations correspond to a process in which a particle is transformed from one state into another without changing its identity. Thus an electron can go to another energy state by sending out a photon. The initial state and the final state do not differ in electrical charge, and the transitions between the two states by means of the emission of photons can be represented by a 1 χ 1 matrix. Here we encounter an entirely new kind of symmetry which is no longer a matter of "external" space-time symmetries such as reflections, rotations, translations, etc., but instead of "inner" (intrinsic) symmetries of transformations of matter. Another inner symmetry is isospin-symmetry which establishes a connection between the nuclear particles proton and neutron and the nuclear forces. Both particles possess the same spin and almost the same mass, so that they - as W. Heisenberg recommended - can be conceived of as two

Introduction

9

possible states of a particle, the nucleón. Transitions from one state to another are described by means of the so-called SU(2) group. While neutrons and protons are the only particles with strong interaction which are stable for a long time, with today's high-energy technology a multiplicity of very short-lived particles with strong interactions ("hadrons") can be produced. This "zoo" of hadrons which was discovered in the fifties more or less by chance, was finally derived from a unitary symmetry structure. Since then all hadrons are built up out of sub-elemental "quarks" (so far only indirectly confirmed). Their strong interaction can be described by means of an SU(3) symmetry. This theory is built up according to the model of quantum electodynamics and is called quantum chromodynamics since the strong force does not come into play between electric charges but between so-called color charges of the quark. Today a fourth fundamental force of nature, weak interaction, is distinguished from gravitational energy and the electromagnetic and strong interactions. While gravitational energy and electric and magnetic phenomena are well known through everyday experience, nuclear forces and weak interactions can be observed only by means of the modern technologies. Thus the weak interaction is responsible for the ß-decay. This force proves to be especially critical for the discussion of left-right symmetry ("parity") in nature, which goes back to Leibniz and Kant. Namely, experiments at the end of the fifties confirm that for the weak interaction in the case of ß-decay of 60 Co - in contrast to the other three basic forces - neither parity (P) nor reversal of charge (C = charge) is a symmetry operation; rather, it is only the combination PCT with the operation Τ (T = time) for reversal of time (PCT theorem). After H. Weyl, already in 1918, had attempted a unification of the electromagnetic forces with gravitation, S. Weinberg, A. Salam, S. Glashow et al. succeeded in 1967 at uniting the electromagnetic and the weak interactions. Both forces derive from the so-called SU(2) χ U(l) symmetry, which nevertheless is present only in extremely small spatial ranges and is broken already in spacings in the size range of the nuclear radii. While the interactions of gravitation and electromagnetism are spatially limitless and therefore are transmitted by massless particles (gravitan, photon), the weak interaction (as well as the strong one) extends only for short distances. Therefore the breaking of the SU(2) χ U(l) symmetry becomes observable when the intermediary particles (except for the photon) suddenly take on large masses. So far, the uniting of allfour fundamental forces in one symmetry group is carried only by assumption in mathematical models. Thus the SU(5) group, which is the smallest simple group which combines the SU(3) symmetry and SU(2) χ U(l) symmetry, proves to be especially interesting for describing strong, weak and electromagnetic interaction. This theory predicts a tiny

10

Introduction

extension in which there are no fundamental differences between quarks and leptons or between the strong, weak and electromagnetic interactions, but instead only one kind of matter and only one fundamental force. In organic evolution the SU(5) symmetry would have existed a fraction of the first second after the big bang. The rest of the spatial-temporal evolution of matter consists of breaking of the basic symmetry and the appearance of partial symmetries with varying particles and fundamental forces - a cosmic kaleidoscope whose symmetries depend on spatial orders of magnitude and temporal developmental phases. Certain rules of conservation then become time-dependent, so that the decay of the proton is among the most spectacular prognoses of this theory and is sought after in expensive experiments. Finally the theory of supergravitation strives for a modern theory of the Aristotelian "materia prima" with super-symmetry, in which all four fundamental forces are indistinguishable. The evolution of symmetries is pursued further from chemical and biological points of view. For example, if one views a crystal in the atomic size realm, only the symmetry of the individual atoms becomes distinct. On the larger scale, the binding forces appear, breaking atomic symmetry but building up the new molecular symmetry of the crystal lattice. The old problem of left-right symmetry was investigated already in the previous century for crystals in relation to polarized light, and led to stereochemistry. In biochemistry the determination of a chain direction is often central, for example, to reach an unequivocal gene coding for the DNA molecules. The symmetry principles and their violations in the macromolecular realm are today a widespread research field. It seems to be characteristic that organisms prefer the middle realm of the transition from highest symmetry (e.g., crystals) to perfect chaos (e.g., gases). In the last century, Pasteur had advanced the thesis that dissymmetry was typical for life. We find this opinion reflected in literature in The Magic Mountain, by Thomas Mann: Hans Castorp, gazing at snow crystals, surmises: "Life shuddered before this exact correctness." Indeed the dynamics of life processes can be described by means of symmetry breakings, as in the case of cell division. On the other hand, it is precisely living creatures, as self-reproducing systems, that display particular temporal developmental symmetries, which show themselves in the course of generations as the periodic recurrence of the cyclical course of growth of individuals. In today's biology one also speaks of the "cell cycle" and the "hypercycle." In addition, the mathematical chaos theory suggests the possibility of deriving apparently "chance" (stochastic) evolutionary developments (e.g., of genes) from simple determining rules which are set by the symmetry law of self-similarity (automorphism).

Introduction

11

Morphological symmetries of plants and trees are striking to anyone. Movement in all directions in the isotropic medium of water is made possible by the central symmetry of many sea organisms, while the arrow form of the fish is expedient for a goal-oriented movement. Under specific environmental conditions symmetrical forms offer selective advantages, which have been imitated and further developed by modern technology (e.g., in building cars, airplanes and rockets). The bilateral symmetry of higher animals seems to solve the problem of optimal mobility with simultaneous balance of forces, while this organizational principle is followed only partially in the anatomy of the inner organs. Thus, although we have two lungs, we have only one left-leaning heart. In the macroscopic realm also it comes down to a layering and breaking of varying symmetries. Thereby, in the cosmic evolution of the symmetries, we have arrived at man. The 5th chapter discusses to what degree human knowledge, inquiry and action are determined by symmetry principles. Symmetries serve the orientation of our perception and the organization of our imagination. Here is the aesthetic basis of symmetry in representational art and music. But symmetry structures underlie our thinking and knowing, as can be demonstrated by the epistemological categories ("substance," "causality"). Closely connected is the role of symmetries in the psychology of research. Important discoveries have frequently been made because they were predicted on the basis of a theory's assumption of symmetry (e.g., Dirac's prognosis of an anti-particle to the electron) or because breaks of symmetry (e.g., planet theory) had to be accounted for. Even the new computer generations of artificial intelligence take over symmetry assumptions as simplified strategies for solving problems. In epistemology one tries to establish the simplicity of a natural law by means of characteristics of symmetry, bearing in mind the historical principle of epistemology that the simple must be the true. The points of contact between the symmetry discussion and the epistemological question of the reduction of the natural sciences are central. Can physics, chemistry and biology be derived from a unitary theoretical framework with common basic principles? Mathematical structure theory is opening up new possibilities for that now, but also cautious limitations and precise definitions. The emergence of new chemical and biological structures is linked with new breaks of symmetry. The complexity of macroscopic dynamic systems also demonstrates that there are limits to a program of reduction to atomic and molecular building blocks. Vis-avis this "view from below, " whose success in atomic physics and molecular biology is by no means contested, one should not exclude the "view from above " to macroscopic wholes which we know from the everyday. In the "geometry of fractals" and the mathematical catastrophe theory, new theory frameworks exist which reveal surprising symmetries in the apparently

12

Introduction

chaotic multiplicity of macroscopic systems. The "view from below" and the "view from above" prove to be complementary research approaches which supplement rather than exclude each other. Under the conditions of modern physics, the philosopher confronts the pointed question of whether the symmetries are only epistemological projections onto nature which as heuristic principles prove to be useful for forming theories in the natural sciences, or whether they can be understood as principles of self-organization in nature. In the application of symmetry principles in natural science and technology the evolution of nature seems to perpetuate itself under new conditions. But first, a part of this nature (namely, we humans) also seems to be in a position to intervene over the long term, and destructively, in the natural cycles and symmetries. Therefore we face a requirement for a humane conception of nature that would allow humans to act in harmony with nature precisely because of our knowledge of natural science and of our technological possibilities. In this sense, the traditional approaches of natural philosophy become meaningful again for an education in the natural sciences, which conceives of research as a part of human history and nature. There is still the question about the unity of nature and science with art in the present time. The unity of the experience of nature, art and religion in the mythos of early peoples is surely lost. The Antique unity of the mathematical teaching of harmony, natural philosophy and art has also dissolved since the Renaissance. In art history (e.g., in Classicism), to be sure, there were always resonances to the Antique conception of art. But it has the effect of reciting old texts; it reflects a merely partial taste, and it no longer mirrors the cosmos and its laws. Art and science in modern times have differentiated themselves from each other into distinct media of life experience. The varied multiplicity of contemporary artistic attempts corresponds entirely to the complexity of the modern life world. In contrast to the Antique-Medieval life world, whose aesthetic forms depended on the potentialities of their handwork, we live today in a civilization that is determined by industry, technology and science, which co-determine our action, thinking and sensibility. The "Bauhaus" of the twenties was an artistic movement of modernity which tried to develop a new world of form under the conditions of technology and industry. The community of handworkers and artisans in the cathedral associations of builders and artisans of the Middle Ages and the artistengineers of the Renaissance led Gropius to the thought of uniting art and technology again under the conditions of modern industrial society. Architects, painters, graphic artists, sculptors, form designers, etc. were to work in coordination and by division of labor, seeking forms that would provide practical and functional solutions to people's needs, whether it was a mat-

Introduction

13

ter of furniture, dishes, homes, office buildings, factories, streets or leisure facilities. The measure, the "logos," of this art is the human being with his needs in the technical-industrial life world. In post-modern times there is a renewed "loss of the center," with breaks of symmetry more prominent than unity through symmetry. At the end there remain the words of Heraclitus about the hidden harmony of opposites which determines the dialectic of nature, science and art.

1. Early History of Symmetry Regular patterns and symmetries are used in all known cultures. They recur continually as ornaments on finery, cult objects, and everyday objects. Regular forms in crafts and architecture prove to be more stable, more economical in the use of materials, more distinct, simpler, easier to reproduce and to hand down to succeeding generations and - not least - of great aesthetic charm. Since the earliest times nature itself has manifestly been a model, evincing regularity in sundry forms and occurrences - from the minerals and plants, to the anatomy of living beings, to the regularly recurring stellar constellations. The old high cultures, as well as the still extant cultures of various ethnic groups, e.g., Asia, Africa, North and South America, use certain symmetrical patterns to give order to nature and their life-world. Modern natural science and technology were not the first to achieve this. For that reason, in the following there will be no discussion of "prescientific" cultures - nor the attendant admonition that "the primitives" need to undergo development. On the contrary, the cosmogenies that these peoples have are no less functional for the orientation of their lives than the contemporary conceptions of natural science and technology are for us.1 In Greek mathematics and philosophy of nature, principles of symmetry assumed a commanding position, and they became a decisive historical precondition of modern natural science.

1.1 Symmetries in Early Cultures Anyone who seeks out the Navajo Indians in the North American Southwest is astonished by the symmetry forms that govern their culture.2 What is so striking, is not so much the regular patterns and ornamentations of their artful textiles or ceramics, but rather the Navajo's use of symmetries in the ordering of their rituals and myths, in short, their Weltbild (image of the 1

2

Cf.. C.R. Hallpke, The Foundations of Primitive Thought, Oxford, 1979, Chapt. 3; for the origin of world images, cf. P. Berger/T. Luckniann, The Social Construction of Reality, Garden City, NY, 1966. Cf. G. Reichard, Navaho Religion. A Study of Symbolism, 2 vols., New York, 1950; additionally, the classical study by W. Matthews, Navaho Legends. Memoirs of the American Folk-Lore Society 5, New York, 1897.

16

1. Early History of Symmetry

world). In this context "Weltbild" is to be understood literally, as the representation of their life-world and mythologies of nature, not as a philosophical doctrine that has been derived from particular principles. Here we have in mind the sand paintings of the Navajo, which represent a Weltbild that has a particular ceremonial intention. These paintings are produced from pulverized sandstone in red, yellow or white, the pigments of cornmeal, plant pollens, or flower petals.

Fig. 1

Figure 1 shows the sandpainting of the "rainbow people", who are meant to be mythological representations of rain and light. In the centrallysymmetric hub of the cosmos there is the source of life - water, bordered by four rainbow bands in the four directions of the heavens or the four di-

1.1 Symmetries in Early Cultures

17

rections of the wind. 3 The four sacred plants - maize, beans, pumpkin and tobacco - grow out of the center. Two masculine (round-headed) and two female (angular-headed) rainbow people are situated behind each of the four rainbow bands. The enclosing circle represents the goddess of the rainbow, who protects the life-world of the Navajo. Two flies serve as messengers or sentinels. This sandpainting displays an abundance of superimposed symmetries. The centrally-symmetric square has all the reflection symmetries of the diagonals and lateral bisectors. Therefore the center, with water as the basis of life, produces a statically resting effect, which is emphasized by its black color. However, the surroundings of this center display only rotational symmetries. Thus the foursome groups of rainbow people in the four directions of the sky can be joined to each other by quarter-turns of the circle around the center. The feather decorations and the outstretched arms have the effect of small directional arrows and provide an impediment to reflection symmetry at the diagonals and lateral bisectors of the square, which would otherwise convey an impression of resting stasis. Therefore the rainbow people travel around the center in the direction of the sun. This dynamic impression is further underscored by the rainbow goddess, whose arc, with its inscribed head and feet, has the effect of a torque vector. Therefore the message of this world picture is clear: the element water is at the center, and all natural and life processes revolve around it. Along with reflection symmetries and rotational symmetries, color symmetries play a great role in all early cultures. The Navajo employs four basic colors - white, blue, yellow, black (and sometimes red), which can be interchanged with four directions of the sky or the wind and then have varying meanings in the different myths and tales. The rituals, which continue for days with their many sandpaintings, hymns, and dances are varied multifariously and applied to every possible individual case. In the world of the Navajo, however, symmetry does not have a separate aesthetic, religious or technical purpose. Their central concept is called "hózhó", which is often translated as beauty, but cannot be separated from health, happiness and harmony.4 The life and culture of the Navajo is based on a unity of experience which is expressed as "hózhó." "Hózhó" is the intellectual concept of order, the emotional state of happiness, the moral value of the good, the 3

4

K.A. Nowotny, Beiträge zur Geschichte des Weltbildes. Farben und Weltrichtungen, Horn/Vienna, 1970, p. 195. Cf. G. Witherspoon, Language and Art in the Navajo Universe, Ann Arbor, 1977, p. 154 ff; E.P. Hatcher, Visual Metaphors: A Formal Analysis of Navajo Art, The American Ethnological Society Monograph 58, St. Paul, 1974, 74ff.; for the philosophy of the Navajos, cf. also C.K. Kluckhohn, The Philosophy of the Navajo Indians, in F.S.C. Northrop, Ideological Differences and World Order, New Haven, 1949, pp. 356-384.

18

1. Early History of Symmetry

biological condition of health and well-being and the aesthetic charm of balance, harmony and beauty - a projection of wishes, ideas and experiences which is found also in other cultures.

Fig. 2

The cult in the North American Southwest and in Mexico goes back to the same historical roots. The formal similarity of the Navajo sand painting to the Aztec world map in Figure 2 (Codex-Fejérváry-Mayer) is indeed striking. 5 The picture shows the cosmos as a flower-shaped construct that is centrally-symmetrically arranged according to the five regions of the world.

5

E. Seler, Codex Fejérváry-Mayer, Eine altmexikanische Bilderhandschrift des Free Public Museums in Liverpool, Berlin, 1901, 1-3; for interpretation, cf. also Κ.Α. Nowotny, Tlacuilolli. Die mexikanischen Bilderhandschriften, Stil und Inhalt. Mit einem Katalog der Codex-Borgia-Gruppe, Berlin, 1961 ; also H. Biedermann, Altmexikos heilige Bücher, Graz, 1971.

1.1 Symmetries in Early Cultures

19

The four quarters of the 260-day cycle of the Aztec year (arranged in 5 χ 52 days) are inscribed in it. The interpretation of the map, however, is less peaceable than it is with the Navajo. At the center stands the fire god Xiuhtecutli, who receives streams of blood from the four sides. The representation of the centrally symmetrical cosmos is at the same time the depiction of a cult site on which a sacrificed human was chopped up and the bleeding pieces of the corpse were spread out. Xiuhtecutli is the first of the "Nine Lords of the Night", who are apportioned in pairs to the individual directions of the wind and represent the nine hours of the night: in the upper trapeze, Itzli (the god of the sacrificial knife) and Piltzintecutli (a subsidiary form of the sun god); to the right, the maize god Cinteotl and Mictlantecutli, the god of death; below, the water goddess Chalchiuhtlicue and Tlazolteotl, the earth and moon goddess; to the left, the earth god Tepeyollotli and the rain god Ttaloc. The points on the trapezes symbolize the days of the twenty 13-day-long weeks of the 260-day cycle of the year.6 Typical plants are, e.g., the blossoming trees, thorn trees, cocoa trees and the fig cactus. Here, also, an abundance of superimposed symmetries appears. Even if one were to consider only the geometrical form of the central square, the four lateral trapezes and the loops that bisect the corners, complete reflection symmetry and rotational symmetry would be apparent. Whereas the plants at least partially possess reflection symmetry, the individual images of the gods convey breaks of symmetry and dynamics. Similar forms of symmetry are found in all of the representations of Weltbildern (world-pictures) in early cultures, although in their content they are often interpreted entirely differently. An example is the Weltbild of the Jaina from the Indian sphere of influence. In addition there are miniatures of the Jaina Kälpasutra, which were not documented until the 15 th and 16th century, but go back to very old sources. Represented here is the Samavasarana which the gods erect for every Jina. 7 It is a round or square space, surrounded by three circular walls with four gates to the regions of the world. The Jina sits in the center and meditates or preaches, magically quadrupled on lion thrones under a tree. The tranquillity and composure which these images radiate is achieved formally by means of central symmetry and reflection symmetry. This impression is strengthened by the fourfold point reflection of the Jina at the midpoint of the four corners of the miniature. The inner

6

7

Additionally W. Cordan, Popol Vuh. Mythos und Geschichte der Maya, Düsseldorf/Köln, 1975, 183. Cf. W.N. Brown, A Descriptive and Illustrated Catalogue of Minature Paintings of the Jaina Kälpasutra. Smithson, Inst. Freer Gallery of Art. Oriental Studies, Washington, 1934.

20

1 · Early History of Symmetry

wall consists of jewels and is decorated with pinnacles of rubies; the middle one is made of gold. In various cultures symmetry characteristics are used cabbalistically, i.e., with words or letters, in order to gain insights by means of geometrical arrangements and combinations. A noteworthy example of Indian cabbalistics is the Scricakra. It consists of a diagram made of 43 triangles, called Meru (Figure 3).8 It is surrounded by an 8-petaled lotus and a 12-petaled lotus, which are again enclosed by four circles. It is characterized by four T-formed structures at the sides of the outer square frame. Instructions are indicated for the two lotus blossoms, and most especially for the potential combinations inherent in the Meru diagram. Proceeding from the outside to the inside, one distinguishes a 14-pointed star, an outer 10-pointed star and an inner 10-pointed star. The center is a triangle, which is also the structural principle of the diagram. It is interpreted either according to the nature mythology of the three Vedic lights - the moon, the sun and fire; or linguistically in accordance with the sounds of various syllables; or anthropologically in accordance with the trinity of thought, voice and body.

8

T.A. Gopinatha Rao, Elements of Hindu Iconography, 1/2, Madras, 1914, Iñanarnavatantra X 39; cf. also Κ.Α. Nowotny, (see fn. 3) p. 100.

21

1.1 Symmetries in Early Cultures

Fig. 4

Although no historical explanation can be cited, the Buddhist diagrams from India show a strong similarity to Chinese mirrors from the early Han period.9 These mirrors with their characteristic T-, L- and V-formed corners ("TLV mirrors") are interpreted unequivocally as cosmological. The animals of the wind directions (dragon, bird, tiger, and a turtle with snakes twisted around it) are frequently portrayed with a swarm of legendary animals and demons. Later these mirrors are further developed into compasscards, which employ - along with the four cosmic animals - 12 cyclical animals (in analogy to the 12-day week), 28 star pictures as constellation figures, etc. Cabbalistic speculations with symmetries are found as late as the late European Middle Ages, e.g., in the "Ars Magna" of the Catalonian philosopher Raimundus Lullus (Figure 4), who was demonstrably under Arabic influence. Leibniz in particular regarded him highly.10

9

10

On that subject, O. Karlbeck, Notes on some Early Chinese Bronze Mirrors, China Journal of Science and Arts, 4, 1926; B. Karlgreen, Huai and Han, The Museum of Far Eastern Antiquities. Bulletin. Stockholm, p. 13, 1941. R. Lullus, Opera ea, quae ad adinventam ab ipso artem universalem scientiarum artiumque omnium... pertinent etc. Argentorati 1617; also L'Ars compendiosa de R. Lulle. Avec une etudé sur la Bibliographie et le fond Ambrosian de Lulle par C. Ottaviano, Paris 1930.

22

1. Early History of Symmetry

S SE

/,

Chhicn sky

Tut

S

sea

«ri

Chen thundei Nt;

Ν Fig. 5

Fig. 6

The use of symmetries in China has a special charm. Very early on, they were interpreted in the framework of philosophy of nature. In the "Book of Changes" (the I Ching), from the 8th century B.C., four pairs of natural opposites - forces and elements such as heaven-earth, fire-water, lakemountain and thunder-wind - were symbolized by eight triagrams arranged according to reflection symmetry (Figure 5). They were also represented on coins (Figure 6).11 According to a later interpretation in the "great treatise" (Ta Chuan), these symmetries derive from the duality of light ( ) and dark ( ), Yang and Yin. The complicated 64 hexagrams made out of and appear in the so-called Wen Sequence (Figure 7). In this sequence the hexagrams are identical in pairs, allowing for a hexagram to be turned 180° (except for pairs (1,2), (27,28), (29,30), (61,62), which in each case are formed by an exchange of and ). By contrast, the circular and square sequence of the Fu-Hsi Sequence is to be read in the directions of the arrows that are inscribed for each sequence (Figure 8). The use of it as an oracle book for divining decisions by combinations of Yes ( ) and No ( ) approaches contemporary ideas of information theory, but remains hypothesis. A marginal observation: When Leibniz developed his dual system, he was influenced by the Yang-Yin dualism, which he learned about from French missionaries to China. In analogy to Greek science, spherical symmetries 11

I Ching - Das Buch der Wandlungen, German translation by R. Wilhelm, Jena 1924, repr. Düsseldorf/Köln 1973.

23

1.2 Symmetries in Antique-Medieval Mathematics

JL

Èmm m.

*2L

ι»

fe*

¡L

Μ

κ

JL

ȟ

JL

¿L

«JL

»

»

_I MΗ»

. VM«h η

j*

Fig. 7

played a great role in Chinese astronomy and cosmology. And here we have arrived at the scientific use of symmetries whose rigorous development assumed a central position in Greek mathematics.

1.2 Symmetries in Antique-Medieval Mathematics Regular patterns and figures were recognized intuitively very early. In many cultures the conviction was formed early on that these symmetries can be

24

1. Early History of Symmetry

Fig. 8

traced back to particular combinations of size and number. Proportional relationships were investigated in all areas of life. The Chinese "Mathematics in 9 Books" (Chiù chang suan shu), presumably from the second century B.C., offers a graphic example. It is an encyclopedia of mathematical information for geodesists, architects, merchants, artisans, government officials, etc. For example, it deals with instructions for construction and measurement, rules for calculating the equivalent exchange of goods, tax liabilities proportional to rank and villeinage, harvest delivery quotas, grain prices, etc. 12 According to the Chinese conception, symmetry also had to 12

Data, sources and authors of a "Mathematics in Nine Books" are not known precisely. The annotated version by the Chinese mathematician Liu Hui (263 A.D.) is extant. According

1.2 Symmetries in Antique-Medieval Mathematics

25

do with rightful proportional relationships, which in a large central apparatus of state and government like the Chinese imperial realm had to be regulated in detail. There is much similarity here to the known mathematical documents from Babylon. 13 Particular forms and proportional relationships are designated for cultic and ritual purposes. In Egypt pyramids served as monuments to the dead. In the Indian Salvasutras the symmetry of the Hindu altars was calculated with the use of Pythagorean number triads. In the astronomy of these cultures periodic celestial motions were registered and then viewed astrologically in connection to the course of lives on earth. But in Greek mathematics something happened that was completely new. Symmetries were made the systematic object of mathematical research. It is probable, to be sure, that the early Pythagoreans drew their basic mathematical knowledge from Egyptian and Babylonian sources. There, however, individual proportions remained related to technical-practical purposes. They were not based on proofs but, at best, determined by approximative reckoning procedures. Yet the Pythagoreans made the mathematical concept of harmony the central theme of their philosophy, which is based on geometry, arithmetic, music and astronomy. In Plato's time a general mathematical doctrine of proportions was developed, and it remained the mathematical basis of the concept of symmetry until the beginning of the modern era.

1.21 Geometry In all known cultures the circle is the symbol of perfection or of eternal recurrence. While it displays infinitely many symmetries resulting from random rotations and reflections at the diameters, the regular polygons inscribed in it possess a finite number of symmetries. If one connects the vertices of regular polygons with the center, one derives directional indicators that are useful for geodetic and astronomical orientation. The technical application of the spoked wheel comes later. Appropriate connections of the vertices render aesthetically charming star patterns which are also often used as ritual symbols. In architecture, centrally symmetrical edifices still play a great role.

13

to Liu Hui, the author of this book was a high official of imperial finance in the 2 nd cent. B.C. In analogy to Euclid's "Elements," an abundance of revised versions existed subsequently. Cf. A.P. Juschkewitsch, Geschichte der Mathematik im Mittelalter, Leipzig 1964, 23 ff. K. Vogel provided a German translation of Mathematics in Nine Books, Braunschweig, 1968. See also B.L. van der Waerden, Geometry and Algebra in Ancient Civilizations, Berlin/Heidelberg/New York/Tokyo 1983, 36ff.

26

1. Early History of Symmetry

Accordingly, the Pythagoreans set themselves the objective of constructing regular polygons with mathematical precision using the compass and the ruler. This, then, is not only a matter of near-regularities which are found approximatively by trial and error and which could be altogether adequate for technical purposes. It has come to be a matter of mathematical symmetry, which is provable and exists independently of technical application and perception, as an ideal form, as Plato will later say. The Pythagorean doctrine of the regular polygons has been handed down in Book IV of Euclid's "Elements. " l 4 It deals with the construction of the 3-, 4-, 5, 6- and 15-angled polygons. The first theorem of Book I presents the construction of a regular (equilateral) triangle with a given side AB (Figure 1)

D

C Fig. 1

Fig. 2

Describe a circle around A and another around B, each with radius AB, and connect the point of intersection C with A and B. The fact that the triangle ABC is regular - no matter how elementary the conclusions may be has to be derived and proven from the premissed definitions, postulates and axioms. As radii of the circle around A, AB and AC are equal, as well as BA and BC as radii of the circle around B. It is axiomatically given that if two magnitudes are equal to a third, they are equal to each other. Therefore the sides of the triangle are equal. The second theorem in Book IV demonstrates how a regular triangle can be inscribed into a given circle. The construction of a regular rectangle in a circumscribing circle is proven in the sixth theorem (Figure 2). In the given circle draw two perpendicular diameters AC and BD. On each side of the diameters there is a 14

Euclid, Die Elemente (trans. C. Thaer), Leipzig 1933, repr. Darmstadt 1962/1971; in English The 13 Books of Euclid's Elements, translated from the Text of Helberg with Introduction and Commentary by T.L. Heath, 3 vols., New York, 1956.

27

1.2 Symmetries in Antique-Medieval Mathematics

semicircle. According to Thaies' theorem, therefore, the square ABCD has right angles in its four corners. The four sides AB, BC, CD and DA are likewise equal. The two triangles DEA and CED have the same base DA or CD since they possess the same enclosed right angle at E and the circle radii AE, DE and CE as equal triangle sides. Analogously for the triangles AEB and BEC. The construction of the pentagram occupies an exceptional position. It served the Pythagoreans as an esoteric sign and possessed magical significance right into the Middle Ages in astrology and alchemy. Even as late as he appears, Faust is said to have exorcised Mephistopheles with it. The point of departure for the Pythagorean-Euclidean construction is a product equation which proves to be equivalent to what was later called the "Golden Section" (cf. 1,22).15 The 11th theorem of Book II of the "Elements" proves: A given straight line can be divided into two segments in such a way that the rectangle consisting of the straight line and one of the segments is equal to the square of the remaining segment, i.e., x 2 = a (a-x) in modern algebraic notation (Figure 3). This theorem allows one to construct an isosceles triangle ABC whose base angles are each twice as large as the angle at the top (Book IV, 10th theorem). The bisectors of the angles divide the facing sides of the triangle in accordance with the golden section (Figure 4). If one inscribes into a given circle a triangle A'B'C' similar to ABC (i.e., with the same angles) with the corresponding angle bisectors (Figure 5), one derives the desired regular pentagon (Book IV, 11th theorem). C

X X a—χ

E

C

C

a

A

A Fig. 3

15

Β Fig. 4

Fig. 5

A reconstruction of this connection is discussed by B.L van der Waerden, Die Pythagoräer. Religiöse Brüderschaft und Schule der Wissenschaft, Ziirich/Munich, 1979, 346ff.

28

1. Early History of Symmetry

To construct a regular hexagon, take a given circle with center M and radius MD and describe around D a circle with the same radius MD (Figure 6). If one lengthens the connecting lines through the points of intersection of the two circles through the center M, then according to the 15th theorem one derives the desired regular hexagon with equal sides and equal angles at the corners. The last theorem from Book IV provides the regular 15-sided polygon. Again the proof is elementary: In a given circle, draw the sides AB and AC of an inscribed regular pentagon and an inscribed regular triangle (Figure 7). The arc over AB is 1/3 of the total circle and therefore contains three angles of the 15-sided polygon, while the arc over AC amounts to exactly 1/3 of the total circle and thus must contain five angles of the 15-sided polygon. The difference of both arcs BC must therefore contain two angles. Therefore, bisecting the arc BC at E yields the side EC of the 15-sided polygon.

Fig. 6

Fig. 7

Book IV introduces the construction of the 15-sided polygon which Proklos worked out on the basis of astronomy, since the distance between the poles of the ecliptic and of the equator was assumed to be equal to a side of the regular 15-sided polygon - a marked symmetry in the Antique geocentric model of the celestial spheres. The number of sides of the regular polygons that have been described up to this point can each be doubled by bisecting the angles, so that, by the Pythagorean-Euclidean methods, regular 3·2 η -, 4·2 η -, 5·2 η - and 15·2η- polygons can be constructed. What further regular polygons can be constructed with the compass and ruler? The search for answers to this question runs

1.2 Symmetries in Antique-Medieval Mathematics

29

parallel to the historical development of the other famous construction problems of Greek geometry such as doubling the cube, squaring the circle and trisecting an angle. 16 No way was found to construct the regular heptagon with the compass and the ruler. But the method of extending the Euclidean means of construction by the so-called intercalated ruler yielded a means of constructing it that has come down to us via Archimedes. Plato had condemned extending the means of construction beyond the compass and the ruler by supplementary mechanisms of motion and drawing tools, since he feared an incursion of materialism into the art of geometry which dealt with the realm of the Ideas. In this view geometry, in the Pythagorean tradition, had to do with the unchanging forms, which could be derived from the excellent forms of the circle (compass) and the straight line (ruler). According to this conception, motions that are undertaken with a physical intercalated ruler on the plane, to fulfill certain conditions, belong to the realm of physics and - as Aristotle will further sharpen the argument cannot be mathematized. In actual fact, the great mathematicians of Antiquity did not allow themselves to be swayed by this philosophical argument, and they solved their problems of construction, in part, on the basis of ingenious inspirations (e.g., by postulating conic sections). But they did not adduce empirical testing and experimentation with original tools of measurement as supplementary arguments for their proofs. The latter, rather, were only material models for certain presupposed proportional equations that were appended as further postulates, as the ruler and the compass were only material models for the mathematical definitions of the straight line and the circle. 17 Let us look, for example, at the regular heptagon. The Archimedian construction for the heptagon was transmitted by the Arabic mathematician Täbit ibn Qurra in the 9th century A.D. 18 In a regular heptagon, the angle dimensions entered in Figure 8 correspond to the peripherial angle α = 180°/7 over the side of the heptagon. First we establish two equations of proportion for the heptagon and inquire into procedures for construction. 16 17

18

Cf. Κ. Mainzer, Geschichte der Geometrie, Mannheim/Vienna/Ziirich, 1980, 32ff. In that connection, A.D. Steele, Über die Rolle von Zirkel und Lineal in der griechischen Mathematik, in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Β 3,1934-1936., 287-369; Κ. Mainzer, Axiomatischer Konstruktivismus und Ontologie: Zum philosophischen Selbstverständnis der griechischen Mathematik, in H. Stachowiak (ed.), Pragmatik Vol. I, Hamburg, 1985, 126-138. C. Schoy (ed.), Die trigonometrischen Lehren des persischen Astronomen Abu'1-Raihân Muh. ibn Ahmad al Bîrûnî, Hannover, 1927, 74-84; cf. also J. Tropfke, Geschichte der Elementarmathematik Vol. 1, Arithmetik und Algebra (completely revised by Κ. Vogel, Κ. Reich, and H. Gericke), Berlin/New York, 1980,429ff.

30

1. Early History of Symmetry

Since trianges AHD and HDC are similar, it follows that AD:HD = HD:DC. Since triangle DHB has similar base angles, then HD:BD. Therefore (1)

AD · DC = BD 2 .

Now triangles DHB and HBC are also similar. Thus CB:HB = HB:BD. Because HB = HC = CA, it follows that (2)

C B - D B = AC 2 .

In order to construct the heptagon, a line AB must be divided by two points C and D and a line BC must be divided by two points A and D in such a way that equations (1) and (2) are satisfied. Then point H of the heptagon side BH is determined by describing a circle around C with AC and around D with BD. Now Archimedes divides segment BC using an intercalated ruler (Figure 9). For this the square BC is inscribed with the diagonal BQ. Then the straight line PA must be shifted around Ρ in such a way that the triangles TPQ and ACE have the same area. When this obtains, the straight line BC fulfills conditions (1) and (2): Because of the equal areas, PQ-TL = AC CE. Because PQ = BC and TL = LQ = DC, it follows that BC-DC = AC CE and BC:AC = CE:DC. Because of the similarity of triangles ACE and PLT, CE:LT = CA:LP or CE:DC = AC:BD. Thus it follows that BC:AC = AC:BD, i.e., equation (2) is satisfied. Equation (1) follows from the similarity of triangles TLP and TDA. Then, namely, TL:DT = LP:DA and DC:BD = BD:DA. It is easy to see how the solution may also be found by means of conic sections. The Arabic mathematician and physicist ibn al-Haitam offered such a

31

1.2 Symmetries in Antique-Medieval Mathematics

Β

D C

A

Fig. 9

solution around 1000 A.D. (Figure 10). He posited ρ = BD as a fixed magnitude and determined the position of the points C and A. For χ = DC and y = CA, equation (1) gives a hyperbola (x + y)x = p 2 , which goes through point Τ with χ = ρ and y = O and has the asymptotes DM and DS. From equation (2) with (p + x)p = y 2 one derives a parabola through Β with the parameter P·

Modern times developed the equation x 3 + 2px 2 = p 2 x + p 3 by eliminating y. Thus, on the basis of a general doctrine of proportion, which will be discussed in Section 1.22, the observations of the symmetry of regular polygons led to a theory of algebraic equations, the solutions to which were still being sought in geometrical constructions in the time of Descartes 19 . The symmetries of regular polygons were clearly also of considerable practical interest. That is documented by the practical geometry of the Me19

Cf. Κ. Mainzer, see Note 16, 92ff.

32

1. Early History of Symmetry

dieval mathematician Abü-al-Wafa, who wrote an influential book "What a Craftsman Needs to Know about Geometric Constructions." 20 . The second and third chapters of this book are about the construction of polygons with 3,4,5,6,8 and 10 angles. For the construction of the heptagon, what is chosen as an approximative side is half the side of the equilateral triangle inscribed in the same circle. The calculations for a side of the regular nonagon have trigonometric significance. That is, the angle functions for the 20° angle can be derived from them in order to calculate the angle functions for I o by combining the edge angle functions for 72° (pentagon) and 60° (hexagon). Ptolemy applied corresponding constructions for chord tables in astronomical calculations. 21 By extending the Euclidean means of construction, the Arab mathematicians of the 11th century derived constructions of the regular nonagon. 22 Our initial question: which regular polygons can be constructed with the compass and the ruler, was answered fully by the young C.F. Gauss at the end of the 18th century. He found that a regular η-sided polygon can be constructed with a compass and a ruler if, and only if, the uneven prime factors of η are differing Fermât prime numbers p k = 22"1 + l. 2 3 In analogy to the remaining construction problems from Antiquity, this definitive answer was first possible in modern mathematics after the mathematicians had learned how to formulate geometric problems algebraically. Thus Gauss, too, achieved his ingenious initial success, not by geometrical experimentation, but by a basic number-theoretical and algebraic analysis of the problem. 24 Indeed, the problem of constructing regular polygons exemplifies the applicability of the modern mathematical concept of symmetry, which can be defined generally and exactly by the algebraic Galois theory. Thorough discussion of this subject will follow separately (cf. Chapter 2.2). Fermat's prime numbers are p 0 = 2 1 + 1 = 3, pi = 2 2 + 1 = 5, p2 = 2 4 + 1 = 17, p3 = 2 8 + 1 = 257, p 4 = 2 16 + 1 = 65 537. Since 7 is not one of Fermat's prime numbers, in principle a heptagon cannot be constructed with a compass and a ruler. Since the prime factors of 9 are not different, the same conclusion holds for the regular nonagon. This negative argument was abso20

21

22 23

24

H. Suter, Das Buch der geometrischen Konstruktionen des Abu'l Wefâ, in Beiträge zur Geschichte der Mathematik bei den Griechen und Arabern, Erlangen, 1922 (Abhandlungen zur Geschichte der Naturwissenschaften und der Medizin 4 ), 94-109. C. Ptolemaios, Opera quae extant omnia (ed. J. L. Heiberg), Leipzig, 1898, Book 1, Chap. 10. J. Tropfke provides an overview, see Note 18,434ff. Cf. also J. Tropfke, Geschichte der Elementarmathematik, Vol. 4, Ebene Geometrie, Berlin/Leipzig, 1923, 257f. For the number-theoretical context, cf. Α. Scholz, Einführung in die Zahlentheorie, Berlin, 1939, 32. Details in Chap. 2.2.

1.2 Symmetries in Antique-Medieval Mathematics

33

lutely new compared with the geometry of Antiquity. New also was Gauss' positive deduction that it must be possible in principle to construct the regular 17-sided polygon with a compass and a ruler. But this conclusion was not reached by means of experimenting. Instead, the constructibility of the regular 17-sided polygon was, in a sense, predicted on the basis of the algebraic analysis of the problem, just as, at almost the same time, a new planet was predicted on the basis of the Newtonian theory of gravity and was indeed also discovered. In 1832 F.J. Richelot constructed the regular 257sided polygon. J. Hermes worked for ten years on the construction of the 65 532-sided polygon. 25 The geometric constructibility of regular polygons is now reduced to the number-theoretical question of whether Fermât pk numbers can also be prime numbers for certain greater values of k. Such problems in number theory today depend extensively on the efficiency of modern computers. Therefore the symmetry of regular polygons is an "evergreen" which has been newly investigated in every phase of mathematical history - from the elementary constructions with a compass and ruler in Antiquity, via algebraic number-theoretical analyses in modern times to the calculability problems of modern computers. The star polygons, derived from the regular polygons, were investigated in the High Middle Ages, possibly reflecting a special aesthetic interest, such as that expressed in the lovely rosette windows of Medieval cathedrals. In the "Geometria Speculativa" of T. Bradwardine (1290-1349) the star polygons are treated systematically for the first time, although individual problems (e.g., the sum of the angles) were solved by J. Campanus (f 1296). 26 In the first section of his book Bradwardine derives star polygons by lengthening the sides of regular polygons (beginning with the regular pentagon) to the points of intersection (Figure 11). Especially well-known are the pentagram made from the pentagon as a secret sign of the Pythagoreans, and the Star of David made from the regular hexagon. J. Kepler was also interested in the star polygons. 27 The 19 th -century Swiss mathematician L. Schläfli measured the density of a polygon with ρ sides by the number d of the points of intersection which a central ray that does not intersect any corners, has with sides of the polygon. Thus for ρ = 25

26

27

For the historical development, cf. J. Tropfke, see Note 23, 18 Iff. J. Hermes mentions F. Klein, Vorträge über ausgewählte Fragen der Elementargeometrie (elaborated by F. Tägert), Leipzig, 1895, 13. T. Bradwardine, Geometria speculativa, Paris 1511 ; for the historical impact of this book, cf. J.P. Juschkewitsch, see Note 12, 396. J. Kepler, Gesammelte Werke, Vol. 6, Harmonice Mundi. Weltharmonik (translated and introduced by M. Caspar), Munich/Berlin, 1939.

34

1. Early History of Symmetry

Fig. 11

5 and d = 1 the result is the pentagon, for ρ = 5 and d = 2, the pentagram. If d > 1, the points of intersection of the sides do not qualify as corners. Polygons with d > 1 are therefore the star polygons. Generalized regular polygons are then characterized by rational numbers p/d > 2 without common divisors of ρ and d. 28 Regular polygons and star polygons can be realized optically by means of the reflections of a kaleidoscope. It is presumed that such an instrument was first described historically in the "Ars magna lucis et umbrae" (1646) of A. Kirchner, who is in the tradition of R. Lullus with his philosophical speculations about mathematical symmetries, and who had a great influence on Leibniz as the typical representative of his Baroque Age. 29 More will be said about that later (cf., Chapter 2). Along with the regular symmetry figures in the plane, the symmetrical bodies of space have fascinated human beings from of old. In pre-Greek times some of these bodies already had cultic and religious symbolic value because of their regular construction and their crystalline structure. The

28 29

Cf. H.S.M. Coxeter, Regular Polytopes, New York, 1963, 93f. A. Kirchner, Ars magna lucis et umbrae, in decern libros digesta ..., Rome, 1646; Amsterdam, 1671.

1.2 Symmetries in Antique-Medieval Mathematics

35

Pythagoreans30 were acquainted with the regular tetrahedron composed of four regular triangles, the cube composed of six regular squares and the dodecahedron composed of twelve regular pentagons (Figure 12). A specimen of the dodecahedron made from steatite is extant from the Etruscan time (500 B.C.). But a complete derivation of all five possible regular solids was first handed down in the last (XIII) book of Euclid's "Elements", which dates back to the Greek mathematician Theaetetus (415-369 B.C.).31 Therefore the octahedron with eight regular triangles and the icosahedron with twenty regular triangles were probably also first constructed by Theatetus (Figure 12). The last theorem of the "Elements": that these are the only regular solids in Euclidean space, is already a significant mathematical insight. The proof is this: It is generally required of a regular polyhedron that all its corners, edges and surfaces be indistinguishable. Further, all surfaces should be regular polygons. This definition suffices to justify the five mentioned Platonic solids as the only regular bodies. First, a regular polyhedron will not possess any invaginated corners and edges. Since not all corners and edges could in30

31

For the knowledge of the regular solids among the Pythagoreans, cf. B.L. van der Waerden, see Note 15, 362 f.; cf. also E. Sachs, Platonische Körper. Zur Geschichte der Elementenlehre Piatons und der Pythagoreer, Berlin, 1917. Euclid, see Note 14; Κ. Mainzer, see Note 16, 52 ff.

36

1. Early History of Symmetry

vaginate, some corners or edges would be distinctive - contrary to the definition. Therefore, also, the sum of the polygonal angles that come together at one corner must be smaller than 2π. Otherwise these polygons would lie in one surface and invaginating edges would go out from this comer. Further, at least three polygons must come together in one corner. Beyond that, for the sake of regularity all angles of the polygon must be equal. Therefore they must all be smaller than 2π/3. In the regular hexagon the polygonal angle amounts to an even 2π/3. Since the angles for η > 3 increase in the regular n-angled polygon, only regular 3-, 4- and 5-angled polygons can be chosen as surfaces of regular polyhedra. In the case of the regular 4-angled polygon, the square, which has only right angles, no more than three squares can come together in a corner without exceeding the angle sum of 2π. In the case of the regular pentagon, no more than three pentagons can meet in a corner. A regular body is by definition already completely determined if the number of surfaces abutting in a corner and their number of corners is known. Therefore there can be, at the most, only a single regular polyhedron that is bordered by squares and similarly only one bordered by regular pentagons. By contrast, three, four or five equilateral triangles can come together in a corner since it takes six triangles to yield the corner angle sum 2π. The regular (equilateral) triangle can thus appear as a surface in three different polyhedra. Altogether, therefore, five possible regular polyhedra emerge:

Number of Polyhedron

Bordering polygon

Corners

Edges

Surfaces

Surfaces meeting at a comer

Tetrahedron

Triangle

4

6

4

3

Octahedron

Triangle

6

12

8

4

Icosahedron

Triangle

12

30

20

5

Cube

Square

8

12

6

3

Dodecahedron

Pentagon

20

30

12

3

Euclid shows in Book XIII that these five possibilities are also constructible. In addition, the regular bodies are inscribed into cubes, as, in the plane, polygons are inscribed into circles. In these constructions the irrational proportional relations from Book X play a great rôle. These will be addressed further in connection with the doctrine of proportions ( c f . 2.22). However, while there are infinitely many regular polygons in the plane (which Euclid confirms by the application of his procedure of exhausting the area of

1.2 Symmetries in Antique-Medieval Mathematics

37

the circle using polygons with increasing numbers of corners), the number of polyhedra is limited. In Antiquity and the Middle Ages the symmetries of the regular solids gained great significance in the philosophy of nature, alchemy and astronomy ( c f . Chapter 1.3). The increased interest is revealed by the history of mathematics: the two books added later to Euclid's Elements are thoroughly occupied with characteristics of the "Platonic" bodies. Solid stars are examined by analogy to the star figures in the plane. To every regular solid a reciprocal one can be assigned which is enclosed by the planes of the polygon at every corner of the original polyhedron. 32 For that reason the edges of the reciprocal polyhedron are centrally perpendicular to those of the original. Figure 13 shows the octahedron as a reciprocal polyhedron to the cube and vice versa, and the reciprocal polyhedron of the regular tetrahedron as an equal tetrahedron. In nature the combination of the two reciprocal tetrahedra appears as the twin crystal. L. Pacioli, in his book "De divina proportione" (1509) characterized the combination of two tetrahedra as "octaedron elevatum." 33 J. Kepler rediscovered it 100 years later and called it "stella octangula." 34 In the Platonic tradition regular bodies are viewed as elements of matter and of the universe (cf., Chapter 3.1). Pappus reported in the 5th book of his "Collectio" that Archimedes had also occupied himself with semi-regular polyhedra.35 According to Hero of Alexandria, some of these semi-regular polyhedra were even discovered by Plato. This may be an expression of the interest, on the part of the Platonic philosophy of nature, in tracing the multiplicity of phenomena back to geometric forms. In fact the semi-regular polyhedra do exhibit forms of solid bodies that were already familiar in the everyday as crystals, precious stones or building stones. These polygons are called semi-regular since each is bounded by various regular polygons. According to Pappus, Archimedes cited 13 semi-regular polyhedra with the following boundary surfaces: Pi : P2: P3: P4: P5: 32 33

34 35

4 triangles, 4 hexagons 8 triangles, 6 squares 6 squares, 8 hexagons 8 triangles, 6 octagons 8 triangles, 18 squares L. Schläfli, Gesammelte mathematische Abhandlungen, Vol. 1, Basel, 1950, 215. L. Pacioli, De divina proportione, Venice, 1509/Milan, 1956, Tables XIX, XX; cf. H. S. M. Coxeter, Unvergängliche Geometrie, Basel/Stuttgart, 1963, 197f. J. Kepler, see Note 27, Vol. 6, Book II. Pappos, Collectiones, 3 vols. (F. Hultsch, ed.), Berlin 1876/1878,5 th book, 352-358, cf. the presentation by T. Heath, A History of Greek Mathematics II, Oxford, 1960, 98ff.

38

1. Early History of Symmetry

Pô: P7: Pg: P9: Pio: Pu : P12: P13:

12 squares, 8 hexagons, 6 octagons 20 triangles, 12 pentagons 12 pentagons, 20 hexagons 20 triangles, 12 decagons 32 triangles, 6 squares 20 triangles, 30 squares, 12 pentagons 30 squares, 20 hexagons, 12 decagons 80 triangles, 12 pentagons

Pi, therefore, has 8 surfaces, P2, P3 and P4 have 14 surfaces, P5 and Pô 26 surfaces, P7, Pg and P9 32 surfaces. Pio has 38 surfaces. Pu and P12 have 62 surfaces. P13 has 92 surfaces. The fact that the semi-regular bodies were of technical-practical interest, is documented in the previously mentioned work on practical geometry by the Arab mathematician Abü-al-Wafa in the first millenium A.D. 36 In the 12th chapter he gives simple constructions of P2 and P7. However, the constructions of three further semi-regular solids are erroneous. Systematic constructions of the solids were indicated by J. Kepler in his work "Harmonice mundi." 37 We mention only a few examples here: The procedures sometimes remind one of the construction of regular polygons in the plane. Thus one can construct an octahedron from a square by symmetrically cutting off appropriate parts from the corners of the square. Thus Pi emerges from a tetrahedron if angles are cut off in such a way that hexagons remain as surfaces of the tetrahedron (Figure 14). Pio is the so-called frustum cube with 6 squares as boundary surfaces (like the cube), whose corners, however, are each bounded by 4 regular triangles 36 37

See Note 20; also cf. J.P. Juschkewitsch, see Note 12, 276. J. Kepler, see Note 27.

1.2 Symmetries in Antique-Medieval Mathematics

39

(thus a total of 8-4 = 32 triangles). P13 is the frustum dodecahedron with 12 pentagons as boundary surfaces (like the dodecahedron), the corners of which, however, are bounded by 4 regular triangles (therefore altogether 20-4 = 80 triangles) (Figure 14).

1.22 Arithmetic and the Doctrine of Harmony In many early cultures proportional relationships are described by means of numbers. But the Pythagoreans, as far as we know, were the first to want to base characteristics of harmony and symmetry on specific numerical relationships. Their representations of numbers by figures made of pebbles as well as the related number mysticism, became popular. Thus they differentiated the triangle's numbers 1 + 2 + 3 + ... + n, the square's numbers n2,

40

1 • Early History of Symmetry

the numbers of the rectangle n(n + 1), the numbers of the pentagon, etc., as follows: 38

• · ·

· · ·

· · · ·

· · · · ·

Already in the old Pythagorean time playing with figurai representations of numbers led to the discovery of some remarkable laws of numbers, e.g., that triangle numbers can be represented as halves of rectangle numbers, and therefore 1 + 2 + 3 + .. . + η = (1/2) n(n +1). The scientific number theory of the Pythagoreans is set down in Books VII, VIII and IX of Euclid's Elements. Some of the sections, such as the doctrine of the perfect numbers, are not at all elementary, but lead quickly to basic problems of number theory. With the Pythagoreans the tetraktys (quaternity) of the numbers 1,2,3,4 occupies a special position since it "begets the number ten" arithmetically, forms a regular triangle geometrically, is assigned musically to the four strings of the lyre, namely Hypate, Mese, Paramese, and Nete, and their properties correspond to the harmonious sounds of the musical fourth (4:3), the fifth (3:2) and the octave (2:1).39 In the Pythagorean conception, the harmony of nature is expressed in the unity of arithmetical, geometrical and musical proportions. Euclid calls such proportions "logos" (λόγος). In this sense the logos is the measure of all being. Pythagoras demonstrates in his music theory why the numbers 12, 9, 8,6 are excellent. To that purpose he uses the monochord, an instrument with only one string, which is divided into twelve equally large intervals. It is possible, namely, to express in whole numbers half, two-thirds and threequarters of the number 12, thus the shortened lengths 6,8,9 of the whole string 12, which correspond to the octave, fifth and fourth (Figure 1). Further, these numbers form the proportions 12:9 = 8:6. Here 9 is the "arithmetic mean" between 12 and 6, i.e., the differences 1 2 - 9 and 9 - 6 are equal. The number 8 is the "harmonic mean" between 12 and 6, i.e., the 38

39

Nichomachus of Gerasa, Introduction to Arithmetic (translated by M. Luther d'Ooge, commentary by F.E. Robbins and L.C. Karpinski), New York, 1926, repr. New York, 1972; also cf. Theon of Smyrna, Expósito rerum mathematicarum ad legendum Platonem utilium (ed. Hiller), Leipzig, 1878. On the Pythagoreans' doctrine of harmony, cf. B.L. van der Waerden, Die Harmonielehre der Pythagoreer, in: Hermes 78, 1943, 163-199.

41

1.2 Symmetries in Antique-Medieval Mathematics

I

1

0

1

1

1

1

1

1 6

1

1

1

8

9

1

1

1 12

Fig. 1

differences 1 2 - 8 and 8 - 6 are related as 12 to 6. In general the arithmetic mean m between two numbers a and b is defined in the Pythagorean way as a - m = m - b, and in the modern way as m = - (a -f b) ; the harmonic mean h is (a - h) : (h - b) = a : b and expressed in the modern way as h =

2ab

. a+ b Corresponding to the Pythagorean unity of arithmetic, music and geometry, the "geometric mean" g is distinguished as the third ratio, expressed classically as a : g = g : b, and in modern terms as ab = g 2 or g = \/ab. The designation "geometric" results from the fact that g is the side of a square that has a surface equal to that of the rectangle ab 40 . The arithmetic, geometric and harmonic means constitute the three Pythagorean ratios called μεσάτης, i.e., proportional ratios of three magnitudes, the middle one being determined by the other two on the basis of proportion. At a later time further ratios are added, by means of equivalent formulations of the Pythagorean ratios and exchange of the component parts. Strictly speaking, however, their derivation and proof require a general doctrine of proportion which will be discussed later. What was considered to be the most perfect proportion was the one that consists of two numbers a,b as well as their arithmetic and harmonic means, thus a : m = h : b, e.g., 12 : 9 = 8 :6

41

The famous Golden Section, which Pythagoras is said to have taken over from the Babylonians or Zarathustra, and which was considered for centuries to be simply the aesthetic standard, came to our attention earlier in the pentagram of the Pythagoreans. The product equation x 2 = a(a - x), used in 1.21 (Figure 3), is equivalent to the proportional equation a : χ = χ : (a - x). It is also said that the diagonals in the regular pentagon are divided "according to the extremes and the middle" (Figure 2), i.e., the ratio of the whole diagonal a to the greater part χ is equal to the ratio of the greater part χ to the smaller part a - x. In that connection, see the regular pentagon ABCDE in Figure 2, into which all the diagonals are inscribed. In the center the diagonals produce a smaller regular pentagon A ' B ' C ' D ' E ' . In a regular pentagon, sides and diagonals are parallel to each other in pairs by reason of symmetry. Thus the triangles AED and BE'C have parallel sides and are therefore similar. 40 41

Nichomachus, see Note 38, 134. Nichomachus, see Note 38, 118.

42

1. Early History of Symmetry

A

E

Β

D

C Fig. 2

Thereby, AD : AE = BC : BE'. The relationship BE' = BD - BC holds, because BC = AE = DE', since EA is parallel to DB and DE is parallel to AC. Thus it is established that for the pentagon: diagonal : side = side : (diagonal - side). There are many indications that precisely this symbol of the order of the Pythagoreans made their philosophy fundamentally questionable. What is under consideration here is the discovery of incommensurable straight-line proportions - presumably by the Pythagorean Hippasus of Metapontum in the 5th century B.C. - which is said to have set off a shock in Pythagorean circles.42 Ultimately this discovery called into question the assumption on which the philosophy of the Pythagoreans was originally based, namely that all proportions of magnitude could be expressed in ratios of whole numbers - like the harmonies on the monochord. In this sense harmony and whole-number rationality coincide in the philosophy of the Pythagoreans. For that reason the discovery of proportions of magnitude that are not in the ratio of whole numbers also seemed to them to be an incursion of the irrational, which according to legend brought the punishment of the gods upon the discoverer.

42

K. von Fritz, The Discovery of Incommensurability by Hippasus of Metapontum, in Annals of Mathematics XXXVI, 1954, 242-264; about the following presentation, cf. Κ. Mainzer, Real Numbers in: H.-D. Ebbinghaus et al. (eds.), Numbers , New York/Berlin/Heidelberg/London/Paris/Tokyo/Hong Kong/Barcelona 1990, 27-53; there also, detailed discussion of the sources.

1.2 Symmetries in Antique-Medieval Mathematics

43

In the example of geometry, the investigation of straight-line proportions started out in an age-old practice of measurement. A straight line a was measured by imposing a unit of measure e on the line m-times after each other: a = e+... + e = m · e m-times Two straight lines ao and ai are called commensurable if they can be measured in this way by the same unit of magnitude e: a 2, only 5 regular ("Platonic") polyhedra exist in 3-dimensional space {cf. Sections 1.21 and 1.32). This had already been proven by Theaetetus. Moreover, if we consider, additionally, the finite number of proper rotation groups around a center in space, we find only three new groups which leave unchanged or invariant (i) the regular tetrahedron, (ii) the cube or the octahedron, and (iii) the dodecahedron or icosahedron, respectively. For case (ii), inscribe an octahedron into a cube in such a way that the corners of the octahedron meet the corresponding sides of the cube at the centerpoints of the six square surfaces. Conversely, a cube can also be inscribed into an octahedron (Figure 12). Then compare the analysis of the corners, edges and surfaces of the Platonic solids in the table in Section 1.21. Every rotation that turns the cube back into itself also leaves the octahedron invariant and vice versa. Therefore the group for the octahedron is the same as for the cube. Analogously it can be shown that the dodecahedron and the icosahedron are described by means of the same group. The regular solid that corresponds to the regular tetrahedron is the tetrahedron itself. This gives us three new groups of proper rotations - group Τ of the tetrahedron, group W of the cube or octahedron and group D of the dodecahedron or icosahedron, with 12, 24 and 60 operations respectively. Corresponding to the finding by Theaitetos about the uniqueness of the Platon26

For the following derivations cf. also H.S.M. Coxeter, Unvergängliche Geometrie, Basel/Stuttgart, 1963, pp. 338ff.; H. Weyl, Symmetrie (see Note 5), pp. 120ff.

2.1 Symmetries of Ornamental Patterns and Crystals

151

Fig. 12

ic solids in space, it can be shown that groups C n (n = 1, 2, ...), D n ' (n = 2,3, · · ·). T, W, and Ρ are the only proper rotation groups in space. This list has to be supplemented by the number of improper rotations in space, analogously to the plane rotation groups. An improper rotation in space is nothing but a rotation reflection, i.e., the combination of a reflection and a rotation around an axis that is perpendicular to the mirror. A rotation reflection can also be grasped as a rotation inversion, i.e., as a combination of a point reflection or inversion at the center O (which brings every point Ρ back to P' on the extension OP' of line PO with PO = OP') and a rotation around an axis through the reflection point. A finite group of motions, which contains not only proper rotations, has such a proper rotation group as a subgroup of index 2. A moment's thought reveals that a finite group of motions with η rotations must still contain exactly η improper rotations. For, if the group consists of η rotations S¡ and m rotation reflections T¡) then the η + m motions can be represented by the combinations S¡Ti and T¡Ti : The η motions S¡Tj represent the rotation reflections T¡ and the m motions T¡Ti represent the rotations S¡. Therefore m = n. Now we distinguish one case from another by whether point reflection Ζ belongs to the group of motions or not. In the first case the η rotation reflections are simply S¡Z = ZS¡. The group that results is the direct product Γ = Γ χ ΖΓ of group Γ of proper rotations and group ΖΓ, produced by Z, which is none other than C2 or Di. Thus the list of proper rotation groups still needs to be supplemented by the finite groups C n , D n ', T, W, and P. In the second case, point reflection Ζ does not belong to the group. A group consisting of η rotations and η rotation reflections that does not contain Z, is isomorphic to a rotation group Γ' of order 2n (i.e., with 2n elements) which contains a subgroup Γ of order n. The 2n transformations S¡ and TjZ of a rotation group of order 2n have, namely, the same multiplica-

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tion table as the group which consists of S¡ and T¡ : from S¡Tj = T k , it follows

that S ; TjZ = T k Z, and if T;Tj = S k , then T¡ZTjZ = 1(Z?1S = T¡Tj = S k . Therefore our finite groups need to be supplemented by corresponding pairs of rotation groups. Each pair is a mixed group Γ ' Γ , which consists of all the rotations of the (smaller) group Γ with the rotations of Γ' which are multiplied by Z. The list of the proper rotation groups yields the pairs C2„Cn, Dn'Cn, D 2 n 'D n ' (n = 2,3, ...) and WT. The result is the group WT because the tetrahedral group Τ is a subgroup of index 2 of the octahedral group W. By analogy to the situation in a plane, we can now ask which of the finite point groups of motions leave space lattices invariant. In the 2-dimensional case we encountered 10 point groups. In the 3-dimensional case we obtain the 32 crystal classes. 27 By analogy to the 2-dimensional case the limitation of the finite groups arises through the restriction to only 2-, 3-, 4-, and 6fold rotation axes. Thus we obtain the rotation groups Ci, C 2 , C3, C4, C6, D 2 ', Os', D 4 ', Dg', T, W. The dodecahedral group Ρ is eliminated because Ρ has more than six axes. The direct products of these 11 groups need to be supplemented by the groups generated by Z, that is, C i , C2, C3, C4, Cö, D2', D3', D4', Í V , Τ, W. Further, there are the mixed groups C2C1, C4C2, C6C3,

D 4 'D 2 \ BeOS', D 2 'C 2 , D 3 'C 3 , D 4 'C 4 , D 6 'C 6 , WT. Here Ci = Ci χ Q Z = CiZ. The 32 crystal classes are of considerable significance for crystallography. In fact there is at least one crystal in nature corresponding to each class (up to C6C3). Crystallographers customarily arrange them according to the following 7 crystal systems. The classification corresponds to the metrical points of view according to which the 14 Bravais lattices also were summarized: a) b) c) d) e) f) g)

Triclinic system: Monoclinic system: Orthorhombic system: Rhombohedral system: Tetragonal system: Hexagonal system: Cubic system:

Ci,Ci C2, C2, C2C1 D2', ÉV, D2'C2 C 3 , C3, D3', D3, D 3 'C 3 C4, C4, C4C2, D4', D4', Ö4'C4, D4O2' C 6 , C 6 , C 6 C 3 , D 6 \ D 6 ', D 6 'C 6 , D 6 'D 3 ' T, T, W, W, WT

In the plane the 10 point groups have been supplemented by the possibilities of two independent translations to preserve the 17 ornament groups. The corresponding groups in three dimensions are the 230 discrete groups of movements with three independent translations. They are the central research theme of mathematical crystallography. The first 65 consist of proper movements. The mathematician C. Jordan first enumerated them in 1869, 27

Cf. also M. Klemm, see Note 8, § 10; A. Speiser, see Note 6, § 33.

2.1 Symmetries of Ornamental Patterns and Crystals

153

but only in 1879 was their application to crystallography described. 28 The simplest group contains only translations. The remaining 64 contain in addition rotations and screw axes, i.e., combinations of translations and rotations. Of these, 22 appear as 11 so-called enantiomorphic pairs, which are mirror images of one another, i.e., one member of the pair contains a left-, the other a right-handed screw. 29 Examples in nature include the left- and right-rotating quartz, known since the early days of mineralogy. However, the deeper molecular significance of left- and right-handed screws has become known only since the time of Pasteur, who also pointed out its relationship to the optical activity of many materials (e.g., levo- and dextrorotatory tartaric acid). These analyses were fundamental for the applications of symmetry to stereochemistry, which commenced with the molecular hypothesis of J. van't Hoff in 1874, and will therefore be treated at length in Chapter 4.41. If one drops these practically important distinctions, from grouptheoretical considerations, one obtains only 54 cases, consequently altogether 219 groups. 30 The remaining 165 groups contain, in addition to proper movements, also improper ones, such as reflections, rotation-reflections and glide reflections. These 230 groups can likewise be arranged according to the metrical viewpoint into the 7 classes previously used to categorize the 14 Bravais lattices and the 32 crystal classes. As a whole, all the groups were first described by the Russian crystallographer Fedorov (1890), and also independently by the German Schoenflies (1891) and the Englishman Barlow (1894). 31 From the point of view of the history of science, it is notable that in this period of time the discovery was in the air to some degree and Fedorov, Schoenflies and Barlow came to the same conclusion independently, to some extent proceeding from different considerations. An analogous situation had resulted, a few years earlier, in the discovery of the periodic system of the elements. In any case, mathematical crystallography provides an example of the way in which mathematical (in this case group-theoretical) analyses are motivated by discoveries in the natural sciences. As an example of the 230 space groups, here we will treat the sodium chloride crystal NaCl ("table salt"), since it also played a central rôle in the

28

29

30 31

Cf. in that connection H. Hilton, Mathematical Crystallography and the Theory of Groups of Movements, Oxford, 1903, p. 258. Cf. also, C.J. Bradley/A.P. Cracknell, The Mathematical Theory of Symmetry in Solids, Oxford, 1972. Cf. also J.J. Burckhardt, Die Bewegungsgruppen der Kristallographie, Basel, 1947, p. 161. E.S. Fedorov, Symmetry of Crystals (originals 1888-1896), American Cryst. Assoc., 1971; A. Schoenflies, Krystallsysteme und Krystallstruktur, Leipzig, 1891.

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X-ray investigations ofM. von Laue?2 In 1912 neither the nature of X-rays nor the atomic structure of crystals was known with any certainty. M. von Laue and his co-workers Friedrich and Knipping solved both problems by means of the following experiment. A narrowly attenuated beam of X-rays passes through a crystal cube (Figure 13). In von Laue's experiment the NaC1 crystal had cube faces parallel to the screen. A system of black spots with symmetry group D4 formed around the penetration point of the direct beam on a photographic plate Ρ which was set up behind the crystal (Figure 14).

ífírifrl "TfrT Fig. 14

Fig. 15

Von Laue now could establish the atomic structure of the crystal as well as the wave nature of X-rays on the basis of the diffraction pattern. The NaCl crystal is constructed alternately of sodium cations and chloride anions. In Figure 15, black points represent sodium ions, white points chloride ions. A sodium lattice is placed within a chloride lattice. Equivalent particles of chloride ions and sodium ions exchange places in the translations a = b = c along the corners of the cube or through translations (a + b)/2, (a + c)/2 and (b + c)/2 along the diagonals of the faces. Hence the translation group of this structure is Bravais lattice (14) in Figure 11. The structure of NaCl remains unchanged not only in the case of these translations, but also in operations of the point group (crystal group) W. 32

M. von Laue, Zur Geschichte der Röntgenstrahlinterferenzen, in M. von Laue, Aufsätze und Vorträge, 2 1962, pp. 110-117.

2.1 Symmetries of Ornamental Patterns and Crystals

155

Thus far we have restricted the finite movement groups from different crystallographic points of view. Now, by analogy to the plane, we ask which spatial point groups can be realized with a kaleidoscope. This aspect is also of extreme interest from the standpoint of cultural history, since - as already mentioned - the reflection effects of the kaleidoscope became the symbol of an entire epoch and found their expression in the philosophy and art of the Baroque. For Leibniz the world consisted of such crystalline reflection units, the so-called monads , in which all other shapes and figures are reflected in a typical manner. 33 This particular aspect of Leibnizian philosophy of nature has proven to be of great current interest in modern elementary particle physics, and it will be explored especially in Chapter 4.3. In what follows the mathematical/group-theoretical background will be explained first. Of the point groups in the plane, the dihedral groups D n can be realized in the kaleidoscope on the basis of their reflection characteristics ( c f . 2.11). Which spatial point groups can be produced by a kaleidoscope? Naturally, those which are generated by reflections, namely D n 'C n (n > 1), D 2 n ' D n ' (n odd), D n ' (n even), WT, W, and P. In that connection, Di 'Ci is the group generated by a single reflection. D 2 'C2 is the group of order 4 which is generated by two mutually perpendicular reflections. The remaining groups D n 'C n correspond to the groups of n-cornered pyramids. D 2 is a group of order 8 which is generated by three mutually perpendicular reflections. The remaining groups D2 n 'D n ' and D n ' correspond to the groups of η-fold prisms or the double pyramids (as their reciprocals). WT describes the perfect symmetry of the tetrahedron and arises from the rotation group Τ by the inclusion of the reflections. In that instance one should also compare the reflection possibilities of the twin crystal of two reciprocal tetrahedra in Figure 13 (Chapter 1.21), which were already investigated by Pacioli and Kepler. The complete symmetry characteristics of the remaining Platonic solids are described by W, and P, which in contrast to the tetrahedron possess a central point. In the plane some of the 17 ornament groups, which can also be produced by folding and cutting out ("cutting with scissors"), can be realized by means of a kaleidoscope. Analogously we can investigate space groups which are realizable in the kaleidoscope. 34 For that purpose one should 33

"Or cette liaison ou cet accommodement de toutes les choses creées à chacune et de chacune à toutes les autres, fait que chaque substance simple a des rapports qui expriment toutes les autres, et qu'elle est par consequent un miroir vivant perpetuel de l'univers." (G.W. Leibniz, Monadologie § 56) in G.W. Leibniz, Vernunftprinzipien der Natur und der Gnade. Monadologie (French/German), H. Herring, ed., Α. Buchenau, transi.; Hamburg 2 1960, p. 50.

34

A.V. Shubnikov/V.A. Koptsik, see Note 9, p. 201.

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2. Symmetries in Modern Mathematics

imagine, as in the Baroque castles, illuminated spaces with walls, ceilings and floors which are completely covered with mirrors. In Figure 16 (1) a kaleidoscope for an orthorhombic system is presented which corresponds to the Bravais lattice (4). A kaleidoscope for a tetragonal system in which two of three mutually perpendicular directions are the same, consists of a prism with an isoceles triangle as its base (Figure 16 (2)). That corresponds to a Bravais lattice of type (10). For a hexagonal structure with three equal directions in a plane, which can be transformed by rotation, there are two kaleidoscopes: a prism with an equilateral triangle as its base (Figure 16 (3)) and a prism with a right-angled triangle with angles of 30° and 60° (Figure 16 (4)). These four types correspond to the four plane figures which can be produced by a kaleidoscope ( c f . Figure 9). For cubic structures with three mutually perpendicular directions there are three more kaleidoscopes. Each of these seven kaleidoscopes generates one of the seven symmetry classes of the 230 discrete space groups.

(1)

(2)

(3)

(4)

(5)

It is obvious to interpret Leibniz' doctrine of monads by means of this mathematical result. In Euclidean space the crystal symmetry structures are described by 230 space groups. From metric viewpoints they can be subdivided into 7 classes which can be generated by kaleidoscopes, i.e., by definite reflecting entities ("monads"). One could therefore state succinctly

2.1 Symmetries of Omamental Patterns and Crystals

157

that the crystalline symmetry of the (Euclidean) space is generated by these mathematically defined monads or is reflected by these mirror centers. For planes we have referred to the number-theoretical relationship of the discrete groups with the theory of quadratic forms. This approach to the question can also be extended to the three dimensions of space. Analogous to the question of how the 10 ornamental point groups operate on plane lattices, the problem now becomes how the 32 crystallographic point groups operate on space lattices. The 13 arithmetical groups of the plane now correspond to 73 of space.35 In conclusion we pose the question of which discrete symmetries exist in n-dimensional Euclidean spaces with higher dimensions (n > 3) than the Euclidean infinitive space. Which regular ("Platonic") bodies are there, for example, in 4-dimensional space ? These considerations are by no means mere toying with thoughts of possible worlds, but can be mathematically-physically and psychologically motivated. In the history of mathematics n-dimensional spaces have been investigated from different points of view since approximately the middle of the 19th century, though their possibility had been commented on philosophically already prior to that time by, for example, Kant and Herbart. Physically they have demonstrated themselves to be extremely fruitful in the application of configuration-, phase- and later Hilbert-spaces, for example, in the kinetic theory of gases and in quantum mechanics. Psychologically the question arises as to whether our intuition is in fact restricted to three dimensions as Kant asserted and higher-dimensional spaces can only appear as arithmetical calculations.36 In three dimensions we are familiar with the picture of the cube as the analog to the two-dimensional square. Can we speak just as graphically about the 4-dimensional cube as the analog to the 3-dimensional cube? 37 Conceptually the 4-dimensional hypercube can be generated in four steps: In the first step two (0-dimensional) points are connected to form a 1dimensional line segment; in the second step the corners of two such parallel segments separated by the same segment length are connected to form a two-dimensional square; in the third step the corners of two such parallel squares separated by the same segment length are connected to form a 3-dimensional cube; in the fourth step the corners of two such parallel

35 36

37

Cf. also M. Klemm, see Note 8, § 12. Cf. also K. Mainzer, Geschichte der Geometrie, Mannheim/Vienna/Ziirich, 1980, pp. 126ff., 150f. In that connection also see K. Mainzer, Philosophische Grundlagenprobleme und die Entwicklung der Mathematik, in: Grazer Philosophische Studien. Intern. Z. Analyt. Philosophie 20, pp. 184ff. (1983).

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2. Symmetries in Modern Mathematics

cubes separated by the same segment length are connected to form a 4dimensional hypercube. The first three steps are visually familiar; the fourth step can be carried out only conceptually. The properties of the hypercube can be exactly determined conceptually: Since a cube has eight corners and the hypercube arises from the connection of the corners of two cubes, the hypercube possesses 16 corners. Furthermore, the hypercube has all 12 edges of the two cubes from which it arises, and in addition the edges which arise from the connection of the 8 pairs of points from the 16 corners, thus 2 · 12 + 8 = 32. Analogous considerations show that the hypercube has 24 surfaces (squares) and 8 cubes as 3-dimensional boundary "surfaces." A thought experiment in the tradition of Helmholtz shows how one can obtain visual impressions of such bodies. A 2-dimensional being, for whom only perceptions of length and breadth on the surface of a plane are possible, would be quite able to construct for itself a picture of a 3-dimensional cube: If the cube were to dip from the air into a planar water surface, this creature could perceive the 2-dimensional cross-sectional surfaces. If the cube dips into the water repeatedly from different angles and directions, the 2dimensional observer gains increasing amounts of information in order to make an image for itself and become familiar with the higher-dimensional object. Such intuitive learning processes are today objectifiable by computer graphics and no longer left to introspection, which is difficult to check. Thus a computer film by T. Banchoff and C.M. Strauss generates exact 3dimensional figures, which, via rotation of the hypercube in 4-dimensional space, appear as "sections" of our 3-dimensional visual space. The observer can generate all these impressions successively on a video screen through the manipulation of knobs and thereby in a learning process obtains visual and kinesthetic familiarity with the hypercube - which cannot be visualized to begin with. 38 Just as in 3-dimensional Euclidean space there are four additional regular ("Platonic") solids besides the cube, so L. Schläfli in 1855 was able to demonstrate exactly five more regular bodies, in addition to the hypercube, for 4-dimensional Euclidean geometry. 39 Besides the corners, edges and surfaces, volume elements appear as boundary units of these bodies. Analogous to the regular polygons as boundary surfaces of the 3-dimensional Platonic solids, there are regular polyhedra as "boundary volumes" of the 38

39

The film by T. Banchoff and C. M. Strauss won Le Prix de la Récherche Fondamental au Festival de Bruxelles in 1979. For the model concept, cf. also H. Freudenthal, The Concept and Role of the Model in Mathematics and Social Sciences, Dordrecht, 1961. L. Schläfli, Ges. mathem. Abhandlungen, Vol. 1, Basel, 1950; cf. also H. de Vries, Die Vierte Dimension, Leipzig/Berlin, 1926.

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2.1 Symmetries of Ornamental Patterns and Crystals

4-dimensional Platonic solids. One speaks of an η-cell if it is bounded by η polyhedra. In the 3-dimensional instance the octahedron was dual to the cube, and the dodecahedron to the icosahedron, while the tetrahedron was dual to itself. In 4-dimensional space the points are dual to the volumes and the straight lines to the planes. Thus the following duality relationships emerge: 4-dimensional space

1. 2. 3. 4. 5. 6.

5-cell 8-cell 16-cell 24-cell 120-cell 600-cell

Number and kind of bounding polyhedra

Number of corners

5 8 16 24 120 600

5 16 8 24 600 120

tetrahedra cubes tetrahedra octahedra dodecahedra tetrahedra

Duality

dual to itself 1

dual to each other dual to itself

1

dual to each other

According to this table the tetrahedron corresponds to the 5-cell, the cube to the 8-cell ("hypercube"), the octahedron to the 16-cell, the dodecahedron to the 120-cell and the icosahedron to the 600-cell. The 24-cell is an additional body in 4-dimensional space which is dual to itself. In contrast to the tetrahedron, however, it is also centrally symmetric with a middle point. Are there models of the 4-dimensional Platonic bodies in 3-dimensional visual space? To answer that question we will first imagine 2-dimensional creatures ("inhabitants of flatland") who want to make concrete models of the conceptually defined 3-dimensional Platonic solids. Mathematically this amounts to projections onto a plane of the regular polyhedra of 3dimensional space. These projections depend upon the choice of the center of projection and the plane of the picture. In Figure 12 (Chapter 1.21) parallel projection was employed, in which the center of projection is removed to infinity. In that case parallel edges in 3-dimensional space indeed remain parallel in the plane as well. To be sure, sometimes partial overlaps of the surfaces occur. This disadvantage can be set aside if one moves the projection center very close to one of the lateral faces. Figure 17 gives such projections for the tetrahedron (1), the cube (2), the octahedron (3), the dodecahedron (4) and the icosahedron (5). 40 In these figures the center of projection is located directly perpendicularly above a lateral face. One would receive the exact same impression if a face of the Platonic solid were removed and one looked through this opening into the interior. 40

D. Hilbert/S. Cohen-Vossen, Anschauliche Geometrie, Darmstadt, 1973, p. 129.

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2. Symmetries in Modern Mathematics

A (1)

Fig. 17

The 4-dimensional Platonic solids are now to be analogously represented in 3-dimensional visual space. Parallel projections prove to be unsuitable, since the "boundary solids" are represented by polyhedra which have to cover up and interpenetrate one another. To be sure, the projection procedure of Figure 17 provides visual models in 3-dimensional space, at least for the 5-cell (1), 8-cell (2), 16-cell (3) and 24-cell (4) (Figure 18).41 The "boundary solids" are represented by polyhedra, of which one is singled out to be filled up by the others. In model (4) of the 24-cell one can still recognize how the big octahedron is filled up by 23 smaller octahedra so that in 41

See Note 40, pp. 133 ff.

2.1 Symmetries of Omamental Patterns and Crystals

(4) Fig. 18

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2. Symmetries in Modern Mathematics

all 24 octahedra appear. With the 120- and 600-cells, however, even these projections become difficult to survey. The projections in 3-dimensional space are mathematically extremely interesting. In the 2-dimensional plane the projections produce graphs, i.e., systems of points and lines, which in a higher dimension prove to be symmetry structures. Analogously one obtains bodies of 3-dimensional space which in a higher dimension present themselves as symmetry structures. In the recent past the analogous 4-dimensional groups which we already investigated in 2- and 3-dimensional space have also been calculated. Thus, analogous to the 10 ornamental point groups and 17 ornament groups of the 2-dimensional plane and the 32 crystallographic groups and 230 movement groups in 3-dimensional space, there are 271 point groups and 4895 space groups in 4-dimensional space.42 What now is the situation for symmetries in arbitrary n-dimensional (Euclidean) spaces with η > 4? As far as the Platonic solids are concerned the situation is simplified. For η > 4 in fact now only three regular bodies are possible: 43 n-dimensional space, η > 5.

1. 2. J.

(n+l)-cell 2n-cell 2"-cell

Number and kind of boundary cells of dimension η — 1

Number of corners

n+1 2/1

n+1 2" 2η

2"

n-cells (2η - 2)-cells n-cells

Duality

dual to itself 1

dual to each other

In 3-dimensional space, since η + 1 = 4, 2n = 6, and 2 n = 8, the (n + 1)cell, the 2n-cell and the 2 n -cell correspond to the tetrahedron, the cube and the octahedron; in 4-dimensional space they correspond to the 5-, 8- and 16cell. In the 3-dimensional case therefore the dodecahedron and icosahedron, and in the 4-dimensional case the 24-, 120- and 600-cell, constitute special instances of Platonic solids which have no analogs in higher dimensions. 2.13 Color symmetries and symmetries of music To conclude our discussion of discrete symmetries in the plane and in space we turn to symmetries of color and music. At first this may seem to be 42

43

H. Brown/R. Bülow/J. Neubüser/H. Wondratscheck/H. Zassenhaus, Crystallographic Groups of Fourdimensional Space, New York, 1978. Cf. D. Hilbert/S. Cohn-Vossen, see note 40, 128.

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163

a purely aesthetic interest, but color symmetry is very useful in physics (cf. Chapter 4.3). Once again it evokes the old Pythagorean idea of a single symmetry structure in mathematics, art and nature, which runs through this treatise like a red thread. The great significance of color symmetries in the natural sciences will become clear as soon as we consider the following fundamental concept. It is a matter of assigning non-geometrical properties to the points of a geometrical symmetry structure that is governed by one of the well-known discrete groups. These non-geometrical characteristics can be colors - in wall ornamentations and plane ornaments, for instance, but they can also be physical characteristics such as points, atoms, ions, etc. of a crystal structure. Historically, of course, the symmetrical use of colors is ancient. The systematic group-theory analysis of color symmetries is more recent, however, and goes back above all to Russian mathematicians and crystallographers like N.V. Belov and A.V. Shubnikov, among others. The use of methods of group theory in crystallography is almost a national trait of Russian mathematicians and scientists, and has been so since the great success of Fedorov at the end of the 19th century. 44 The mathematical analysis of color symmetries has been given quite an impetus ever since the sixties when three degrees of freedom ("colors") were attributed to quarks as elementary building blocks of matter in elementary particle physics and thus the strong interactions could be traced to a single symmetry structure. Chapter 4.3 will go into that thoroughly. In general color groups can be regarded as groups of motions that do not involve a geometrical mapping, but instead a change of color. The simplest case of the two-colored or black-white groups involves only two colors or characteristics. One example is the black-white symmetry of the chessboard, which turns back to itself when you turn it 90° around the center, when black and white are reversed. Naturally there can be other characteristics than black and white, for example the shift from positive (+) to negative (-) magnetism or the alternation of shapes in ornaments such as round or cornered, etc. In the case of the chessboard the C4 group of motions - which leaves the colorless chessboard unchanged when you turn it 90° - is extended by the amount of operation ε of reversal of color, i.e., gs = εg means the sequential execution of a rotation and of a reversal of color or vice versa. If 1 is the group unit or identity, we can characterize the color reversal by ε = v T , since the doubled sequential execution of a color reversal leaves the color unchanged, i.e., ε 2 = 1. Therefore in our example C4 is expanded

44

A.V. Shubnikov/N.V. Belov, Colored Symmetry, Oxford/London/New York/Paris, 1964; cf. also A.L. Loeb, Color and Symmetry, New York, 1978.

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2. Symmetries in Modern Mathematics

by the amount of the permutation group {1, ε}. Hence we characterize this 2-colored rotation group as C ^ 2 ' as well. The 2-sided border ornaments that are like reliefs, in Section 2.11, which had arisen from the one-sided border ornaments in the plane as a result of additional spatial motions ("flippings"), can be described now by 2colored translation groups in the plane. In general let a spatial group of ndimensional space be given (with an n-dimensional translation lattice). If this spatial group is embedded in (n + l)-dimensional space (for instance a line group in the plane or a plane group in space) then a reversal of characteristics (sign change, change of up and down, of behind and before, etc.) can be associated with the symmetry operations in n-dimensional space in the (n + 1) dimension. This reversal can also be interpreted in n-dimensional space as a color reversal. In the art of Arabic ornamentation one finds 2-sided and 2-colored border patterns. 45 In modern elementary particle physics this operation is often also called antisymmetry. It describes the change of a particle into another particle. An example of higher color symmetries is the 17 ornament groups from Section 2.11 (Figure 7). The Islamic ornament artists found a successor in the Dutch graphic artist M.C. Escher (1898-1971). Like no other artist of this century, Escher expressed the variety of mathematical structures in representational art. The significance of ornamentation for Escher 's art can scarcely be exaggerated. He does not imitate historical models such as Islamic art nor does he perform an artistic conversion of mathematically previously defined and examined structures. On the theme of ornamentation Escher avows: I wander around like a motherless child in the garden of regularly filled spaces. No matter how satisfying it may ever be to have a domain of one's own - solitude is not so agreeable. In this case it really seems to be impossible as well. Every artist, or rather, every human being in order to avoid the word art as much as possible in this connection - possesses highly personal characteristics and bad habits. But the regular partition of space is not a tick, or bad manners, or a hobby. It is not subjectivity, but objectivity. With all good will on my part, I cannot believe that it has never occurred to anyone to do something so obvious as depicting figures that complement each other as well as their meaning, function, and purpose. For, when we cross the threshold of the primary phase, the game takes on more value than a merely decorative one. Long before I discovered in the Moorish artists of the Alhambra a kinship with the regular distribution of space, I had already discovered that in myself. At first I had no idea of how to design my figures systematically. I did not know any rules of the game and - almost without knowing what I was doing - 1 tried to fit congruent spaces together and to give them the shapes of animals... Later I succeeded in sketching new motifs with less effort than be-

45

Cf. E. Müller, see Note 11.

2.1 Symmetries of Omamental Patterns and Crystals

165

fore, but it was always a tense effort, a real "mania" that possessed me and it took great effort on my part to get free of it. 46

Escher's art fascinates us chiefly because it intuitively expresses a piece of higher mathematics without any previous theoretical, mathematical analysis - and does this at a time when abstract algebraic-geometrical structures are at the center of interest in mathematics. Another example in art history is the geometric analysis of figures and colors in cubism. These are instances of cultural-historical connections between mathematics and art that make the situation seem less hopeless than it does in Snow's thesis about the "Two Cultures." Pythagoras' notion that mathematics, art and nature are one seems to revive again and again in different historical circumstances. An example in art is Escher's 3-colored network pattern "Lizards" (Figure 19).47 If you disregard the colors, this is an example of an ornament (17) (Figure 7). Disregarding the translations, one gets rotation symmetry C6Each rotation of 60° alternates the colors black-red-white. Thus mathematically, group C(¡ and cyclical group {1, ε, ε 2 } are extended so that ε = v^T, ε 2 = (λ/ϊ) 2 , ε 3 = (s/T) 3 = 1 and 1 is the unity or identity. Hence the 3-color rotation group of 60° is also characterized as C^ 3 ). Solid-state physics provides a physical application of Cß color groups. Magnetic moments are assigned to atoms in order to describe magnetic configurations . Frequently there are interactions between neighboring magnetic atoms (for example in a crystal of iron atoms), forcing a parallel alignment of all the magnetic dipoles in a macroscopic domain itself without an external field. The alignment and coupling of all the magnetic dipoles does not break down until a temperature is reached which is characteristic of the material - the so-called Curie temperature. Materials that have this property - iron, for instance - are therefore called ferromagnetic (for example, cobalt and nickel). 48 Now we take a look at group C(, crystals. Magnetic moments of atoms are interpreted by arrows that rotate clockwise by the angle ε or in the opposite direction. There are not enough classical operations of this group to describe all the magnetic configurations. In Ce the magnetic moments re-

46

47

48

M.C. Escher, Regelmatige vlakverdeling, Utrecht, 1958; German translation in B. Ernst, Zauberspiegel des M.C. Escher, Munich, 1978, p. 41. M.C. Escher, Lizards, figure in A.V. Shubnikov/V.A. Koptsik, see Note 9, Figure 228; cf. also C.H. Macgillavry, Symmetry Aspects of M.C. Escher's Periodic Drawings, Utrecht, 1965. Cf. R.P. Feynman, Vorlesungen über Physik, Vol. II/2, Munich/Vienna, 1974, Chapter 36: Ferromagnetismus; C. Kittel, Einführung in die Festkörperphysik, Munich/Vienna, 1973, Chapter 16: Ferromagnetismus und Antiferromagnetismus.

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Fig. 19

main unchanged (Figure 20). 49 If one interprets the magnetic moment by O

the operation of antisymmetry, i.e., as the rotational angle ε = 180 , one obtains the configuration of the 2-color group CV 2) . The magnetic moment remains unchanged in every other position. In the 3-colored C 6 groups the magnetic moments are interpreted as rotation angles of ε = 120°, by which the rotations g of crystal symmetry C5, the symmetrical group S n is not solvable. That is why the centurieslong search for solution formulas for fifth degree equations remained unsuccessful: For n>5, the general nth degree equation cannot be solved by roots (Abel Theorem). 70 67 68

69

70

See E. Artin, see Notes 65, 70. G. Verriest, Leçons sur la Theorie des Equations selon Galois, Paris 1939, 34; G. Verliest, Oeuvres mathématiques d'E. Galois, publiées en 1897, Paris 1961, 327. A subgroup U is defined as an invariant subgroup of the group G precisely when it is identical to all of its conjugated subgroups, i.e. for o e G is o U a ~ ' = U , Galois calls the key requirement la decomposition propre, e.g. in Oeuvres mathématiques d'E. Galois, Paris 1897, 25,26. See E. Artin, see Note 65, 78.

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The most notable difference between Galois and his predecessors is that he was not seeking mathematical formulas, but insight into algebraic principles.71 The structure of the solutions of an equation can be read from the structure of a permutation group which clearly corresponds to the equation. Galois speaks of a "simplification intellectuelle" [intellectual simplification], by means of which "l'ésprit saisit promptement et d'un seul coup un grand nombre d'opérations" [the mind quickly and suddenly grasps a large number of operations].72 This appraisal is typical of considerations of symmetry in the history of science. For example, analyses of symmetry are frequently in order when the problem is to define a common structure in a multiplicity of apparently immense and disconnected individual results. That is true for the development of equation theory with its many individual results and methods of calculation, and for the development of elementary particle physics in our century. Therefore when Galois laments that there is no mention in the textbooks and publications of his time of the major theoretical ideas, and that the heart of the theory is suffocating under a chaotic heap of isolated results, this criticism must be seen in historical perspective. A common base structure or unity in multiplicity (Leibniz) can only be brought into relief if the greatest possible multiplicity is known in advance. There is a risk, of course, that we will not see the forest for the trees, but there is also a risk that we will not know what a forest is if we have never seen a tree.

2.22 Sample Applications The following section shows how the Galois theory solves in one fell swoop ("d'un seul coup") a series of classic problems from arithmetic and geometry which date back to antiquity. First, we will show that the incommensurability of diagonal and side of a square, i.e. the irrationality of y/l is defined by the Galois group of the equation x2 - 2 = 0 73 . x 2 - 2 is a polynomial over Q with the roots c*i=\/2 and a2=-OL\=-V2. The smallest field which comprises all the solution roots is therefore Q ( ± λ/2). Over this extended field of Q the polynomial can also be represented as the product of linear factors, namely x2 — 2 = (χ — λ/2)(χ + λ/2). We can therefore speak of the factorizing field of the polynomial. We claim: The Galois group of x2 - 2 comprises the permutations 71

72 73

See the analysis by H. Wussing, Die Genesis des abstrakten Gruppenbegriffs, Berlin 1969, 74 f. E. Galois, Manuscrits de Evariste Galois (publiés par J. Tannery), Paris 1908, 25. See H. Weyl, Symmetrie (see Note 5), 137 f.

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1 = ί α ι α Λ Vai α 2 J

and

σ = ( aα ι aα 2 Υ V 2 i J

Therefore, it must be shown that each algebraic relation of the roots, i.e. each polynomial R(xi, X2) in two unknowns xj and X2 with rational coefficients and R(oii, 0C2) = 0 is invariant under both permutations. In the case of identity, that is trivial. It remains to be shown: If R (αϊ, (X2) = 0, then R(a 2 , ai) = 0 also. On account of the definition of R, R(ai,-a0 = R(ai,02) = 0, i.e. the polynomial R (x, -x) of an unknown χ vanishes for χ = αϊ. If we divide this polynomial by x2 - 2, we get the expression R(x,-x) = (x2 - 2) · S(x) + (ax + b), with a linear remainder ax+b and rational coefficients a and b. Since αϊ is a solution of x2 - 2, we get 0=R(ai,-ai)=0-S(ai)+(aai+b), i.e. 0 = aai + b. If a and b are not 0, then αϊ can be represented as a rational number αϊ = - b/a. But that contradicts the irrationality of y/2. Therefore, a and b are zero, i.e. R (x, -x) = (x2 - 2)-S(x). Since x2 - 2 vanishes for oti and 0C2, it is also true for R(x, -x), i.e. R(a2,oci) = R(a2,-oi2) = 0. The result, which comprises the Galois group of x2 - 2 the permutations 1 and σ, is therefore equivalent to the irrationality of \ f l . We shall now show how the Galois theory solves construction problems involving a compass and straight edge "d'un seul coup". First, it should be noted that construction with a compass and straight edge consists of a finite sequence of construction steps in a plane:74 1) Selection of any point on the plane, 2) Construction of the straight lines connecting any two constructed points, 3) Construction of a circle from previously constructed center and peripheral points, 4) Construction of the intersection of two constructed straight lines or a previously constructed straight line with a previously constructed circle, or two previously constructed circles. A solution using a compass and a straight edge then consists of constructing on a specified object with specified conditions ai, ..., ar a desired object with conditions xi,..., xr in a finite number of steps. If we introduce a Cartesian system of coordinates into the plane, then the specified requirements ai,..., ar can be interpreted as numerical coordinates of the positive x-axis. Since segments with rational coordinates can be specified as constructable, in accordance with the geometrical requirement there is algebraically a number field Fo = Q (ai, ..., ar), the elements of which can be expressed as quotients of polynomials in ai,..., ar with rational coefficients of Q. Constructed lines and circles by means of the specified set of points correspond to linear and quadratic polynomials with coefficients in Fo. Algebraically, a step-wise field extension FoC...cFiCFi+iC...CF n with xi,...,x t eF n then 74

See K. Mainzer, see Note 36, 40f.

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181

corresponds to a geometric construction of the desired segments xi, ... xt from ai,..., ar in η steps. The construction is considered feasible if the segments Xi,..., xt can be plotted on the positive x-axis. To extend the field, F¡ = Fj + i for the construction steps 2, 3) and also 1), since in the selection of points after 1), rational coordinates can be introduced. For construction step 4) over F¡, square roots of elements from F; can occur during the calculation of coordinates of intersections with circles, so that for a degree (F¡+i/F¡) of the field expansion FjCFj+i, i.e. for the dimension of the vector space F¡+i over field F¡: (FÍ+I/F¡) = 2. Therefore the degree (Fn/Fo) of field extension F 0 CF n is always a power of 2. In general, according to the Abel and Galois theory, a geometric problem with specified components ai,..., ar and desired components xi,..., xt can be solved using a compass and straight edge if and only if the desired quantities are roots of polynomials with coefficients of Fo = Q (aj ... ar) and the (smallest normal) field extension E of Fo with xi, ..., xt € E as degree has a power of 2.75 As our first example, let us consider the construction of a regular nsidedpolygon using a compass and straight edge, a problem which has kept mathematicians busy since Euclid (See Chapter 1.21). The solution of this construction problem was the first mathematical "coup" of the 19-year-old Gauss.76 This is a special case of the Galois theory. The problem is to construct an η-sided polygon which can be inscribed in a circle of radius 1. Algebraically, it is the polynomial χ11 - 1 over the base field Fo = Q with η roots which form the corners of the regular η-sided polygon. The coordinates of 2π . 2π the corner points are Xi = cos — and X2 = sin —. According to Gauss, the complex number ε = xi + i · X2 can be represented in the plane by the Cartesian coordinates xi and X2- In this sense, it is a question of the construction of the nth root of unity ε using compass and straight edge. The factorizing field E of xn - 1 is the smallest (normal) field expansion of Fo which contains exactly the η roots of xn - 1 for the corresponding corners of the η-sided regular polygon. To answer the question whether or which regular polygons can be constructed using compass and straight edge, we must investigate the degree (E/Q) of the field expansion of Q by E, as indicated in our general remarks. The decomposition of η into various prime number powers is n=piVl -p2V2 •...p r Vr for the prime numbers pi = 2, p2 = 3, ... .It can then be shown that the degree of extension (E/F) is given by the Eulerian function φ(η) = 75 76

See E. Artin, see Note 65, 84 f. Discovered on March 30, 1796. A generalization is described in C. E Gauss, Disquisitiones Arithmeticae (1801), § 365.

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Pi Vl_1 (Pi - 1)P2 V2_1 (P2- 1) ...pr V r _ 1 (Pr- 1)· If Pi = 2, then the exponent Vi is arbitrary. But if p¡ is odd, then ν ι must be equal to 1, and p¡ - 1 a power of 2, i.e. 2 m . Then p¡ = 2 m + 1. If m = ab, a > 1 and a odd, then the polynomial x ab + 1 = (x b ) a + 1 is divisible by x b + 1, the number 2 m + 1 is divisible by 2 b + 1, and 2 m + 1 is not a prime number. Then m must be a power of 2. For Pi, therefore, only numbers of the form 22* + 1 can occur. Therefore precisely those regular η-sided polygons can be constructed with compass and straight edge for which η = 2V pi p2... p r , whereby p¡ are various prime numbers of the form 2 2 + 1. Precisely in this case, namely, φ(η) is a power of 2. For k = 0 , 1 , 2 , 3 and 4 we get the prime numbers 3,5,17,257 and 65537. For k = 5, the number is by the way divisible by 641. The actual constructions are uninteresting from an algebraic point of view, since although they relate to the geometric symmetry of regular η-sided polygons, they do not relate to the algebraic symmetry of the equation solutions of xn - 1. It is worth noting that the geometric symmetry of the η-sided polygon, namely the respective dihedral group D n , predicts nothing regarding the feasibility of construction, since there is a D n for every regular η-sided polygon. But the feasibility of construction can be determined from the algebraic symmetry resp. the equation solutions, i.e. the solution of the Galois group of the degree of field extension of the splitting field of xn - 1. In this sense, the algebraic symmetry is more fundamental than the geometric symmetry. The famous age-old problem of trisecting an angle is one immediate consequence. Even the mathematicians of antiquity knew that any arbitrary angle can be trisected by means of a suitable extension of the tools which could be used for the construction.77 But the Galois theory was the first to show that theoretically, not every angle can be trisected using a compass and straight edge. For example, an angle of 60° can be constructed using a compass and straight edge, since a regular hexagon can be constructed using a compass and straight edge (See 1.21). But the trisecting of this angle would make possible the construction of a regular 3 - 6 = 1 8 sided polygon, which contradicts the criteria for the construction of regular hexagons because 18 = 21 · 3 2 . The famous Delian problem can also be explained. According to legend, this construction problem was posed by Apollo, who required that an existing, cube-shaped altar be doubled in volume while retaining its cubic shape. Archytas of Tarentum, among others, indicated a geometric solution to the problem, with a suitable expansion of the tools used for construction.78 But 77

78

See also K. Mainzer, Axiomatischer Konstruktivismus und Ontologie: Zum philosophischen Selbstverständnis der griechischen Mathematik, in: H. Stachowiak (ed.), Pragmatik, Vol. 1, Hamburg 1965, 130 ff. See K. Mainzer, see Note 36, 37 ff.

2.3 Symmetries and the Invariance of Geometric Theories

183

the Galois theory was the first to show that such a construction is theoretically impossible using only a compass and straight edge. If the existing altar has sides with a length of 1, then for the equation x 3 - 2 = 0, the root χ must be constructed. Therefore Fo = Q and the splitting field of the polynomial is E = Q(\/2). Since the polynomial x 3 - 2 in Q cannot be broken down into linear factors, i.e. it is irreducible, the degree of field extension (E/Fo) = 3. But then a construction using a compass and straight edge is impossible. The deficiencies of the old arithmetic and geometric theory are apparent from the centuries of unsuccessful attempts to solve certain algebraic equations or geometric construction problems. It was impossible to explain this unsatisfactory situation with the old geometric concept of symmetry, and there even seemed to be "gaps" in this theoretical concept. Only Galois's group theory approach revealed a more fundamental and more general symmetry structure, in which these apparent "gaps" turned out to be necessary consequences. We encounter this dialectic of scientific conceptions in many historical situations in which observations of symmetry led to breakthroughs. In the words of Heraclitus: "... έκ των διαφερόντων κάλλιστη αρμονία" - "from the most diverse arises the most beautiful harmony."

2.3 Symmetries and the Invariance of Geometric Theories Symmetry was explained for the first time by the Galois Theory as grouptheory invariance of algebraic laws. This line of research has been extended by F. Klein, among others, to include geometry. While Galois was concerned with the symmetry of a finite number of discrete elements, namely the roots of algebraic equations, Klein attempted to derive the symmetry of an infinite number of elements of continuous geometric manifolds. Because, since the 18th Century, various other theories such as non-Euclidian, affine and projective geometry have arisen in addition to Euclidean geometry, it seems necessary to investigate the apparently diverging and specializing directions of research in geometry from a common point of view. Thus, for the first time in mathematics, group theory was the key to the unification of different theories, a trend which continues in modern physics as the program of the unification of natural forces. The Norwegian mathematician S. Lie was able to transfer the Galois theory from algebraic equations to differential equations. Thus, in addition to discrete groups, there are continuous ("Lie") groups, which have proven to be a central mathematical tool of physics.

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2.31 Klein's "Erlanger Program" The old definition of "geometry" in the sense of "measuring the earth" became inadequate to cover specializations in research as early as the 19th Century. Only with F. Klein's "Erlanger Program" of 1872, with the concept of "geometric invariants" which remain unchanged with metric, affine, projective or topological transformation groups, among others, did it become possible to organize the various directions of research into a hierarchy of theories. Coordinate transformations of course played a large role in analytical geometry as long ago as the 17th and 18th Centuries. In two-dimensional analytical geometry, for example, geometric expressions concerning points of a plane are translated into analytical expressions of coordinate values, so that functions χ and y for each point Ρ correspond to the real values χ = x(P) and y = y(P). The transformations x' = ax + by + e and y' = cx + dy + f can be summarized as

whereby the matrix

must be orthogonal with

The Cartesian geometry of the plane then consists of those expressions and characteristics which are invariant under these transformations. Since the successive application of Cartesian transformations in turn leads to more of the same, they form a group which clearly characterize the invariant properties of Cartesian geometry. As examples of Cartesian transformations79, let us investigate rotation, reflection, and inversion, which can be represented by corresponding matrices. First of all, let us consider the rotation of a coordinate system (x, y, z) around the z-axis (Figure 1). Let the point in the (x, y) plane of the coordinate system be given with a distance r from the origin O, whereby φ is the angle of the vector r with the x-axis. The polar coordinates of Ρ are then χ (Ρ) = r cos φ and y (Ρ) = r sin φ. Therefore, if the axes are rotated around O by an angle θ, Ρ gets the new coordinates x' = r cos (φ - Θ) = r cos φ cos θ + r sin φ sin Θ, and y ' = r sin (φ - Θ)

79

With regard to the history of the concept of Cartesian coordinates, see also K. Mainzer, see Notes 36, 111 ff.

2.3 Symmetries and the Invariance of Geometric Theories

185

= r sin φ cos θ - r cos φ sin θ. Therefore, χ' = χ cos θ + y sin θ and y' = -χ sin θ + y cos θ with the transformation f x ' ) _ [ y ' J ~~

/ cos θ sin θ \ y — sin θ cos θ J

f x ) [ y J

or in three dimensions x'\ / cos θ sinG 0 \ y' = - s i n θ cosÖ 0 z'J

\

0

0

1/

We can easily demonstrate that the matrix is orthogonal, its trace 1 + 2 cos θ and its determinant +1. In an analogous manner we can find the rotations around the y and ζ axes, or around any axis through the origin.

Fig. 2

For reflection, a point Ρ in the (x, y)-plane of the coordinate system is reflected on P' (Figure 2). For the coordinates of P', we then get x' = r cos (2Θ - φ) = r cos φ cos 2Θ + r sin φ sin Θ, and y' = r sin (2Θ - φ) = r cos φ sin 2Θ - r sin φ cos Θ, i.e. the transformation ( * ' ) _ ( cos29 sin26\/x\ \y' ) - y sin20 - c o s 2 6 ) ^y^l or in three dimensions

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2. Symmetries in Modern Mathematics

/x'\ /cos20 sin20 0 \ / x \ Í y' J = f sin20 — cos26 0 J f y J . We can easily demonstrate that the matrix is orthogonal, its trace 1 and its determinant -1. For an inversion, the axes of Ρ are inverted. This transformation can therefore be represented by the inversion matrix

which is orthogonal, the trace -3 and the determinant -1. According to F. Klein, the investigation of a geometric theory generally consists of the following algebraic problem: "There is a manifold, and in it there is a transformation group; we must investigate the elements belonging to the manifold with regard to those characteristics which are not changed by the transforations of the group. 80 In short: "There is a manifold, and in it there is a transformation group. Develop the invariant theory relating to the group." In his "Erlanger Program", Klein describes what he calls the "principal group": "Geometry can .. in any way be concerned only with those characteristics of threedimensional figures which are independent of the position in space assumed by the figures, and of the absolute size of the figures. Nor (without the assistance of a third body) can it make a distinction between the characteristics of a body and those of its mirror image. These theorems characterize a group of three-dimensional transformations - which may be designated the principal group - whose transformations leave all of the geometric characteristics of a figure untouched. This group consists of the 6-fold infinite number of motions, of the infinite number of similarity transformations and of the transformation by reflection on a plane."81

Thus similarity transformations are the key concept for Euclidean geometry, because there, figures can be enlarged or reduced arbitrarily without changing shape. Consider a triangle, for example, which can be arbitrarily enlarged or reduced without changing the angles. Only in Euclidean geometry is the group of motion a genuine subgroup of the similarity group. As represented by Euclid's "Elements", Euclidean geometry acted for almost 2000 years as a monolithic block, the manifesto of an author supplemented by numerous commentaries over the centuries. The development of Euclidean geometry is altogether comparable to that of the Bible, 80

81

F. Klein, Das Erlanger Programm (Vergleichende Betrachtungen über geometrische Forschungen), in: Ges. math. Abh. Bd 1., Berlin 1921,463. F. Klein, see Note 80, 318

2.3 Symmetries and the Invariance of Geometric Theories

187

which was also surrounded by a crowd of commentators. Following centuries of discussion concerning the independence of Euclid's parallel postulate, it was found only in the 19th Century that the "Elements" contained two independent geometric theories which are independent of one another in terms of their axiomatic principles and basic concepts. These are absolute and affine geometry. This development is also altogether comparable to the commentaries on the "Book of Books". Here again, we have learned to distinguish between the different traditions, stylistic elements, apostles, etc. and to make our evaluations accordingly. In short, absolute geometry arises from Euclidean geometry, without the acceptance of Euclid's parallel postulate, according to which, through a given point, there is only a single line parallel to a given straight line. We move from absolute geometry to Euclidean or non-Euclidean geometry by adding either Euclid's parallel postulate or one of the non-Euclidean versions. On the other hand,the parallel line plays a central role in affine geometry. For Euclid, the affine theorems are those which remain unchanged after parallel projection from one plane into another. Affine geometry, which was first noted by L. Euler 82 , is of major importance for physics, since its theorems are valid not only in Euclidean geometry, but also in Minkowski space-time geometry, which is the basis of Einstein's Special Theory of Relativity. Analytically, the affine geometry of the plane can be characterized by the transformations y ' ) = ( c d ) ( y ) + ( f ) with an invertible matrix Klein only later added the affine group to his Erlanger Program as a supergroup of Euclidean geometry. 83 In contrast to affine geometry, there is no parallelism in projective geometry. Nor do lengths or angular measurements play any role. Although the geometricians of antiquity (in particular Euclid) were interested primarily in lengths, Pappus (4th Century BC) proved theorems which are part of projective geometry. The historical origin of projective geometry is the problem of perspective. Projective geometry was given added impetus by the architect G. Desargues (1591-1661), the philosopher B. Pascal (1623-1662), and the artists A. Dürer and L. da Vinci. Kepler was the first to speak of "infinitely distant points" which are assigned in projective geometry to the 82

83

L. Euler, Introducilo in analysis infinitorum, Lausanne 1748, Volume 2, Chapter XVIII, Article 442, later adopted by A. F. Möbius, Ges. Werk Bd. 1 (R. Baltzer edition), Leipzig 1885, Preface, XII. Klein later evaluated this oversight (1921) as the result of one-sided tradition (See Note 81, 320). See also H. Wussig, see Note 71, 143.

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infinitely distant intersections of parallel lines.84 But an independent theory was launched only in the early 19th Century by the French School around J. V. Poncelet, and was continued in Germany by J. Steiner, V. G. C. von Staudt and others.85 O

Let us begin with the basic definitions of central and parallel projection. Let ε and ε' be any two planes in space and O any point which does not lie on ε or ε'. A central projection from ε to ε' with O as the projection center is executed by defining as the image of each point Ρ of ε the point P' on ε', which lies with Ρ on the same straight line through O. We speak of a parallel projection when all projecting lines run parallel. The projection of a straight line g in ε to the straight line g' in ε' starting at a point O is defined accordingly (Figures 3 and 4). Each transformation from one figure to another by central and parallel projection or a finite series of projections is called a projective transformation. The projective geometry of the plane or of the straight line consists of the totality of those geometric properties which remain valid and unchanged over any number of projective transformations of the figures to which they relate. In contrast, metric geometry consists of the system of those geometric properties which relate to the sizes of figures, and remain invariant only under rigid motions. The following are simple projective characteristics: (1) A point (a line) is transformed under projection into a point (a line).

84 85

J. Kepler, Collected Works, Vol. 2, Munich 1937, 92. See Bibliography and description in K. Mainzer, see Note 36, Chapter 5.1.

2.3 Symmetries and the Invariance of Geometric Theories

189

(2) The incidence of a point and a straight line is invariant under projective transformations, i.e. A lies on g, and therefore after the projection A' also lies on g'. (3) The colinearity is invariant under projective transformations, i.e. if three or more points lie on a line, then their images also lie on their lines. (4) The congruence of lines is invariant under projective transformations, i.e. if three or more lines intersect in a plane, then their images also intersect, etc. On the other hand, in general the measurements of lengths and angles and the relationships between such values are changed by projections. For example, isosceles and equilateral triangles are transformed by projections into triangles with sides of different lengths and proportions between the sides. Therefore the triangular shape is preserved, and thus the term "triangle" is projective, while the terms "equilateral" and "isosceles" triangles are not projective, and belong to metric geometry.

O

Fig. 6

It can be easily demonstrated as follows that the proportion between two line segments AB and BC is generally also not a projective characteristic: Three points A, B, and C on a line g can be assigned projectively to any other three points A', B' and C' on another line g' (i.e. in particular those

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with AB:BC = A'B':B'C'). To do that, we rotate g' around C' until this line is parallel to g (Figure 5). On this line g" parallel to g, A" and B" are plotted by parallel projections. Since the extensions of B ' B " and A'A" intersect in O (otherwise g' coincides with g" and there is a parallel projection) g' and g" are projected centrally. If, on the other hand, we project four points A', B', C' and D' in g' on A, B, C, D in g (Figure 6), their cross ratio remains unchanged: CA

DA _ C A

ETA

7

CB ' DB ~ C B ' D'B' ' The proof can be presented using elementary resources of geometry. The cross ratio is therefore a ratio of ratios of line segments which can also be transferred to lines and planes. Poncelet's decisive discovery is that the invariance of the cross ratio for projective geometry can be used in place of the invariance of the length of the line segment in metric geometry. The introduction of infinitely distant points as imaginary intersections of parallel lines makes the distinction between central and parallel projection superfluous. Parallel projection is now a special case of central projection. Important characteristics of projective geometry are the duality principle, the projective definition of metrics and of configurations, which will not be discussed here in any further detail. 86 From the point of view of invariance, the introduction of coordinates is meaningful. 87 In Cartesian geometry, the points of a plane are defined by two perpendicular coordinates. But such an analytical definition of points by pairs of real numbers only works for conventional points and fails for the infinitely distant points of projective geometry. The plane ε is therefore initially selected parallel to the (x, y)-plane of a cartesian system of coordinates with the coordinates (x, y, z) at a distance ζ = 1. All points Ρ of ε then have the coordinates (x, y, 1). If we now select O as the projection center, then each point Ρ of ε defines a single line through O and vice versa. The lines through O parallel to ε define the infinitely distant points (Figure 7). The conventional Cartesian coordinates of a line g though O and Ρ with (xo, yo, 1) are designated the homogeneous coordinates of P, since they define Ρ except for factors λ φ 0 with coordinate triples (λχο, λyo, λ). This three-dimensional definition of points in a plane is not unique, but only with respect to an arbitrary factor λ. However, the defining lines can be distinguished, since the point (0, 0, 0) common to all the projecting lines is excepted. Moreover, the infinitely distant points of ε can now be defined by the lines parallel to ε with coordinates (x, y, 0). Therefore a point in a plane 86 87

See K. Mainzer, see Note 36, Chapter 5.12, 5.13, 5.14. See bibliography and description in K. Mainzer, see Note 36, Chapter 5.15.

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2.3 Symmetries and the Invariance of Geometric Theories

ζ (λχ0, λγ0, λ)

χ

y Fig. 7

does not change if its homogeneous coordinates (x, y, z) are multiplied by an arbitrary factor λ Φ 0. For ζ φ 0, they define a conventionals point, and for ζ = 0 an infinitely distant point. Following the analytical definition of the points of a projective plane, we must now analytically define the conventional and infinitely distant lines in projective planes. For that purpose, we first consider geometrically that all lines which connect O with the points of a line in ε lie in a plane through O. This plane through O in cartesian geometry has the analytical representation ax + by + cz = 0, whereby the constants a, b and c are not all 0. A line in ε therefore consists of all points (x, y, z) which satisfy such an equation. Since the infinitely distant points all satisfy the equation ζ = 0, it is defined as the analytical representation of the lines at an infinite distance from ε. The constants (a, b, c) are considered as homogeneous coordinates of a line in the plane ε, since the equation (^a)x + (Àb)y + (λο)ζ = 0 with λ φ 0 is satisfied by the same triples (x, y, z) as ax = by = cz = 0. In 1829, J. Plücker (1801-1868) provided an analytical explanation of the duality principle: When homogeneous coordinates are used, the symmetry of points and lines occurs in the equations ax + by + cz = 0, since the triples (a, b, c) can be interpreted as line coordinates, and (x, y, z) as point coordinates, and inversely, (a, b, c) as point coordinates and (x, y, z) as line coordinates. It is known that in projective geometry, the duality principle is valid only if we also accept the concept of infinitely distant points and lines. In the conventional, non-homogeneous equation of the straight line of Cartesian geometry, there is no symmetry of points and lines. In ax + by + c = 0, the pair (x, y) and the triple (a, b, c) are not interchangeable. According to the analytical description of point, line and duality principle, the projective transformation of a plane ε to another ε' can now be explained analytically. For that purpose, we select the system of linear equations x' = aix + biy + cjz, y' = a2X + b2y + C2Z and z' = a3X + b3y + C3Z with homogeneous coordinates ( x \ y', z') of the points from ε' and (x, y, z) from

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2. Symmetries in Modern Mathematics

ε. The projective geometry of the plane then becomes an algebraic theory of linear homogeneous equations. The group of projective transformations turns out to be exceedingly rich, and comprises more than just the affine and euclidean group. NonEuclidean (hyperbolic) geometry can also be defined with projective metrics, as Klein has shown by the following model of this geometry.88 If, for example, we interpret axioms as formulas, then basic concepts such as "point", "line", "plane", and relations such as "incidence" and "congruence" are nothing more than variables which can be interpreted by certain semantic images. A formal non-Euclidean theory then consists of the conventional axioms for incidence, congruence, the axioms of order, continuity and the negation of the parallel axiom, which requires either an infinite number or no parallels through a point Ρ to a line (on which Ρ does not lie). The basic concepts and relations of such a non-Euclidean theory are designated "non-Euclidean point", "non-Euclidean line", etc. According to Klein, they must be replaced by Euclidean terms so that the formal axioms of the non-Euclidean theory are transformed into genuine Euclidean theorems.

For Klein's Model of hyperbolic geometry, we interpret non-Euclidean points by points inside a given circle, non-Euclidean lines by the chords of this circle (not counting the corner points). The points of the circle and the external points are therefore not included in the points of the system. The non-Euclidean geometry is interpreted by the inside of the circle. It can be easily verified that the Euclidean axioms of incidence and order are satisfied by this system. One exception is the axiom of parallelism, since there are obviously an infinite number of non-Euclidean lines through Ρ which do not intersect the non-Euclidean line UV (Figure 8), i.e. are parallel in the nonEuclidean sense. 88

See K. Mainzer, see Note 36, 170 ff.

2.3 Symmetries and the Invariance of Geometric Theories

193

The Euclidean axioms of congruence specify that linear quantities are invariant with respect to congruent displacements in the plane. From projective geometry, we know that a circular plane can be transformed into itself by an infinite number of projections, whereby the points on the periphery of the circle and inside the circle are changed only in terms of their position, but remain respectively on the periphery of the circle or inside the circle. For a suitable interpretation of non-Euclidean congruence, such a segment must be defined in the model so that it remains invariant under projection. For two points A, B, the cross ratio (UVBA) on the chord UV remains invariant. The obviously possible definition of the segment AB by the cross ratio UVBA is not correct, since the addition of segments AB + BC = AC usually does not equal (UVCA) for the cross ratio (UVBA) + (UVCB). The non-Euclidean segment AB is defined by the logarithm of the cross ratio log (UVBA), which for Α φ Β is a cross ratio greater than 1, and therefore gives a positive integer: d(AB) = log (UVBA). The segment addition can be traced back to the logarithm laws. If we allow A to wander (Figure 8) toward U, then d(AB) increases beyond all limits Hm^d(AB) = oo. Therefore the non-Euclidean lines are of infinite non-Euclidean length, although measured in a Euclidean fashion they are the finite segment of a Euclidean chord. Instead of hyperbolic geometry, we also speak of two-dimensional Lobatschevski space L 2 , for which there is now a projective model in the two-dimensional Euclidean space E 2 . According to H. Poincaré, a conformai model of L 2 can be indicated in 2 E . A straight line ("geodesic line") is then represented by a curve in the circular disk, both ends of which are perpendicular to the edge of the circle. The distance between points A and Β is measured in turn by the logarithm formula of the cross ratio. The advantage of the Poincaré Model over the Klein Model is that angles and circles are correctly reproduced in the conformai model. Therefore shapes are reproduced with practically no distortion. It should also be noted that figures become very small and tightly compressed as they approach the edge of the circle in an infinite process. The shape of a figure is therefore less distorted, the smaller it is reproduced in the model. But here it becomes clear that the Euclidean similarity postulate, according to which figures can be enlarged or reduced by any amount without changing shape, is theoretically no longer valid for non-Euclidean hyperbolic geometry. The above-mentioned Dutch artist M. C. Escher illustrated the conformai model of hyperbolic geometry in his picture "Circle Limit III" (1959) (Figure 9). He explained his picture as follows: "Now we have only rows with "through traffic"; all the fish which belong to a row have the same color and swim after one another, head to tail, along a circular path from edge to edge.

2. Symmetries in Modern Mathematics

The closer they get to the center, the larger they become. Four colors were necessary to make each row stand out clearly against its surroundings. Like all these rows of fishes, which rise at an infinite distance like rockets perpendicular to the edge of the circle and then fall back again, not a single component ever reaches the borderline. Because on the other side is the "absolute void". And yet this round world cannot exist without the surrounding void -not merely because an inside requires an outside, but also because the strictly geometrical, intangible centers of the arcs from which the system is constructed lie in the "void"." 89

A direct relationship can be established between the conformai and projective model. In Figure 10, the conformai point P^ is uniquely mapped on the corresponding projective point P p by the transformation:

OP =(

' TTWOP'

89

B. Ernst, Der Zauberspiegel des M. C. Escher, Munich 1978, 109.

2.3 Symmetries and the Invariance of Geometrie Theories

195

Fig. 10

The differences between the individual models therefore do not relate to the characteristics of "intrinsic" hyperbolic geometry.90 A distinction must be made between hyperbolic geometry and spherical geometry, which is an additional non-Euclidean geometry. In the twodimensional case, let us consider a two-dimensional spherical surface in the three-dimensional Euclidean space E 3 . Non-Euclidean lines are now described by great circles, and non-Euclidean points by diametrically opposite pairs of points (as the Euclidean intersections of the great circles). Thus there are also no parallels to a specified "line" (= great circle), through a "point" (= diametric pair of points) lying outside, since all the great circles intersect on a sphere. From two-dimensional spherical geometry S 2 we get the elliptical geometry P 2 , by identifying in pairs the diametrically opposite points of a great circle. In visual terms, a sphere is cut in half, and when the surfaces of the great circles are folded around the diameter, care is taken that opposite points overlap. The result in visual terms a sphere with an incision, the surface of which represents P 2 . To visually illustrate "life" in the two-dimensional spherical world, M.C. Escher produced several models of a sphere, the great circles of which are populated by various creatures, e.g. the "Sphere with Angel and Devil" of 1942 (Figure II). 91 Analogous to hyperbolic geometry, conformai and projective models of spherical and elliptical geometry can be introduced. The stereographic pro90

91

See also R. Penrose, The Geometry of the Universe, in: L. A. Steen (ed.), Mathematics Today. Twelve Informal Essays, New York/Heidelberg/Berlin 1978, 83-125. J. L. Locher (ed.), The World of M. C. Escher, New York 191, 86.

196

2. Symmetries in Modern Mathematics

Fig. 11

jection of S 2 from the north pole Ν of the sphere onto the Euclidean plane E 2 (Figure 12) gives a conformai model, since the angles and circular shapes remain intact, and figures are reproduced in similar small figures. If the projecting line becomes the spherical tangent in N, then Ν must be assigned an "infinite point" which does not belong to E 2 . It is historically noteworthy that Ptolemy recognized the conformity of stereographic reproduction in his "Planisphaerium" as long ago as 150 A.D. 92 If the sphere touches the surface E 2 at one point, then the projecting line through the center of the sphere provides a projective model of P 2 in E 2 . Specifically, the projecting line now strikes two diametrically opposite points on the surface of the sphere, which are mapped on one point in E 2 . In the case of a sphere and an ellipse, conformai models can also be converted to projective models. The non-Euclidean theories of L 2 , S 2 and P 2 can naturally also be introduced independently of Euclidian models. Since the parallelism axiom is 92

See also K. Mainzer, see note 36, 71 f., 173.

2.3 Symmetries and the Invariance of Geometric Theories

197

equivalent to the theorem of the sum of the angles in a triangle, different surface areas can be derived for triangles in the Euclidean and non-Euclidean geometries. If we designate the angles of a triangle α, β and γ, which must be measured in radians (i.e. 180° = π), and if the sides of the triangle are selected as the straight segments (geodesic lines) of the respective geometry, then for the difference Δ > 0 of the sum of the angles α + β + γ of π, Δ = π - (α + β + γ) for L 2 and Δ = (α + β + γ) - π for S 2 or Ρ 2 . While in Euclidean geometry, any segment can be made arbitrarily larger or smaller, in the non-Euclidean case there are absolute quantities. If we start with a unit sphere, then the total surface area of S 2 equals 4π, and for P 2 equals 2π. For L 2 , π is the upper limit for the surface area of all possible triangles. In our discussion of differential geometry we will come back to additional "intrinsic" characteristics of non-Euclidean geometry (See. 2.33). At this point, we should take special note of the projective characterization of nonEuclidean geometry. A. Cayley summarized the importance of projective geometry emphatically in the motto: "Projective geometry is all geometry". 93 But he did not thereby take into consideration the most general of all geometries.94 That is topology, which is characterized by the group of continuous transformations. As an example of characteristics which are left invariant under transformations, let us consider polyhedrons. 93

94

A. Cayley, Collected Mathem. Papers 2, Cambridge 1889, 592. Cayley calls projective descriptive, as was common at the time. For information on the history of this discipline, see G. Feigl, Geschichtliche Entwicklung der Topologie, in: Jahresber. Deutsch. Mathem.-Vereinigung 37,1928,283: and K. Mainzer, see Note 36, Chapter 5.5.

198

2. Symmetries in Modern Mathematics

A polyhedron is called a simple polyhedron when its surface can be continuously deformed into a spherical surface, i.e. simple polyhedrons do not have "holes", like a torus, for example. The Euler formula for the simple polyhedron is thus : E - Κ + F + 2 for the number of corners E, the number of edges Κ and number of surfaces F. We can easily verify this formula for the Platonic bodies, for example, but it covers a great deal more than just the polyhedrons of metric geometry with straight edges and plane surfaces. It even remains valid if we imagine the surface of a regular polyhedron made of rubber, which can be deformed arbitrarily, as long as it is not torn. That is because only the number of corners (points), edges (lines) and surfaces is important for this formula. Length, surface area, linearity, cross ratio and other concepts of metric, affine or projective geometry are not left invariant under topological transformations.

2.32 Continuous Lie Groups The analytical formulation of geometry since Descartes has followed analytical mechanics since D'Alembert and Lagrange, among others. (See Section 3.31). Problems of motion in physics were translated into equations of motion, which in general have the form of differential equations. Under some side conditions, therefore, the solution of motion problems in physics meant the solution of differential equations. A transfer of the Galois Program from algebraic equations to differential equations was therefore also of interest in terms of physics. Then it was possible, to a certain extent, to determine the characteristics of the solutions to these equations, i.e. including the solutions of corresponding problems of motion, by means of symmetry observations. On the subject of this program, S. Lie wrote in 1874: "In the theory of algebraic equations, the question before Galois was only: Can the equation be solved by radical signs, and how? But since Galois, the question has also been: What is the simplest way to solve the equation by radical signs? ... The time has come, I think, to take a similar step forward in differential equations."95

To explain the relationship between continuous groups and the concept of symmetry let us first consider a simple example. A rotation of a plane coordinate system around its origin in a counterclockwise direction by an angle θ can be considered a symmetry of the plane, because it leaves the relationships between distance and angle invariant. Simple trigonometric calculations show (see also the examples in Chapter 2.31) that the point Ρ with coordinates (x, y) is transformed into point P' with coordinates (x cos θ - y sin θ, χ sin θ + y cos θ) (Figure 1). 95

S. Lie, Ges. Abhandlungen, Vol. 5, Leipzig/Kristiana, 1924, 586.

2.3 Symmetries and the Invariance of Geometric Theories

199

(x cos θ — y sin θ, χ sin θ + y cos θ)

\ \

\

Fig. 1

These rotations form a group. If, for example the rotation by the angle θι is followed by an additional rotation by the angle Θ2, then the result is a rotation by the angle θ] + Θ2. It can easily be verified that this rule of composition satisfies the group axioms. For example, the rotation by the angle 0 can be used as the unit element 1. If σι is the rotation by the angle Θ, and σ 2 is the reverse rotation by the angle 2π-θ, then σι σ2 = 1 = σ2σι. The group of rotations is continuous, since it is a function of a continuous parameter Θ. In particular, the circle with a radius r = a can be understood as the location of a point rotating continuously around O from (a, 0), whereby (r,0) is transformed into (r, θ + t) with a continuously changing t. The discrete groups of regular polygons are therefore imbedded in the continuous group of the circle. It describes the perfect symmetry of the circle which so fascinated ancient and medieval philosophers and scientists. The line 6 = 0 can also be understood as the location of the transforms of (a, 0) with a continuous stretching, which supplies all (r, 0) for all positive r. The continuous stretching is a genuine similarity transformation (See 2.11). As a result of a suitable composition of continuous rotation and stretching, we get a continuous rotation and stretching, by means of which the logarithmic spiral can be generated (Figure 2). Historically, it was discovered by Descartes and discussed in a letter to Mersenne in 1638. 96 But on account of its transcendent character, Descartes did not count it as part of "algebraic" geometry, but instead considered it a part of approximate mathematics, whose figures can of course be approximated by mechanisms of motion, but 96

See also J. Vuillemin, Mathématiques et métaphysique cartésiennes, Paris, 1960, 35 ff., 51 ff.

200

2. Symmetries in Modern Mathematics

cannot be defined by algebraic formulas. Jacob Bernoulli was so fascinated by its symmetry that he had the inscription "Eadem mutata resurgo" chiselled on his tombstone in the Basel Cathedral. In fact, this motto expresses the symmetry of the logarithmic spiral, since by means of continuous rotation and stretching, it can be transformed into itself.97 Let μ be the ratio of magnification for the rotation by 1 radian, i.e. the angle which in the unit circle corresponds to the arc of length 1. Then μ 2 is the ratio of magnification for 2 radians, ..., μ π for π radians, ..., μ 1 for t radians. If t changes continuously, then the point (r, Θ) is transformed into (μιΓ, θ + t). We therefore get the parameter representation r = μ ^ , θ = t, or the equation r = 8μ θ . Therefore, dr/d0 = r log μ for the logarithmic spiral. This curve shares with the straight line and the circle the symmetry characteristic that it is transformed into itself by a continuous group of similarity transformations. The line and circle are the extreme cases of the logarithmic spiral, and are formed when, in the composition rotation + dilatation, one of the two components coincides with identity. As will be discussed below, the spiral, as a form of symmetry in nature, plays an important role, in particular in the morphology of plants and animals (See Chapter 4.4). Goethe, for example, spoke of the spiral tendency of nature. The Golden Spiral described in Chapter 1.22 (Figure 5) is also an approximation of the logarithmic spiral.98 Like the spiral in the plane, the circular helix can be 97 98

See also H. S. M. Coxeter, see Note 26,160 f. In Figure 5 of the Golden Spiral (Chapter 1.22), the rotation and stretching which transforms OE into OC, transforms the point (r,0) into (ττ,θ) + π/2) in polar coordinates with pole O, whereby τ is the ratio of the Golden Section. If we select OE as the unit, so that E has the coordinates (1,0), then C has the coordinates (τ, 1/2 π), A the coordinates (τ 2 , π),

2.3 Symmetries and the Invariance of Geometric Theories

201

introduced in space by continuous helical motion, the technical application of which was discovered as long ago as Archimedes (See Chapter 1.41). In 1874, S. Lie began to classify continuous transformation groups. He speaks of r-parameter groups, if their transformations are a function of r continuous parameters α ϊ , . . . , ( ν 9 9 For transformations with only one variable χ, he makes a distinction between the 1-parameter groups with translations x' = f (χ, α) = χ + a , the 2-parameter groups with the linear transformations x' = f (χ,γ,δ) = γχ + δ, which are formed by the combination of the translation with the similarity transformations, and the 3-parameter transformation groups, which comprise all linear transformations of the form >

*

a

s\

ax + ß

In general 100 , the transformations Xk —> Xk' are defined by functions Xk' = fk(xi;ai, ...,α,.), which are a function of r real parameters, and which can be differentiated by all variables xj and α;. We then speak of an r-parameter local Lie transformation group, if 1) the transformations Xk'=fk(xi ;0Cj) form a transformation group, 2) the identical transformation Xk'=fk(xi;0)=Xk corresponds to the parameter values ctj = 0 for all i with 1 < i < r, 3) there are two transformations xk'=fk(xi;a¡) and Xk'=fk(xi';oti') in the vicinity of identity result in a compound transformation xk'=fk(xi ';Oi') =fk(fi(xm;ai);ai')=fk(xi;0i") with the analytical parameter functions θί"=φί(αι,αι')·

In this case, "local" means that the corresponding characteristics are present at least in the vicinity of identity. Since the parameters values (Xj = 0 correspond to the identity transformation, for very small values óotj, we get a transformation which is close to identity. In this case, we speak of infinitesimal transformations Xk~>Xk'=fk(xi ';8a¡), and distinguish them from finite transformations Xk—>Xk'=fk(xi ;oc¡).

99 100

the corner opposite A the coordinates (τ 3 ,2/3 π) in a new square over AF, etc. Analogously, G has the coordinates (τ - 1 , -1/2 π), I the coordinates (τ - 2 , -π) = (τ - 2 , π), etc. Overall, we get a series of points with polar coordinates r = τ" and θ = 1/2 πη, which satisfy the equation r = τ2®/*· Moreover, all these points lie on the logarithmic spiral r = μ® with μ = τ2/π. S. Lie, Über Gruppen von Transformationen (1874), in: S. Lie, see Note 95, 1-8/ See also L. P. Eisenhart, Continuous Groups of Transformation, New York 1961; R. Hermann, Lie-Groups for Physicists, New York 1966. The parameters r specified in this definition must be essential, i.e. it must not be possible to express them by less than r parameters.

202

2. Symmetries in Modern Mathematics

In 1883, Lie designated a group continuous, if all of its transformations are generated by "an infinite number of repetitions of infinitesimal transformations " 101 . This designation is based on a proposal offered by Leibniz, according to which the continuous modification of a parameter can be introduced as the successive addition of infinitesimal elements. Strictly speaking, calculation with infinitesimally small numbers was without a basis in logic at the time Lie was writing, but the approach is nevertheless suggestive and illustrative. In A. Robinson's non-standard analysis, however, these concepts can be precisely justified, today. 102 Lie divides continuous groups into finite and infinite, depending on whether their transformations are a function of infinitely many (continuous) parameters or of functions. As an example of a continuous and finite group, he cites all movements of a plane. Lie's theory of continuous groups became the "Galois Theory" of differential equations when he used it in 1888 to characterize their solutions of equations: If the transformations x¡'=f¡(xi, X2, ..., x n ; « ι , . . . , α,·) with i = 1, ..., η form a finite, continuous group, then by differentiation by x¡ and elimination of the parameters, we can set up a system of differential equations which defines the functions f¡ and therefore has as solutions the functions f¡ with the integration constants a p . On account of the group characteristic, along with the solution systems x¡'=f¡(xi,..., x n ; ßi,..., ß r ) and x¡'=f¡(x,a), xj'=fi(fi(x,a),..., fn(x,oc); ßi,..., ß r ) is also a solution system. In the case of a continuous and infinite group 103 , Lie once again starts with a range of transformations, which is defined by a system of differential equations of the form , 3χι'

θ 2 χι'

and has the group characteristic, i.e. in particular with xi'=f;(xi,...,x n ) and xi'=gi(xi,.. .,x n ), x¡'=gi(fi(x),..., f n (x)) is also a solution. But now the general solution is not a function of a finite number of constants, but of functions. The theory of continuous and finite groups was investigated in the 1890s in France by H. Poincaré and E. Cartan, among others, and attracted a great deal of attention in physics (in particular in connection with the Theory of Relativity). We should also mention the Lie Algebras of Transformation Groups. In general, an n-dimensional real algebra is defined as an n-dimensional real 101 102

103

S. Lie, Über unendliche kontinuierliche Gruppen (1883), in S. Lie, see Note 95, 314. See also K. Mainzer, Grundlagen in der Geschichte der exakten Wissenschaften, Konstanz 1981, 23 ff. S. Lie, Theorie der Transformationsgruppen, Erster Abschnitt. Leipzig/Berlin 1999, repr. Leipzig/Berlin 1930, 6.

2.3 Symmetries and the Invariance of Geometric Theories

203

vector space V n over the field R of real numbers, for whose elements a, b from V n the multiplication (ab) is also defined, and for all a, b, c from V n and λ, μ from R the distributive law a(Àbn^c)=X(ab)-^(ac)

(Xb-i^c)a=X(ba)-i^(ca) is satisfied. A real, non-commutative, non-associative algebra is called Lie Algebra if the algebra composition also satisfies the requirements of 1) antisymmetry and 2) Jacobi identity, i.e. 1) (ab) = - (ba) 2) ((ab)c) + ((bc)a) + ((ca)b) = 0. For each Lie Group, a Lie Algebra can also be defined with certain common characteristics ("structure constants") and vice-versa. 104 The Lie Algebras are of major significance for the investigation of transformation groups in physics. 2.33 Differential geometry and symmetrical spaces The differential geometry of C. G. Gauss, E. Riemann, E. Carten and others forms the basis for the symmetries of Einstein's Theory of Relativity. Using the example of Gauss's Theory of Surfaces105, we shall first compile several illustrative results of differential geometry. The coordinate system on a surface x¡ (ui, U2) which is generated by the curves ui = const, and U2 = const., is called a Gaussian coordinate system. Curves on the surfaces (e.g. distances on the curved surface of the earth) with a < t < b can now be described by surface coordinates ui = u¡(t), U2 = U2(t) and by spatial coordinates x¡ = Xj(ui(t), U2(t)). Since partial differentiation gives dxj

3xj dui

dxi du2

dt

3ui dt

3u2 dt

such a curve has the arc length:

104 105

See also L. P. Eisenhart, see Note 100, 53 (3. Theorem von Lie). See C. F. Gauss, Disquisitiones generales circa superficies curvas, in: Werke, Göttingen 1863-1903, Vol. 4, 217 ff, C. F. Gauss, Allgemeine Flächentheorie, Leipzig 1912; for a modem description, see W. Blaschke, Über die Differentialgeometrie von Gauss, in: Jahresbericht der Deutschen Mathematiker-Vereinigung 52 1942, 61-71.

204

2. Symmetries in Modern Mathematics

=

ÍV[áxi 0, therefore, a sphere with χ 2 + ζ 2 = 1 is imbedded in the flat space with ds 2 = K" 1 (dx 2 + dz 2 ). For the coordinates, in general, x 2 < 1. Since for each χ there are two points, according to the two roots for ζ in χ 2 + ζ 2 = 1, the n-dimensional sphere volume is: Vn = 2 J

v/gdx! ... dx n .

2

x ϋμ, vp—>vp=vp, whereby ΰ μ leaves invariant the points in the sense of isotropy and does not characterize any directions. With the prerequisite of this isometry group, the metric of the entire space can be determined as ά8 2 =§ ρσ (ν)άν ρ άν σ +ί(ν)§μν(ϋ)άιι μ άυ ν , whereby gp 0, Κ < 0 or Κ = 0 for the subspace.

2.34 Representation Theory and Hilbert Spaces The theory of Hilbert spaces and the theory of representations of discrete and continuous groups forms the mathematical basis for the symmetries of quantum mechanics. First of all, therefore, let us summarize several definitions and results from the theory of vector spaces and Hilbert spaces. In three-dimensional Euclidean space, the position of a particle is determined by a vector r = (x, y, z) relative to the origin of a coordinate system with the

214

2. Symmetries in Modem Mathematics

three Cartesian coordinates x, y, z. For η particles, we need 3n coordinates, i.e. a 3n-dimensional vector space. Historically, Kant 121 in his early writing in 1747, as well as the Kantian Herbart 122 in his graduation thesis, had already recognized the possibility of spaces with an arbitrary number of dimensions. Lagrange bases his analytical mechanics on a 4-dimensional space, in which he adds time as the fourth dimension to the three space coordinates. The year 1844 is important for the mathematical analysis of n-dimensional spaces. A. Cayley's "Chapters on the analytical geometry of η dimensions" had appeared in 1843, and was followed in 1844 by H. Grassmann's First Edition of "Ausdehnungslehre". Grassmann's objective was, as we would say today, to introduce vector algebra on an affine basis. Nowadays, we say that vectors from affine geometry result from translations which transform a point A into a point χ (A) = B, which is designated the endpoint of the vector drawn from A. The set V of η-tuple real numbers χ = ( χ ι , . . . x n ) ("vectors") for which an addition χ + y = (xj + yi, ..., x n + y n ) and multiplication λy = (λχ η , ..., λχ η ) with λ e IR is defined, satisfies the conventional axioms of a real vector 123

space. If on V, the scalar product (x, y) = xiyi + ... + x n y n is also defined, then V is called the Euclidean vector space IR". The scalar product provides a scale for the length or norm of a vector x€ IRn with ||x|| 2 =(x,x)=xi 2 +.. .+x„ 2 and for the angle α between two vectors x,ye IR2 with c o s a = ΤΓ-ΤΠΓΠΤ· IWIIIyll Two vectors are called orthogonal or are perpendicular to one another if (x, y) = 0, i.e. because cos α = 0, α = 90°. Unit vectors ej=(l, 0, ..., 0), ..., e n =(0, ... , 0 , 1) have the length 1 and are orthogonal because (e¡, ej)=ôy (with ó¡j=0, if ϊφ] and ôy=l, if i=j). They 121

122

123

I. Kant, Gedanken von den wahren Schätzung der lebendigen Kräfte... (1747), I § 10 (Ak. Ausg). J. F. Herbart, Sämtliche Werke (ed. G. Hartenstein), 13 vols., Leipzig 1950-1893, Vol. 2, 203. A given vector space over the field F (e.g. of real numbers) is a set V, for whose elements an addition and a scalar multiplication by the scalars from F is defined. The elements (vectors) from V satisfy regarding the addition the axioms of an Abelian group. For the multiplication of the elements χ and y from V with the scalars λ, μ from F, the following axioms apply: (1) λ ^ ) = λ χ - ( ^ (2) (λ+μ)χ=λχ+μχ (3) λ(μχ)=(λμ)χ (4) 1·χ=χ See also H.-J. Kowalsky, Lineare Algebra, Berlin 1967; N. Bourbaki, Eléments de mathématique, Fase. VI Algèbre, Chap. 2 Algèbre linéaire, Paris 1967.

215

2.3 Symmetries and the Invariance of Geometric Theories

form an orthonormalized base for IRn, i.e. each vector xelR" can be noted η

clearly as a linear combination of the base vectors with χ = ^ x ¡ e ¡ and the i=l

coefficients Xi=(e¡, χ). An n-dimensional complex vector space Cn can be obtained in an analogous manner with complex scalars λ€C —oo

with J f*(x)f(x)dx < oo.

124

125

126

A vector space V over the field C of complex numbers is also called a linear space. The addition Ox = Ö is also valid for the zero element 0, whereby 0 is the zero element from C. See also D. Hilbert, Grundzüge einer allgemeinen Theorie der Integralgleichungen, Leipzig 1912; G. Hamel, Integralgleichungen, Berlin/Göttingen/Heidelberg 1949, W. Schmeidler, Lineare Integralgleichungen, Leipzig 1950; G. Hamel, Lineare Operatoren im Hilbertschen Raum, Stuttgart 1954. See also S. Grossmann, Funktionalanalysis I, Π, Frankfurt 1970; M. Reed/B. Simon, Methods of modern mathematical physics I: Functional Analysis, New York 1972.

216

2. Symmetries in Modem Mathematics

If f and g are square integrable, so can f + g and λί. The set of these square integrable functions forms a complex vector space over C with the inner —oo product (f,g) = J f*(x)g(x)dx, which is designated L2(IR). The Hilbert space of complex valued square integrable functions is of denumerable infinite dimension. Namely, there exist denumerable infinitely many functions fi orthogonal to one another and normalizable to 1, with (f¡,fj)=ó¡j, so that each function f of the Hilbert space can be noted as a linear combination of these base vectors, i.e. f = ^ c ¡ f i with complex coefficients c¡ = (f, f¡). i=l Historically, the linear integral equations which interested mathematicians after the turn of the century were understood by means of operators on Hilbert spaces. The linear operators on Hilbert spaces are of interest in physics because they describe changes of the Hilbert space functions (i.e. in terms of physics, the states of the quantum systems). In general, the mapping Τ : V —• V of a vector space is called linear if (2)

T(x+y)=T(x)+T(y) and Τ(λχ)=λΤ(χ)

for all vectors x, y and scalars λ. On an n-dimensional vector space, the linear operators can be represented by an η χ η matrix. If the unit vectors e¡ as above are represented by columns from 0 and 1 (at the i-th place) then we get

(3)

Te¡ =

/0\

/ T u T 12 T2i

Tin \

VTnl

Tnn/

...

/1\ 0 = T,

+T2i Vo y

1

Vo /

/Tii\ T 2i

VT,

/o\ /0\ 1 0 + ...+T n 0 VI/ \0/

Jl«-j· j=l Without going into additional detail here, it should be noted that inverse operators and products of operators ("composition") can be formed. Of major interest in terms of physics are, in particular, the unitary ("adjoint") and Hermitian ("self-adjoint") operators. The operator T* adjoint to Τ is defined by

2.3 Symmetries and the Invariance of Geometric Theories

217

the fact that it must satisfy the equation (x, T*y) = (Tx,y). On an orthonormalized basis, T* can be represented by the adjoint matrix of T. Unitary operators are also frequently defined by the equation T*T = E with the unit operator E or T* = T _ 1 . It is easy to see that unitary transformations leave the scalar product invariant, i.e. (Tx, Ty) = (x, T*Ty) = (x, y). In particular, orthonormal bases are in turn transformed into orthonormalized bases. The Hermitian operator is defined by T* = Τ or (Tx, y) = (x, Ty). The eigenvalue problem of an operator Τ over a vector space is the task of finding a vector χ ("eigenvector") which is mapped by Τ on a constant λ ("Eigenvalue") multiplied by x, i.e. (4)

Tx = λχ.

With the corresponding matrix representation, we get η ^TjiXj = λχ;, j=l

therefore ¿(Tij-^j)xj=0. j=i

These linear homogeneous equations have non-trivial solutions χ (i.e. solutions other than zero), precisely when the determinant of the coefficients disappears. The Hermitian (self-adjoint) operators all have only real eigenvalues, i.e. λ = λ*. The eigenvalues of a unitary operator are complex numbers with the absolute value 1, i.e. | λ | =1. The operators are an important mathematical technique for the study of the symmetry characteristics of vector spaces. Symmetries in vector spaces are described by group transformations of the vectors (e.g. rotations, translations). In the vector space V, let us define a group of transformations G which transform the vectors from V into corresponding vectors from V. Let us also consider the functional space F(V) with functions f, which are dependent only on the vectors χ from V. In terms of physics, V can be visualized as the three-dimensional coordinate space of a body, by means of which determined symmetry operations such as rotations around the origin can be executed. The functional value f(x) can be imagined, for example, as the temperature of a body at the position x. The question arises what temperatures the body has after the rotation. In response, we can say that a symmetry operation G in the vector space induces a transformation T(G) of the temperature function. Mathematically, a transformation Τ is defined which assigns to the symmetry operations G from the vector space an operator T(G) on the functional space with

218

(5)

2. Symmetries in Modem Mathematics

(TXGXfWxMCG" 1 «)

The group transformation G _ 1 inverse to G is a convention which is useful and appropriate for applications in physics. In our temperature example, for example, the definition (5) of the operator T(G) guarantees that the new temperature function f'(x) = (T(G)(f))(x) also represents the temperature of the body at the location χ after the rotation. In Figure 1, Q is the point G~'(x). After the rotation G, the point Q is therefore transformed to the point P, because G(G _1 (x)) = χ. The temperature f'(x) of the body at the position Ρ after the rotation is then the same as at the position Q before the rotation, namely f(G _1 (x)), i.e. f'(x) = f(G _1 (x)). In this sense, a symmetry of the vector space determines an operator on the corresponding functional space. In quantum mechanics, the functional space will be a Hilbert space with corresponding wave and state functions ψ(χ). The question then arises, what operators on the states of the quantum system are induced by symmetries of the coordinate systems. Let us now investigate in general how discrete or continuous groups can be represented by linear operators on vector spaces 127 . For that purpose, let us define a transformation Τ of the elements G of the group in a set of linear operators T(G), whereby (6) 127

T(Gi)T(G 2 )=T(GiG 2 ) and T(E)=1 See H. Boener, Representatives of Groups, Amsterdam 1963; A. Speiser, see Note 6, Chapt. 11-12.

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In the case of a bijective mapping T, the group is represented isomorphically by the set of linear operators. There is a special case if all the group elements are represented by the unit operator. The definition of the linear operators by their matrices, i.e. the representation of groups by matrices is physically very important.128 For example, to be able to use the symmetry of an abstract group in physics, the elements of the group must be quantified. That is what the matrices do. In terms of the history of mathematics, the representation of abstract groups by matrices goes back to work done by G. Frobenius, I. Schur and others129 around the turn of the century. Let ei, ..., e n be a base of the vector space. Then we define, as in (3) (7)

T(G)(ei) = £ T j i ( G ) q . j

The set of matrices T(G) with matrix elements Tj¡(G) is a matrix representation of the group. That can be easily demonstrated by the representation requirement (6). The relationship Tj¡(G) = (ej,T(G)e¡) is frequently appropriate on an orthonormal base. As an example, let us investigate the matrix representation of the dihedral group D3 (See Chapter 2.11) in the 3-dimensional Cartesian vector space with the coordinates x, y and z. In visual terms, D3 can be understood as the symmetry group of an equilateral triangle in the xy-plane, i.e. it consists of the identity element E, the rotations Rj and R2 by 120° and 240° respectively around the z-axis, and the reflections R3, R4 and R5 on the three bisectors of the triangle. First let us present an isomorphic representation corresponding to the operations from D3. In Figure 2, the base vectors e¡ and e2 are visible, while e3 is perpendicular to the plane of the page in the origin. The operators T(R 1 )(e 1 )=er=ie 1 +(|)^e 2 T(R 1 )(e 2 )=e 2 '=-(|)^e 1 -Ie 2 T(R 1 )(e 3 )=e 3 '=e 3 . correspond to the rotation Rj by 120° around the z-axis. Because (ej,T(Ri)ei) = Tj¡(Ri), we get the matrix

128

129

See also J. P. Elliott/P. G. Dawber, Symmetry in Physics, Vol. 1: Principles and Simple Applications, London/Basingstoke 1979, Chapter 4. See the publications by G. Frobenius from the Berliner Sitzungsberichten between 1896 and 1910, e.g. with I. Schur, Über die reellen Darstellungen der endlichen Gruppen ( 1906), 186-208.

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4 T(R,) =

-λ/Ϊ

λ/Ϊ ν 0

"S 0

0

1,

In a corresponding manner, we get, for the other group elements:

a -a Λ

T(R2) =

"5 0

« 1/

0

T(R4) =

\

0

\ T(E)

>T(R 3 ) =

0

-1/

,T(R5) =

0 )/ϊ V 0

^ 4

0 0 - 1 /

/ I 0 0^ = 0 1 0 Vo 0 1 ,

It is easy to see that these operators are an isomorphic matrix representation of D3. For example, the group operation R1R4 = R5, and accordingly T(RI)T(R4) = T(R 5 ). We can also select a representation of D3 in a 1-dimensional vector space with the unit vector e by itself. The reflections are then represented by - 1 , while the rotations are designated by + 1 :

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T ^ ( R i ) = l , T(1)(R2)=1, T(1)(E)=1, T (1) (R 3 )=-1, T(1)(R4)=-1, T (1 )(R 5 )=-1 A distinction must be made between this 1-dimensional representation and the identical representation which corresponds to all group elements 1. Next let us consider the matrix representation of a continuous group. We select the rotation group in the xy-plane around the z-coordinates whose rotations R(0) are a function of the constant parameter θ (See Chapter 2.32). We have already encountered the matrix representation Τ(θ) =

/ cos θ sin0 0 \ - s i n G cose 0 V 0 0 1/

in Chapter 2.31 (where it appears in Figure 1). It is easy to see that the rotation composition R(ei)R(e2)=R(6i+e2) results in the composition Τ(Θ,)Τ(Θ2)=Τ(Θ,+Θ2) of the operators or matrices. So far, we have limited ourselves to representations of symmetries in 3dimensional spaces. We shall now investigate representations in functional spaces, which are a key concept in quantum mechanics. In general, we start, as in (5), with a group of coordinate transformations G in a vector space V. We assume that they leave invariant the functions f of a functional space whose elements depend only on the vectors χ from V, i.e. if f is a function in the functional space, then f-G" 1 for all group elements G must also belong to the functional space. The symmetry operations G then induce operators T(G) on the functional space. We get a representation Τ of the group elements in the sense of definition (5). It can be easily verified that the requirement (6) is satisfied, i.e. T(Gi)T(G 2 )f(x) = T(G 1 G 2 )f(x). We get a matrix representation in the functional space from (7), if we select a base (f¡) in the functional space, i.e. (8)

T(G)fj(x)= fj(G _ 1 x) = fi(x) = XT j i (G)f j (x). j

The example with the functional spaces shows that representations of arbitrary complexity can be introduced for groups. For (finite) groups, however, it can be demonstrated that all possible representations can be constructed of a finite number of representations which cannot be further reduced. Therefore it is always sufficient to study these irreducible representations of a group. For example, D3 has only three irreducible matrix representations,

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namely two 1-dimensional representations and one 2-dimensional representation, from which all other matrix representations can be constructed. But an infinite number of irreducible representations are still possible, as can be easily seen in the example of D3. The space which is spanned by the base vectors e¡ and e2 is irreducible and provides a 2-dimensional representation of D3. But a different selection of base vectors inside the space would provide a different set of matrices T(G). It can be correctly assumed that such a trivial change of the vector space base leaves the essential characteristics of the representation unchanged. Let T(G) be a representation of a group in the vector space V. If A is a mapping of V on a vector space V' of the same dimension, then the operators (9)

T'(G) = AT(G)A" 1

on the vector space V' are also representations of this group. Two representations Τ and T' with the characteristic (9) are therefore called equivalent. In the theory of representation of groups, it is therefore a question of studying the non-equivalent, irreducible representations of groups. All the other concepts of representation theory required will be discussed when we study symmetries in quantum mechanics (See Chapter 4.2). There we will also discuss the contributions to mathematics and physics of E. P. Wigner, H. Weyl, B. L. van der Waerden and others who, starting in 1926, are largely responsible for the application of group theory representation theory in quantum mechanics.

3. Symmetries in Classical Physics and the Philosophy of Nature The symmetries of the laws and theories of physics and of the natural sciences in general became clear only after the application of group theory methods in the 19th and 20th Centuries. In particular, Klein's "Erlanger Program", according to which the objective validity of geometric laws is defined by their invariance under certain groups of transformations, turned out to be the key concept for the mathematical explanation of the objectivity and invariance of the laws and theories of physics. Lie's continuous groups became a valuable resource for classical physics. Old controversies in the philosophy of nature concerning the symmetry of space and time, which had been discussed ever since Aristotle, by Leibniz, Newton and Euler, to Lange and Mach, were explained by group theory, and were given a comparatively clear mathematical form in a common conceptual framework. Their relationship to modern space-time problems then becomes apparent, and the mathematical core of a "philosophia naturalis perennis" crystallizes from antiquity to the modern age, which is concerned, for example, with the homogeneity and isotropy of space and time, or the parity of left and right. But more than just the space-time symmetries of classical physics are becoming apparent. Since antiquity, philosophers of nature have speculated on the first causes and archetypes of matter, from which the variety of phenomena is derived. In Chapter 1 we saw how, as long ago as antiquity and the Middle Ages, this concept of a unity of natural forces was connected with intuitive and speculative considerations of symmetry. After Kepler's celestial physics and Galileo's geophysics had been explained by Newton's theory of gravitation, there was a further unification of the forces of nature in the 19th Century, in which electrostatics, magnetostatics and optics were derived from Maxwell's electrodynamics. The concept of gauge theory in electrodynamics turned out to be the core of this unification in terms of group theory, and was fundamental for the modern unifications of the forces of nature in elementary particle physics. Even in classical mechanics, invariance under groups of transformations leads to important consequences, if we apply the Lagrange formula. For example, if the Lagrange equations of a physics problem are invariant in relation to an η-parameter symmetry group in the Lie sense, then η conservation quantities can be indicated explicitly. Conservation theorems of physi-

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cal quantities, which have a long tradition in the philosophy of nature, as in the case of the conversation of mass, can now be traced to space-time symmetries. These general relationships between symmetry groups and conservation quantities are later found in an analogous fashion in the theory of relativity and quantum mechanics. The assumption of a perfect and purposeful harmony of nature also has a long tradition in the philosophy of nature, and in the 17th Century, Leibniz related it both to extremal principles of physics and a theological theodicy. Leibniz's metaphysical considerations are a fruitful heuristic background for the origin of the calculus of variations and theory of extremal principles, although in the early Enlightenment they were increasingly criticized and derided, and came to a premature end at the court of Frederick the Great in a dispute between Maupertuis, Leibniz's successor as the President of the Prussian Academy, and Voltaire. In the 19th Century, the extremal principles were regarded only as a perfect mathematical formalism for the solution of physics problems, which at best reflects the way of thinking which came up with Laplace's demon. But when Laplace spoke in terms of a "Weltgeist" which knows all the determined sequences of motion of (classical) physics, he intended it only as a metaphor. The modern movement toward secularization has also taken up the concept of the law of nature. With thermodynamics and the irreversible processes of nature assumed in the Second Law of Thermodynamics, it became possible for the first time to explain a development in nature. The question thus arose of how the assumption of an arrow of development in nature is compatible with the time symmetry of the laws of nature - a question which is still being discussed in the philosophy of science. For Darwin's Theory of Evolution, thermodynamics offered a physical framework for the transfer of concepts such as system, development, equilibrium, etc. to animate nature. The harmony and equilibrium of life, which had been discussed metaphysically and philosophically in all cultures, now also became a problem for the natural sciences.

3.1 Symmetries of Space and Time Let us now investigate the natural philosophy principles of modern concepts of space-time. Only a precise statement of these principles in terms of group theory makes clear their relationship to the concept of symmetry. On the other hand, it is now apparent that the interpretations of Newton, Leibniz or Lange in no way represent positions which have become outdated in the

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history of science, but groups of transformations which precisely explain certain aspects of mechanics. 3.11 Pre-scientific Space-Time From the point of view of everyday perception, the symmetry of space and time, i.e. their homogeneity and isotropy, is by no means self-evident. While in Euclidean geometry, space is of the same condition everywhere and in all directions, unchanging and unlimited, our senses give us the impression of the inequality of directions, of the changeability of points in space, and of the limited nature of perceptions. While physics proceeds on the assumption of a time which is constant and uniform, stress and fear can make minutes "seem like an eternity", and hours of happiness can pass "in a few seconds".1 We orient ourselves in the space perceptible to our senses by using terms such as "up" and "down", "near" and "far", and "left" and "right", although these distinctions are geometrically invariant. I can only speak of the "top" or "bottom" side of a rectangle if I see the rectangle drawn on the paper in front of me. And whether a body is designated "near" or "far" depends on whether I can sense it (e.g. heat), touch it or see it. Even the distinction between "left" and "right" is based on various impressions which have to do exclusively with the physical makeup of our sensory organs. For example, in the "Prolegomena", Kant states: "What can be more like my hand or my ear, and identical in every detail, than its image in a mirror? And yet I cannot take such a hand, as seen in a mirror, and put it in the place of the original, because if the original was a right hand, then the hand in the mirror is a left hand, and the image of the right ear is a left ear, which can never take the place of the right ear. Although there are no intrinsic differences, as far as intellect can tell, the differences are inwardly apparent, as common sense tells us, because the left hand and the right hand, regardless of all their identity and similarity to one another, cannot be enclosed in the same limits (they cannot be made congruent); the glove for one hand cannot be worn on the other."2

As an example of the everyday perception of the changeability of bodies (although they remain geometrically rigid), E. Mach brings up the apparent swelling of the stones of a tunnel entrance as a train enters the tunnel, and the 1

An introduction to the psychology of time is given in J. Cohen, Time in Psychology, in: J. Zeman, Time in Science and Philosophy, Amsterdam/London/New York 1971, 153164; J. Zeman, Psychological Time in Health and Disease, Springfield, 111. 1967; M.L. Franz, Zeit. Strömen und Stille, Frankfurt 1981, presents a cultural-historical overview of various interpretations of time in individual cultures.

2

I. Kant, Prolegomena (Ak. Ausg. IV), 286.

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shrinkage of the same objects as it exits.3 As a function of the physiological capabilities of our sensory organs, we can now distinguish initial fragments of spaces and figures, which are only made possible by a subsequent abstract theory of Euclidean space. We begin with the propagation fields of sensory data which are recorded by the individual sensory organs. The sensations of light experienced when pressure is applied to closed eyes (phosphenes), the sensations of touch, e.g. the touch of a hand, and sound manifolds are examples of such fields. The field of vision and the field of touch are initially altogether separate areas; they are those "manifolds, in which the visual or tactile sensory data, if they are simultaneous, are propagated side by side".4 Not only is the field of vision limited, but it appears to have quite narrow limits. Tests have shown that the size of an image no longer decreases significantly if it is projected on a surface whose distance from the eye increases beyond 30 m. The narrow limits of the field of vision can be demonstrated by our perception of panoramic paintings. The field of vision overall appears to us to be a receding series of strips which already have a primitive center-edge orientation system. Ideal geometric constructs such as point and line cannot be detected in the field of vision as visually morphological shapes. But there are intuitive processes in the field of vision which prepare these ideal geometric constructs. Imagine, for example, a bright spot in the field of vision against a dark background. If the size of the spot decreases and the color remains constant, no matter how much it is reduced in size, it will not disappear altogether. Rather, it approaches a certain limit which can be designated a visual "point" in the field of vision. Of course, all figures grow indistinct if their size drops below a certain level, and to that extent continuous processes in the field of vision can be experienced visually only to a limited extent. But the colored spot does not disappear, in the sense that a "colored point", without apparent extent but both localized and qualified (with a very definite shade of color) remains.5 Such colored points are what Berkeley and Hume see when they speak of the minimum (visible). In "A Treatise of Human Nature", Hume writes: "The table before me is alone sufficient by its view to give me the idea of extension. This idea, then, is borrow'd from, and represents some impression, which this moment appears to the senses. But my senses convey to me only the impressions of colour'd 3 4

5

E. Mach, Erkenntnis und Irrtum, Leipzig 1917, 22. O. Becker, Beiträge zur phänomenologischen Begründung der Geometrie und ihrer physikalischen Anwendung, Tübingen 2 1973 (first printed in "Jahrbuch für Philosophie und phänomenologische Forschung" [ed. E. Husserl], Volume VI, 1923), 63 f. O. Becker, see Note 4, 79 f.

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points, dispos'd in a certain manner. If the eye is sensible of any thing farther, I desire it may be pointed out to me. But if it be impossible to shew any thing farther, we may conclude with certainty, that the idea of extension is nothing but a copy of these colour'd points, and of the manner of their appearance." 6 Analogously, there are processes in the field of vision which prepare ideal geometric constructs such as lines. For example, imagine a bright strip against a dark background, the "upper" and "lower" extension of which shrinks with a constant lateral extension. Of course this "continuous" process in the field of vision can again be perceived only to a very limited extent. But the colored strip does not disappear, because an apparently extension-less but both localized and qualified (with a very definite shade of color) "colored line" remains. Does it follow from these considerations that a field of vision is finite or infinitely divisible? All we can say is that: Although we can pack shrinking line segments as intervals on a line into one another (to determine a point), this operation cannot be reiterated an arbitrary number of times. We soon arrive at figures which merge into one another.7 Likewise, between two visible points on a line in the field of vision, there is always a third point such that the three points are completely separate from one another. Therefore it is impossible for our sensory experiences in the field of vision alone to have prepared the geometric idea of continuity. But on the other hand, we cannot combine a specified segment of infinitely many points, because a visual point in the field of vision is not seen as a fixed, precise construct, but is only indicated by a visual process of limitation. The morphological construct indicating the point has a certain extension, but one which is not determined in a fixed manner and is not comparable to constructs with finite extension. In the area of auditory field, we can also demonstrate visual preparations of certain geometric constructs. For example, a line is experienced as a sound of approximately identical pitch and timbre, but different volume. If a segment of this line becomes increasingly concentrated as a certain variation of the volume of a sound, a sound of a very definite intensity is registered which is comparable to the visible point in the field of vision. In contrast to the field of vision, however, two points do not exist simultaneously as sounds. When we hear musical chords, for example, we are experiencing fusion phenomena. 8 6

7 8

D. Hume, A Treatise of Human Nature (ed. E. C. Mossner), London 1969, 82 f. (Section III: Of the other qualities of our ideas of space and time). O. Becker, see Note 4, 84. O. Becker, see Note 4, 85.

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Fig. 1

Fig. 2

Of course, in the tacticle field there are sensations on the skin which prepare geometric constructs such as point and line, e.g. as a "pain in one spot" or as a stabbing pain. But the physiological makeup of the skin differs in important ways from Euclidean geometry. If we distinguish not only the quality of the stimulus, but also the point stimulated by any additional sensation, we can demonstrate the major anomalies which the localization of the skin has in relation to metric, Euclidean space. For example, the distance which can be clearly distinguished between two points of a compass touching the skin is 50 to 60 times smaller on the tip of the tongue than in the middle of the back. Different parts of the skin exhibit major differences in their sensitivity to touch. A compass, the tips of which enclose the upper lip and the lower lip between them, seems to close if we move horizontally toward the side of the face: 9 If the tips of the compass are placed on the tips of two neighboring fingers, and the compass is moved across the palm of the hand toward the underarm, the tips seem to coincide completely. The figures show the actual path in dotted lines, and the apparent path in solid lines. Of course the figures which touch the skin are different, but the skin's sense of touch obviously takes a back seat to the sense of vision.

9

E. Mach, see Note 3, 339. Mach's analyses go back to Ε. H. Weber, Über den Raumsinn und die Empfindungskreise in der Haut und im Auge, in: Ber. d. kgl. sächs. Gesellsch. d. Wissenschaften, mathem.-naturw. Cl. 1852, 85f.

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Overall, therefore, in the individual areas of sense of the sensory organs, preparations are made for the geometric ideal constructs. But in many respects, we are still missing important characteristics which are ascribed to the geometric ideal constructs, such as, and above all, their unlimited approximability. In accordance with E. Husserl and O. Becker, the various propagation fields of sensory data must therefore also be called prespatial (or quasi-spatial) fields.10 The primary common feature of the above considerations was the fact that these propagation fields were investigated as separate actions by our bodies. But the human body is primarily a unit which consists of interlinked and harmonious individual functions. It is on one hand the sphere of potential action for the will, whereby an action ultimately consists of movements of the body, and on the other hand the sphere of sensory sensations and emotions (e.g. pain, desire). As a unit, we experience our own bodies in a unique manner "from inside", and not merely as one thing among other things. As a material thing, it comes into being only as a result of what are called dual sensations, which originate for the sense of touch when a part of our body (e.g. the hand) touches another part of the body. For example, we can feel or "sense" our hand both as a movable, sensitive limb of the body "from inside", or we can touch it with the other hand "from outside". The selfinduced sensations of touch aroused simultaneously with this feeling in the felt hand are the basis for the identification of the limb, which is perceived simultaneously "from outside" and "from inside".11 In contrast to the individual fields of propagation of sensory data, let us now consider the "oriented space" as the environmental space at the center of which "I" am constantly located, with all my bodily capabilities. The circumstance in which I, by means of my body, assume a position in the oriented space is then what makes it "space", in contrast to the above-mentioned pre-spatial sensory fields. Those things which are represented in the oriented space are no longer individual sensory data or pre-spatial figures, but things which can be identified on the basis of various sensory data (e.g. visual and tactile) as sense units. For example, we can touch and see the same thing. We then make a distinction between "near" and "far", "here" and "there", as a function of the position in relation to our body as the center of the oriented space. For example, we can designate things "near", whose sensory data 10 11

O. Becker, see Note 4,62. O. Becker, see Note 4, 70; also E. Ströker, Philosophische Untersuchungen zum Raum, Frankfurt 1965, 100 f. (Der Leib als Zentrum des Anschauungsraumes). In particular, the investigations of E. Ströker run parallel to Kant's body constitution in his "Opus posthumous". Here, we should also mention the problem of Self-awareness?.

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can be obtained directly by the body's senses (by touching, seeing, hearing), while those things we designate "far" are included only as spheres available to our will and its actions. Both tactilely and visually, the point of the oriented space at which the body is located is distinguished from all others. That is what we mean when we speak of "here", in contrast to any "there". Likewise, in oriented space, the distance of an object from "me" is different from the distance of two objects from one another. The oriented space is, for example, that space on whose horizon I see the sun "rise" in the morning and "set" in the evening. In oriented space, the train which rushes past me also "expands" and "contracts". In addition to the coordinated sensory experiences of hearing (which registers the one-dimensional flow of sound) and the senses of sight and touch (which register two-dimensional superficial phenomena), the sensation of movement of the limbs (in particular of the arms, hands and fingers) adds something of a third dimension. But in the oriented space, a motion such as walking, in which a rhythmically repeated movement of the limbs is used to achieve constant forward motion in one direction, is initially excluded. The oriented tactile field thus extends only as far as my limbs can reach. 12 But in the oriented space, some processes in the sensory fields can no longer be interpreted as the preparation of geometric ideal constructs, as was possible in the separate sensory fields. For example, what is seen as a point in the field of vision when viewed diagonally can become a small spot when viewed head on as a result of a possible additional movement in the oriented space of the torso, of the head etc., a line can become a surface patch, etc. Obviously, points and lines in the oriented space must be invariant under such possible movements, and may not be expanded. Overall, as before, the oriented space gives the separate sensory fields the impression of limitation, the inequality of directions and the changeability of positions in space. From the oriented space restricted in this manner, we arrive at a homogeneously unrestricted space only when we can leave our respective position at rest, and we can simultaneously reach each point in space by our own movements (e.g. walking). 13 The indistinct "far" objects of the oriented space are thereby brought into distinct proximity. This approach can take place from any given distance and up to any given distance. An approach in this sense is also observation of an object through a telescope, a magnifying glass or a microscope. Therefore a fundamental difference from the conditions in the oriented space exists, in that there are no theoretical limits to 12

13

O. Becker, see Note 4, 73; E. Ströker, see Note 11, 102 f (Der orientierte Anschauungsraum). O. Becker, see Note 4, 73f.

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an approximation, but only empirical limits. The space is now experienced homogeneously and without limitations. Objects can also be experienced as identical or "rigid", while in oriented space (depending on the position in relation to the defined center), images frequently appear to "expand" and "contract". But with the ability to approach the objects arbitrarily, many of the surfaces which appeared smooth in the oriented space turn out to be rough, for example, many uniformly colored surfaces turn out to be spotted, straight lines turn out to be curves, etc. For the continuous contraction processes leading to the visual construction of ideal points from colored points and of lines from colored strips, such phenomena must be excluded. The arbitrary capabilities of movement and action of the body as a unit, and its ability to adopt any given orientation, promote the opinion that we can execute these same movements everywhere and in all directions, and that space can be imagined as having the same properties, unlimited and infinite, everywhere and in all directions. If we continue to change our orientation, e.g. by rotation around the vertical axis, these same changes of positions in space are repeated over and over. Thereby, not only does the uniformity become clear, but also the inexhaustibility, the unlimited repeatability and continuability of certain spatial perceptions become clear. For example, our spatial perceptions of course gradually approximate geometric space,14 but are unable to completely achieve it in this manner. Therefore it is necessary to define geometric figures such as lines, points, planes, etc. as basic concepts of a geometric theory, for which the physiological phenomena of points and surfaces etc. or corners, edges, surfaces etc. which can be technical produced are only approximated realizations.15 But the homogeneous and isotropic space in which we can move bodies without restriction in all directions does not have any metric. Only with the additional requirement that spatial dimensions, i.e. lengths and angles, be measured with rigid measuring rods, do we move from homogeneous sensory space to metric geometry. Historically, then, the Euclidean parallel postulate or one of its equivalent theorems was also adopted to characterize euclidean geometry for physical space.16 But if we introduce the geometric shapes by saying a priori that they can be (continuously) enlarged or reduced in size arbitrarily, i.e. if they are de-

14

15

16

See in particular E. Ströker, Der Anschauungsraum also Grenzfall gelebter Räumlichkeit, in: see Note l l , 2 0 2 f . With regard to phenomenological space constitution, see in particular E. Husserl, Ding und Raum (1907 Lectures) (ed. U. Claesges), in: Husserliana, Vol. XVI, The Hague 1973. See also K. Mainzer, Geschichte der Geometrie, Mannheim/Vienna/Zürich 1980, 119 ff.

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fined a priori by the similarity group, then the parallel postulate must not also be required. The criterion of arbitrary expandability and reducibility of shapes is equivalent to the parallel postulate ("Wallis's Criterion").17 In this case, however, congruence ("scale") must also be introduced as a derived concept.18 In the next abstraction step, Euclidean geometry is transformed into Cartesian geometry, in which geometric shapes such as points, lines, etc. are designated by (real) numerical coordinates, ordered sequences of numbers, equations, etc. Only now can we define the symmetry group of (euclidean) space IR3. This is the 6-parameter Euclidean group which consists of the 3-parameter translation group and the 3-dimensional rotation group (See also Chapter 2.31). In pre-scientific perception, we experience time in the form of changes of the position where bodies or (idealized) mass points are located. The onedimensional quantity of all space points through which a mass point passes is a geometric path line. The process of the successive, point by point and continuous generation of a curve can be represented geometrically by defining the curve by a function χ = χ (θ) with a real, continuous parameter θ > 0, whereby the curve originates with the constant increase of θ from a point of origin χ = χ (0). Since the selection of the parameter is specified only with the exception of one-to-one and continuous transformations, we can also speak of topological time}9 It designates only the sequence of points of time without a metric. It can be realized by any given continuous movement of a mass point. In particular, therefore, a straight line can also be selected. On the continuum of topological time, a metric is defined by any selected movement process. For that purpose, it is specified that identical Euclidean segments ("Pythagoras") on the path of a mass point are traversed in equal intervals of time. With regard to this metric time, we designate a movement uniform, if the path traverses equal distances in equal intervals of time. 20

17

18

19 20

J. Wallis, De Postulato Quinto et Definitione Quinta, lib. 6 Euclidis, disceptatio geometrica, in: Opera Mathematica II, Oxford 1693, 669-678; F. Engel/P. Stäckel (eds.), Theorie der Parallellinien von Euklid bis auf Gauss, Leipzig 1895, 21-36. P. Lorenzen constructed such a geometry in P. Lorenzen, Elementargeometrie. Das Fundament der analytischen Geometrie, Mannheim/Vienna/Zürich 1984. See also P. Mittelstadt, Klassische Mechanik, Mannheim/Vienna/Zürich 1970, 24. E. Mach emphasized that a uniform movement can only be defined relative to another movement: E. Mach, Die Mechanik, Leipzig 1902, 234: likewise H. Reichenbach, Philosophie der Raum-Zeit Lehre, Berlin 1928, § 17. On the other hand, proposals have been made in the context of protophysics, that linear uniform movements be introduced purely geometrically and with no reference to time, to justify chronometry on this basis.

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The symmetry of one-dimensional euclidean time Τ (= IR) is defined by the 1-parameter translation group. Events occur at a given position at a given time, i.e. they can be represented as points (x, t) with χ G IR3 and t G T. In this case, the direct product can also be written 1R3XT for the 4-dimensional space-time in which the events occur. The symmetry of this space-time in which, in addition to spatial rotations and translations, there are only temporal shifts, is naturally minimal. Mathematically, it is a question of a 7-parameter group consisting of the 6-parameter Euclidean group (with 3-parameter rotation group and 3parameter translation group) and the 1-parameter translation group of time. In this space-time, it is correct to say of two events (xi, ti) and (x2, t2) that they are spatially separate, even if they occur at different times. In this context, the mathematical core of Aristotelian space-time can be stated as follows. 21 We recall (from Chapter 1.23 and 1.33) that space and time in the Aristotelian world are defined by certain regular natural events. For an observer who assumes his position to be stationary, the sun rises in the morning and sets in the evening, and is higher in the sky in the summer than in the winter. On a clear night, the stars seem to be in orbit around him. Astronomers in the era of Aristotle therefore conceived of the universe as an onion, with the stationary earth in the center and the planets on skins orbiting uniformly around the earth. The outermost skin is the sphere of fixed stars which revolves once a day. In this spherical model, all the movements observed in the sky can be simulated by means of a few geometric tricks. In any case, the accuracy is greater than can be perceived with the naked eye. The well-known medieval celestial globes are spatial, three-dimensional models of this world. In addition to the absolutely stationary position of the earth and the orbital motions of the stars and planets, Aristotle distinguishes on the earth only motions of constant velocity: Heavy bodies tend to move downward, and light bodies tend to move upward. Such a world has in no way been made historically obsolete, but approximately within the limits of experience, corresponds to our experiences of nature. 22 But if we also recall how well this

21

22

See P. Janich, Die Protophysik der Zeit. Konstruktive Begründung und Geschichte der Zeitmessung, Frankfurt 1980. See also J. Bacryl/I.M. Lévy-Leblond, Possible kinematics, in: J. Math. Phys. 9 1968, 1605-1614. A group theory definition of Aristotelian space-time symmetry is naturally ideal, and can be used to understand the Aristotelian concept of space and time from a modern physics standpoint. Historically, Aristotelian space theory is a teleologicallyoriented theory of the natural sites of bodies. See also W. Wieland, Die aristotelische Physik, Göttingen 2 1970. See also J. Piaget, Les notions de mouvements et de vitesse chez l'enfant, English translation: The Child's Conception of Movement and Speed, London 1970.

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interpretation of space-time fits the cosmological and religious notions of the time, it is no wonder that this image of the world remained unshakable for almost a thousand years. 3.12 Space-Time Symmetry according to Newton and Kant A comparison with Newtonian space-time shows that while it differs from the Aristotelian system in important respects, the two concepts still have several features in common. Newton relates all events to an absolute space and an absolute time. We should recall the famous Scholium in the "Principia", where he makes a distinction between absolute space and relative spaces: "Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is commonly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. Absolute and relative space are the same in figure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be continually changed." 23

Analogously, for absolute time: "Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration. Relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year."24

Newton's concepts of absolute space and absolute time have occasioned a great deal of puzzlement in the history of science. Absolute space was imagined simply as a giant, empty container, around whose absolutely stationary center the material world is concentrated. In place of the stationary earth as the reference point of the Aristotelian world, there is now an imaginary center of the world. It is as if the era of Copernicus had never happened. Since absolute space was also defined as infinite and indivisible, it also seemed to the theologically-minded scholars of the 17th Century to have quite divine attributes. Newton himself spoke of the "Sensorium Dei",25 God's "senso23

24 25

I. Newton, German translation, Mathematische Prinzipien der Naturlehre (ed. J. P. Wolf). Berlin 1872, repr. Darmstadt 1963, 25. I. Newton, see Note 23. I. Newton, Optik, Vol. 3, German translation and edition by W. Abendroth, Leipzig 1898, 145.

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ry organ", by means of which God perceives the material world. Newton's pious interpretations are in sharp contrast to his methodology, according to which physics must be traced to empirical facts: "Hypotheses non fingo."26 Newton attempted to prove at least indirectly the existence of absolute space, which is imperceptible by definition. In his famous "bucket experiment"21 , his objective was to show how accelerated motions in relation to absolute space can be defined by the inertial forces which are exerted on moving bodies. To do so, a bucket full of water was suspended on a rope, and the rope was twisted by rotating the bucket. When the bucket was released, it began rotating, whereupon the following phenomena were observed: During the first phase, only the bucket rotates relative to the observer, while the water remains stationary. During the second phase, the water gradually reaches the same speed of rotation as the bucket, and thereby climbs up the walls of the bucket so that there is a curvature to the surface of the water. In this case, according to Newton, the motion executed by the water is not relative to the bucket, but to absolute space, so that the latter is also the only possible cause for the observed curvature of the surface of the water. Mach later criticized Newton's interpretations, noting that the effects of the bucket experiment are not caused by an imaginary "Sensorium Dei", but by a rotation of the water relative to distant but real cosmic masses. 28 I. Kant (1769) in his publication entitled, "Von dem ersten Grunde des Unterschiedes der Gegenden im Räume" [On the first principles of the difference between regions in space], attempted an additional proof of the existence of absolute space. Kant bases his proof on the difference between right and left. According to Kant, observation shows that the internal relationships between the individual parts of the left hand to one another are identical to those in our right hand. Still, there must be a fundamental difference which prevents us from replacing one hand with the other. But, argues Kant, if this difference cannot be explained by the different position of the parts in relation to one another, then it can only be explained by their different position in relation to absolute space. 29 26

27 28

29

I. Newton, Philosophiae naturalis principia mathematica, Cambridge, 2 1713,484. For information on Newton's methodology, see also J. Mittelstrass, Neuzeit und Aufklärung. Studien zur Entstehung der neuzeitlichen Wissenschaft und Philosophie, Berlin/New York 1970,287-306,312-319. Also M. Jammer, Das Problem des Raumes. Die Entwicklung der Raumtheorien, Darmstadt 1960,118. I. Newton, see Note 23, 29. E. Mach, Die Mechanik, historisch-kritisch dargestellt, Leipzig 9 1933, repr. Darmstadt 1973, 222 ff. I. Kant, Von dem ersten Grunde des Unterschiedes der Gegenden im Räume, in: Ak. Ausg. II, 377: "Because the positions of the parts of space in relation to one another presuppose

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Mathematically, however, Kant's fiction of absolute space is incorrect as an explanation of the difference between left and right. Whether two congruent systems (e.g. triangles) are left-oriented or right-oriented (Figure 1) can be explained by a purely combinatorial consideration: The direction of rotation is determined by a permutation in relation to linear independent vectors. 30 The left side of a straight line can also be replaced by its right side, by rotating it 180° in a plane. A screw with a right-hand thread can be transformed into a screw with a left-hand thread, and a screw with a lefthand thread into a screw with a right-hand thread, by moving the object in question of three-dimensional space into 4-dimensional space. Β

Β

Fig. 1

Mathematically, therefore, the left-right difference ("parity") is therefore invariant. 31 Only in modern elementary particle physics and molecular biology are there good reasons for the assumption that basically, parity need no longer exist in nature. But absolute space plays no part in these modern investigations. The mathematical core of these methodological, theological and metaphysical discussions of absolute space can be explained as follows on the basis of group theory, and gives us a certain understanding of the symmetry of space-time. The decisive assumption is that an objective distinction can be made between two events, whether they are simultaneous and whether

30

31

the region, along which they are located in such a relationship, and in the most abstract sense, the region does not exist in the relationship of one thing in space to the other, which is indeed the definition of position, but in the relationships of the system of these locations to the absolute world space." See H. Weyl, Philosophie der Mathematik und Naturwissenschaft, Munich/Vienna 1982, 113. Every system of η linear independent vectors defines a "direction". Two systems define the same direction if they proceed from one another by means of an even permutation. An uneven permutation changes the direction into its opposite. The even permutations form a subgroup with index 2 within the group of permutations. For two bases (e¡) and (e¡') of an n-dimensional vector space, the direction can also be determined by the concept of determinants. If (e,) is transformed in (e¡') by e¡' = a^ei + ... + a ni e„, then the coefficients a^ have a non-zero determinant. The directions of the two bases are therefore either the same or different, depending on whether the determinant is positive or negative. See also K. Reidemeister, Raum und Zahl, Berlin/Göttingen/Heidelberg 1957, Chapter 4.

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they take place at the same location. The set of events can therefore be understood as the direct product IR3xT of the Euclidean space IR3 and the quantity Τ of points in time. In Euclidean three-dimensional space, there is a metric which can be realized by straight edges. The time Τ is understood as onedimensional Euclidean space, whose time coordinate t is defined by a linear transformation t' = t + b and which can be measured by standard clocks. The quantity of all events at a constant point in time t = t(e) simultaneous with an event e forms a three-dimensional space layer. Figure 2 shows a model of this space-time with the layers of simultaneous events. In the three-dimensional limitations of our field of view, the time Τ is represented by a vertical axis and the three-dimensional layers of simultaneous events by two-dimensional parallel planes. They represent respectively the present, which separates the past from the future with t > t(e). The parallel layering of space-time by (maximum) subsets of simultaneous events also defines the causal structure or the correlation of action of the world. For example, if we shoot bullets with different velocities from a world point 0 in all directions, they only reach world points which are later than 0. An event which occurs at 0 therefore only influences events which occur in the future. The past no longer belongs to my sphere of influence. Leibniz says, in the "Initia rerum mathematicarum metaphysica": Of two elements which are not simultaneous, if one includes the reason for the other, then the former is considered preceding, and the latter following." 32 The characteristic of space-time, that future and past of an event e have a common border, namely the present, is expressed by the Newtonian assumption that there are arbitrarily rapid time transfers. One method (although it can only be realized to a limited extent) of instantaneous time transfer from a location A to a location B, is if a rod from A to Β is given a jerk in A, which is immediately transferred to B. The psychological reason for the belief in this type of simultaneity probably has something to do with the fact that in the everyday world, we naturally place the things we see at the center of their perception. The observer thus extends his time to the entire world which occurs in his range of perception. Another important assumption by Newton is his belief in an absolute point of rest. In Figure 2, such a point is represented by a vertical line Pi, since the space coordinates remain the same and only time advances. Psy-

32

G. W. Leibniz, Initia rerum mathematicarum metaphysica, in: Hauptschriften zur Grundlegung der Philosophie, Volume 1., ed. E. Cassirer, translated by Α. Buchenau, Leipzig 1904, 53 (Mathem. Sehr. VII, 18). On the history of the concept of space as understood by Leibniz, see also W. Gent, Leibnizens Philosophie der Zeit und des Raumes, in: KantStudien 31 1926, 61-88; M. Jammer, see Note 26, 126 ff.; F. Kaulbach, Die Metaphysik des Raumes bei Leibniz und Kant, Cologne 1960.

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Fig. 2

chologically, this belief has particularly strong roots in the everyday world, which since prehistoric times has assumed that the earth is permanent and stable, and that "everything revolves around it". In Figure 2, the diagonal line P2 represents a uniform motion and the curve P3 an accelerated motion. The motions in A and Β are parallel. The parallel layering of the space-time model expresses quite precisely the Newtonian interpretation of an external world in which time passes independently of the observer (absolute), whose instantaneous status is determined by the instantaneous distribution of matter in a layer with t = t(e), and whose causal history is defined by the sequence of space layers. The transformation group which leaves this structure of Newtonian space-time invariant consists of the direct product of the group of dilations, rotations and translations in the euclidean space IR3 and the affine group of time T. H. Weyl called this Lie group the "elementary" symmetry group of space-time.33 We shall designate it the Newtonian group G new below. 3.13 Space-Time Symmetry according to Leibniz and Huygens While the metric and the corresponding causal structure of Newtonian space-time were generally accepted in the 18th and 19th Centuries, Newton's assumption of absolute rest and motion were questioned almost from the beginning. As it happened, one notable criticism came from a theolo33

See H. Weyl, Raum, Zeit, Materie. Vorlesungen über allgemeine Relativitätstheorie, Berlin 1923, repr. Darmstadt 1961, 142; see also J. Ehlers, The Nature and Structure of Spacetime, in: J. Mehra (ed.), The Physicist's Conception of Nature, Dordrecht/Boston 1973, 73.

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239

gian, which is all the more surprising, because English theologians in the 17th and 18th Centuries appealed to the great Newton as "Defensor fidei" [Defender of the faith] against an atheism motivated by natural science, which was gaining strength in the early stages of the Enlightenment. This particular criticism came from Bishop Berkeley, who detected behind Newton's fiction of absolute space the theological danger of pantheism, in which God is identified with nature. According to Berkeley, space may only be interpreted as relative, "otherwise there would be something beside God which would be eternal, uncreated, infinite, indivisible, immutable. "34 Although Berkeley proved to be an extremely sharp-witted critic of the principles of natural science, he did not oppose them in the sense of fighting a rear guard action, in the manner of the Inquisition persecuting Galileo. Rather, he directed his criticism at those who took their modern belief in progress to be an improvement over theology, and who would replace religion with a naive belief in the natural sciences.35 Berkeley is therefore as up-to-date now as he was then. Fictions such as Newton's absolute space which are not based on empirical evidence offered a good opportunity to lead those "ad absurdum", who expressed their belief in natural sciences in theorems such as, "I only believe what I can see, feel, taste and smell." And Berkeley's theology - it should also be noted - is thoroughly modern, since he frees God from the uncomfortable position of serving as a stopgap to make up for an ignorance of the natural sciences and attempts to reveal God as the creator disclosed by faith. Leibniz also criticizes Newton's amalgamation of theology and natural science and in his famous controversy with Clarke demonstrates the problems generated by equating the omnipresence of infinite space with the omnipresence of the infinite God. For Leibniz, space is only a system of relations between bodies, which has no metaphysical or ontological existence. According to Leibniz, the relationship of positions is sufficient to define space: "A concept of space can be derived approximately as follows. We observe that different things exist simultaneously and find in them a certain order to their coexistence, which is more or less simple according to their relationship. This is their relative position or distance

34

35

G. Berkeley, Treatise on the Principles of Human Knowledge, German translation by F. Überweg, ed. Α. Klemmt, Hamburg 1979, 92. See the title page of G. Berkeley, The Analyst, Dublin/London 1734: "The Analyst, or, a Discourse addressed to an infidel Mathematician, wherein it is examined wether The Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith." His advice to advocates of the natural sciences who criticized religion came from Matthew 7:5: "First cast out the beam out of thine own eye; and then shalt thou see clearly to cast out the mote out of thy brother's eye."

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from one another. Now, if one of the elements changes its relationship to a majority of the other elements, without any change in the elements themselves, and if a new arrival assumes the relationship to the others which the first element had, then we say that it has replaced the first one, and this change is designated a motion, which is ascribed to that element in which the direct cause of the change resides. Now, if several or even all of the elements proceed according to given rules of the change of direction and velocity, then the position relationship which each element has in relation to any other can be determined; we could even indicate the position of each element in relation to all the others, as long as it had not moved at all or occurred in any way other than it actually did. But if we assume, or make the hypothesis, that among these coexisting bodies there are a sufficient number of them which do not experience any change in relation to one another, then we would say of elements which are in a relation to these fixed elements like that previously ascribed to other bodies, that they are now in the "place" of these others. But the essence of all these places is called space.36

To use Newton's terminology, therefore, Leibniz considers only relative reference systems. In his third letter against Clarke, he claims that no spatial position and no point in time can be characterized absolutely: "Consequently, on the assumption that space is something in itself, in other words that it is more than the mere arrangement of bodies in relation to one another, it is impossible to give a reason why God has placed bodies -assuming that they retain their distances from one another and their positions in relation to one another - at one determined point and not another; in other words, "why everything has not been reversed by switching East and West."37

Accordingly, with regard to time: "... Then if would of course be impossible to find a reason why things - on the assumption of their fixed identical sequence - should have been placed in one time rather than in another."38

Therefore Leibniz bases the relativity of all points in space and time on his metaphysical principle of sufficient reason ("principium rationis sufficientis"), according to which nothing is or happens in the world without sufficient reason. Even God has to obey this commandment of reason, in order not to contradict His own creation. This famous didactic play of modern natural philosophy, in which it is apparently only a question of relativity and homogeneity, of space and time, not only has metaphysical implications, but for the 17th and 18th Centuries it also had also shattering theological and political implications. A God who must obey the laws of reason is, in political terms, a constitutional monarch and may not intervene in the lives of human beings merely on the basis of despotic whims. Clarke seems to be

36 37 38

G. W. Leibniz, Hauptschr., See Note 32, 182. G. W. Leibniz, See Note 32, 135. G. W. Leibniz, See Note 32, 136.

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on the scent of this conclusion, when he attacks Leibniz with a thoroughly inquisitional tone: "Against all those who claim that in an earthly government, things could completely go their own way without the intervention of the King, we may suspect that they would really like to push the King aside altogether. Indeed, the theory that the world does not require the constant guidance of God, the supreme ruler, is in fact an attempt to banish God from the world."39

Fig. 1

The mathematical core of Leibniz's argumentation leads to a new spacetime symmetry which is different from Newton's. For that purpose, we must abandon the concept of absolute rest and motion (rotation). For the spacetime model in Figure 1, that means that there are no preferred motions ("straight lines") or axes and no parallelism (except in the layer of simultaneity at a fixed point in time). In contrast to Newtonian space-time, therefore, Leibniz's space time is not affine. Overall, it has less structure. Its symmetry is defined by a transformation group which requires not only 3 parameters, but 6 arbitrary real time functions for the angular and translation velocity. Thus it is not a finite continuous group in the Lie sense. Since it leaves the arbitrary continuous motions invariant, it is designated the kinematic group of space time, after H. Weyl.40 Below, we speak of the Leibniz symmetry group Gyn- Obviously, for Newtonian symmetry, G new C Gk¡n. Since the concept of simultaneity is unchanged, Leibniz's space time has the same causal structure as Newtonian space time. Methodologically it follows: If in Leibniz's space time, no form of motion is characterized, then to investigate concrete systems, such as the plan39 40

G. W. Leibniz, see Note 32, 123. H. Weyl, see Note 33, 146; see also J. Ehlers, see Note 33, 74.

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ets, it can only be a question of selecting the coordinate system so that the time functions of the mass points satisfy the simplest possible laws. In this Leibnizian context, for example, the decisive contribution by Copernicus is the indication that a coordinate system exists for which the laws of planetary motion assume a significantly simpler form than if -as for Aristotle and Ptolemy - they are related to the stationary earth. In fact, that was only a kinematic question. Newton (building on the foundation laid by Kepler) was the first to indicate dynamic reasons for the "real" motion of the planets around the sun and not around the earth. As long as we are dealing only with kinematic questions, Leibniz's symmetry accurately describes the space-time of (classical) physics. Therefore Leibniz indicates only a kinematic principle of relativity. But then how does he explain dynamic effects such as the occurrence of centrifugal force during motion in a circle? In fact, Leibniz and especially his physics teacher Huygens saw this problem. For example, Huygens attempted to explain centrifugal force on a revolving disc by the relative motion of different parts of the disc.41 The relative motion of these parts, however - speaking in terms of group theory - could be transformed away, if the reference system selected is the one which has the same origin and the same angular velocity as the rotating disc. Relative to this rotating system of coordinates, the parts of the disc are at rest. Of course the pressure which the centrifugal forces exert is not thereby eliminated. After a geometric and a kinematic principle of symmetry, therefore, a dynamic principle of symmetry is not forthcoming.

3.14 Space-Time Symmetry of Classical Mechanics Newton and Leibniz were both correct in their criticism of one another's work. Newton's fiction of an absolute pole of rest in the universe cannot be proven by any observation or by any experiment. Newton's space-time therefore has "too much structure". Leibniz's space-time, on the other hand, has "too little structure", since the absolute rotational motions characterized by Newton require a dynamic explanation. But to do that, do we have to invoke the monster of absolute space? L. Euler took the first step toward eliminating this monster by noting the close relationship of absolute space to Newton's law of inertia "lex inertiae". As early as 1748, C. MacLaurin had pointed out: "This persistence of a body in a state of rest or uniform motion can only exist in relation to absolute 41

See the article found in the posthumous papers of C. Huygens, published by D. J. Korteweg and J. A. Schouten in: Jahresbericht der Deutschen Mathematiker-Vereinigung XXIX 1920, 136; M. Jammer, see Note 26, 132 ff.

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space, and becomes comprehensible only if we assume it." 42 Euler therefore attempted, like D'Alembert in his "Traité de dynamique" and Kant in the "Metaphysische Anfangsgründen der Naturwissenschaften" to demonstrate a priori the necessity of the law of inertia by the principle of sufficient reason, to thereby indirectly prove the existence of absolute space. 43 Since Euler, in contrast to Newton, has access to an analytical formulation of mechanics with equations of motion, for him the invariance of a theory in relation to transformations also has an algebraically precise significance. Historically, however, we must bear in mind that G. Galileo, I. Beeckman and R. Descartes had already made initial formulations of a principle of inertia and relativity. 44 In 1885, L. Lange took the decisive step with his introduction of inertial systems.45 According to Lange, relative to such "inertial systems", the law of inertia retains its physical significance even without the assumption of an absolute space. For example, if we assume that three mass points are spun off from the same origin, and are then left to themselves, i.e. no force is exerted on them, the corresponding system of coordinates, relative to which the three points describe three different straight lines, is defined by Lange as an "inertial system". Then, according to Lange, the law of inertia is equivalent to the claim that every fourth mass point, left to itself, will likewise move along a straight line relative to this system. The short definition generally goes: An inertial system is a coordinate system in which Newton's law of inertia applies. Whether, technically and empirically, such inertial systems can be proven is another question. Historically, the astronomers of the 19th Century had already indicated the astronomical fundamental system as empirically good realizations of an inertial system 4 6 It is defined by the fact that the average rotational motion of the galaxies relative to this system is zero. Inertial systems are also called "laboratory systems", to emphasize that the effects

42

43

44

45

46

C. MacLaurin, Account of Sir Isaac Newton's philosophical discoveries, London 1748, Book 2, Chapter 1, Section 9. L. Euler, Theoria motus corporum solidorum seu rigidorum, Rostock/Greifswald 1765, 32. For Euler's influence on Kant, see also H. E. Timerding, Kant und Gauß, in: KantStudien 28 1923, 16-40. Beeckman's principle of inertia for translation and rotational motions of 1613 can be found in: Journal tenu par I. Beekman de 1604 à 16341-IV, ed. C. de Waard, The Hague, 1939-1953, I, 24, 256; see also E. J. Dijksterhuis, Die Mechanisierung des Weltbildes, Berlin/Göttingen/Heidelberg 1956, 366-370; J. Mittelstraß, see Note 26, 330 ff. L. Lange, Über die wissenschaftliche Fassung des Galileischen Beharrungsgesetzes, in: Ber. kgl. Ges. Wiss., Math-phys., Kl. 1885, 333-351. H. Seeliger, Über die sogenannte absolute Bewegung, in: Sitzber. Münchener Akad. Wiss. 1906, 85.

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observed experimentally in the laboratory can only be described with reference to these systems. A frequent complaint against the Newtonian formulation of the law of inertia was that in the structure of its mechanics, it would be circular to speak of "force-free" motions, if no forces are defined. It was therefore proposed that the inertial system and the mass be introduced as inertial resistance to accelerations by means of impact kinematics.47 If we assume inelastic impact processes, then in relation to the earth we have velocities ui, U2 before the impact and their combined velocity ν after the impact. For an inertial system, it is then required that the ratio of the changes in velocity ui - ν and U2 - ν must be constant in repetitions for all Ui, U2, i.e. the momentum theorem for impact processes is defined: If the ratio of the change in velocity is constant, a ratio of the masses of the impacting bodies can be defined by the inverse ratio of the changes in velocity. Observable deviations from the constancy of the ratio of the changes in velocity for impacts can be explained by interference factors. In any case, it is mathematically decisive that four-dimensional spacetime, in addition to causal structure and metric, has an affine geometry, whose ("time-like") straight lines (i.e. those which do not lie in the layers of simultaneity) represent free motions (Figure 1). This requirement correctly expresses the law of inertia in the language of geometry: A mass point which moves freely, without external influences, experiences a uniform translation. In the space-time model illustrated in Figure 1, its "world line" (Weyl) is a straight line with linear time functions x¡ = x¡(t) as space coordinates. The inertial systems with right-angle coordinate systems, an initial momentum and a length and time unit are connected by transformations of the Newtonian group Gnew· However, two inertial systems in no way need to be at rest relative to one another, but the one can execute a uniform translation in relation to the other. The space coordinates x¡ and x¡' of two inertial systems I and I' are therefore connected by a transformation of the "Leibniz" group Gkjn. It should thereby be noted that the x¡' are transformed into linear time functions, if linear functions of time are used for x¡. In addition to the transformations of the Newtonian group Gnew> therefore, the only new transformations are x,' = Xj + v¡t and t' = t, whereby the constants v¡ are the components of the translation velocity of the inertial system I in relation to I'. In the model illustrated in Figure 1, the uniform motions (e.g. Pi, P3) are distinguished from accelerated movements P2. But no absolute rest, i.e. verticals, are defined.

47

See also P. Lorenzen, Zur Definition der vier fundamentalen Meßgrößen, in: J. Pfarr (ed.), Protophysik und Relativitätstheorie, Mannheim/Vienna/Zürich 1981, 30.

3.1 Symmetries of Space and Time

245

However, the parallelism of the 4-dimensional vectors of the 4-dimensional space-time model is defined (i.e. A||B and C||D). With regard to the metric of this space-time, for each layer of simultaneity, i.e. for each three-dimensional Euclidean space of simultaneous events, the Euclidean metric applies. Therefore the Euclidean metric is only defined for the four-dimensional vectors of the space-time which lie in one layer of simultaneity. If two four-dimensional world points A and Β do not lie in the same layer of simultaneity, there is a certain time difference between them, namely the time component of the four-dimensional vector ÁB. It is determined by the selection of the measurement unit for the time, and is a linear form t(AB) of the Vector AB. A is earlier than, simultaneous with or later than Β if and only if t(AB) > 0, t(AB) = 0 or t(AB) < 0. Using the terminology of H. Weyl, we also speak of the Galilean metric of space-time in Figure 1, which is determined 1) by the linear form t(r) of the duration of the displacement r, and 2) for t(r) = 0 by the Euclidean metric. 48 It is immediately apparent that in addition to the law of inertia, Newton's famous 2nd Law ("Mass times acceleration equals force") is invariant under the above-mentioned transformations. The mass is a scalar quantity independent of the inertial system (See Chapter 3.21). If we differentiate the transformation x¡' = x¡ + v¡t according to time, then it is not the velocity dxj' dxj , , . d2x¡' d 2 x¡ , . , . — = h v¡, but the acceleration - —2 r = - τ2- , which is a space vector dt dt dt dt independent of the inertial system. Overall, therefore, the forces with which bodies act on one another do not change if the bodies are given a common uniform translation. It is correct that Newton's Second Law implies the law 48

H. Weyl, see Note 33, 149.

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3. Symmetries in Classical Physics and the Philosophy of Nature

of inertia. It should be noted, however, that the law of inertia alone defines the affine space-time structure of Newtonian mechanics. The other laws of mechanics do not restrict or expand this space-time geometry. We can now explain precisely what it means that the laws of mechanics are objective laws of nature, and apply "always and everywhere", or as they said in the 17th and 18th Centuries - "eternally". They are objective in the sense of invariance in relation to a transformation group which is designated the Galileo group Ggai below.49 Each inertial system represents a potential laboratory in which the laws can be verified. Their form does not change if the test results from one laboratory are recalculated for the inertial system of another laboratory according to the transformations from Ggai. Obviously, their symmetry structure is more special than the "Leibniz" group Gidn, but more general than the "Newton" group Gnew, i.e. G n e w C Ggai C G|dn. The Galileo group therefore characterizes classical mechanics correctly in the sense of Klein's "Erlanger Program". Finally, its transformations should once again be summarized individually and some of its group theory characteristics noted:50 1) The transition from an inertial system I to a system I' shifted in space around the vector a¡ is given by the transformation x¡ ' = x¡ + a¡ and t' = t. This space transformation is obviously a function of three parameters, namely the components of the space-time constant vector a¡. 2) The transition from a system I with the coordinates x¡ to a system I' with a rotated coordinate system is given by the transformation x¡' = a^Xk and t' = t with orthogonal matrix a^aik = δπ = a^ay and the Kronecker symbol δϋ = 1, if i = 1 and 6¡i = 0 otherwise. In vector notation, it is also abbreviated Γ = Rr with the orthogonal rotation matrix R, whose components and other conditions we have already investigated in Chapter 2.31. The rotations in three-dimensional space are also a function of three independent parameters. 3) The transition from a system I to a system I' displaced by the constant time interval b is given by the transformation t' = t+b and x¡' = x¡, which is a function of one parameter, the constant b. 4) The transition from a system I to a system I' displaced in relation to it at a constant velocity v¡ is given by the transformation x¡' = x¡ + v¡t and t' = t. These transformations are a function of three parameters, namely the space-time constant components v¡ of the vector of the relative velocity of I compared to I'.

49 50

See also H. Weyl, see Note 33,147; J. Ehlers, see Note 33, 75. See also P. Mittelstadt, see Note 19, 47 f.

3.1 Symmetries of Space and Time

247

From transformations 1) - 4) we can also indicate the most general form of a Galileo transformation, namely r=Rf+vt+a and t' = t + b, which is a function of a total of 3 + 3 + 1+ 3 = 10 parameters, namely 3 parameters for a, 3 parameters for R, 1 parameter for b and 3 parameters for v. The Galileo transformations, with reference to the sequential execution of transformations, form a continuous 10-parameter Lie group: Since the elements are a function of a, R, b and v, we can also write the general group element as σ=σ(a, R, b, v)=(a, R, b, v). The identity transformation is the identity element ι = (0, 1,0, 0) of the group. The group operation has the form σ'·σ=(Γ, R', b', v')(a, R, b, v)=(a'+R'a+bv', R'R, b'+b, v'+R'v). Obviously, the sequential execution of two transformations from Ggai again results in a transformation from Ggai. For each group element a=(a, R, b, v) an inverse element a _1 =(-R~ 2 (a-bv), R - 1 , -b, -R _ 1 v) can be indicated, which satisfies the requirement σ · = σ _ 1 · σ = t. Several interesting subgroups can now be identified in Ggai·51 Corresponding to the transformations 1) - 4) there are the following subgroups·. 1) the 3-parameter (Abelian) group Gt of the space translations, 2) the 3-parameter group Go of rotations in space, 3) the 1-parameter (Abelian) group G t of the time translations, 4) the 3-parameter (Abelian) group G 0 of the pure Galileo transformations. Additional examples of subgroups are the Euclidean group G E = G J X G D , which all contain space translations and rotations, and the subgroup U = G t xG 0 from the time translations and the pure Galileo transformations. U is important from a-group theory point of view, since it is the maximum abelian invariant subgroup of Gga]. It can thus be shown that the Galileo group Ggai is not "simple" in the sense that it cannot be broken down into other groups. It can be proven that the Galileo group has the product representation GGAI = ( G D X G T ) X U . We say that it is the semi-direct product of an (Abelian) group U with the (semi-direct) product of an (Abelian) group GT with GoIt should now be noted that the Galileo group Ggai and thus the space-time symmetry of classical mechanics is significantly more complicated than the symmetry of the Lorentz group which we first encountered in connection with electrodynamics.

51

For proof, see also M. Hamermesh, Group Theory and its application to physical problems, Reading, Mass. 1962.

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3. Symmetries in Classical Physics and the Philosophy of Nature

3.2 Symmetry and the Classical Physics of Forces Which forces can act in Galilean space-time? Newton's 2nd Law of classical mechanics is a general pattern for laws of force. It can be verified experimentally by employing special force functions (e.g. of gravitation, electrostatics, magnetostatics, electrodynamics), to calculate their mechanical effects as changes of momentum. These force functions are defined mathematically with field vectors which are introduced by means of differential or integral equations. The standard mathematical method used to solve these equations is indicated by the potential theory of Lagrange, Gauss, Poisson, etc. This mathematical theory played a key role in 19th Century physics, since it explained the transition from a physics of action at a distance to modern field physics. For the first time, an old notion of the philosophy of nature is transformed into a concept of physics, which we have already encountered in Chinese Taoism and hellenistic Stoicism: The microcosm does not consist of atomic bodies in empty space, between which there are actions at a distance, but of a number of continuous, overlapping fields. The symmetry of electromagnetic fields in turn is the core problem for the unification of the various forces in Newtonian physics. Maxwell's electrodynamics, for example, are not Galileo-invariant, so that a revision in 4dimensional vector analysis becomes necessary. The symmetry of electrodynamics is in no case a simple question of mathematical elegance. Only the symmetry or covariance of its equations makes clear the fundamental consequences of electrodynamics. Magnetic and electrical phenomena are only components of a new, general physical quantity, the electromagnetic field tensor. We can now introduce an invariant (Lorentz-invariant) function of electromagnetic force. But to calculate its mechanical effects, the concept of momentum must be modified relativistically. The unification of the forces of nature is therefore only possible at the expense of a revision of Newton's theory of nature.

3.21 Newton's Program of Forces Newton took the step from geometry and kinematics to dynamics by introducing the concept of forces. We therefore require the measurement of length, time and mass and the related concept of the inertial system.52 Newton's Second Law indicates a method by which the change in the momentum of a body under different influences can be determined: The change in mo52

For the history of the term "inertial system", see K. Mainzer, Inertial Systems, in: J. Mittelstrass, Enzyklopädie Philosophie u. Wissenschaftstheorie II, Mannheim 1984,237-238.

3.2 Symmetry and the Classical Physics of Forces

249

mentum — (mv) in time is equivalent to the force F. In the world in which we live, we initially sense force as pressure or tension in our muscles. But force - and this represented one of Newton's central insights over his predecessors - not only causes a modification of the quantity of momentum, but also of the direction of momentum.53 As a laboratory system, let us consider a suitable inertial system with space coordinates x¡ and time coordinate t. k force function is then defined as a function F, which is a function of a position vector r = (χι, X2, X3), velocity vector ν, time coordinate t and (sometimes) a parameter λ (for mass or charge) of a moving body, and satisfies the functional equation: (1)

F(X > ? I v,t) = ^(mv).

We can abbreviate by writing F = ma with the acceleration a =

dv d i '

since the mass m is independent of the time. It is initially a question of defining that functional solutions of equation (1) are called "force functions". It then becomes a technical-empirical question of whether there are laboratory systems and moving bodies which approximately satisfy equation (1). An inertial system connected to the earth, or better yet to the sun, is a suitable candidate for that purpose. Newton's program of natural forces consists of finding force functions for the equation (1). Galileo's function of free fall or Newton's formula for gravitation are suitable examples54, as are the electrostatic, magnetic and electrodynamic forces which will be discussed in greater detail below. From (1), we get a differential equation of motion in the notation of Newton or Leibniz: (1')

F(^r,?,t) = n Ä = mF.

53

For the central role of Newton's concept of force, see. I. B. Cohen, Newton's Second Law and the Concept of Force in the Principia, in: R. Palter (ed.), The Annus Mirabilis of Sir Isaac Newton 1666-1966, Cambridge, Mass./London 1970, 143-185. The program of forces in the 17th Century is investigated by R. S. Westfall, Force in Newton's Physics. The Science of Dynamics in the Seventeenth Century, London/New York, 1971; W. Kutschmann, Die Newtonsche Kraft. Metamorphose eines wissenschaftlichen Begriffs, Wiesbaden 1983.

54

I. Newton, Philosophiae naturalis principia mathematica, London 1687, De Munde Systemate Liber Tertius, 405.

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3. Symmetries in Classical Physics and the Philosophy of Nature

By successive integration of this equation, we get solutions for the velocity and position of particles as functions of time. To solve such an equation, we must know a formula for the force F, its dependence on the position r, on the velocity r, the time t (if a function of time) and (sometimes) of the parameter λ (mass or charge) of the particle. Up to now, we have defined force functions only relative to our concrete laboratory system. In the next step, we must demonstrate their independence of the choice of the inertial system in the sense of Galileo invariance. In general, several forces act on a body in motion. Newton's program of forces can be generalized to the task of finding force functions Fi and F2 which - as we say - can be linearly superimposed, i.e. they fulfill the equation (2)

F i + F 2 = mF.

It is again a technical-empirical question of whether there are forces which, under suitable conditions, satisfy Equation (2). For example, consider an iron body in free fall which is deflected by a magnetic force. In general, Newton's program of forces consists of the task of finding forces F j j ( l < j < n ) of n-1 particles, which act on the i-th particle (i φ j) in a system of η particles. The distances r¡ of the η particles with masses m¡ are determined by the equation system η (2') mtf = £ F i j with corresponding side conditions. In short, we can say: Newton's program of forces consists of the task offindingsuitable forces which satisfy the linear superposition principle.55 To solve differential equations of forces, the so-called potential theory was developed beginning in the late 18th Century. In the following chapters, we will examine this mathematical theory, which is important for the concept of symmetry. Leibniz took the first step with his introduction of the concept of work. Leibniz spoke at the time of "living" force, and initiated a discussion of the "correct" concept of force, which ultimately deteriorated into a dispute concerning the words used for various physical quantities. In any case, Leibniz mathematically defined the work which is performed by an external force F, to move a particle from position 1 to position 2 on a curve C, by:

55

On the subject of the superposition principle, see also P. Mittelstaedt, see Note 19, 66.

251

3.2 Symmetry and the Classical Physics of Forces 2

(3)

Wi2 =

J

Fdr.

ι

Leibniz

56

knew even then that:

2

2

f=·^ y

F d r

2

?dv =

m

/

dF

v d t

m fd

,,

= τ y dîv

d

m. , t

= τ -

v

·

'·

If we designate the scalar Τ = mv2/2 the kinetic energy of the particle, then the work equals the change of the kinetic energy: (4)

W 12 =T 2 -T,.

Conservative forces are those whose work is independent of the distance over which the particle is moved between 1 and 2. Friction forces are therefore an example of the opposite of conservative forces. In the case of conservative forces, the line integral along a closed distance equals zero, i.e. the work (5)

Wi2 = φ Fdr = 0.

On the basis of Stokes's Theorem57, we get the differential equation (5')

rot F = 0.

from the integral equation (5). Thus, for purely mathematical reasons, the vector analysis leads to the conclusion of the existence of a scalar potential V with (6)

F = - grad V

which is also called potential energy. It therefore follows that: (7)

W 12 = T 2 - T 1 = V 1 - V 2 .

The time-dependent value (8) 56

57

E = T 2 + V 2 = Ti + Vi For the criticism Leibniz's derivation, see K. Mainzer, G. W. Leibniz: Principles of Symmetry and Conservation Law, in: M. G. Doncel, A. Hermann, A. Pais (eds.), Symmetries in Physics 1600-1980, Barcelona 1986, 69-75. Stokes's Theorem says that the line integral of a vector F along a closed curve C equals the surface integral of the rotation of F along the surface S, which is enclosed by C, i.e.

c

c

252

3. Symmetries in Classical Physics and the Philosophy of Nature

is called the total energy of the particle. Therefore, in the case of conservative forces, we get the law of conservation of energy.58 This law makes possible a mathematically correct expression of one principal feature of Leibniz's philosophy of nature. According to this philosophy, the world is a closed system of energy ("force"), which is of course converted from potential energy into kinetic energy (and vice versa). But in the final balance, the total energy of the world remains the same. Consider, for example, Leibniz's example of a hoisted weight, the potential energy of which is converted continuously into kinetic energy during a free fall. Here, in fact, Leibniz is able to reconcile old Aristotelian considerations with modern mechanics: The potential energy of the body recalls the "potentia" which is realized in the motion of the "actus". On the other hand, Newton emphasizes the friction forces in nature and the related continuous loss of energy, and therefore rejects the law of conservation. But behind the mathematical-physical formulation of the law of energy in the 17th and 18th Centuries there were also metaphysical and natural philosophy considerations. In Leibniz's closed world, no external intervention is necessary, while Newton requires God as an "energy supplier", who every once in a while must give a boost to His creation. Mathematically it is apparent that the knowledge of potential energy makes it possible to calculate the forces. In the following chapter, the calculation of potentials will be demonstrated as a general pattern for the solution of differential equations.59

3.22 Theory of Gravitation The first application of Newton's program of forces was his famous law of universal gravitation: Every body of mass M in the universe attracts every other body of mass m with a force F which, for any given body, is proportional to the mass of the respective body, and varies inversely as the square of the distance between the bodies:

58 59

See also H. Goldstein, Klassische Mechanik, Wiesbaden 4 1976, 4. An additional law is Newton's principle "actio = reactio": Whenever two bodies interact, the force F2i which body 1 exerts on body 2 is equal and opposite to the force F12 which body 2 exerts on body 1 : F12 = -F2i . A logical consequence of Newton's 2nd and 3rd laws is the law of the conservation of momentum.

3.2 Symmetry and the Classical Physics of Forces

253

The force of gravitation is a central force which acts along the line connecting the two mass points. The value of the constant γ was determined for the first time in Cavendish's classic experiments. 60 It is well-known that Newton's law was discovered by a derivation from Kepler's second and third laws of planetary motion. Newton generalized the law of gravitation for all bodies in the universe.61 It was confirmed by a long series of successful applications, such as the explanation of Kepler's laws, the ebb and flow of tides, and the prediction of unknown planets.62 Since Newton refrained from offering any explanation of gravitational forces on account of his empirical methodology, the image in his "Principia" is of empty absolute space, in which bodies with mass can act on one another by arbitrarily rapid transmissions of force. The only thing that could possible serve as a medium or carrier for these intangible forces at a distance was the "Sensorium Dei" of the otherwise empty absolute space, which gave somewhat mystical overtones to Newton's theory of gravitation, in spite of its apparent confirmations. Continental physicists and philosophers such as Leibniz and Huygens therefore sought a theory of matter which could explain the effect of gravity. But Newton also commented on the interactions between his hypothetical corpuscles of matter: "It also seems to me," he writes in Optics "that these particles possess not only inertia and thus are subject to the passive laws of motion which quite naturally originate from this force, but they are also moved by active principles, such as gravity, or the origin of fermentation and the cohesion of bodies. I do not consider these principles to be concealed qualities, which are caused by the specific form of things, but as general laws of nature, according to which the things are formed." 63 He makes a distinction between empty (absolute) space, particles and forces, the quality of whose interaction is a function of the distance between the particles: "From the relationship of the particles, I would rather conclude that the particles of the bodies are all mutually attracted with a force which is very great when the bodies are in direct contact and results in chem60

61

62

63

For modem experiments, see also R. D. Rose, Η. M. Porter, R. A. Loury, A. R. Huhltau, J. W. Beams, Phys. Rev. Lett. 23 655 (1969). See also C. A. Wilson, From Kepler's laws, So-called, to Universal Gravitation: Empirical Factors, in: Arch. Hist. Ex. Sci. 6, 1969/1970, 89-170. I. Β. Cohen, The Newtonian Revolution, with illustrations of the transformation of Scientific Ideas, Cambridge, Mass. 1980; M. Jammer, Concepts of Force. A Study in the Foundations of Dynamics, Cambridge, Mass. 1957, New York, 1962; K. Mainzer/J. Mittelstrass, Isaac Newton, in: J. Mittelstrass (ed.) Enzyklopädie Philosophie und Wissenschaftstheorie 2, Mannheim/Vienna/Zürich 1984, 997-1005; E. MacMullin, Newton on Matter and Activity, Notre Dame Ind. 1978; P. M. Rattansi, I. Newton and Gravity, London 1974. I. Newton, Optik (See Note 25), Volume 3, 143.

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3. Symmetries in Classical Physics and the Philosophy of Nature

ical action at a small distance, but which at greater distances has no notable effects." 64 According to Newton, therefore, there is only one force which can be differentiated qualitatively. This Newtonian theory of a unified theory efforces of matter is related by R. Boscovich in 1758, in his "Theoria Philosophiae Naturalis " 65 to Leibniz's theory of monads. According to Boscovich, matter consists of identical points without extension, which have no properties other than inertia and the capability of mutual interaction by means of forces, the quality of which is a function of the distance between them. Boscovich's points of matter are therefore three-dimensional versions of Leibniz's monads. Like Leibniz, a law of continuity of the transmission of force is assumed. If the particles had a finite extension, then there would be a discontinuous change of the density of the matter on their surface. In the event of absolute contact between the particles of matter, the velocities would also have to change discontinuously. The force between two particles is therefore described by a continuous function of the distance between them, which for very short distances tends toward infinite repulsion, and for increasing distances becomes repulsive and attractive in alternation, and finally, for macroscopic bodies, makes the transition into a law of attraction in inverse relationship to the square of the distance, according to the pattern of Newton's law of gravitation (Figure 1). Force

Repulsion Distance

Attraction

Fig. 1

Therefore Boscovich attempts to explain with a unified continuous force function all interactions of matter, from the gravitation of macroscopic bod64 65

I. Newton, Optics (See Note 25), Volume 3, 135. R. G. Boscovich, Philosophiae naturalis theoria reducta ad unicam legem virium in natura existentium, Vienna 1758, Zagreb 1974, Engl/Lat. A Theory of Natural Philosophy, Chicago/London 1922.

3.2 Symmetry and the Classical Physics of Forces

255

ies, through the aggregates of matter and chemical affinities and cohesions, up to the attractive and repulsive forces of magnetism and electricity.66 The only confirmed mathematical-physical theory which existed was Newton's theory of gravitation, which meant that Boscovich's hypothesis remained speculative. But it was still the first step toward replacing Newton's ominous empty space as a medium for the transmission of forces at a distance with a theory in which matter is continuously filled by centers of force. Kant continues this trend in the theory of matter in his "Metaphysische Anfangsgründe der Naturwissenschaften" (1786).67 For example, Kant rejects the assumption of point singularities such as Boscovich's monads as the carriers of the effects of forces, and defines matter itself as a continuous dynamic force field, the stability of which is guaranteed by the equilibrium between attractive and repulsive forces. From the original mechanical atomistic explanation of matter, a physics of continuous fields of matter is now initiated with the assumption of central forces. Thus an outlook was once again established in natural philosophy, which had come to dominate mechanistic atomism both among the Stoics and in Chinese Taoism. It should also be noted that Schelling 's philosophy of nature stands in this tradition, when in the first draft of his natural philosophy in 1799, partly under the influence of Leibniz, he writes: "Therefore there must be something in experience which, although it is not itself in space, would still be the principle of all filling of space" 68 Schelling speaks of "monads of nature", which are pure action, with the tendency to fill space. Thus he arrives at the assumption of a formless, continuous fluid, which at least intuitively comes very close to the modern concept of a field. This transformation of natural philosophy from mechanistic atomism to dynamic field theory was promoted by 18th Century developments in the history of science. We should recall the physics of fluids of Newton, D. Bernoulli and D'Alembert. The high-water marks of these developments are Daniel Bernoulli's and Euler's books on hydrodynamics. 69 Euler in particular rejects a mechanistic molecular theory of fluids. For Euler, hydrodynamics becomes a field theory in which the fields of motion of a fluid are determined by the velocities of the fluid at each point, and can be described overall by partial differential equations. The theory of differential

66 67

68 69

R. G. Boscovich, see Note 65, 15. I. Kant, Metaphysische Anfangsgründe der Naturwissenschaften, in: Ak. Ausg. IV, 2. Hauptstück. Metaphysische Anfangsgründe der Dynamik. F. W. Schelling, Sämtliche Werke III, Munich 1858, 21. For the role of hydrodynamics in the derivation of the field theory of matter, see also M. B. Hesse, Forces and Fields. The Concept of Action at a Distance in the History of Physics, New York 1962,192.

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3. Symmetries in Classical Physics and the Philosophy of Nature

equations, which underwent rapid development in the 18th Century, is the mathematical tool for the study of the behavior of continuous fields. The standard methods for the solution of such equations were supplied by the potential theory with its methods for the calculation of fields and potentials. After Newton's theory of gravitation had been transformed by the potential theory into a mathematical field theory, analogous laws and concepts could be introduced for electrostatics and magnetostatics.70 The potential theory was therefore a useful heuristic tool for the development of new physical field theories. It also turned out to be the mathematical framework in which the laws of various theories of physics have the same mathematical form. First let us investigate the field theory formulation of Newton's theory of gravitation. Consider the gravitation effect of a test body with mass m and a constant distribution of mass p(r") in a volume V' (Figure 2):

70

For the theory of gravitation, J. L. Lagrange shows that the components of attraction can be expressed mathematically by partial differential quotients. (Mém. de Berlin 1777,155). P. S. M. de Laplace applied the method to the attraction of a continuous mass (Théorie des attractions des sphéroïdes et de la figure des planètes, Mém. de Paris 1782). The Laplace Equation was used in the 1782 experiment and in the 'Traité de mécanique céleste". The Poisson Equation was introduced by D. Poisson in 1812 (Bulletin de la société philomatique t. 3, 388). But C. F. Gauss was the first to supply a correct proof of the equation in: Allgemeine Lehrsätze in Beziehung auf die in verkehrtem Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs-and Abstoßungskräfte (1840). Ostwald's Klassiker der exakten Wissenschaften Nr. 2, Leipzig 1889, 12 ff. J.-C. Maxwell includes a short application of potential theory in his mechanics textbook "Matter and Motion" (1877), New York 1952, 114 f. For a modem application, see S. Flügge, Lehrbuch der theoretischen Physik I, Berlin/Göttingen/Heidelberg 1961, § 23.

3.2 Symmetry and the Classical Physics of Forces

257

For the force of a volume element dV' at the position Γ with a mass p ( r ) d V ' on a test mass m at the position r, Newton's law of gravitation gives the following infinitesimal element: |r — r | z

|r — r |

The vector notation shows that the force acts opposite to the unit vector -ρ—^-j-. The total force of the distribution of mass on the test particle is indicated by the integral function (v\ (2)

dV'p(r)(r-r) v(mA = - γ — F(m,r) π ι //-dV'pÇfXi — — |3 J r-r

which is a function of the position vector r and the mass m of the test particle. The vector g(r) of the gravitation field of a distribution of mass p(r) can now be defined by (3)

F(m,r) = mg(F).

Since gravitation is a conservative force, we get (4)

F(m,r) = - grad V(r)

with the potential energy V(r) of the test mass m in the gravitation field. We can easily calculate the following equation for the potential energy: (5)

Vm

vm

fW'PF)

In the same manner, we define the potential 2 at' 2 But if we differentiate the transformation equations, we get the classical addition theorem of velocities. A simple calculation then shows that the wave equation is not Galileo-invariant. Naturally, that is true not only for electromagnetic waves, but also for sound waves, for example. But for sound waves, the lack of invariance is not surprising. Sound is transported by a medium, e.g. air. Sound waves are compressions or dilutions of air or another medium. The wave equation indicated above obviously corresponds to a reference system, relative to which the medium is at rest. But the decisive difference from electromagnetism is that waves such as sound waves are phenomena of a medium, the characteristics of which are determined by classical mechanics. The existence of clearly distinguished reference systems can be traced to the multilayered movements of the carrier medium (e.g. air). On the other hand, the fiction of an ether as a "medium" for electromagnetic waves is a genuine ad hoc hypothesis, since it is not intended to explain anything other than the propagation of waves. If the equations of electrodynamics must be accepted as correct because they have been convincingly confirmed experimentally, there remain only two possibilities for the space-time symmetry of this theory: a) Galilean symmetry relates only to classical mechanics, while for electrodynamics, a clearly distinguished reference system must be hypothesized, in which the ether is at rest. b) There is a common space-time symmetry principle for classical mechanics and electrodynamics, but it cannot be Galilean. Such an assumption would require a modification of the equations of classical mechanics. On the basis of a series of experimental results which argued against the existence of an ether, Einstein selected the second possibility. We will discuss the realization of this program in detail in Chapter 4.1. In the following

3.2 Symmetry and the Classical Physics of Forces

281

section, we shall restrict ourselves to the space-time symmetry of electrodynamics. It can be derived from two of Einstein's requirements: 106 a) Relativity postulate: The laws of physics and the results of experiments in a given Euclidean reference system are independent of translation motions of the reference system. We can also say: There are an infinite number of Euclidean reference systems ("inertial systems") which move in a straight line and at a constant velocity relative to one another, and in which the laws of physics and the results of experiments are indistinguishable. From the age of Copernicus through the laws of mechanics, and up to the Michelson-Morley experiment, the history of physics has met this requirement, which makes the search for motions relative to the ether superfluous. b) Postulate of the constancy of the speed of light: The speed of light is independent of the motion of its source. While this postulate was altogether speculative in Einstein's day, it is now an empirical fact which has been confirmed experimentally. To find the suitable transformation group, let us consider two inertial systems I and I' with space coordinates x, y, ζ and a time coordinate t, or x \ y \ z' and t'. As an actual laboratory system I which meets our requirements, let us select the solar system, which firmly connects three cartesian coordinates x, y, ζ and the Galilean time coordinate t with the sun. The coordinate axes are parallel, and I' moves relative to I at the velocity ν in the positive z-direction. At the time t = t' = 0, the coordinate origins may coincide. In I, we assume a light source at rest, which is therefore moved relative to I' in the negative z-direction at a velocity v. If the light source flares up and down rapidly at the time t = t' = 0, it follows from Einstein's second postulate that an observer in both systems I and I' sees the propagation of a spherical light source from the origins at velocity c. The wave front reaches the point (x, y, z) in I at the time t, which is determined by the equation c 2 t 2 (x2 + y 2 + z 2 ) = 0. Likewise, the wave front in I' is determined by c 2 t' 2 - (x' 2 + y' 2 + z' 2 ) = 0. It follows from Einstein's relativity postulate that spacetime is homogeneous and isotropic. Therefore the connection between the

106

See also A. Einstein, H. A Lorentz, H. Minkowski, Das Relativitätsprinzip. Eine Sammlung von Abhandlungen, Leipzig/Berlin 1913, 5 1923, repr. Darmstadt 1974. It is of historical interest that in 1887, W. Voigt introduced a system of coordinate transformations (Ges. Wiss. Gött. 4, 41 1887, repr. Phys. Z. 16 1915, 381), which is quite similar to the Lorentz transformations. The addition of the velocities from Voigt's transformations is identical to Lorentz's result. But the electrodynamic consequences are different, since the deflection of the electrons which are emitted by radium in the presence of an electrical and magnetic field and which were investigated experimentally by W. Kaufmann (Phys. Z. 4 1902, 54; Gött. Nachr. Math. Phys. Kl. 90,1903) can of course be explained by Lorentz, but not by Voigt. See also A. G. Gluckman, Coordinate Transformations of W. Voigt and the Principle of Special Relativity, in: Amer. J. Phys. 36, Nr. 3, 1968, 226-231.

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3. Symmetries in Classical Physics and the Philosophy of Nature

two coordinate quantities must be linear. The two quadratic forms in I and I' are then related to one another by (1)

c 2 t' 2 -(x' 2 +y' 2 +z' 2 )=X 2 (c 2 t 2 -(x 2 +y 2 +z 2 ))

where λ = λ(ν) designates the possible scale change between the two reference systems. By means of a suitable selection of the orientation of the axes and investigations of the inverse transformation of I' to I, it can be demonstrated that λ(ν) = 1 for all velocities v. It turns out to be mathematically appropriate to note the coordinate values of space-time by the coordinates of a 4-dimensional (flat) manifold ( "fourcoordinates "): (2)

(x a )=(ct, x, y,z)=(ct, r).

Space-time coordinates in I' then result from I by means of the Lorentz transformations (3)

ν x 0 ' = γ ( χ 0 - βχι) with β = - , β = |β| χ,' = 7 ( x , - ß x 0 ) a n d x 2 ' = x2 x 3 ' =x 3

γ=(1-β2)."1

The inverse transformations are accordingly (3')

χ0 χι x2 x3

= = = =

γ(χ 0 ' + βχΓ) γ(χι' + βχ 0 ') x2' x3'

In the laboratory system I (e.g. the sun), the metric ("Minkowski metric ") (4)

ds2 = g a pdx a dxp = c 2 dt 2 -dx 2 -dy 2 -dz 2

is defined with the metric tensor (5)

It remains invariant when the Lorentz transformation is used ("Lorentz invariance"). After we have already introduced four-coordinates (2), four-vectors (6)

(a«) = (a«, a x , a y , a z ,)

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3.2 Symmetry and the Classical Physics of Forces

with Lorentz-invariant "length" act2-ax2-ay2-az2=act 2-ax 2-ay 2-az

(7)

2

can also be defined by analogy to the vectors in three dimensions. To simplify the following calculations, the length and time units are selected so that the speed of light c = 1. By analogy to the 3-dimensional differential operators, we must now define 4-dimensional operators. With the conventional 3-dimensional Voperator, we get the Lorentz-invariant forms:107 (8)

Scalar product:

a ^ ß = atbt — a · b

Gradient operator:

V a = ( , — V) dt "dt 3a c t -> V a ad = — + V · a at

Divergence: Laplace operator:

, • = V V a

a

=

a—2 , a 2

The wave equations (13) from Chapter 3.25 can now be written in a simplified manner with the Laplace operator: (9.1) ϋ 2 φ = ^ εο (9.2) D2Â = - . εο On the right side of both equations, the electric charge density ρ and the current density j= (jx, jy, j z ) result in a four-vector (10)

(j e ) = (p,j)

("Four-current").

Since εο is a universal constant with the same unit of charge unit in all reference systems, ja/eo is also a four-vector. The 4-dimensional Laplace operator is Lorentz-invariant. Therefore the potentials φ and A = (Ax, Ay, Az) on the left sides of equations (9.1) and (9.2) also result in a four-vector (11)

107

(Αα) = (φ,A)

("Four-potential").

An explicit proof appears in R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. II, Reading, Mass. 2 1966, 25-35 ff. See also J. D. Jackson, see Note 77, 547 ff.; S. Weinberg, Gravitation and Cosmology. Principles and Applications of the General Theory of Relativity, New York, etc. 1972, 41 ff.

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3. Symmetries in Classical Physics and the Philosophy of Nature

The wave equations (9.1) and (9.2) can now be written in Lorentz-invariant fashion • 2 A a = —. εο We must bear in mind that it was possible to derive the wave equations (13) in Chapter 3.25 from the Maxwell equations only with the assumption of the Lorentz convention (12), i.e. in 4 dimensions (12)

(13)

V a A a = 0.

The space-time symmetry of the Maxwell equations can now be determined on the basis of the wave equations. We can also say that the Maxwell equations are written in covariant ("Lorentz-invariant") form. The fields Ejind Β in three dimensions are noted by the 3-dimensional potentials φ and A with the 3-dimensional V-operator (See (6) and (8) in Chapter 3.25) as follows: (14)

Ε=

- ^ - ν ψ

Β = V χ A. In 4-dimensional vector notation, the six components of the electric and magnetic field É = (E x ,E y ,E z ) and Β = (B x ,B y ,B z ) can be expressed by the elements of an (antisymmetrical) second- order tensor (15)

F a ß = V a Ap — VßA a

with α, β for 0, 1, 2, 3 and t, χ, y, ζ. For that purpose, we need only recall definition (11) of the four-potential A a and the gradient operator from (8) and note F a p as a matrix with its dual form F a p: -B x —Bv - B z \ /o Β, -Ey Faß = By —Ez 0 Ex -Ex 0 / \BZ The space-time symmetry of the Maxwell equations is now apparent. The homogeneous equations (4.1*) - (4.2*) in Chapter 3.25 can be combined with F a ß to /O —EX —Ey - E z \ Ev 0 —B-z By (15') F a ß = 0 -Bx Bz Ey \ E Z -Bv Bx 0 )

(16)

V«F aß = 0.

3.2 Symmetry and the Classical Physics of Forces

285

With F a p and the four-current j a from (10), the non-homogeneous equations (4.3*)-(4.4*) in Chapter 3.25 get a covariant form: (17)

V a F a ß = |iojß.

The foundation for the space-time symmetry or covariance of the field equations in electrodynamics is established with the definitions of the fourcurrent j a , of the four-potential A a , the tensor F a p and the wave equations (12)-(13) or the Maxwell equations (16)-(17). The introduction of the new notation does more than merely provide a more elegant mathematical formulation. In terms of physics, the unification of magnetic and electric fields is completed only by the covariant field tensor F a p. The magnetic and electrical phenomena are then simply components of a single physical quantity, namely of the electromagnetic field. Thus, for the first time, we have the precise mathematical-physical explanation of a unified field, which had been speculatively discussed since the Tao of Chinese natural philosophy and the Pneuma of the Stoics, through the force fields of Leibniz, Boscovich and Kant, and up to the unification theories of the Romantic philosophy of nature. But from the Lorentz-invariance of electrodynamics, it follows that Newton's program of unifying all natural forces in the context of Galilean symmetry cannot be realized. What must be changed in Newton's program of forces to preserve the symmetry of the forces of nature? In Chapters 3.23 and 3.24 on electrostatics and magnetostatics, fields were introduced as the causes of forces. Therefore we should first indicate a covariant formulation of the Lorentz force function (5) F = (F x ,F y ,F z ). The 3-dimensional velocity vector is v = (v x ,v y ,v z ) dx dy dz with vx = — ,v yv = - and v z = —-. dt' dt dt Since the Minkowski metric ds from (4) is Lorentz-invariant, there is also a Lorentz-invariance of the proper time dx, which is defined by (18)

ds = c dx.

Therefore a Galilean time element dt differs from dx by dx = γ - 1 dt with the factor γ from (3). If we then differentiate the four-coordinates x a from (2) according to dx, i.e.

286

3. Symmetries in Classical Physics and the Philosophy of Nature

we get the Lorentz-invariant four-vector (19)

(ν α ) - (γο,γν).

A simple calculation now supplies the Lorentz invariant formulation of the Lorentz force (5) from 3.25 as (20)

F a = qv ß F aß .

Newton's program of forces requires that the force function F a must be used in the functional model (1) for laws of force. The problem then arises that the left side of model (1) with the force F a is of course Lorentz-invariant, but the formula on the right side for the calculation of its mechanical effects is not. It is therefore the task of electrodynamics to modify Newton's model of laws so that it is totally invariant. With four-coordinates x a , we first introduce the four-dimensional momentum (21)

P a = mova

with the rest mass m —xn, on account of =— = mx k k 2 dxk we get the following equation: d2xk (2)

d ΘΤ =

For the left side of equation (1), let us consider the simplest case, in which the force functions can be expressed by an location-dependent potential function V = V(x k , t), i.e. (3)

dV F i (x 1 ,t) = - g r a d V = - ^ - ( x 1 , t ) .

If we now define the Lagrangian function L = L(x k , x k , t) = Τ -V(x k , t) and consider that ΘΤ J av = 0 and = 0 dxk dxk from (1), we get the Lagrangian equation of motion (4)

— _ -o 8xk dt 3xk We also get such an equation of motion if a force function is derived from a generalized velocity-dependent potential. In this case there is a potential function V = V(x k , x k , t) with (5)

„ Fl

9V J W = -gn,dV = - - +

d8V - - .

For the Lagrangian function (6) 109

L = L(x k , x k , t) = T(x k ) - V(x k , x k , t) See also P. Mittelstaedt, see Note 19, 87 ff.

3.3 Symmetry, Laws of Conservation and the Principles of Nature

289

we then get the Lagrange equation of motion 3L dXfc

d 3L dt dXk

Λ

As an example, let us consider the force of gravity F(m, r) from Chapter 3.22, which a mass at a defined location exerts on the test mass m at the location r. In this case, the Lagrangian function is (8)

L=^-V(r)

with a position-dependent potential function (See (5) in Chapter 3.22) and a Lagrange equation of form (7). Another example is the Lorentz force F(q, r, r, t), which exerts an external electrical and magnetic field on a point charge q (see (5) in Chapter 3.25). With the potentials A and Φ, from which the magnetic and electrical field can be derived (See (6), (8) in Chapter 3.25), a generalized velocitydependent potential V can be formed, which provides a Lagrangian function of the form (6) and a Lagrange equation of the form (7). In addition to the formal and technical advantages of this representation, which were indicated above and need not be discussed here in any further detail, the Lagrangian functions call for several basic philosophical comments on the physical concept of a system. For example, if a physical system is characterized by the Lagrangian function L = Τ - V, and if an equation of motion of the form (7) is applied, then L is not clearly determined by this equation of motion. A Lagrangian function of the form (9)

dt with any given function Ω which is a function only of Xk and t, also supplies such an equation of motion. In other words, the Lagrange equations of motion are invariant under transformations L —> L'. In this context, we can speak once again of gauge transformations or "gauging " (See 3.25) and designate the Lagrangian equation of motion as gauge-invariant. It is decisive, however, that as a result of such gauge transformations of the Lagrange function or of the physical system, the mechanical paths x¡(t) are not changed. But in the context of mechanics, only these paths are observable, so that the gauge transformation of a system therefore has no empirically observable consequences. The physical system which a certain equation of motion realizes is therefore definitely determined by this equation of motion only with respect to certain gauge transformations. Since the gauge transformations do not depend on a finite number of parameters, but on an arbitrary function, they form an infinite continuous group in the Lie

290

3. Symmetries in Classical Physics and the Philosophy of Nature

sense. Therefore we can also say: The observable characteristics of a physical system possess the "intrinsic" symmetry of the gauge group. In the example of electrodynamics (See (9) in Chapter 3.23), the gauge transformations of the magnetic and electrical potentials A and φ were also not empirically observable phenomena.110 The Lagrange equations of motion of mass points can be derived from what is called a variation problem. This proposal goes back to W. R. Hamilton ( 1805-1865) and is therefore also called Hamilton's principle.111 Hamilton starts with the question of what properties characterize a distance r = r(t) traversed during a finite interval of time tj < t < t2, compared to other possible and locally varied trajectories r(t)+6r (Figure 1).

Fig. ι If L = L(xk(t), Xk(t), t) is the Lagrangian function of the problem in question, with the independent variable t and the dependent variables xk(t) and Xk(t), then the solution of Hamilton's problem proceeds to an investigation of the functional t2 W(x k (t)) = J L(x k (t),x k (t),t)dt ti of the possible mechanical paths xk(t) between times ti and t2. Thus W(xk(t)) is also called the Hamilton's action integral}n Hamilton's principle then says: Between two times ti and t2, the motion of a system of mass points proceeds so that for the actual paths xk(t), the action integral assumes an extreme value, i.e. the variation is t2 (10)

δ J Ldt = 0. tl

110 111

112

See also Note 101. See also W. Buchheim, William Rowan Hamilton und das Fortwirken siner Gedanken in der modernen Physik, in: NTM Heft 12, 5. Jg. 1968,19-30; 6. Jg. Η. 1 1969,43-60. See also P. Mittelstaedt, see Note 19, 98 ff.

3.3 Symmetry, Laws of Conservation and the Principles of Nature

291

The Lagrange equations of motion are the direct result. This approach can be generalized for systems with a finite number of degrees of freedom. A system with η degrees of freedom is characterized by a Lagrangian function L(q k , qk, t) with generalized coordinates q k (t) and generalized velocities qk(t). Its development over time is described by the Lagrange equations: (11) 3q k

dt 3q k

In this case, therefore, the physical events correspond to the solutions q k (t) of a system of second-order differential equations. The event which actually occurs corresponds to a special solution which is unambiguously defined by initial conditions for q k (0) and qk(0)· In place of Newtonian causality, which spoke of "forces" as the cause of effects or phenomena, there is now a formal system of equations which unambiguously determine the development of a physical system over time in the configuration space defined by position and time coordinates. 113 In the alternative Hamiltonian representation, the velocity coordinates are replaced by momentum coordinates. 114 In the absence of magnetic fields, the momentum of the k-th mass point is given by -

dL

Pk = m k v k , in the other case by pk = ^—. oqk

From the Lagrange equation of motion (11), we get the differential „

ν-1 dL

3L

.

3L "λ

= Σ

k

Pkdqk + X Pkdqk +

k

-^dt. 01

From d ( £ p k q k ) = £ q k d p k + ^ P k d q k k k k

we

§ e t t h e difference

d(XPkl and I a I =+1. While the first requirement guarantees that the direction of time is conserved, the second requirement selects the continuous transformations. In other words, it is impossible to jump from a$+l or from | a | = 1 to I a I =+1 by a constant change of parameters. For identity, a$=+l and |a|=+l. The full homogeneous Lorentz transformations have, as an additional subgroup, the rotations with a$=l, a$=a®=0 and ap=R a p, whereby α, β stands for 1, 2, 3 and R 0 p is a unimodular orthogonal matrix (with | R | = 1 and R'R =1). The fact that the metric Minkowski tensor η α β is invariant with respect to space-time rotations is an expression of the isotropy of the Minkowski world; the fact that it is invariant with respect to spacetime translations x a —» x a + a a expresses the homogeneity of the Minkowski world. The restricted (discrete) Lorentz transformations contain the space reflections with a{]>l and | a | = - l , and the reversal of time with a$ χ'. A general gravitational field is therefore described in the language of tensor analysis by apseudo-Riemannian manifold M with local Minkowski metric, which replaces the local Euclidean metric in a Riemannian manifold (See Chapter 2.33), and can be represented in the corresponding tangential space

With Grossmann's assistance, Einstein used the general field equations, in which the gravitational potentials gHV were to be combined with the previously disregarded causes of the gravitational field, the gravitating masses. Since these field equations, as an objective law of physics, had to be invariant in relation to any given transformations >ϋμ, they had to be formulated mathematically with covariant tensors. But derivations of tensors, e.g. gμv, generally do not lead to tensors in return. As long ago as 1869, Christoffel had used the expression Γμν to define "covariant differentiation", so that the derivation is again a tensor. There11

See also J. Ehlers, The Nature and Structure of Spacetime, in: J. Mehra (ed.), See Note 9, 82 ff.; J. Audretsch, Ist die Raum-Zeit gekrümmt?, in: J. Audretsch/K. Mainzer (ed.), See Note 1.

4.1 Symmetries in the Theory of Relativity

355

fore Einstein attempted to formulate the field equation with the gμv and their covariant derivations to thereby guarantee their covariance. In fact, according to Christoffel and Riemann, from the metric tensor gμv and its first and second derivations, a new tensor R^VK ("RiemannChristoffel tensor") can be introduced. Thereby is the Ricci form := an of the Riemann-Christoffel tensor, Rλμvκ gλδRμvκ d Κ:=8μκΚμκ a scalar. R|xVK characteristically expresses the action of the gravitational field. Specifically so that gμv is equivalent to the Minkowski metric η α β of the special theory of relativity, R^VK must be equal to 0 (and at one point, the matrix g ^ must have three positive eigenvalues and one negative eigenvalue). Mathematically, R^VK characterizes the 4-dimensional pseudo-Riemannian manifold with the metric tensor gμv in the same manner as the curvature Κ characterizes the 2-dimensional Gaussian surfaces. For example, the metric ds 2 in n-dimensional manifolds is invariant with respect to R^VK, as in 2-dimensional surfaces with respect to K. For R^VK in the 2-dimensional case, „ βμν = 8μν

R1212

,η 0 Rl212 Τ and R = 2 r . gllg22 — gf 2 gllg22 - g f 2 R R1212 With Κ = — — = the Gaussian curvature Κ is clearly defined by the Riemann-Christoffel tensor. The designation "curvature tensor" for R^VK results from this special case. Geometrically and intuitively, this designation is therefore justified for the Riemann-Christoffel tensor only in the 2dimensional case. 12 Variable curvature tensors which change from point to point of a manifold are an indication of inhomogeneous gravitational fields, which change at defined space-time intervals (e.g. the fluctuations in the sun's gravitational field). The following analogy can be used to describe the relationship between Gaussian-Riemannian differential geometry and Einstein's theory of gravitation: Just as, locally, there is a Cartesian coordinate system with a Pythagorean metric for a Gaussian coordinate system with the metric gμv (i.e. in the "infinitely small"), an inertial system can be indicated for all co-

12

On the philosophical problem of "curvature", see also H. Reichenbach, Philosophie der Raum-Zeit-Lehre, Berlin/Leipzig 1928, Braunschweig 1977, § 8, § 34-42; likewise Α. Grünbaum, Philosophical Problems of Space and Time, Dordrecht/Boston 1973; M. Jammer, Das Problem des Raumes. Die Entwicklung der Raumtheorien, Darmstadt 1960, 176 ff.; B. Kanitscheider, Vom absoluten Raum zur dynamischen Geometrie, Mannheim/Vienna/Zürich 1976, Chapters VI, X.

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4. Symmetries in Modern Physics and Natural Sciences

ordinate points of a gravitational field locally, in which the laws of the special relativity theory apply. We shall now describe a heuristic method of setting up the covariant gravitational equations by means of the general covariance principle. 13 The principle of general covariance even predicts that at each point Ρ in an arbitrarily strong gravitational field (i.e. mathematically in a pseudoRiemannian manifold) with gravitation potential g a ß (i.e. mathematically with the corresponding metric tensor), a local inertial system can be found with Minkowski tensor η α ρ, so that (4)

g

a ß

( P ) = ^

ß

a n d ( ^ )

x p

= 0.

The partial derivations of the metric tensor disappear at the point P, since the affine connection is zero. It should also be recalled that the indices α, β and γ stand for the coordinate indices 0, 1, 2, 3 of the inertial system. In the vicinity of P, the gravitational field is very weak. In the nonrelativistic case, the Poisson equation (See (8) in Chapter 3.22) (5)

Δφ = 4πγρ

defines a weak, static gravitational field. In this case, it can be shown that the Newtonian potential φ and the mass density ρ are given approximately by the time-time component goo of the metric tensor and the energy density Too for non-relativistic matter, i.e. goo——(l+2χμ, t—>t=t, in which the physical condition remains unchanged, e.g. form invariance applies for the gravitational potential gμv and the energy-momentum tensor Τ μ ν of matter, i.e. gμv=gμv and Τμν=Τμν. Mathematically, therefore, the universe is portrayed as a 4-dimensional space-time manifold, whose 3-dimensional "spatial" subspaces are isotropic and homogeneous. That was the assumption of the "cosmological principle". In terms of differential geometry, therefore, it was the assumption of an isometry group which - in purely mathematical terms - makes it possible for us to define the "cosmic" metric of the 4dimensional universe. In 1935/36, H. P. Robertson and H. G. Walker indicated the conventional standard form of this metric. This form results from the metric described in Chapter 2.33 for manifolds with isotropic subspaces, by means of a suitable selection of new coordinates, and specifically by / (-g(v))2dv=t and

362

4. Symmetries in Modern Physics and Natural Sciences

the conventional spatial polar coordinates u'=r sin θ cos φ, u 2 =r sin θ sin φ and u 3 =r cos Θ. From the metric on page 213, we get the proper time dr 2 dx2 = dt2 - R 2 ( t ) ( - — z+ r W + r 2 sin2 6dcp2) 1 — kr with R(t) := y/f(y). The 3-dimensional spatial metric of the isotropic universe can then be easily indicated for constant t by 3

grr =

3

gee = ^ ( t ) , 3 g w = r 2 sin 2 0R 2 (t) and

3 §μν

- 0

for μ φ v. Because R(t)= v /f(v) and f(v)-1 K(t) | ~ 1 , the 3-dimensional curvature scalar is 3 K(t)=k R" 2 (t). For k = -1 or k = 0, the space is infinite. For k = +1, the 3-dimensional spatial volume of the universe is V3 = 2u 2 R 3 (t)and for the geodetic circumference U3 = 2π R(t). For k = +1, therefore, the 3-dimensional spatial universe can be understood as the surface of a sphere having a radius R(t) in 4-dimensional Euclidean space. The world radius R(t) thereby increases in the expanding universe with the time t. At each instant, the universe is therefore finite (because of V3) but unlimited (because of U3). This visual interpretation of R(t) as an expanding world radius is of course only possible for k = +1. Still, in each case, R(t) defines the scale of the geometry of space and is therefore generally called the cosmic scale factor. Up to this point, the geometric description of the universe follows exclusively from the cosmological principle. R(t) remains an unknown timedependent function. To be able to verify the symmetry characteristics of the universe in terms of physics, the "radius" R(t) must be calculated. For that purpose, assumptions concerning the material characteristics of the universe are necessary, like those expressed in Einstein's gravitational equations (See (9) in Chapter 4.12). Therefore the Robertson-Walker metric must be defined as the solution of the gravitational equations, which was done for the first time by A. Friedmann in 1922. On the assumption of the cosmological principle, i.e. the assumption of a homogeneous and isotropic universe, we get Friedmann's standard models for the three possible values k = ± 1 or 0, by means of which spatial curvature is defined. Mathematically, these standard models are described by the development R(t) of the universe by means of a first-order differential equation R2 + k = ^ p p R 2

363

4.1 Symmetries in the Theory of Relativity

which can be derived from Einstein's gravitational equations, whereby the function ρ for energy density is a function of time. In each case, this is a 4dimensional space-time manifold whose 3-dimensional homogeneous "spatial" subspaces expand temporally isotropically. Figure 2 illustrates these subspaces expanding in time as "spatial cross sections" for the three possible cases of k. In the case k = -1, each spatial cross section is a 3-dimensional Lobachevski geometry L 3 with negative curvature. For k = 0, the spatial cross sections are 3-dimensional Euclidean spaces E 3 . For k = 1, they are spherical or elliptical spaces S 3 or P 3 . t

In each case, there is an initial singularity, in which the space-time curvature is infinite. Cosmologically, this is designated the Big Bang. According to this theory, the universe initially expanded very rapidly, and then continued to expand somewhat more slowly. In the case k = 1, the expansion reverses to a collapse, which represents a new singularity. We then speak of a closed universe. For k = 0 o r k = - l , the expansion continues, but more rapidly in the case k = -1. Once it has been formed, the universe remains in existence and unlimited in both cases. We can therefore also speak of an open universe.

364

4. Symmetries in Modern Physics and Natural Sciences

Historically, this assumption is represented in the Greek philosophy of nature, but there it is applied to a static, finite universe. The notion of a spatially symmetrical and temporally unlimited universe was propagated in the West by Giordano Bruno, in particular. For k = 1, there is a beginning and an end, a notion which was interpreted in the Christian tradition as creation and the end of time. In Figure 2, the curve for k = 1 is a cycloid. It is generated by a fixed point on a wheel, which in this case is rolling down a straight line, namely the time axis. Actually, therefore, it is a periodic curve which reproduces the same arc pattern in unlimited fashion. Cosmologically, we have the model of a universe which is repeatedly expanding and collapsing. Philosophically, it suggests Nietzsche's theory of the "eternal recurrence of the same". In Figure 2, numbers are noted along the curves for k = ± 1. These are R(tn) values for the parameter ("deacceleration parameter") qo = —R(to) -τ——R (to) toward the present time to, which can be calculated from the "radius" R(t), i.e. from the solutions of Einstein's gravitational equations. We thus get a first value which plays an important role in the empirical-physical verification of the cosmological assumptions of symmetry. 18 The curvature is essentially a function of the current energy density. The universe is open or closed, depending on whether po is less than or greater than a specified critical value pk- For example, the universe is open and po is less than p k if the parameter qo is less than 0.5, It is closed and po is greater than pk if the parameter qo is greater than 0.5. Thus the empirical importance of qo becomes clear. Unfortunately, the areas of the curves in Figure 2 currently known from observations are too small to decide whether the universe is open or closed. If the universe is open, then its age is equal to the reciprocal of the Hubble constant H. This reciprocal 1/H is also designated Hubble time. On the other hand, if the universe is closed, its age is only two-thirds of Hubble time. At the moment of the initial singularity, the density must have been infinite. A relic from this early hot phase of the universe is the 2.7°K background microwave radiation which was predicted as early as 1950 and discovered in 1965. It has the spectrum of a Planck black body and is completely isotropic. Therefore, the microwave background radiation can also be considered a reliable confirmation of the Friedmann standard models with their symmetry assumptions. Finally, other cosmological principles were also proposed which were used to define the metrics of other cosmological models. We should recall H. Bondi's "steady state" model of 1948 and the increased assumption of 18

S. Weinberg, see Note 6, 481 ff.

4.1 Symmetries in the Theory of Relativity

365

a spatially and temporally homogeneous and isotropic universe. We should also recall K. Gödel's assumption in 1949 of a homogeneous, non-isotropic universe.19 But if the cosmological principle of the Friedmann model is correct, then the question is 1) whether the symmetry of the universe has existed since the big bang, and 2) how it can be explained in terms of physics. Presence

Fig. 3

The problem of the first question is illustrated in Figure 3. It shows the "horizon" of an observer as that part of the universe which can influence the observer. Since signals cannot travel faster than the speed of light, an observer O in the present can only know of events which take place at a distance which could have been travelled by light since the Big Bang. In Figure 3, A and Β are two events between which there can be no causal connection, since the "horizons" of these events do not overlap. Therefore it would be possible that in the beginning, the universe was non-uniform (see wave in Figure 3), so that an expansion in B' began earlier than in A'. But the cosmic background radiation from A and Β reaching the observer O has the same intensity, and the universe is expanding at both points at the same speed. 19

S. Weinberg, see Note 6, Chapter 16.

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The cosmological principle and the theory of relativity no longer suffice to explain this regularity and symmetry of the universe. Modern cosmogony is merging with quantum mechanics and elementary particle physics into a theory of physics in which the evolution of the universe is explained by the step-wise origin of elementary particles, atoms, molecules, etc. Against this background, it can then be shown how, in the individual phases of development, some of the currently-known basic physical forces of strong, weak and electromagnetic interaction initially prevailed, until the current structure of the universe with its macroscopically predominant gravitational force arose. Modern cosmology therefore regards the universe as a gigantic high energy physics laboratory which requires a unified theory of natural forces for its complete explanation. We will return to this discussion following the chapters on quantum mechanics and elementary particle physics.

4.14 Symmetry and the Unity of Gravitation and Electrodynamics Up to this point, the global and local symmetries of the theory of relativity have been considered as characteristics of an independent theory of physics. Historically, however, the theory of relativity was also the impetus for new ideas on a unification of various physical theories. These include the attempts by G. Mie, D. Hilbert, H. Weyl and others to develop a unified theory of gravitational force and electromagnetic force which, well into the 1920s, were considered the only two basic forces of nature. These attempts, of course, soon turned out to be misguided. But they are of historic and heuristic importance from the standpoint of symmetry. For example, starting in 1915, the writings of D. Hilbert show how the idea of a unified theory of matter was closely related to Hamilton's principle. Thus a historical line of development becomes clear in the search for a "world formula", which in modern times extends from Leibniz, through Euler, Hamilton and others, and into the 20th Century. H. Weyl, in his proposal, used the mathematical technique of gauge symmetries, and although of course that failed in the concrete case of gravitation and electromagnetism, it later led, on the basis of quantum mechanics, to successful combinations of the newly-discovered physical forces. The first attempt at a unified theory of matter in the 20th Century dates back to G. Mie. In his 1912-1913 publication on "The Principles of a Theory of Matter" 20 , he attempted to define a link between the existence of electrons and gravitation. According to Mie, atoms consist of electrons which 20

G. Mie, Grundlagen einer Theorie der Materie (I), in: Ann. Phys. Leipzig 37 1912, 511534; (II) 39 1912, 1-40; (III) 40 1913, 1-66.

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are held together by a weak positive charge. Mie imagines the atoms surrounded by "atmospheres" which, because the atoms are electrically neutral, do not act on electrical fields, but on gravitational fields. The constant transitions which Mie assumes between the electrical action and the gravitational effect recall to some extent the Boscovich model of matter. D. Hilbert, in his 1915 and 1917 publications on "The Principles of Physics"21 established a unified mathematical theory of matter, in which the approaches adopted by Mie and Einstein are taken into consideration. Hilbert's proposal is also of great interest from the point of view of scientific theory, since it applies to physics the axiomatic method which Hilbert had previously explored in mathematics. Hilbert begins with a 4-dimensional space-time continuum, the points of which are characterized by four "world parameters " χ μ (μ = 0,1,2,3). The following are allowable physical factors which determine χ μ : 1) Einstein's 10 gravitational potentials gμv, which form a (symmetrical) tensor with general transformations of χ μ , and 2) Maxwell's 4 electromagnetic potentials Α μ , which form a vector with general transformations of χ μ . According to Hilbert, the physical processes in a world with these factors are determined by two axioms: The first axiom ( "Mie 's axiom of world function ") formulates a general Hamilton principle from which the laws of physics can be derived. The Hamilton function H ("world function") is a function of the gravitational potentials with their first and second derivations, and of the electromagnetic potentials with their first derivations. Hilbert's Axiom I then requires (1)

δ J H^/gd 4 x = 0,g = |gnv|,d4x = dx°dx'dx 2 dx 3 ,

i.e. the variation of the Hamilton integral vanishes for each of the 14 potentials gμv and Αμ. Axiom II ("Einstein's axiom of general invariance") requires that the Hamilton function H remain invariant with respect to any given transformations of the coordinates χ μ . As usual, Lagrange equations can be derived from the Hamilton principle. According to the definition of H, Hilbert derived 14 Lagrange equations for the 14 potentials, namely 10 gravitational equations for the 10 gravitational potentials g ^ and 4 electromagnetic equations for the 4 electromagnetic potentials Α μ . After Einstein's gravitational equations have been defined, Hilbert gets Maxwell's equations of electrodynamics in the Mie version, on the basis of a general mathematical theorem of variational theory. Accordingly, in general for the η Lagrange equations which can be derived 21

D. Hilbert, Die Grundlagen der Physik, in: Nachr. Kgl. Ges. Wiss. Gött. 1915, 395-407; 1917, 201; Mathem. Ann 92 1 1924.

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from a variational principle of form (1) with η quantities of an invariant Η over a 4-dimensional space-time continuum, four of these ones will always be a consequence of the remaining n-4. In this sense, therefore, electrodynamics can be derived from Einstein's theory of gravitation. In terms of philosophy of science, therefore, the theory of gravitation would be the more fundamental physical theory to which electrodynamics can be reduced. But a prerequisite is the assumption that the world is actually as defined by Hilbert's axioms, in particular his general variational principle. This Hamilton principle in turn acts as a "world formula", as in the discussion of Leibniz, so that we could also speak of the Leibniz-HamiltonHilbert principle. But there is a significant difference in terms of method between the variational principles of the 17th and 18th Centuries and the Hilbert version. In the tradition of Leibniz, the assumption of "world formulas" as variational principles was related to ontological requirements concerning natural processes. Hilbert, on the other hand, assumed a formal axiomatic, i.e. of the selected physical sets, the only characteristics used are those which are formally defined by the axioms. In a similar manner, Hilbert, in his "Foundations of Geometry", rejected any ontology of geometric objects (e.g. point, line, circle) in Euclid's sense. This rejection of ontology is clear in Hilbert's famous formulation, according to which we can imagine whatever we want among the basic concepts of a system of axioms, provided that our ideas satisfy the formal requirements of the axioms. The old criterion of evidence, according to which axioms are characterized by an immediate understanding of their truth, cannot be accepted in a formal system of axioms. How could that even be expected of a principle as general as (1)? Instead, for a formal system of axioms, in addition to consistency, Hilbert requires the correctness and completeness of the theory, i.e. eveiy theorem which can be correctly derived from the axioms is true ("correctness"), and conversely, every true theorem can be derived from the axioms of the theory ("completeness"). In contrast to mathematics, "physical truth" also requires experiment and measurement. Hilbert celebrates his derivation of Einstein's gravitational equations and the Maxwell-Mie equations of electrodynamics as the greatest triumph of the axiomatic method.22 Mathematical elegance, methodical simplicity and beauty were for him the motivation for a unified theory of matter. Such a unified theory would be the secularized version of an ideal of natural philosophy, which since the days of Pythagoras had linked harmony and beauty to mathematical regularity. Of course Einstein accepted that Hilbert had derived the gravitational equations independently from the Hamilton variation principle. But he crit22

D. Hilbert, see Note 21 (I), 407.

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icized Hilbert's formal approach. In a letter to Weyl (dated November 23, 1916)23, he called Hilbert's ansatz "childish", and said that Hilbert was "just like an infant who is unaware of the pitfalls of the real world." It was a mixture of physically-justified considerations (such as Einstein's general relativity principle) and risky hypotheses concerning the structure of matter (in the sense of G. Mie) which he, Einstein, could not accept. In Einstein's criticism of Hilbert, we see a precise reflection of Newton's dictum "Hypotheses non fingo", which Newton had invoked in his objection to the hypotheses of natural philosophy concerning the structure of matter put forth by Descartes, Huygens and Leibniz. In place of the ontology of nature of the followers of Descartes and Leibniz, we now have the formal axiomatic of mathematical and physical theories. W. Pauli also praised Hilbert's derivation of the gravitational equations as an independent achievement, but at the same time he criticized Hilbert's presentation, which was unacceptable to a physicist for two reasons.24 First, the variational principle is introduced as an axiom, and second, the field equations are derived by Hilbert not for any given system of matter, but under the special condition of Mie's theory. Pauli wrote his criticism in 1921, already under the influence of quantum theory. It did not seem that Hamilton's principle could be applied in microphysics in the same way physicists were used to using it in macrophysics. Therefore the belief that a universal "world formula" would have the form of the Hamilton principle seemed to have come to its natural end in the theory of relativity. Hilbert's contribution would have been the conclusion of a modern development which had begun with Leibniz.25 In any case, as a result of the development of quantum mechanics in the 1920s, Mie's theory of matter turned out to be altogether obsolete. In the Göttingen school of mathematicians which included D. Hilbert, E Klein, E. Noether and others, invariance principles were seen in close connection with physical laws of conservation. We should recall Noether's results for classical mechanics and special relativity theory.26 In electrodynamics, the conservation of electrical charge can be explained by gauge 23

24 25

26

"To me Hilbert's Ansatz about matter appears to be childish, just like an infant who is unaware of the pitfalls of the real world..." A. Einstein, Letter to H. Weyl dated November 23, 1916, cited by C. Seelig, Albert Einstein, Zürich 1954, 200. W. Pauli, Theory of Relativity, Oxford 1958, 145 (Footnote 277). But in modern textbooks, the axiomatic application of a Hamilton principle is well established. See Chapter 3.32; E. Bessel-Hagen, Über die Erhaltungssätze der Elektrodynamik, in: Math. Ann. 84 1921. A generalization of Noether's work for electrodynamics is in E. P. Wigner, Über die Erhaltungssätze in der Quantenmechanik, in: Nachr. Ges. Wiss. Gött. 1927, 375; for an evaluation of Noether's works, see also H. Weyl, Emmy Noether,

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symmetry. On the other hand, problems arise if we try to understand the laws of conservation of energy and momentum as consequences of the general relativistic covariance principle. It is therefore little wonder that these questions of the laws of conservation in general relativity theory were recognized by the Göttingen mathematicians.27 In pre-relativistic physics, the conservation of a quantity (mass, energy, charge, etc.) is always connected to its location, i.e. we proceed from a clear density of the corresponding quantity at a defined location at a defined time. It can also be said that in a fixed three-dimensional space there is a well-defined quantity of the physical value in question. Our intuitive ideas of substance, which are also reflected by the various philosophers, are determined by the localization of physical values. It can now be shown that the mathematical terms for density of momentum and energy of the general relativity theory do not correspond to this requirement for localization. If we examine this energy density with respect to spatial transformations, it becomes apparent that this value is not transformed like an invariant. For example, in a space point we get numerical values for these quantities which are a function of the arbitrarily selected system of coordinates. But that calls into question the definition of an objective basic concept of physics. With the characteristic of localizability, the law of conservation of energy has lost its absolute character, at least in the context of general relativity theory. As shown in Chapter 4.13, a curved manifold is assumed for the universe as a whole. Therefore predictions concerning the conservation of energy of the universe as a whole seem problematic from the point of view of relativity theory. However, on the basis of quantum mechanics, new laws of conservation of elementary particle physics become possible which open new perspectives on the structure of matter.28 Finally, let us discuss H. Weyl's 1918 proposal for a unification of gravitational theory and electrodynamics, which has of course been refuted in terms of physics, but which made use of the fundamental mathematical technique of gauge symmetry.29 Weyl was oriented toward Einstein's model, which in the general relativity theory had interpreted gravitation potential geometrically as fundamental tensors. In a similar matter, Weyl believed

27

28 29

in: Scripta Mathematica 3 1935, 201-220, B.L. van der Waerden, Nachruf auf Emmy Noether, in: Math. Ann. 1935,469-476; A. Dieck, Emmy Noether, Basel 1970. See F. Klein, Nachr. Ges. Wiss. Gött, Math. Phys. Kl. 1918, 171-189; H. Bauer, Phys. Zeitschr. 19 1918, 163; D. Hilbert, Nachr. Gött. 1917, 477-480; see also the description by J. Mehra, Einstein, Hilbert and the Theory of Gravitation, in: Mehra et al., see Note 9, 137 ff.; E. P. Wigner, Symmetry and Conservation Laws, in: Wigner, see Note 14, 25. See also E. Schmutzer, Symmetrien und Erhaltungssätze der Physik, Berlin 1972,45 ff. H. Weyl, Sitz. Ber. d. preuß. Akad. d. Wissensch. 1918, 465; Mathem. Zeitschr. 2 1918, 384; Anm. d. Physik 59 1919, 101.

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that electromagnetic fields could also be interpreted geometrically, if it were possible to correctly analyze the transport of scales. According to Weyl, it cannot be concluded a priori that a scale of length 1 in a space-time point P, after transport to the point P \ has changed its length to Γ, e.g. because different unit scales are used in Ρ and P'. In general, it can be expected that 1) the relative change 1 dl ™ '"1 (2) - Γ = Τ = Φ is independent of the special scale, and that 2) with a sufficient number of near neighbor points, the gauge can be selected so that 1 = Γ, i.e. φ = O.30 In this case, we get in space-time coordinates (3)

φ = φ μ (Ρ)άχ μ = (po(P)dx0 - 2 .

If the parameters ν and a differ from 0, then the states ψ and ψ' differ by only one phase factor, if the masses mi and ni2 are identical. In the other case, we can derive a superselection rule for states of different masses. Since the identity transformation for all values of a and ν does not change the physical state of a system, the interference terms between ψι and ψ2 must disappear. The mass is therefore a classical observable in Galileo-invariant quantum mechanics. Historically, the existence of superselection rules was recognized only quite recently, namely in 1952 in a publication by Wick, Wightman and Wigner.71 But even now, no definitive answer has been found to the question whether there are universally valid superselection rules, or whether superselection rules occur only under additional special conditions and assumptions in the holistic symmetry of the quantum world. Thus the mass only be70 71

See V. Bargmann, see Note 49. G. C. Wick/A. S. Wightman/W.P. Wigner, The Intrinsic Parity of Elementary Particles, in: Phys. Rev. 88 1952,101-105.

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comes a classical observable if the speed of light in a limit process c —> °° is assumed to be arbitrarily great, i.e. if there is a transition from the Lorentz group to the Galileo group of space-time. But here, the Galileo space-time is an approximative aspect of the quantum world, which may be sufficient for chemists. Temperature and chemical potential then become classical observables, if the transition is made to thermodynamic systems with infinitely many degrees of freedom. 72 Under these additional assumptions, which may seem appropriate for various reasons for a scientific investigation, an abstraction is made from the original EPR correlations of a system with the holistic quantum world. The respective subsystem is, to some extent, isolated under certain epistemological aspects in the quantum world. If the chemist investigates the mass of a molecule under the assumption of Galileo-invariant space-time, then he is making an abstraction from the real EPR correlations between electrons and positrons. The subsystem and the rest of the quantum world are originally a correlated total system, from which EPR correlations are abstracted. But that does not mean that physical interactions between object and its environment are being ignored. For example, when investigating a molecule in an electromagnetic radiation field, the chemist eliminates only the EPR correlations between object (molecule) and environment (field), but in no way the electromagnetic interactions between the two. 73 In summary, then, generalized algebraic quantum mechanics offers a theoretical framework in which classical and non-classical theories can be investigated consistently. For example, traditional thermodynamics is a classical theory. Temperature and chemical potential relate specifically to macroscopic systems with a great many degrees of freedom and are classical macro-observables, which give defined (dispersion-free) values. Mathematically, macroscopic systems result, if we introduce systems with infinitely many degrees of freedom by means of an idealizing limit process. This thermodynamic limiting value results when the number η of particles of a system and its volume V are arbitrarily large (n —• °° and V —> whereby the density n/V remains finite. 72

73

In the Galileo-invariant quantum mechanics, in addition to mass (see Note 70), time is also a classical observable. See also A. Amann, Observables in W*-Algebraic Quantum Mechanics, in: Fortschritte der Physik 34 1986, 167-215. In molecular quantum mechanics, chirality is a classical observable. See also H. Pfeifer, Chiral Molecules - a Superselection Rule induced by the Radiation Field, Thesis ΕΤΗ Zürich No. 6551; ok Gotthard S + D AG Zürich 1980. In the quantum statistical description of thermal systems, the temperature and the chemical potential are classical potentials. See U. Müller-Herold, Chemisches Potential, Reaktionssysteme und algebraische Quantenchemie, in: Fortschritte der Physik 30 1982, 1-73. See also Figure 5 in Chapter 4.41.

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This idealization makes possible a mathematical process to get a handle on calculating the enormous complexity of such systems. One example is the above-mentioned free molecules, which inevitably interact with the rest of the world by an electromagnetic radiation field. Mathematically, this radiation field with its innumerable photons with hidden degrees of freedom can be taken into consideration in the model of an infinite quantum field. From the point of view of symmetry, infinite thermodynamic systems exhibit new characteristics which differ from those of finite systems. For example, if the total energy E and the total number η of particles become infinite, then no actually measurable observables belong to the corresponding Hamilton and number operator. An infinite thermodynamic system - in contrast to a finite system - is invariant with respect to a finite change of energy or the number of particles. With these new symmetries, new physical values such as temperature and chemical potential occur, which have defined values in the sense of classical observables. In the thermodynamic limit value, the phase transitions also occur, which are of fundamental importance for the modern theory of matter, and are discussed in the following chapters. These include, for example, the formation of crystals or the magnetization of a ferromagnetic material which result from the change of appropriate parameters (e.g. temperature). These phase transitions are linked to spontaneous symmetry breaking, in which the symmetrical state becomes unstable and an asymmetrical, but energetically stable state is assumed. The symmetries which are broken are, for example, spatial translation symmetries in crystals, spin-rotation symmetry for ferromagnets or gauge symmetries for superconductors. The new quantity in the case of the ferromagnetic material is the spontaneous magnetization, in the case of the superconductor the phase, e.g. as measurable by the Josephson effect. The symmetry breaks which occur far from thermal equilibrium (e.g. lasers and dissipative structures) belong to this category (See Chapter 4.4). Since characteristic properties of living systems (e.g. structural differentiation and metabolism) can also be described by phase transitions far from thermal equilibrium and the associated symmetry breaking (See Chapter 4.44), new opportunities emerge here for a unified theory of the natural sciences in the context of a generalized quantum mechanics. Many of the old paradoxes (e.g. Maxwell's demons and the understanding of irreversible processes) can thus be resolved. Entropy as a thermodynamic value of irreversible processes is linked to a temporal symmetry breaking which differs from temporally reversible classical and quantum mechanical systems. In a generalized quantum mechanics, therefore, a microscopic entropy operator M is introduced which no longer relates to individual particle paths or trajectories (such as an observ-

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able of classical mechanics) or wave functions (such as an observable of traditional quantum mechanics), but to distribution functions or bundles of trajectories. In terms of physics, the description with operators means that the classical description with trajectories must be given up, on account of the instability and chance nature of complex thermodynamic systems on the microscopic level or on account of quantum correlations of quantum systems. Analogous to the Schrödinger equation, in which the development of a wave function over time is described by the Hamilton operator, B. Misra and I. Prigogine propose a generalized operator L ("Liouville operator ") which represents the temporal development of distribution functions as the states of complex thermodynamic systems. For this approach, it should be noted that the temporal development of complex unstable systems involves a new operator T, which gives time a new significance. Assuming that we know the current age of the system, given by T, then we do not know how it will change in the future. Conversely, if we assume that the rate of change of the distribution functions (i.e. of the states) of the system with time is known, the age of the system is undetermined. The two operators L and Τ are therefore not commutative, just as position and momentum in quantum mechanics are not commutative. This new uncertainty relation of temporal change and instantaneous age is characteristic of all non-equilibrium situations of complex systems and expresses the chaotic behavior of their distribution functions or states. Infinite quantum mechanical systems can therefore have a time arrow, since the entropy operator M(T) on average increases with time and thus points irreversibly into the future. On the other hand, all finite quantum mechanical systems have a periodic time behavior. Therefore they can never tend toward an equilibrium, but after a certain time, allow their wave or state function to return to its initial value. The treatment of classical finite systems (e.g. as in classical mechanics), classical infinite systems (as in thermodynamics) and non-classical quantum systems (as in the von Neumann theory) in the joint theoretical framework of a generalized quantum mechanics also offers new opportunities to explain the measurement problem of quantum mechanics. For example, C. M. Lockhart and B. Misra (see Note 84) consider the measurement apparatus in this theoretical framework to be an irreversible dynamic system which breaks the time reversal symmetry in the sense of a causal system. This opportunity exists only if the apparatus is described as a macroscopic system with an infinite number of degrees of freedom. The measurement problem is then reduced to an interaction process between a non-classical system (quantum mechanical measurement object) and a classical system (measurement apparatus). For the measurement apparatus described in this

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manner, it is decisive to prove that it delivers unrelated final states which are independent of the measurement object. As difficult and complex as such analyses may be, including from the mathematical point of view, speculative goblins such as "Schrödinger's cat" or mysteries such as the "collapse of the wave packet" would dissolve in the thin, high-altitude atmosphere of mathematical abstraction. In philosophy of science, therefore, it would once again be shown that modern natural philosophy interpretations must always be related to concrete mathematical formalism, and can become obsolete if we change to another formalism. Discussions of a philosophy of nature which overlook this relationship between scientific theory and mathematical formalism, i.e. in which fictions such as Maxwell's demons or Schrödinger's cat to a certain extent take on a life of their own and become lodged in people's brains, can only lead us astray. 4.25 Symmetry and Complementarity According to the superposition principle, each system is indissolubly entangled with its environment, the rest of the quantum world. It becomes the object of knowledge, in which, under certain conditions of cognition, the entanglement can be disregarded. Only in this manner can the system be recognized and defined as individuality. But every physical perception is dependent on context, and specifically the conditions of the scientist's location and perception. Philosophically, Leibniz had already emphasized the contextual dependence of perception. He spoke of the different "point de vue " from which the observer perceives reality at a different angle. According to Leibniz, there is even a scale of more or less clear perception, which is a function of the different points of view of the living thing. According to Leibniz, only God is not subject to this type of dependence on context. In connection with the double slit experiment (See Chapter 4.23), N. Bohr ultimately called context-dependent means of describing the quantum world, such as wave and particle, "complementary" , 74 Of course, in Bohr's interpretation, the determination of the path of the particles (electrons) leads to the destruction of wave superposition. However, these two aspects are not mutually exclusive, but complementary, depending on the information being sought. The relationships which Bohr called complementary are explained in terms of physics by the superposition principle, i.e. by the - as 74

See also C. F. von Weizsäcker, Komplementarität und Logik. Niels Bohr zum 70. Geburtstag am 7.10.1955 gewidmet, in: Naturwiss. 42 1955, 521-529, 545-555.

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we now know - real and existing EPR correlations of correlated systems. The symmetrical structure of the quantum world which is defined by the superposition principle is therefore fundamental for Bohr's complementarity. Another example of context-dependent perception: In everyday life, we think of the moon as a system which exists individually. On the other hand, the moon consists of elementary particles which are inseparably linked by an electromagnetic radiation field. The radiation field cannot thereby be separated into regions which correspond to the earth, to the observer etc. For the ancient astronomers, the moon was an ideal sphere; for modern astronauts it is a pock-marked satellite of the earth; for geophysicists it is a reservoir of certain natural resources; for the chemist it is a complicated system of crystals, ores, molecules, etc., and for the poet it is a romantic beacon. Thus the moon only becomes an individual object of perception in the scheme of these perceptual contexts. Even the physicist with his measurement instruments, generally the perceiver, is correlated with his physical environment into an inseparable whole. Only by deliberate acts of perception can distinctions be made between the object perceived, the measurement instrument, the observer, etc. On the other hand, classical physics is based on an a priori separation into the objective world and the observer. To a certain extent, it is irrelevant to the planet whether or not Galileo observes it through a telescope. The state of the classical system "planet" is not changed by this measurement process. The state of the physical system during and after the measurement is clearly determined. Both systems are in separate states during and after the measurement. In the sense of Cartesian ontology, therefore, the world of classical physics can be broken down into two independent systems, the thing perceived ("res extensa") and the thing perceiving ("res cogitans"). On the other hand, the measurement process and - in the broader sense the perception process of quantum mechanics raises novel problems. It was N. Bohr who criticized the Cartesian separation between the thing perceiving and the thing perceived in quantum mechanics. 75 Bohr spoke of a complementary relationship between two systems, which we can now explain as the EPR correlation. Let us observe a quantum system (e.g. an atom), the states of which are represented by the vectors of state of the Hilbert space Let the observable A of the system be connected to a discrete and non-degenerating spectrum ai, a 2 ,..., i.e. Α ψ η = an\|/n with ( ψ η , ψ ι π ) = 8 n m · Let the measurement apparatus be a digital display, the states of which are described by the 75

See also Κ. M. Meyer-Abich, Korrespondenz, Individualität und Komplementarität. Eine Studie zur Geistesgeschichte der Quantentheorie in den Beiträgen Niels Bohrs, Wiesbaden 1965.

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Hilbert space J i j , i.e. the observable Β of the apparatus has digital values bi, b2, ... with Bcpn = bn > of the electron

Right-handed helix portion Fig. 2

Overall, it is apparent that the spatial reflection symmetry (parity) of weak interaction is violated. Mathematically, the parity Ρ transforms the spatial components of the four-coordinates Ρ (χ μ ) = (et, -χ, -y, -ζ). The parity symmetry of a quantum system then means, according to (8) in Chapter 4.22, that the Hamilton operator Η of the system is invariant with respect to the induced operator transformation T(P), i.e. (1) 105

T(P)H T(P)" 1 = H. T. D. Lee/C. N. Yang, Questions of Parity Conservation in Weak Interactions, in: Phys. Rev. 104 1956, 254; C. S. Wu/E. Amber/R. W. Heyward/D. D. Hoppes/R. P. Hudson, Experimental Test of Parity Conservation in Beta-Decay, in: Phys. Rev. 105 1957, 14131415. For the history and examination of the Lee/Yang experiment, see also C. N. Yang, Some Concepts in Current Elementary particle Physics, in: J. Mehra (ed.), see Note 9, 447-453; V. L. Telegdi, Crucial Experiments on Discrete Symmetries, in: e.d., 454-480.

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Analogously, the time-reversal symmetry means that the Hamilton operator of the quantum system is invariant with respect to the time-reversal operator Y (See (16) in Chapter 4.22). As already noted, however, Y is not a unitary representation of the temporal coordinate inversion. As an additional symmetry transformation, the unitary charge conjugation C was also discussed in the previous chapter (C for the English word "charge"). The successive application of all three symmetry operations T(P)CY leads to a famous symmetry, which is known as the PCT theorem (with Ρ for parity and Τ for time-reversal), and was gradually proven by W. Pauli and others between 1952 and 1957. 106 According to this theorem, the Hamilton operator of a (Lorentz-invariant) quantum system is invariant with respect to the combination of parity, charge and time-reversal transformation. For classical systems of physics, this result is trivial, since such systems are more or less invariant with respect to each individual transformation of this type. That is also true for the electromagnetic (and strong) interaction, but not for the weak interaction. From a left-handed particle, for example, the parity transformation T(P) produces a right-handed particle which does not exist in nature. From a left-handed neutrino, however, the successive application of T(P) and C makes a right-handed anti-neutrino which does indeed occur in nature. For the weak interaction, therefore, during the ß-decay, the product T(P)C is conserved, as is Y, but not T(P). It should be noted in passing that several experiments with the decay of K° mesons also indicate a violation of T(P)C and Y with the conservation of the total symmetry T(P)CY. "Superweak" interactions have also been mentioned in this context. Historically, E. Fermi had attempted as long ago as 1934 to describe the energy in weak interactions by analogies to electrodynamics 107 . If we bring a second charge distribution into the field of the first, forces and interaction energies arise. With stationary charge distributions p(r) and p ( f ) , the corresponding Coulomb energy can be derived in electrostatics (See 3.23). If there is a motion of the charge, the charge density p(r) in electrodynamics must be replaced by the respective four-current ] μ (χ μ ). For the weak interaction, the unrestricted range of action of the Coulomb force must be trimmed down. The weak currenty? introduced for this purpose converts the neutron into a proton during the decay process, and generates a lepton pair. In contrast to the electromagnetic interaction, it must change charges. During the

106 ψ Pauli, Niels Bohr and the Development of Physics, London 1957; see also L. C. Biedenharn/M.E. Rose, Phys. Rev. 83 1951, 459; H. A. Tolhock/S. R. De Groot, Phys. Rev. 84 1951, 151; G. Lüders, Ζ. f. Phys. 133 1952, 325; J. S. Bell, Proc. Roy. Soc. London A231 1955, 479; R. Jost, Helv. Phys. Acta 30 1957, 409; see also the textbook presentation by J. P. Elliott/P. G. Dawber, see Note 89,411 ff. 107 E. Fermi, Nuovo Cimento 11 1934, 1; Ζ. f. Phys. 88 1934, 161.

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emission of a photon by an electron, the electromagnetic current does not produce any change in the charge of the electron. The electromagnetic current is therefore neutral, while the weak current is charged. The following aspect was decisive for Fermi's approach: In contrast to quantum electrodynamics, Fermi's quantum field theory of weak interactions raised significant formal problems. For example, it had an infinite number of divergences, which also make observable quantities infinite -in contrast to quantum electrodynamics, in which there were only a finite number of renormalizable infinities. Thus the fate of Fermi's ansatz was sealed early: his theory of the weak interaction proved not to be renormalizable.108 Further progress required a new stroke of genius - namely the idea of tracing weak interaction, like electromagnetism and gravitation, to a local symmetry in the sense of gauge theory. The fundamental overall symmetry ( "isospin symmetry") was first investigated by W. Heisenberg with regard to a special case of strong interaction.109 Heisenberg noticed that the two nuclear particles proton ρ and neutron η are almost indistinguishable from one another, since they have almost the same mass and the same spin. If we overlook the electromagnetic interactions, there is hardly any difference among the strong nuclear forces pp, nn and np. Therefore we could interchange protons and neutrons in the world, without significantly changing the nuclear forces. Heisenberg thus proposed combining the wave functions ψ ρ and ψ η of proton and neutron into a two-component wave function

of the nucleón. Proton and neutron now designate two possible states of the nucleón.

Neutron

Proton Fig. 3

108

109

See also C. Itzykson/J.-B. Zuber, see Note 89, 606 ff; H. Rollnik, Ideen und Experimente für eine einheitliche Theorie der Materie, in: Phys. Bl. 32 1976, 706 ff. W. Heisenberg, Z. f. Phys. 77, 1932,1.

438

4. Symmetries in Modern Physics and Natural Sciences

Geometrically, the nucleón can be represented by two intersecting double arrows, which stand respectively for the proton and neutron components (continuous and broken line in Figure 3). The vertical position of a double arrow indicates the current state of the nucleón. The global symmetry operation, according to which all protons of a system become neutrons and all neutrons become protons, is carried out in the geometric model by an overall rotation of the total isospin space by 90°. Algebraically, these transformations of the 2-component wave function ψ can be represented by unitary 2x2 matrices U, which form what is called an SU(2)-group. (2) Mathematically, the complete group U(2) of unitary transformations in two dimensions is generated by the four Pauli matrices: (3)

In terms of physics, that corresponds to the four cases in which the two components of the wave function are displaced either with the same phase factor e i a or with opposite factors e,ot and e~' a , or the neutron is transformed into a proton, or the proton into a neutron. If we overlook the first case, the result is the "special" group SU(2) of the unitary transformations in two dimensions. In general, for special unitary groups SU(n), there is a mathematical requirement that the trace of the representing matrix be zero. The first case corresponds to the identity matrix I and therefore has the trace SP (1) Φ 0. For particles with spin the three Pauli matrices ôi, Ô2, Ô3 correspond to the matrices by means of which all of the rotation operators of their spin states can be generated.110 In global isospin symmetry, the states of the nucléons are changed in the same manner everywhere and at the same time, i.e. geometrically, the double arrows are rotated by the same angle. In this case, therefore, the phase factors of the SU(2) transformations are locally independent of space-time points χ μ , i.e. SU (2) describes the global isospin symmetry. Local isospin symmetry would mean that phases can be locally fixed differently or - in the geometric representation - the double arrows of the nucléons can be rotated in different positions and at different times by different angles. In the 110

See also J. P. Elliott/P.G. Dawber, see Note 89, 214 ff.

439

4.3 Symmetries in Elementary Particle Physics

TRANSFORMATIONS φ

GAUGE FIELD

Opposite phase factor V —

eS,

Ρ

Κ

JL

>

η

η

Ρ

(2) Neutron transformed into proton

©

INTERACTION

—ρ* η

Proton transformed into neutron Ρ

"μ

Λ"

—ρ"

Fig. 4

sense of local gauge theory, however, corresponding gauge fields must be introduced for that purpose, to preserve the overall symmetry of the system when local changes occur. After local U( 1 ) symmetry had been discussed using the example of electrodynamics, C.N. Yang and R. L. Mills investigated a local SU(2) symmetry for the first time in 1954.111 Corresponding to the three generating transformations of the SU(2) group, for local symmetry three gauge fields must be introduced (Figure 4). In transformation 1) the gauge field p® is uncharged, since the charges of proton and neutron are not changed. In case 2), the gauge field must make a proton out of a neutron. To make possible the conservation of charge during this change in charge, the field is assigned a positive charge. Accordingly, the gauge field p~ in case 3) receives a negative charge. The local gauge group of Yang-Mills SU(2) theory differs fundamentally from the local U(l) group of quantum electrodynamics: It is noncommutative ("non-Abelian), i.e. the successive application of the transformations is generally non-commutative. In QED, a local symmetry transformation corresponds to the phase shift of a matter field (e.g. electron beam), which is a consequence of the interaction of the matter field with an electromagnetic field (See Chapter 4.33). In QED, the phase shift of an electron field applied twice in succession can mean that a photon is first emitted and then reabsorbed. The overall phase shift which results is just as great 111

C. N. Yang/R. L. Mills, Phys. Rev. 96 1954,191; see also H. Rollnik, see Note 108,713 f.

440

4. Symmetries in Modern Physics and Natural Sciences

as for twice the phase shift, during which the electron first absorbs a photon and then emits it. The sequence of events has no effect on the magnitude of the total phase shift, i.e. the U(l) group is commutative. On the other hand, the multiplication of the 2x2 matrices of the SU(2) group is generally non-commutative: the isospin vector after different sequences of transformations generally does not have the same orientation, so that a nucleón, instead of being transformed into a neutron, could also be transformed into a proton.112 Overall, the Yang-Mills theory is a non-commutative local SU(2) gauge theory. Like the U(l) theory, it is characterized by high symmetry. Forces can be explained by the transition from global to local symmetry: the "preservation of symmetry" in the event of local changes was therefore also possible for isospin symmetry. Still, the Yang-Mills theory has a significant shortcoming compared to the U(l) gauge theory of QED, i.e. in spite of its high mathematical symmetry, simplicity, elegance and beauty, it was demonstrably without any application in physics. The SU(2) theory predicts perfect isospin symmetry, which means that there can be no differences between neutron and proton. As noted above, that it not absolutely true for the masses. But here, we can console ourselves by noting that other theories of physics have only an approximate validity, but have still provided new discoveries, explanations, etc. and have thus been interesting in terms of physics. But the following shortcoming is decisive: The Yang-Mills theory proceeds on the assumption of the unlimited range of all the forces it describes. Like the photon of electromagnetic interaction, gauge particles which are massless must therefore correspond to the three gauge fields. The masslessness of these three particles can be convincingly demonstrated mathematically, if the Hamilton operator and the physical states of the system are invariant under the gauge transformations. Two of these massless gauge particles also differ from the photon in that they are charged. But except for the photons of the electromagnetic force (and the gravitons of gravitation), no massless particles occur in nature, with which interactions could be transmitted over unlimited ranges. Therefore the Yang-Mills theory predicts and requires the existence of particles which would change nature as we know it. In philosophy of science, there is no better way to contradict or "falsify" a theory of natural science. Nevertheless, the Yang-Mills theory exerted an 112

A generalization of the Yang-Mills approach appears in R. Utiyama, Invariant Theoretical Interpretation of Interaction, in: Phys. Rev. 101 1956,1597-1607. For a general representation of non-commutative gauge theories, see also C. Itzykson/J.-B. Zuber, see Note 89, Chapter 12.

4.3 Symmetries in Elementary Particle Physics

441

incredible fascination for scientists. At this stage of development, it was certainly not studied because in some positivistic sense it "fit the measurements better". Its attraction initially consisted only of its high degree of mathematical symmetry and its simplicity, as had been recognized analogously in the simpler case of electrodynamics.113 In philosophy of science, it should be emphasized that in this stage, the impetus to research comes from an almost Platonic belief in simplicity and symmetry in nature. Kepler's "Credo spatioso numen in orbe", the belief in a "harmonía mundi" apparently attracted leading physicists to the Yang-Mills theory. Naturally, this attraction should not be interpreted in a philological and literary sense, as if the physicists had read Plato and Kepler and then tried to carry out their scientific program at all costs. A familiarity with Plato's writings, as in the case of W. Heisenberg, was rather the exception. But here again, we see that the orientation toward symmetry in the natural sciences is not a function of certain cultural and educational traditions (e.g. Greek and Western), although that is where we can find it reflected philosophically. Rather, it is based on the general, leading insight that the stability of nature, in spite of its continuous processes of change and forces, can only be explained by invariant structures, like those first elucidated mathematically in the gauge theories. To make the Yang-Mills theory applicable in terms of physics, and to arrive at observable predictions, the range of its interacting particles must be restricted, in contrast to the photon. If we assign them sufficiently large masses, the range of the gauge fields can be made arbitrarily small. The solution of this problem, namely to give the field quanta of the Yang-Mills gauge fields masses, was finally achieved by an in-depth analysis of spontaneous symmetry breaking, which Heisenberg noted was of general significance for physics.114 We shall explain this concept below with reference to several examples. A (mathematically perfect) egg has rotational symmetry and symmetry of reflection with reference to its longitudinal axis. If we stand it vertically on a plate, and leave it to its.own devices, it rolls over on its side and remains lying in some direction: The symmetry of the egg relative to the vertical axis on the table is broken, although the symmetry of the eggshell remains intact. The symmetry breaking is spontaneous, since it was impossible to predict the direction in which the egg ultimately came to rest. In this case, the cause

113

114

See also the evaluation by S. Weinberg, Vereinheitlichte Theorie der elektroschwachen Wechselwirkung, in H. G. Dosch (ed.), see Note 92, 6-15; J. Iliopoulos, An Introduction to Gauge Theories, Lectures given in the Academy Training Programme of CERN 19751976, Geneva 1976, 2, 4. W. Heisenberg, Einführung in die einheitliche Feldtheorie der Elementarteilchen. Stuttgart 1967.

442

4. Symmetries in Modern Physics and Natural Sciences

is the earth's gravitation, which allows the egg to assume an energetically more favorable state: The symmetrical state relative to the vertical axis of the plate was energetically not stable. Another example in which the symmetry of a system is spontaneously broken is the transition of aferromagnet into the magnetic state. As long as the material is not magnetized, no space axis is defined. But if the material is magnetized, one space axis can be distinguished from the other by the position of the magnetized pole, and the symmetry is broken. The electrons and the iron atoms in an iron bar are described by equations which are rotationally symmetrical. The (free) energy of the magnetized bar is thereby invariant with respect to the definition of north and south pole.

a

b Fig. 5

In Figure 5, the energy V is plotted as a function of the magnetization φ, with a high temperature in Figure 5a and a low temperature in Figure 5b. In both cases, the magnet is induced to seek an equilibrium state of the lowest possible energy. In Figure 5a, the equilibrium state at a high temperature is reached when the magnetization reaches the lowest value 0. In this case, the symmetry is conserved. In Figure 5b, with a falling temperature, the vertex of the curve ν(φ) ascends. The positions of the equilibrium state are to the left and right of the axis of symmetry of the curve with a non-vanishing magnetization. Therefore the system will spontaneously assume one of the possible equilibrium positions and thereby break the symmetry of its equations. A characteristic for the spontaneous symmetry breaking of a system is apparently a critical value which can assume a physical boundary condition (in this example, temperature), so that beyond that point, the symmetrical solution of the equation is no longer stable or corresponds to the equilibrium state, and therefore the symmetry is broken. Such symmetry breaking as a result of the cooling of the system is known in particular from chemical crystallization processes which result from completely homogeneous solu-

4.3 Symmetries in Elementary Particle Physics

443

tions with perfect symmetry (See Chapter 4.41). In terms of the philosophy of nature, we will return to the concept of symmetry breaking in a discussion of the variety of nature. In quantum field theory, a distinction is made between the spontaneous breaking of a global symmetry and of a local symmetry.115 The most stable fundamental state of a quantum field is the vacuum. For example, an electron field has the least possible energy when there are no electrons present, which was called the vacuum state in Chapter 4.31. In this case, the Lagrange operator of the quantum field is invariant with respect to a global phase shift in the sense of U(l) symmetry. Likewise, as we have seen, we can speak of the global symmetry of an SU(2) theory. In the case of symmetry breaking, the vacuum state is not invariant with respect to global symmetry transformations. Therefore we can also say that, in the case of symmetry breaking, the vacuum is unsymmetrical, while the overall symmetry of the system (i.e. of its Lagrange operator) is conserved. In the event of a spontaneous symmetry breaking, terms always occur which are interpreted by Goldstone as massless scalar field quanta ("Goldstoneparticles"). But with this information, we have not progressed any farther with our real problem, which is: How can the gauge particles with local SU(2) symmetry be given masses? The problem is solved perfectly if we investigate the spontaneous symmetry breaking of local symmetry,116 In this case, the vacuum state is again unsymmetrical, while the local gauge invariance of the system (i.e. of its Lagrange operator) remains intact. It would therefore be more accurate to speak of a "hidden" symmetry, which is concealed by the asymmetry of the vacuum state. The decisive factor is that the gauge particles ( "vector bosons ") of the gauge fields are given a mass, while the massless Goldstone particles of global symmetry are transformed away. This process is known as the Higgs mechanism and can be understood as follows:117 The gauge fields of the Yang-Mills theory are vectorial. Their field quanta have the spin s = 1, so that theoretically, therefore, we can distinguish three spin states: parallel, antiparallel and perpendicular to the direction of propagation. Since their action before the symmetry breaking is assumed to be unlimited and at the speed of light, the 3rd spin state perpendicular to the direction of propagation is unrealizable. If it were introduced, the field 115

116

117

See also J. Bernstein, Spontaneous Symmetry Breaking, Gauge Theories, the Higgs Mechanism and all that, in: Revise Reports of Modern Physics 46 1974, 7-48; C. Itzykson/J.B. Zuber, see Note 89, 519 ff„ 612 ff.; J. Iliopoulos, see Note 113, Chapter 4. This phenomenon was already known experimentally in plasma physics and for superconducting. That is how the mass of the plasmons and the Meissner effect can be achieved. P. W. Higgs, Phys. Rev. Lett. 12 1964, 132; 13 1964, 321.

444

4. Symmetries in Modem Physics and Natural Sciences

quanta of the gauge fields could no longer move without restriction at the speed of light, and would correspondingly acquire mass. This missing spin state, i.e. with only one spin state, is precisely what is supplied by the scalar, massless Goldstone bosons with spin s = 0. In visual terms, the Goldstone particles are "eaten" by the gauge particles of the gauge fields, which results in the desired massive gauge particles. Without going into mathematical detail, it should be noted that the YangMills theory, expanded to include the Higgs mechanism, is demonstrably renormalizable. 118 That gave renewed impetus for the search for suitable applications. Although the Yang-Mills theory had originally been developed for the isospin symmetry of strong interaction, its first successful application was to the weak interaction. As noted above, only the left-handed helix portion eL~ and μ ^ of the electron and muon participate in the ß-decay of the neutron and muon. Furthermore, only the left-handed helix portions VL of the corresponding neutrino occur in nature. It follows that the left-handed helix portions which participate in the ß-decay of the neutrino can be combined, analogous to the Heisenberg nucleón, in a two-component wave function, which is notated as a left-hand doublet:

With global SU(2) symmetry, the states of the two-component wave functions L are changed everywhere, at the same time and in the same way. For a local SU(2) symmetry, three gauge fields must be introduced corresponding to the three group transformations generated. By analogy to the local isospin symmetry in Figure 4, we designate the neutral gauge field as W? and the I

^

two charged gauge fields as W^. Mathematically the SU(2) combination of the three gauge fields is notated in the following matrix: 119

118 119

ef

VL

er

W°

W " μ"

vL

w+

W«

See also E. S. Abers/B. W. Lee, Gauge Theories, in: Physics Reports 9 C 1973, 1. See also H. Rollnik, see Note 108, 717 ff; Rollnik, Ideen und Experimente für eine einheitliche Theorie der Materie, Rheinisch-Westfälische Akademie der Wissenschaften Vortrag Ν 296, Opladen 1979, 18 ff; H. Georgi, Vereinheitlichung der Kräfte zwischen den Elementarteilchen, in: H. G. Dosch (ed.), see Note 92, 144 ff.

4.3 Symmetries in Elementary Particle Physics

445

Physical examples are illustrated in the following Feynman diagrams: Ve

\Y I η

a

b Fig. 6

Fig. 7

In Figure 6a, for example, a left-handed neutrino VL is assigned a negative charge by W~ and generates a left-handed electron e L . In general, the W + fields transfer weak and electrical charges of +1, the W~ fields charges of -1, while the W° fields are neutral, like the photon. The same gauge fields cause the beta-decay of the neutron and muon. The Feynman diagram in Figure 7 illustrates the beta-decay of the neutron. During the transformation of the neutron into a proton, a W" field quantum is emitted which is materialized in an electron-antineutrino pair. (On account of the PC parity, the right-handed antineutrino is assigned a left-handed neutrino with an arrow in the reverse direction, for which the combination with the W " gauge field is explained). The three gauge fields of local SU(2) symmetry therefore transmit the parity-violating weak interaction. As noted above, the three SU(2) gauge fields only make sense in terms of physics if they act over short distances and if their field quanta are given large masses. In physical terms, therefore, this process can be imagined, analogous to the local isospin symmetry, as a Higgs mechanism during a spontaneous symmetry breaking. But there seem to be insurmountable obstacles to a unification of the weak interaction with the electromagnetic interaction. How can an SU(2) symmetry with massive field quanta and short range and a U(l) symmetry with a massless field quantum (i.e. the photon) and unlimited range be traced to a common symmetry? In 1967, S. Weinberg, A. Salam and C. Ward proposed such a unification, and once again proceeded on the basis of an insightful symmetry hypothesis. 120 They assumed that in a hypothetical initial state, the weak and electromagnetic interactions are indistinguishable, and in this sense form a joint force, which is described by an SU(2)xU(l) symmetry. Corresponding to this symmetry are three gauge fields of SU(2) symmetry of the 120

A. Salam (1967), in: Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8), Ed. N. Svartholm, Stockholm 1968; S. Weinberg, Phys. Rev. Lett. 19 1967,1264; see also J. Iliopoulos, see Note 113,18 ff;C. Itzykson/J.-B. Zuber, see Note 89, 620 ff.

446

4. Symmetries in Modem Physics and Natural Sciences

weak interaction and one gauge field of the U(l) symmetry of the electromagnetic interaction. In the hypothetical initial state, let the field quanta of the four gauge fields be massless and of unlimited range. Corresponding to the U(l) portion is an uncharged gauge field B°, which arranges only transitions between identical states. Corresponding to the matrix presentation in (5), it is therefore notated in the diagonals: (6)

e~ e~

VL

B° B°

vL

The neutral B° gauge field is therefore connected both to the left-handed doublets (4) and to the right-handed electron eR - . Physical examples for the interactions of the neutral B° field quanta with neutrinos and electrons are illustrated in the following Feynman diagrams: eR

vL \

Β

VL

eL

Β b

eR

Β

c

Fig. 8

In the combined SU(2) χ U(l) symmetry, the neutral gauge fields B° and W° from the matrices (5) and (6) are combined into neutral gauge fields andZ 0 . e£7

VL

ef

A0

w-

vL

W +

Z°

(7)

In the first case, the gauge field A0 corresponds to the gauge transformation which multiplies only charged particles e^ and eR by the same phase factor. That is the gauge transformation of the electron wave function, the corresponding gauge field of which is the electromagnetic four-potential A0 (See

4.3 Symmetries in Elementary Particle Physics

447

(2) and (3) in Chapter 4.31). Since this gauge field carries no charge, it is also given the exponent 0 and can be interpreted as a linear combination of the neutral fields B° and W°: (8)

A° = cos©wB° + sinOwW^.

The actual "Weinberg angle" Bw is thereby selected so that the parity violations associated with B® and Wf} offset one another. In the orthogonal combination for (8) (9)

Z° = - sin OwB° + cos ©w

the parity violations of B® and Wj¡ no longer occur. The gauge field Z° therefore transmits a neutral, parity-violating interaction between weak charges, like those borne by neutrinos. S. Glashow was the first to indicate the SU(2) χ U(l) symmetry. But he had no solution to the problem of assigning masses to the field quanta of the W ± and Z° gauge fields, while the photon, as the field quantum of the A 0 gauge field, had to remain massless. But that was necessary for the reasons indicated above, to make the theory applicable in terms of physics. For that purpose, S. Weinberg and A. Salam used the concept of spontaneous symmetry breaking. In the hypothetical initial state of SU(2) χ U(l) symmetry, the field quanta of the gauge fields are initially massless and of unlimited range. There is a spontaneous symmetry breaking if the vacuum in which the gauge fields are propagated is not symmetrical. In this case, scalar field quanta ("Goldstone particles") occur which, according to the Higgs mechanism, are "eaten" by the gauge fields and thus provide massive gauge bosons. For a spontaneous breaking of the SU(2) χ U(l) symmetry, four scalar field quanta ("Higgs fields") corresponding to the four gauge fields are necessary. According to the Higgs mechanism, three scalar field quanta are required to make the charged W* vector bosons and the neutral Z° vector boson massive. The fourth gauge boson is the photon of the electromagnetic interaction, which is massless. Therefore the fourth scalar Higgs field quantum is left over after the symmetry breaking, and should be observable. The assumption of symmetry of the weak and electromagnetic interaction gradually became a verifiable theory of physics. For example, Z° particles were predicted which cause neutral weak currents. As a result of the exchange of such particles, two particles interact without changing their charge. In fact, the neutral weak currents were observed in CERN for the first time in 1973. The Weinberg-Salam theory was also used to calculate

448

4. Symmetries in Modern Physics and Natural Sciences

the masses of the vector bosons W ± and Z°. In 1983, they were confirmed experimentally with great accuracy. 121 To generate the W* and Z° bosons freely, energies of approximately 100 Gigaelectron-Volts are necessary. If such energies are present, particles with a diameter of approximately 10" 16 cm can be investigated. According to the Heisenberg uncertainty relation, the energy is inversely proportional to the diameter. For energies of more than 100 Gigaelectron-Volts and distances less than 10" 16 cm, there would be a perfect SU(2) χ U(l) symmetry, in which the and Z° field quanta would be exchanged as rapidly as the photon. At lower energies, there would be a symmetry breaking. The energies at that point are no longer sufficient to freely generate field quanta. Then the particles can no longer be observed freely and directly, but only as a result of the effects of virtual particles. One example of such an effect is the beta-decay of unstable atomic nuclei. That was also the starting point for the theory of weak interaction. The SU(2) χ U(l) symmetry states which were produced under complex laboratory conditions seem rather artificial. But these are not merely human "inventions", products of particle accelerators, which to a certain extent occur in nature only as a result of mankind's theories (such as the automobile or the airplane, for example). In the context of the physical cosmology, SU(2) χ U(l) symmetry is rather a real state of the universe, which must have existed during a certain stage of development under certain temperature and energy conditions, and is in no way merely hypothetical. From a cosmological standpoint, the SU(2) χ U(l) symmetry is in no way merely a technical projection of mankind in nature. Nature itself is the powerful high-energy laboratory, whose symmetry states can, to some extent, be "imitated" in our human laboratory. But we will have more to say later about the natural philosophy evaluation of elementary particle symmetries. For the mathematical and physical evaluation of the SU(2) χ U(l) gauge theory, the discovery that it can be renormalized, like the U(l) theory of quantum electrodynamics, was certainly decisive. But it does not result in any complete unification of the weak and electromagnetic forces, since it specifies an individual symmetry group for each of the two forces. Each of these two symmetry groups has its own coupling constant, the ratio of which is defined by the tangent of the above-mentioned Weinberg angle. To study the embedding of the weak and electromagnetic forces in a higher symmetry group (4.34), first we must determine the symmetry of the strong forces.

121

D. B. Cline/A. K. Mann/Rubbia, The Detection of Neutral Weak Current, in: Scientific American 231 (Bol. 6) 1974, 108-119; M. Böhm, Zur Entdeckung des W-Bosons, in: Physik in unserer Zeit, May 1983, 92.

4.3 Symmetries in Elementary Particle Physics

449

4.33 Quantum Chromodynamics: Symmetry of the Strong Forces The strong force was initially called the nuclear force, which holds protons and neutrons together in the atomic nucleus. In the 1950s and 1960s, a number of new particles were discovered which interacted with the strong force, were generated and annihilated, and were therefore called hadrons (Gr. άδρός - strong). With more powerful particle accelerators and energies, it became possible to generate an increasing number of hadrons, i.e. the discovery and investigation of these particles turned out to depend on the development of high-energy technology. In the 1950s, particle transformations were already recognized as a characteristic feature of high-energy physics. From the frequency of transformations, conclusions were drawn regarding the strength of the forces which occur during the reactions between the particles. A distinction was thereby made between the strong, electromagnetic, weak and gravitational interactions, the ratio of which was estimated, in that order, at approximately 1 : 10~2 : 10~ 14 : 10~ 39 . But at that time, we were still far from a quantum field theory of the strong force. In terms of the history of science, the physics of strong forces is more of a development plan, which begins with an innumerable empirical manifold of unorganized measurements and particle discoveries (zoo of hadrons). In a second phase, the first common features and analogies are noted, which lead to initial classification models, but remain largely approximate, as ad hoc hypotheses, and in no case provide any physical proof. Only in the final phase are these models traced to the fundamental symmetry of a quantum field theory, whose explanations and predictions are confirmed with great accuracy. 122 Hadrons are classified in two classes on the basis of their spin - baryons such as protons and neutrons, for example, the spin of which is an integer multiple of 1/2 (i.e. 1/2, 3/2, etc.) and mesons, such as the pion, with an integer spin 0, 1, etc. Baryons and mesons can also be distinguished by the baryon number B, in which a baryon Β = +1, an antibaryon Β = -1, and a meson Β = 0. It was a remarkable experimental fact that the baryon number was conserved during an interaction, i.e. the sum of the baryon numbers of all the baryons participating in a strong interaction. Additional classifications were made by charge multiplets, in which hadrons with certain common characteristics (quantum numbers) were combined, overlooking minor differences in mass and electromagnetic dif-

122

See also H. Rollnik, Teilchenphysik I. Grundlegende Eigenschaften der Teilchen, Mannheim 1971, Chapter I; D. H. Perkins, Introduction to High Energy Physics, Reading, Mass. 1972; Particle Data Group, in: Phys. Lett. 50 Β 19741.

450

4. Symmetries in Modem Physics and Natural Sciences

ferences (e.g. charge, magnetic moment).123 That was done according to the pattern of Heisenberg's binary multiplet (doublet) of proton and neutron. Additional examples are the charge triplet of pions π + , π° and π~, the hyperon triplet Σ - , Σ 0 and Σ + or the charge quartet of the Δ resonances Δ", Δ°, Δ+ and Δ++. These classification models are all based on the empirical fact that within a charge multiplet, all integer multiples of the elementary charge e lie between a minimum value Qmin and a maximum value Q max . Each charge within a multiplet can accordingly be expressed by , Q m i n (1) Q=

+ Qmax 2

+ n

'

whereby η designates the distance from the charge center of gravity (Qmin + Qmax)/2 and can assume the values 0, i 1, i 2 , . . . , i 1/2 (Qmax — Qmin)· The charge center of gravity (by analogy to the spin quantum numbers) is introduced as a new isospin quantum number T, the distance η as isospin quantum number T3 with the values -Τ, -Τ + 1,... Τ - 1, T. The sum Qmin + Qmax or twice the charge center of gravity is defined as hypercharge Y. Overall, for the hadrons, the Gell-Mann/Nishijima formula gives (2)

Q = I Y + T3.

For an equivalent description of hypercharge, the characteristic "strangeness" S is introduced, whereby S = Y for mesons and S = Y-l for baryons, i.e. in general S = Y-B. The advantage of S is that the particles discovered first, such as the π-meson and nucleón, have the value S = 0, while "strange" characteristics were ascribed to the more recently discovered particles such as the K-meson, and they therefore receive a "strangeness" S φ 0. _ι Λ M η M r * Δ"

ι 2 M ρ

M X w° w* X X *

Δ°

Δ+

Δ+

• • +

Fig. 1

123

See also H. Rollnik, Teilchenphysik Π. Innere Symmetrien der Teilchen, Mannheim 1971, Chapter ΠΙ; J. P. EUiott/P. G. Dawber, see Note 42, Chapter 11).

4.3 Symmetries in Elementary Particle Physics

451

Fig. 2c

The charge multiplets can be represented visually in the form of geometric diagrams. 124 In Figure 1, the nucleón doublet Ν (baryon), the pion triplet π (meson) and the Δ-resonant quartet are plotted on the isospin axis. The following multiplets can be graphically represented in 2-dimensional T 3 -Y diagrams. In Figure 2a, the baryons with spin 1/2 form an octet. At point Y=0 and T3=0 there are two particles Σ 0 and A 0 . Figure 2b shows the corresponding antibaryon octet. Figure 2c shows the baryon resonance decuplet with ten particles with isospin 3/2. It should be noted that the decuplet in Figure 2c led to the discovery of the Ω-particle (1964) with T3=0, Ύ--2 and spin 3/2. 124

See also C. Itzykson/J.-B. Zuber, see Note 89, 513 ff.

452

4. Symmetries in Modern Physics and Natural Sciences

In spite of this heuristic success, the particle multiplets initially seemed to many physicists to be like the mystical symmetries of cabalistic diagrams, whose simple rules of combination could of course be learned, but the justification for which remained a mystery. M. Gell-Mann and Y. Ne'emann took the mathematically decisive step in 1962, when they recognized the symmetries in the T3-Y plane as (irreducible) representations of a common unitary group.125 The group in question is the special unitary group in three dimensions SU(3), the transformations of which are represented by unitary 3x3 matrices. They can be completely generated by 3 2 - 1 = 8 hermetic matrices (analogous to the 22 - 1 = 3 matrices of SU(2)). Since SU(2) is a subgroup of SU(3), three generating elements of SU(3) from the three Pauli-matrices of SU(2) (see (3) in 4.32) are constructed for three dimensions:

The diagonal sum of these generating elements is again zero. As operators, they in turn satisfy certain commutation relations, and therefore define the Lie algebra of the group SU(3). The mathematical theory of these Lie algebras has been known since 1894, when E. Cartan succeeded in indicating a classification of all (semi-simple) Lie algebras.

In terms of physics, however, it is decisive that eight operators can thus be defined, with whose eigenvalues the hadron multiplets can be constructed in the plane. In addition to the two operators^Y and T 3 for hypercharge and isospin, these are the six slide operators T±, V±, Û±, whose eigenvalues are understood as a quantum number. The operator V± increases or decreases the quantum number T3 from Τ 3 by ¿L the quantum number Y from Ϋ by A

¿

1. The operator U± increases or decreases T3 by 1/2, Y by 1. The operator T± increases or decreases T3 by 1 and Y not at all. Figure 3 illustrates the action of the slide operators. The units on the Y-axis are < / | χ units of the T3-axis, so that the slide operators form equilateral triangles.

125

See also W. Greiner/B. Müller, see Note 42, Chapter 7.

4.3 Symmetries in Elementary Particle Physics

453

Fig. 3

The multiplet representations in the T3-Y plane are defined by these eight quantum numbers. With an allusion to the Buddhist theory of wisdom, physicists have also spoken of the "eight-fold way" which is necessary to recognize the fundamental symmetry behind the multiplicity of particles. 126 But in terms of natural philosophy, we must be clear that the "eight-fold way" in this stage of research by no means redeems the Platonic program, and completely traces the manifold nature of phenomena to a symmetry. With the individual multiplets, it is even assumed that their particles have exactly the same mass, which is in no way the case. For the time being, therefore the SU(3) symmetry of the hadron multiplet remains only an approximate description. Specifically, an SU(3) multiplet is constructed from T-multiplets parallel to the Y-axis, V-multiplets along the V-lines and U-multiplets along the Ulines. From the algebra of the eight operators, it can be determined that the SU(3) multiplets can be understood as regular, but not necessarily equilateral hexagons of the shape illustrated in Figure 4. A particle can be understood as a state of the multiplet. In finite multiplets there is always a maximum state V|/max with maximum T3-value (outside right in Figure 4), from which the edge of the multiplet can be generated by ρ applications of the Voperator, and q applications of the Τ-operator. A SU(3) multiplet is therefore also characterized by the formula D(p,q) = [m], with the coefficient of state m. Figure 4 illustrates mathematical examples.

126

M. Gell-Mann/J. Ne'eman. The Eightfold Way, New York 1964.

4. Symmetries in M o d e m Physics and Natural Sciences Singlet 4 ® I

CHOP):l - m

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DI1PM31-V*

ih

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λ

λ

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DKtf WÇ] / -1/

/

^

OtfMS]

\

1

f

-Λ W30M1O] \

-1

Dt2,2M27;

J

k 2

WWfJ]

/ 1 \

-y

γ Jt

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D(7,31-[192] Fig. 4

4.3 Symmetries in Elementary Particle Physics

455

In philosophy of science, the SU(3) symmetry in this stage is at best a conceptually economical, aesthetic and more or less accurate (approximate) description of the variety of hadrons, but not a physical justification. The following question remains, above all: While mesons occur only in singlets and octets, with 1 and 8 states ("particles") respectively, barions occur only in singlets, octets and decuplets are found (See Figure 2). In nature, therefore, apparently only these three representations of SU(3) symmetry occur, while mathematically, multiplets with 3, 6,15 etc. members (See Figure 4) are also possible. Gell-Mann and G. Zweig provided the explanation in 1963 with the suggestion that all hadrons can be reduced to a few elementary components. 127 This was to be realized by the two multiplets [3] and [3] with 3 particles ("quarks") or antiparticles ("antiquarks"). Mathematically, these are the two simplest non-trivial SU(3) representations, if we designate the singlet [1] a trivial representation. If, with Gell-Mann, we assume that each meson is composed of one quark and one antiquark, and each baryon of 3 quarks, then with the possible components of quarks, the characterization of the multiplets observed in nature is comprehensible. Then there are exactly 3 2 =9 possibilities for a meson and 3 3 =27 possibilities for a baryon to be composed of quarks. The decompositions are 9=1+8 and 27+1+2-8+10, which correctly correspond to the singlets, octets and decuplets found in nature. Mathematically, the SU(3) multiplets can be formed from (tensor-) products of the triplet representation and vice-versa. 128 For the construction of mesons, let us visually and graphically explain the reduction of the product (4)

[3] [3] = [8] Θ [1].

At each endpoint of a vector of the first multiplet [3] all vectors of the second multiplet [3] are plotted. The endpoints give the potential terminal states in the T3-Y system. The result is a diagram with a multiply-occupied center, which can be broken down into an octet and a singlet. Accordingly, the breakdown (5)

[3]®[3]®[3] = [1]®[8]θ[8]®[10]

for the construction of the baryons can be proven. In general, all the known hadron multiplets can be traced to the triplets of the quarks.

127

128

For historical background, see also S. L. Glashow, Quarks mit Farbe und Flavor, in: H. G. Dosch (ed.), see Note 92,16-30; H. Fritzsch, Quarks, Urstoff unserer Welt, Munich 1981. See also W. Greiner/B. Müller, see Note 42, Chapter 8.

456

4. Symmetries in Modern Physics and Natural Sciences V,

[3]

[β]

® I"31

®[1]

Fig. 5

If Gell-Mann's explanations are not to remain mere mathematical speculations, then the existence of the particles which satisfy the two triplets must be demonstrated. The Λ-hyperon with spin 1/2 was interpreted as a singlet with Y = T3 = 0. For the quarks of the two triplets, there were physically remarkable characteristics from the mathematical representation which initially cast doubt on their existence. The representations [3] and [3] each contain an isodoublet with Ti = +1/2 and T 2 = -1/2, and an isosinglet with T3= 0 (Figure 4). If we interpret a particle as a state ψ ν with ν = 1,2,3 of the triple [3], and correspondingly an antiparticle as a state ψ ν of [3], then the eigenvalue equations (6)

Ϋ 3 ψ ν = Τψ ν .

apply. The hypercharges Yi = 1/3, Y2 = 1/3 and Y3 = 2/3 result as eigenvalues of the Y-operator (7)

Ϋψ ν = Υ ν ψ ν .

For the antiparticles ψ ν , we get the respective opposite hypercharges -5,-5 and On account of (2), it then follows for the charges: (8)

0 ψ ι = ( ^ Υ + Τ 3 )ψι = ( ^ + ^)ψι = ^ ψ ι A

,1«

,11

1N

11

Q¥2 = ( - Υ + Τ 3 )ψ2 = 1 lie outside the unit circle and are attracted by the infinite. The situation changes drastically if a complex number is selected for c, e.g. c = -0.12375 + 0.56508Ì. The real portion and the imaginary portion of a complex number are used as the two coordinates of the points in the Gaussian plane. In Figure 10b, two zones separate the plane, as in Figure 10a. But the inner attractor is not zero. Moreover, the boundary is in no way as smooth as in the circle, but completely irregular. If we were to magnify the boundary, the boundary would remain just as irregular as it appears on the larger illustration. Figure 10b resembles a rocky coastline, and Mandelbrot introduced a wider public to such figures for the first time in 1967 in an article entitled, "How Long is the Coast of Britain?" 237 In the history of mathematics such figures had first been investigated by the French mathematicians G. Julia and P. Fatou, who were interested in drawing very precise maps during the First World War. 238 Mandelbrot called such geometric objects with extremely irregular boundary surfaces "fractals ". The characteristic of such a figure is the completely irregular, random and chaotic boundary of the fractal, although it is completely determined by a simple quadratic equation. 239 We should also emphasize the symmetry characteristic of self-similarity, since the same structure is conserved at any arbitrary magnification. It is as if we were looking at a coastline which, first photographed from an airplane and then magnified step by step at certain points, again and again shows a deeply-indented structure. Nor can it be precisely determined, relative to the observer, where the exact boundary between sea and land runs between the reefs, projecting cliffs and water lapping against the sandy shore.

237

238

239

Β. B. Mandelbrot, How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, in: Science 156 1967, 636. P. Fatou, Sur les équations fonctionnelles, in: Bull. Soc. Math. (Fr.) 47 1919, 161-271; 48 1920, 33-94, 208-314; G. Julia, Sur l'itération des fonctions rationelles, in: Journal de Math. Pure et Appi. 8 1918,47-245. See B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco 1982.

554

4. Symmetries in Modern Physics and Natural Sciences

a

b

Fig. 10

4.4 Symmetries in Chemistry, Biology and the Theory of Evolution

555

Figure 10c shows the famous Mandelbrot set. If we begin with xo = 0 and apply (25) to complex parameters c, then the black portion of Figure 10c contains those points which are not pulled into infinity in the course of their iteration. Here again, note the extremely irregular edge, but also the reflective symmetry of the figure. If we magnify the edge, we encounter an incredible variety of shapes which recall the morphology of marine flora and fauna, patterns seen in surface magnifications of materials, turbulence patterns in fluids, or simply an amazingly refined work of art. Under magnification, new structures appear again and again in the chaotic peripheral and background areas of the images. And everywhere we discover that the seed is the Mandelbrot set, which retains self-similarity in an infinite preformation. 240 This symmetry in the chaos of the variety of shapes is typical for the geometry of non-linear dynamical systems. We immediately think of the genetic organization of higher organisms, in which each cell contains in itself the invariant genome and the complete spectrum of possibilities of development, but only a small portion of that development is actually realized. Of course, the Mandelbrot set is only a mathematical model. The microcosm of an actual organism can ultimately not be investigated to arbitrary depths. But even in the mathematical model, the decisive characteristics of non-linear dynamical systems can be studied, namely symmetry, chaos and evolution. The geometry of fractals opens up a variety of shapes which remind us of the complex shapes of nature. In nature, as a rule, we do not find ideal spheres, triangles and rectangles, but - and suddenly the scales fall from our eyes - we see fractals everywhere: clouds, horizons, mist, a wave breaking on the shore, complex blades of grass, rough patterns on the surface of the skin etc. Instead of Euclidean geometry with its right angles and smooth circles, the study of dynamical systems shows us new geometric paradigms, the variety of which continues Goethe's metamorphoses and D'Arcy Thompson's Platonic morphogenesis of nature. Everything seems related to everything else. It is only a question of position, which shapes emerge clearly from the totality of the whole and which seem blurred and indistinct. The study of non-linear dynamical systems therefore gives us a "bird's eye" view of the totality of reality, as we actually experience it every day in the incredible complex interrelationship of its parts. We do not observe reality as a collection of isolated components or organisms, cells, molecules, atoms and elementary particles, but as a hierarchically ordered structure. R. Thom, one of the founders of the modern geometry of chaos and fractals, has this to say on the subject: 240

See the computer images generated by H.-O. Peitgen/P.H. Richter, see Note 236, 9 ff.

556

4. Symmetries in Modem Physics and Natural Sciences

"And finally, the selection of those phenomena which we consider scientifically interesting is doubtless arbitrary. Modem physics constructs giant apparatus to visualize states which last no longer than 10~ 23 seconds. Undoubtedly we not are not wrong to want to use all the technically available means to take an inventory of all experimentally accessible states. Nevertheless, we can legitimately ask a question: A set of familiar phenomena (so familiar they go completely unnoticed) also have a complex theory: The lizards on an old wall, the shape of a cloud, the spin of a falling leaf, the head of foam on a glass of beer... Who knows whether a somewhat more fundamental mathematical reflection on such minor phenomena would not ultimately prove more profitable to science?" 241

There is still symmetry in fractals, in imperfections, in the mists of the future, in chaos and in ordinary, everyday things, if we base our observations on the appropriate geometry. This view of things seems modern in an age in which many people have become skeptical about the benefits of spending money on technology, and the belief in perfection has become passé. But Thorn's defense of a new mathematical phenomenology of natural shapes should not cause us to observe nature, as it were, only on its undisturbed surface and in its ecologically correlated totality and, with Goethe, to take the field against the molecular and atomic view of nature as a "gloomy, empirical-mechanical-dogmatic torture chamber". Modern quantum mechanics teaches us about the complex interrelationships of nature even on the subatomic level. The forms of nature such as elementary particles, atoms, molecules, cells, organisms, populations, stars, star systems and galaxies arise from the complex variety of phenomena when we as observers introduce a certain position, certain means of measurement and observation, scales and conceptual abstractions. Of course reality does not have an arbitrary macroscopic depth like the self-similarity of geometric fractals. But the mathematical models of fractal geometries remind us that even in chaos, there is a hidden symmetry which is revealed to us as we make corresponding refinements to our means of observation. In spite of all the variety of forms, the non-linear equations of dynamical systems are based on a radical simplification: The chaos of fractals is at least theoretically deterministic. Harmony in chaos is therefore prestabilized, to use Leibniz's terms. But this restriction is not mandatory for dynamical systems. For physical, chemical or biological applications, stochastic equations may seem suitable, in particular when we are attempting to explain macroscopic phenomena by molecular processes. In philosophy of science, the hierarchically ordered structures of nature are understood in the following stages of theories of natural science (Figure 11). We begin with a comprehensive quantum field theory of elementary particles, and proceed through the quantum chemistry of atoms and 241

R. Thom, Stabilité structurelle et morphogenèse, Paris 1972, 26. See also R. Thom, Paraboles et Catastrophes, Paris 1984.

4.4 Symmetries in Chemistry, Biology and the Theory of Evolution

557

molecules to the biochemistry of macromolecules, and finally to the biology of organisms and their populations. Each of these stages requires radical abstractions and broken symmetries which lead to new theoretical concepts. THEORY

OBJECTS

SYMMETRIES

Quantum field theories

Elementary particles, fundamental forces, etc.

Logical symmetries of quantum systems: e.g. Aut ( # )

Dynamic symmetries: e.g. SU(2)xU(l), SU(3) forces

ζ o Ρ υ ΰ

α

§

Quantum chemistry Chemistry

Atoms, molecules, bonds, etc.

Symmetries of nuclear structure, orbitale, crystals

Biochemistry

Macromolecules, proteins, enzymes, etc.

Homochirality

Thermodynamics

Open systems with metabolism

Dissipative symmetry structures

Biology

Organisms

Functional symmetries

Ecology

Population

Ecological equilibrium

te

ο ω χΗ

Kinematic space-time symmetries: e.g. Galileo and Lorentz group en < 2 Μ Η β 90 tri > Ξ ζ α

Fig. 11

That is apparent even in the constitution of the individual elementary particles which resulted from symmetry breakings of the hypothetical common original symmetry. To determine the chemical structure of atoms and molecules, abstractions were made from certain quantum mechanical correlations, after we disregarding relativistic space-time. In the biochemistry of macromolecules, characteristic symmetry breakings occurred which could be traced to symmetry breakings in the subatomic range. During the transition to complex systems with greater degrees of freedom, like those which are investigated in thermodynamics, a temporal symmetry breaking occurs, with which the irreversible development processes of macroscopic systems in chemistry and biology become possible. The differ-

558

4. Symmetries in Modem Physics and Natural Sciences

enee between past and future is accordingly not only a subjective result of human consciousness, but can be objectively described by an entropy operator for infinite quantum systems in the context of a generalized quantum mechanics (See Chapter 4.24). But such irreversible processes are no longer the same as temporally reversible or symmetrical development equations like the Schrödinger equation. Algebraically, in the place of the unitary group of dynamics, there is the weaker structure of a semigroup, which excludes the reversal of time and velocity. The resulting uncertainty relation between temporal growth and instantaneous age (See Chapter 4.24) is characteristic for all non-equilibrium situations of complex systems and expresses the chaotic behavior of their states. Dissipative structures are the result of such complex non-equilibrium systems. The evolution of organisms and populations provides examples of complex system developments with broken time symmetry. To the extent that man is a part of this development, his distinction between past and future can be explained. But the development of structure from cells to the organism is just as related to symmetry breaking as is the population dynamic of plants and animals. The morphology of the macroscopic surfaces of nature can be described approximately by the geometry of fractals. Mathematically, the development of macroscopic, dynamic, non-equilibrium systems is described by non-linear equations of evolution. The nonlinearity seems at first glance to be an insurmountable obstacle to the linearity of quantum mechanical development processes on account of the superposition principle. But in a weak theory reduction, non-linear evolution equations can be retained by approximate decorrelation assumptions (factoring of expectation values, abstraction of higher-order correlations etc.). In this sense, the spontaneous symmetry breaking of non-linear systems can be understood at least theoretically in terms of quantum mechanics, as initial realizations of this program for models of the laser in quantum optics show. That opens up the possibility of understanding important characteristics of animate systems in a common mathematical theoretical framework of physics, chemistry and biology. But that is by no means a claim to be able to provide a complete explanation of animate systems, nor does it guarantee that the complexity of such systems can be mastered at all. The following pragmatic aspects must be considered in any case. By considering these abstractions, hierarchically higher-level theories can be reduced to the principles of the lower theories. Therefore it would be logically possible to describe a given stage in the language of a hierarchically lower stage, i.e. quantum chemistry by quantum field theory, biochemistry by quantum chemistry, biology of organisms by biochemistry, etc. But such a description generally becomes much too complex, perhaps even incomprehensible, and does not do much to inspire research. The use of the

4.4 Symmetries in Chemistry, Biology and the Theory of Evolution

559

respective theoretical language is an enormous simplification, the terminology of which is adapted to the respective observations, models, experiments etc. In the course of such considerations, it can also be appropriate to use a teleological language to describe complex systems. The pluralism of theories with the individual layers of theories is therefore not only heuristically desirable to make possible discoveries, inspirations etc., but is essential for the scientist's practical work. Therefore we are not faced with opposite and antithetical approaches, whether we begin the investigation of nature macroscopically "from above" with the correlated totality of dynamical systems, or reveal the individual subatomic, atomic and molecular layers "from below" using the methods of quantum physics, quantum chemistry and biochemistry. "Holism" and "atomism" have historically been falsely understood to be opposites, although they are related to one another. With N. Bohr, we could speak of a complementary relationship which can be explained in terms of natural science under the conditions of the stages of theory described above. In this sense, a unified theory of nature can be identified in the multiplicity of theories and languages which are related to one another. Using both methods of science, the view "from above" and "from below", the symmetries and symmetry breakings of nature become clear, provided that we are clear about the requirements and dependencies of the respective theoretical point of view. But now we have come to the final chapter, in which we shall discuss the conditions of our perceptions when dealing with the symmetries in nature.

5. Symmetry and Philosophy In the last chapter the rôle of symmetries in the natural sciences was discussed historically and systematically. In conclusion, the philosophical foundations of the concept of symmetry are to be systematically examined. The first two sections concern the epistemological presuppositions of symmetry. Organizational principles that are oriented to symmetries and to breaks of symmetry are present even in human perception and intuition. Structures of symmetry underlie human thinking and cognition as well, as evidenced by the epistemological categories of substance and causality. Philosophy of science examines the presuppositions of the process of scientific research in the narrower sense. Frequently, important discoveries have been made because they were predicted on the basis of assumptions of symmetry (e.g., Dirac's prediction of an anti-particle to the electron) or because breaks of symmetry called old assumptions into question. Historically symmetries were also linked to ideas about the simplicity of natural laws, which could not be precisely defined in a generally rigorous way, but proved to be a fruitful heuristic guide for research. Computer-based search procedures used in artificial intelligence for problem-solving, also start, in part, from assumptions of symmetry in order to develop simplified solution strategies. The symmetry principles that provide the basis for mathematicalscientific theory formation are to be differentiated from this heuristic, and frequently intuitive, application. There it is a matter of precisely defined group structures (e.g., logical symmetries, space-time symmetries, gauge groups) that establish the logical-mathematical structure of a theory. Technology applies symmetries as successful principles of construction. After this discussion, the philosopher faces the more critical question of whether symmetries are only epistemological projections onto nature that merely provide a basis for mathematical-scientific theory formation, or whether they can be understood as self-organizational principles of nature. Thus, along with the application of symmetries in epistemology and philosophy of science, their old interpretation within the philosophy of nature as ontological form - should be tested once more. The chapter closes with a look at art as a possible expression of symmetries, and breaches of symmetry in the present.

562

5. Symmetry and Philosophy

5.1 Symmetries in Intuition and Perception Our attention is particularly drawn to patterns of symmetry in perception. Symmetries are patterns of order in a chaotic manifold of points and spots that are registered on our retina only as light stimuli. From time immemorial, philosophers and psychologists have been concerned with how it is that these points become ordered and structured so as to produce the perception of form. Here we recall some of their names. G. Berkeley traced the perception of corporeal form to the experiences of the tactile sense.1 In Kant's view, intuition and perception are determined transcendentally. According to Kant, the figures and forms of phenomena are generated by the a priori forms of intuition. Thus forms are traced to an a priori faculty of human consciousness and do not pre-exist in things (as with Aristotle) or behind phenomena (as with Plato). Although forms are generated according to fixed constitutive rules, Kant asserts that intuition works entirely spontaneously in order to determine the "unity of the manifold." 2 Whereas Kant limits the forms and shapes of phenomena to the rules of Euclidean geometry, Helmholtz in the 19th century, for instance, extends them to projective and non-Euclidean geometry.3 At the same time E. Mach, in his book, The Analysis of Sensations (1900),4 determined that form should be recognized as independent of other phenomenal qualities. Thus a tree is perceived as a whole, and not as an accumulation of points of color, branches or leaves. It is noteworthy that in the art of this epoch, perception of forms and shapes was traced to the observer's "impressions", most distinctly in pointillism. In pointillist pictures the process by which the impressions of form come into being can be repeated by the observer in an outright experimental way: it is only at a certain distance from the picture that form and shape emerge spontaneously from the manifold of points of color.

1

2

3

4

G. Berkeley, An Essay towards a New Theory of Vision, Dublin, 1709, German: Versuch einer neuen Theorie der Gesichtswahrnehmung (ed. R. Schmidt), Leipzig, 1912; D.M. Armstrong, Berkeley's Theory of Vision. A Critical Examination of Bishop Berkeley's Essay towards a New Theory of Vision, Melbourne, 1960; D.W. Hamlyn, Sensation and Perception. A History of the Philosophy of Perception, London, 1961, 104-116; G.J. Stack, Berkeley's Analysis of Perception, The Hague/Paris, 1970. I. Kant, Kritik der reinen Vernunft. Axiome der Anschauung, Β 202ff., Antizipationen der Wahrnehmung, Β 207ff. Cf. also K. Mainzer, Geschichte der Geometrie, Mannheim/Vienna/Zürich, 1980, pp. 180ff. E. Mach, Beiträge zur Analyse der Empfindungen, Jena, 1886, 1900 (Die Analyse der Empfindungen und das Verhältnis des Physischen zum Psychischen ); cf. also K.R. Popper, A Note on Berkeley as Precursor of Mach, in Brit. J. Philos. Sci. 4, 26-36 (1953).

5.1 Symmetries in Intuition and Perception

563

Mach was one of the pioneers of the school of gestalt psychology, to which, e.g., M. Wertheimer, W. Köhler and K. Koffka belonged.5 The signal initiative of the gestalt psychologists is the assumption that a form is not merely the additive sum of isolated constitutive elements as building blocks, but that it represents, as a "gestalt", a superordinate whole. The whole is more than the sum of its parts. Gestalt psychology frequently distinguishes figure and ground.6 Accordingly, in contrast to ground, the figure is structured and has a significance. It is not possible to perceive both at once. If an equal significance is attributed to both, an indecisive relation between figure and ground may result. In general, small surfaces are apprehended more as figure, and large ones as ground. It is noteworthy that symmetrical patterns are perceived more as figure. Certain fixed images reveal a combination of figure-ground relation and symmetry. In Figure 1 at first it is not obvious whether the dark or the bright surface is the ground or the figure. Given the supplementary information that the white part is to be viewed as the figure and the dark part as the ground, one spontaneously recognizes a rotationally-symmetrical vase. The reverse assignment gives rise spontaneously to the impression of two faces arranged according to reflection symmetry. So the two perceptions: "vase" and "faces" are at first equally valid and in that sense symmetrical. Yet this symmetrical state is labile since perception insists on a judgment between both equally valid pieces of outside information. The situation can be compared to Figure 5b in Chapter 4.32. A sphere lying symmetrically on the midpoint of such a curve is in a labile state of balance, and a tiny change ("fluctuation") toward one side or the other will send it down the hill to the left or the right, causing it to assume a stable but asymmetrical position. In that situation we spoke about spontaneous symmetry breaking which obviously was triggered by an external, secondary physical condition. In the case of perception, spontaneous symmetry breaking of equally valid perceptual contents results from a piece of information which the observer has which is independent of external secondary conditions. The optical-physical perceptual image is the same for all observers, although they can recognize different shapes. In an externally undecided situation the observer "makes his own image", so to say. Spontaneous symmetry break-

5

6

W. Köhler, Die physischen Gestalten in Ruhe und im stationären Zustand. Eine Naturphilosophische Untersuchung, Braunschweig, 1920; idem, The Task of Gestalt Psychology, Princeton, N.J., 1969; Κ. Koffka, Principles of Gestalt Psychology, London/New York, 1935; M. Wertheimer, Produktives Denken, Frankfurt, 1957, 1964. G. Kaniza/W. Gerbino, Conversity and Symmetry in Figure-Ground Organization, in M. Henle (ed.)Vision and Artifact, New York, 1976, pp. 25-32.

564

5. Symmetry and Philosophy

Fig. 1

ing in perception is clearly an example of the fact that perceptions and their associated cognitions depend on the context of the observer. In the case of animals this context can be determined by imprinting - for example, in order to make the shape of an enemy or a member of the same species, etc., recognizable. With humans the level of training is an additional factor which, for example, allows the layman to recognize a combination of wires and glass, and the expert to recognize an electronic tube. Finally, the dependence of perceiving on context can also be predetermined by a particular world view or scientific theory. In the philosophy of science N. Hanson, T.S. Kuhn, et al., discussed this insight as perception's "dependence on theory".7 Kepler and Tycho Brahe 7

N.R. Hanson, Patterns of Discovery. An Inquiry into the Conceptual Foundations of Science, Cambridge, 1965,1 Iff.; T.S. Kuhn, The Copemican Revolution. Planetary Astronomy in the Development of Western Thought, Cambridge, 1966; L. Fleck, Entstehung und Entwicklung einer naturwissenschaftlichen Tatsache, Basel, 1935, reprinted, Frankfurt, 1980. K. Lorenz designated spontaneous cognition or split-second insight as fulguration, and thereby established an etymological connection to the Medieval concept of fulguratio as the (divine) act of new creation. Cf. Κ. Lorenz, Die Rückseite des Spiegels. Versuch einer Naturgeschichte menschlichen Erkennens, Munich, 1973, 47ff. From the viewpoint of modem system theory, the sudden emergence of new structures and system characteristics

5.1 Symmetries in Intuition and Perception

565

had the same impressions of the sun, but, for one of them, the observed effects of motion - on the basis of a heliocentric theory - were caused by the movement of the earth, whereas for the other one - on the basis of his geocentric world view - they were caused by the sun's movement around the earth. However, traditional gestalt psychology came under reproach: its assumptions were said to be based on subjective sensations and interpretations, and therefore could not be treated objectively. Here the automated pattern recognition of information-processing machines offers new possibilities for simulating human perception and clarifying the meaning of symmetry. 8 Reading machines provide an example: they decipher writing and recognize letters by certain characteristics of form, such as curves, dashes and angles. These characteristics features are coded as numbers, so that whole letters correspond to particular number codes. For example, the machine uses photocells to scan the text, translates the recorded distinguishing features of the form into number codes, which are compared with the recorded number codes for letters. If erroneous details are present, the machine can decide in favor of whatever detail is closest to the erroneous one. If one enters the number codes into a system of coordinates, the language of "near" and "far" can be defined in precise geometric terms by means of an appropriate metric. If several possibilities are equally far from the erroneous item, the situation is unresolved. 9 In this regard, consider a system of coordinates for two characteristic features that are each positioned according to a coordinate axis (Figure 2). In the example let the characteristics be two curves ) and ( with number codes 1 and 2 respectively, which can only be combined as )( for X and () for O, corresponding to number codes 12 and 21. Now if the machine records the erroneous combination 11, then the only two possible letters O and X are equidistant. What is present is a symmetrical situation which could be resolved only with a supplementary piece of information and, thereby, a break of symmetry.

8

9

is mathematically describable and can be explained - at least in the case of the emergence of physical, chemical and biological patterns - by means of phase transitions. Cf. in that regard, Chapter 4. W.S. Meisel, Computer-oriented Approaches to Pattern Recognition, New York, 1972; K.S. Fu, Syntactic Methods in Pattern Recognition, New York, 1974; R. Duda/P. Hart, Pattern Recognition and Scene Analysis, New York, 1973; U. Grenander, Pattern Synthesis, Berlin/Heidelberg/New York 1976; idem, Pattern Analysis, Berlin/ Heidelberg/New York, 1978; T. Kohonen, Associative Memory, Berlin/Heidelberg/New York, 1977. K.S. Fu, Note 8, cf. also H. Haken, Erfolgsgeheimnisse der Natur, Stuttgart, 1981, 206 ff.

566

5. Symmetry and Philosophy lrst feature

0

χ

2nd feature

1

2

Fig. 2

For the human reader the context in which a decision can be reached is, as a rule, achieved by referring back to a wholeness such as a word or a sentence that rules out one of the possibilities to begin with. Besides syntatical wholenesses, context of meaning can be drawn upon, in which case, a symmetry breaking will take place in a very complex way. Analogously, a central problem in the recognition of images is that of correcting and completing inexact, erroneous and missing pieces of information by comparing the recorded fragments with reference forms. Gestalt psychology treated such cases - on the basis of sparse data - as spontaneous cognition of a whole, namely the shape of a face. In philosophy of science we encounter this phenomenon on a new level at which a general hypothesis arises from a finite amount of data through a "spontaneous insight" or "brainstorm." The functions of associative memory are already simulated in practice in an automated form in the case of "wanted" photographs when similar faces are systmatically produced on the basis of known characteristic features and compared with the "wanted" card file. Again, in cases of decisions between symmetrical alternatives, it is only the symmetry breaking on the basis of the particular context that allows for the situation to be resolved. Recognition of spoken language is also essential for the dialog between human and machine. Here, too, an important task is to complete and perfect erroneous and incomplete information by comparison with patterns of reference. There are striking analogies between recognition of patterns and shapes in the various forms of perception and the emergence of patterns in ther-

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567

modynamic, chemical and biological processes 10 that we encountered in Chapter 4.44. In the latter case, macroscopic patterns emerged spontaneously from microscopic molecular interactions when certain secondary conditions ("critical values") were fulfilled. Perception obviously also involves interactions of elementary units (tones, light reflections, fragments of form, etc.) that give rise to a spontaneous whole ("gestalt") when there is a certain context and, thereby, certain critical values are reached. However, the process of perception involves critical values for human consciousness, i.e., evaluative and associative processes that take place when certain pattern fragments are present. Insofar as computer simulations allow for it, in the case of recognition of patterns in perception one can also speak of lawful processes that follow (statistical) rules and can be explained by them, just as in the case of molecular interactions, e.g., in the emergence of physical and chemical patterns. In the case of pattern recognition in nature as well, such processes can be predicted on the basis of well-known laws and secondary conditions. To be sure, in the case of pattern recognition in perception, that is possible today only to a limited degree (e.g., in formal languages). In particular, a spontaneous "gestalt formation" is explicable in higher cognitive processes (e.g., in the heuristic development of new ideas for solving scientific problems) only in a limited way under very reduced conditions ( c f . Chapter 5.31). In addition to computer programs (software) for recognition of forms and patterns, computer hardware also finds its counterpart in the human physical brain. At first glance the human brain seems to be structured according to reflection symmetry. Indeed, the bilaterally symmetrical structure of many animals (cf., Chapter 4.43) extends to a corresponding structure of the nervous system. Higher animals exhibit a symmetry breaking in that the two halves of the brain do not by any means retain the same function: that is, copying the left or right domain of the external world via organs of touch, sight and hearing. 11 The two halves of the brain are specialized, not only in perception, but also in higher functions of human intelligence (e.g., calculating). The cerebral asymmetry associated with this specialization is now considered to be a result of evolution. As mirror-image duplication was increas-

10

11

H. Haken (ed.), Pattern Formation by Dynamic Systems and Pattern Recognition, Berlin/Heidelberg/New York, 1979; H. Primas, Pattern Recognition in Molecular Quantum Mechanics. I. Background Dependence of Molecular States, in Theoret. Chim. Acta 39, 127-148 (1975). B. Preilowski, Vergleichende Neuropsychologie: Untersuchungen zur Gehirnasymmetrie bei Menschen und Affen, Konstanz, 1985; HJ. Jerison, Evolution of the Brain and Intelligence, New York, 1973.

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ingly dispensed with, the freed-up capacity could be used by the right and left halves of the brain for new specializations. To be sure, in animals there are also indications of isolated functional and morphological asymmetries of nervous systems and (where present) brains. But asymmetry and associated specializations are nowhere so clearly pronounced as in humans. It is also noteworthy that human thought-models of the various brain functions depend on the particular historical stages of technological development. One has only to recall the different depictions from Descartes until now of hydraulic and mechanical through electromechanical and electronic models up to the microprocessor and hologram models that have been proposed in modern times. Since technical progress is also a product of the brain, we receive the characteristic symmetry of the human brain as a system that mirrors or reflects itself.

5.2 Symmetry as a Category of Cognition Symmetries do not occur only as characteristics of perception. They are closely associated with our categories of thought. Just as with analyses of perception, in this case we have become less dependent on speculation. Logical-mathematical methods make it possible to sharpen the general categorical boundary conditions for scientific theories. Traditional systems of categories, e.g., those of Aristotle, Descartes, Spinoza, Locke, Kant, et al., are well-known. They constituted objects in what were often differing ways or arranged them divided into classes or related them in causal connections. Distinctions were made between the categories of space, time, substance, causality, etc. As a rule, systems of categories were assumed to be unchanging and unique. But they too were subject to changes based on new scientific theories, as discussed in Chapter 4. In particular, the traditional determinants of the categories of substance and causality had to undergo a revision based on modern physics (the relativity theory and quantum theory). In turn the new categorial framework conditions for space, time, substance and causality provided the same aid to orientation as they formerly did, for instance, in classical physics or as in the case of Aristotle. In modern physics we make general assumptions about what presuppositions must be fulfilled for a physical object or a physical reciprocal action before we can describe and explain the particular experimental finding. In this sense as well, a priori categories underline modern physics - however, with the restriction that neither their uniqueness nor their unrestricted validity is asserted and that instead their fallibility is taken into account.

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The experience of history of science shows that scientists are very reluctant to abandon a categorical framework. Figuratively speaking, it is not enough for it to "rain" into the house of categories in just a few examples ("locally"), i.e., for falsifications, paradoxes, etc. to occur. It is possible for the roof of the house of categories to be patched ad hoc while other parts of it continue to function successfully and be "watertight." But if one leaves the house, then, extending the metaphor, at first one is altogether "out in the rain", that is, the uncertain. In some cases such a brave step leads to scientists' "mass move" to a new categorial building. In the following it will be shown that the categories of substance, causality, and reciprocal effect are by no means the "summa genera" and "universalia", as they have frequently been characterized in philosophic tradition. Rather, they prove to be derivatives of a more general category - the concept of symmetry as a mathematical structure defined precisely in group theory, which, however, in physics, despite all its generality, represents - and there should be no doubt about this - an a priori, yet also fallible, boundary condition for the formation of scientific theory.

5.21 Symmetry and the Category of Substance The question of substance is as old as philosophy itself: what is the enduring and constant in the flux of phenomena? For Democritus it was the absolute opposites of the "emptiness" of space and the "fullness" of impenetrable atoms. The flux of phenomena derives from different arrangements and motions of atoms in empty space. For Plato it was the Idea that, for example, as ideal geometrical form or shape, remains constant behind the changing phenomena and can be seen only with the "eyes of the mind." For Aristotle the unchanging is the form in matter. Here matter or substance denotes the determinable, and form that which determines. Therefore in a hierarchy of things Aristotle distinguishes degrees of being formed from the abstraction of an unformed "first matter", as pure possibility, through the primal elements fire, water, air and earth, which are determined by the prime quality, on to the complex forms of natural things. Thus forms constitute, first of all, a merely possible ("potential") matter to a concrete ("actualized") thing. They are constant or "substantial" insofar as many objects can assume the form of a sphere, the state of hardness, the shape of an animal, etc. Leibniz later defended the sustantial forms of Aristotelian physics against the atomism of early mechanics. According to Leibniz the enduring form of things consists in the law, which endures through all changing states of things and remains unchanging ("invariant").

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"The fact that a certain law endures and includes in itself all future states of the subject that we think of as identical: that is what constitutes the identity of the substance." 12 In his famous wax example, in the second Meditation, Descartes poses the question of what is the enduring thing about wax when it changes through heating or cooling. According to Descartes, the substance of a thing is nothing other than its geometric extension which can only be thought, but not perceived. The object of perception on the other hand is the changing characteristics of the wax ("hard", "tough", "liquid", etc.).13 In Leibniz's sense it would be necessary to speak of the laws of extension or geometry, which remain invariant in the flux of phenomena, rather than to speak of extension. Leibniz further criticizes Cartesian ontology by saying that geometry is not sufficient to explain nature. Rather, it is necessary to take into consideration the invariance of the laws of force (dynamics). Thus via Leibniz one actually gains entrance from the old Aristotelian concept of substance to the scientific concept of invariance, which is oriented to natural law. D. Hume introduces the strongest criticism of the concept of substance in modern times. According to Hume, only qualities and their changes can be observed, while a substance independent of these characteristics and temporally constant, as a carrier of these qualities is a mere human fiction that does not correspond to anything observable.14 Mach radicalized Hume's critique insofar as he reduced perception to sense data or elements of sensation. Accordingly, the objects of reality are constituted by us from sense data and from the point of view of usefulness. Obviously, the concept of law is what is short-changed in this early empirical critique. Kant raises the objection to Hume that, although substance is certainly not anchored "in things", but in the consciousness of the knower, it is there as a necessary category that must be presupposed, in order to be able to constitute things in experience at all. Therefore the task of this category is to synthesize phenomena into objects of experience in order to be able to apprehend a stone as a stone, an animal as an animal and a person as a person, and not just as a manifold of sense data. The decisive point of Kant's category of substance is to make the "persistent" comprehensible as a modality of temporal determination: "In the phenomenal realm, that which is constant 12

13

14

G.W. Leibniz, Philos. Sehr. II (ed. C.I. Gerhardt), Berlin/Leipzig 1875-1890, repr. Hildesheim 1960/61, p. 264. R. Descartes, Meditationes de prima philosophia/Meditationen über die Grundlagen der Philosophie, Latin/German (ed. L. Gäbe), Hamburg, 1977. D. Hume, A Treatise of Human Nature. Being an Attempt to Introduce the Experimental Method of Reasoning into Moral Subjects, London, 1739-40, German, repr. Vol. 1 (R. Brandt, ed.) Hamburg, 1973, Part I, Sect. VI. Of Modes and Substances.

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in existence, i.e., substance, corresponds to time, which itself is unchangeable and constant, and only by substance can the sequence and simultaneity of phenomena be temporally determined." 15 In order to decide whether something endures or not, temporal determination or measurement must be used. However, establishing the temporal invariance of a characteristic is not sufficient for concluding that its carrier, i.e., substance, is invariant. Thus in physics it is not sufficient to refer to the conservation principles for such characteristics as momentum. According to Kant, things are, instead, subject to the basic principle of "universal determination." 16 It seems obvious to interpret Kant to be saying in this passage that it can be decided whether any and all characteristics pertain to a given object or not. Leibniz characterized the conjunction of all possible ascribed or excluded characteristics as the "perfect" concept for characterizing a substance.17 Independently from the historical interpretation of Kant or Leibniz, this kind of description in any case corresponds to a categorical precondition for an object in classical physics. There, at least in principle, every physical characteristic can be measured as exactly as is desired, so that a "universal" or "perfect" determination of the object of experience is always assured. The identification of a characteristic in such a case must be objective, i.e., independent of a particular observer. The objectivity of a classically physical object comes to expression in the relativity principle of classical physics, according to which those observers are to be regarded as equivalent whose coordinates are interchangeable by a transformation of the Galileo group. Therefore the objectivity of recognition is derived mathematically from the existence of a symmetry group. The symmetry group is therefore the key to defining precisely the concept of a classical physical system of objects. In this sense it is possible to define the category of substance for an exactly determined realm of application, namely classical physics, i.e., to solve the epistemological problem of how to constitute, in the flux of observations, an unchanging substance which is the carrier of the particular changeable characteristics. Thus the question as to the substance category of a physical theory is not a philosophical or physical pseudo-problem or an 15 16

17

I. Kant, Kritik der reinen Vernunft, Β 183. I. Kant, see Note 15, Β 599. Also, investigations on the natural philosophical and epistemological concept of substance by C.F. von Weizsäcker, Kants 'Erste Analogie der Erfahrung' und die Erhaltungssätze der Physik (1964) in ibid., Die Einheit der Natur, Munich, 1971, 383-404; Aufbau der Physik, Munich, 1985, 567ff. G.W. Leibniz, Discours de Métaphysique, French/German (translated by H. Herring), Hamburg 1958, § 8; cf. also Κ. Mainzer, Metaphysics of Nature and Mathematics in the Philosophy of Leibniz, in N. Rescher (ed.), Science and Mathematics in the Philosophy of Leibniz, Pittsburgh, 1988.

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open question for philosophical interpretation, but instead a well-defined task of basic theory of mathematical physics. Kant would consider this task to be one for the special metaphysics of natural science, which, however, we here assume to be precisely defineable by logical-mathematical methods. In classical mechanics we can visually imagine a massive body moving on a curved trajectory (e.g., a planet). We remember Hamilton's formulation of classical mechanics, according to which the state of a system is determined by location and momentum in phase space. According to it an object system is at any time t, a definite carrier of the characteristics designated by the observables of location and momentum. The concepts location, momentum and time that are employed in this system, are determined by their behavior with respect to the Galileo transformations. They are also called Galileo objects.18 In the case of curved trajectories, location and momentum can also be called accidental characteristics since they are temporally variant. These are to be distinguished from essential characteristics (e.g., mass, charge, geometric shape), which, like mass in classical mechanics, are temporally invariant (cf., Chapter 4.24). 19 They establish the bearer of the accidental characteristics as the object type. The object type can be determined mathematically to be the representation of the basic symmetry group, i.e., in the case of classical mechanics, as the (transitive) representation of the Galileo group on the phase space. Thus different classes of objects are characterized by different representations (e.g., the class of objects of a particular mass m). Therefore, what we describe epistemologically as the constituting of a substance is, in the case of physics, the classification of objects ("representations") on the basis of the corresponding underlying space-time symmetry group. Moreover, in order to individualize a concrete object, the behavior of the accidental characteristics must be determined. In the example, not only the mass of the planet must be known, but also the curve of the orbit, i.e., its momentum and location coordinates at every point in time. But the corresponding orbital law is an example of the use of the category of causality, which we shall discuss in the next section. 18

19

Cf. in that connection C. Piron, Foundations of Quantum Mechanics, Reading, MA, 1976; J.M. Levy-Leblond, Galilei Group and Nonrelativistic Quantum Mechanics, in J. Math. Phys. 4, 776-788 (1963); ibid., Galilei Group and Galilei Invariance: in E.M. Loebl (ed.), Group Theory and its Applications, Vol. II, New York, 1971, pp. 221-299; G. Ludwig, Foundations of Quantum Mechanics I, Berlin, 1983. On the distinction between "accidental" and "essential" characteristics, cf., P. Mittelstaedt, Sprache und Realität in der modernen Physik, Mannheim/Wein/Zürich, 1986, pp. 220ff: Formal-logical nomenclature for classically physical objects are listed there with the indication operator.

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If there can never be more than one object system, at most, at any spacetime point then it is impenetrable, figuratively speaking. In this case the identity of the individual object is guaranteed for an arbitrarily great time interval.20 The example of the planet demonstrates that the recognition of an individual object is a complicated methodical process. Any given material data are classified ("constituted" ) on the basis of a space-time symmetry group in order to be able to use causal laws to individualize a single object from among the types of objects thus constituted ("individualization"). Hence, an immediate perception does not by itself provide any objects. While constitution and individualization of objects take place spontaneously in everyday perceptions, in physics it can be precisely defined by logic and mathematics. That suggests assigning this category of substance to quantum mechanics as well, by analogy to classical mechanics, in order, for example, to be able to make electrons and other elementary particles be objects of experience. In quantum mechanics, as well, a space-time symmetry group (the Galileo group or the Lorentz group) is available for constituting quantum physical object systems whose characteristics can be apprehended objectively. The object type can be determined mathematically as a representation of the underlying symmetry group, i.e., in the case of quantum mechanics, as a (projective) representation of the Galileo group or Lorentz group over the corresponding Hilbert space. If the representation is irreducible, then the corresponding object systems are not further reducible either, and are elementary. Thereby the objects are classified again according to invariant ("essential") characteristics such as rest masses, spin, charge, etc. The operators ("observables") that correspond to these characteristics possess numerical values as eigenvalues that can be employed for characterization of the particular object system. Thus by means of definite values of, e.g., mass m and spin s, the class of electrons can be determined: (m = 9.109... · 10"31 kg, s = 1/2).21 Now, however, according to the notions of classical mechanics, in a subsequent step individual objects must be able to be individualized out of these substance categories which are defined up to invariance with respect to a definite symmetry group. Classical mechanics also referred back to the category of causality, in the form of orbital laws and an assumption of impenetrability for the bodies. For an orbital law, however, the location and momentum vectors must be specifiable at any point in time with any exactitude, 20 21

For the precise definition according to formal logic, cf. P. Mittelstaedt, see Note 19, p. 224. Cf. also G.W. Mackey, Induced Representations of Groups and Quantum Mechanics, New York, 1968; ibid., The Theory of Unitary Group Representations, Chicago, 1976.

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at least in principle. But as is well-known, that is in principle impossible in quantum mechanics since, for example, observables of location and momentum are incommensurable quantities. The maximal knowledge that can be achieved by means of a quantum system is established by its state vectors. Even the Schrödinger equation indicates a deterministic causal law for these states, a law by which every future state can be unequivocally determined. However, indications about magnitudes of location and momentum are only approximately possible. Their peculiarities will be examined again in the section about the category of causality.22 As was shown in the section on quantum mechanics, there is an essential restriction of the category of substance based on the principle of superposition. Namely, if this principle is valid without restriction, then the states of quantum systems, once interacting with each other, are entangled and in principle inseparable. In this case, therefore, quantum systems cannot be individualized. Mathematically, the unrestricted validity of the principle of superposition comes to expression in the logical symmetry of the quantum systems, i.e., in the automorphism group of pure states of the corresponding Hilbert space. According to Wigner, the group of unitary operations is a representation of this symmetry group. The representation group of the Galileo space-time group is a subgroup of it. In this sense, space-time is a special case of logical symmetry. Individualization of a quantum system, as has been shown, is possible only through restriction of the principle of superposition and an associated symmetry breaking. In this case, classical characteristics ("observables") or rules of superselectivity are intoduced and allow for a step-by-step individualization of the objects. Thus, in the transition from the subatomic realm to the molecular, for the first time structures with a spatial form emerged as a result of symmetry breaking - namely molecules, whose building blocks, like quasi-electrons, became possible only through additional abstractions and associated symmetry breakings. Above the molecular level, the process of the emergence of form is repeated through break of a symmetry in alternation with the formation of new symmetries in the stepwise higher organization of matter, over and over into the biological realm. Usually symmetry is broken in this process because of dynamic instability. Examples that were discussed in Chapter 4 include the appearance of laser rays, ferromagnets, crystal formation, chemical reactions with the formation of spatial patterns, optical asymmetry in pre-biotic evolution, cell differentiation, the rise of plant structures, animal bodies and populations, etc. Figure 11 in Section 4.44 gives an overview. This constitution of a hierarchy of forms clearly recalls ideas in philosophy of nature from Aristotle and Leibniz. In this section, to be sure, the con22

Cf. also J.M. Jauch, The Quantum Probability Calculus, in Synthese 29, 131 (1974).

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stituting of categories of substance by symmetry breaking is described as an epistemological achievement. Therefore, substance is always defined precisely in relation to particular mathematical-scientific theories. The question as to whether and in what sense it is possible to form a transition from cognitive categories to ontological assumptions, is to be taken up in a later section. 5.22 Symmetry and the Categories of Causality and Interaction Effect Historically Aristotle was the first to set forth causality as a basic category. Aristotle distinguished four causes of things: the causa materialis is related to the substance of a body, the causa formalis, to its form, the causa finalis, to its function or end, and the causa ejficiens, to its efficient cause. Under the influence of modern mechanics, the causa efficiens has become the primary interest. It is discussed in modern metaphysics and epistemology as the "principle of sufficient reason" ("Satz vom Grund"), for instance, in the Leibnizian assumption that same causes have the same effects. Leibniz also established that all past and future events are linked by a continuous chain of cause-and-effect. Nothing happens without reason, and everything is unequivocally determined. In the case of the category of substance, it is Hume again who attacks the idea of a necessary conformity of law as a mere fiction. It is only the continuing experience of regularly occurring events that engenders in us the idea that this course of events occurs of necessity. Thus for Hume there are only habits of perception that associate events happening to the knower in a temporal sequence and cause him to address them as "cause" and "effect." 23 As is known, Kant admitted to Hume that the law of causality cannot be deduced from the concepts of cause and effect. Nevertheless, we have an attitude of expectation of a causal order of events before we have any specific concrete experiences. Even for animals this orientation can be essential for life, for example when they assume that the movement of a bush may be caused by a potential enemy. In the case of human recognition, this means that causality proves to be a condition for the sheer possibility that things and occurrences will emerge at all in our experience. "The principle of causal connection in the sequence of phenomena applies to all objects of experience, ..., because it is itself the basis for the possibility of such an experience." 24

23

24

D. Hume, An Inquiry Concerning Human Understanding, German: Eine Untersuchung über den menschlichen Verstand (ed. and trans. H. Herring), Stuttgart, 1967, seventh part, Of the Idea of Necessary Connexion. I. Kant, see Note 15, Β 247.

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Kant's "Principle of time sequence according to the law of causality" then reads: "All changes happen according to the law of the connection between cause and effect."25

In analogy to Kant's category of substance, here again the point is that the basic principle of experience is derived from a mode of determining time, namely, from the sequence of time. In his third principle of experience, Kant supplements his "principle of simultaneity according to the law of interaction: " "All substances, insofar as they can be perceived in space as simultaneous, are in thoroughgoing interaction."26

After persistence (substance) and temporal sequence (causality), this category corresponds to the temporal mode of simultaneity. In classical physics these principles can be applied directly. In the Hamiltonian formulation the constant causal chain is determined by differential equations according to which every future and past event can be calculated unequivocally through the magnitudes of location and momentum. By means of these laws a curved trajectory is graphically and geometrically determined in phase space. In that connection the previous section already mentioned the rôle that the category of causality plays in constituting an object of experience. If two phenomena perceived at different times can be interpreted in the sense of a causal sequence in such a way that one phenomenon follows as the effect of the other phenomenon, then both can be viewed as temporally changing characteristics of an object, as long as the point in time of the cause antecedes the point in time of the effect. It is the laws of causation that allow an individualization of bodies along space-time paths. The causal structure of classical physics was defined precisely in Chapter 3.1. The layers of simultaneity defined there, structure space-time according to future, present and past and permit signal transmission at any speed. The objectivity of the structure of causality is guaranteed by the Galilean principle of relativity, that is, the space-time symmetry of the Galileo group. In the theory of special relativity the constancy of the speed of light restricts the effective realm of that which acts causally - in contrast to classical physics. Einstein's special relativity principle together with the constancy of the speed of light results in a new structure of causality which is graphically described by the light-cone of Minkowski geometry. This affects the concept of simultaneity as well, since interactions can occur now only up to 25 26

I. Kant, see Note 15, Β 232. I. Kant, see Note 15, Β 256.

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the speed of light. To be sure, temporal events, which thus occur at a speed less than or equal to the speed of light are still unequivocally determined causally. As is well-known, this presupposition changes in quantum mechanics. The states of a quantum system can be unequivocally determined to be future and past through a corresponding Schrödinger equation. But the characteristics of the system are observables whose eigenvalues do not necessarily have to be actualized at every point in time. Thus in quantum mechanics only the following methods for the individualization of objects are left. One can unequivocally determine objects by measuring location at different times, but not continuously, because there is no causal law available that interconnects the locations. On the other hand one can exactly determine objects by causally-linked states at any point in time. But then there is no unique correlation in the sense that the previously examined object can be assigned to a subsequent state. In principle several objects can be assigned to the later state, so that recognition and identity are not guaranteed. 27 In practice the experimental data in elementary particle physics come from photographic plates of cloud chambers, bubble chambers, etc. Then the manifold of darkened points, as points of observation, must be joined by means of trajectories of elementary particles which interact in different ways. Such a procedure is successful in practice, since the measured data yield only indefinite values for location and momentum which make it possible to constitute the objects at least approximatively. 28 While previously only space-time symmetries were enlisted for the determination of substance and causality, the situation changes fundamentally in the investigation of interactions. Namely the gauge groups establish the "inner" characteristics of a physical interaction, and are thereby fundamentally distinguished from the space-time symmetry groups, which encompass only invariance with respect to the "external" alteration of spacetime systems of reference. The gauge groups establish which characteristics of the interaction are objective, or invariant, and thereby observable and measurable. In the case of electromagnetic interaction (cf., Chapter 3.25) the field was an observable magnitude that is invariant with respect to the gauge transformations (9), in contrast with the non-measurable potentials. 27

28

A thorough discussion of these two procedures is to be found in Mittelstaedt, see Note 19,228ff.; cf. also G.C. Hegerfeld/S.N.M. Ruijenaars, Remarks on Causality, Localisation and Spreading of Wave Packets, in Phys. Rev. D 22, 377-384, 1980. On the simultaneous measurement of imprecise values of location and momentum, cf. also P. Busch, Indeterminacy Relations and Simultaneous Measurements in Quantum Theory, Intern. Journ. of Theoretical Physics 24,63-92 (1985); W.K. Wootters/W.H. Zurck, Complementarity in the Double Slit Experiment: Quantum Nonseparability and a Quantitative Statement of Bohr's Principle, Phys. Rev. D 19, 473-484 (1979).

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As was shown in the previous chapters, physical interactions can be introduced as gauge fields that arise in a transition from a global symmetry to a local one. As we said, the gauge field compensates the local changes and thereby preserves symmetry. The introduction of gravitational force in the general theory of relativity was an example of the transition from a global symmetry (Lorentz-invariance in Minkowski space) to a local symmetry. It is possible that all known interactions can be derived from an original force that can be understood in terms of an original symmetry. The epistemologica! boundary conditions under which interactions can occur in the modern quantum field theories are established by the dynamic symmetries of the gauge groups. Which interactions are determined singly by which gauge groups is a question for the elaboration of physical theory and experimental confirmation in particular cases.

5.3 Symmetry in Philosophy of Science and Philosophy of Nature Philosophy of science investigates the criteria and methods by which scientific observations, hypotheses and theorems are judged. According to this traditional conception, philosophy of science has to limit itself to the justification of professed assertions and theories ("context of justification") and to present a methodology of accepted criteria for - as an example - when an observation and measurement are unambigous, when a hypothesis or theory is scientific, confirmed, or contradicted. By contrast, the creative aspect of scientific work - namely the question as to how the scientist found a new theorem, a proof, a new experiment, a new technical tool, a new theory, etc., in the first place - is bracketed out. However, the investigation of this scientific process of innovation (which traditional philosophy of science excludes under the rubric of "context of discovery" as "psychological" and "irrational" [Frege, Carnap, Popper, Reichenbach, et al.] proves to be unavoidable for an understanding of the scientific research process in which successful problem-solving depends upon both aspects. A historical and systematic analysis shows that ideas of symmetry and symmetry breaking occur as research principles under both aspects. Philosophy of science tended to associate intuitive, aesthetic and heuristic ideas with the principles of symmetry, but modern scientific theories give rise to symmetry structures that are defined mathematically by means of group theory. Traditional research principles now acquire a precisely defined and rigorous meaning. Thus the complicated structures of theories be-

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come transparent and can be classified in their disparate contexts. The symmetry structures and the partial structures that emerge from them as a result of symmetry breaking, assume a central place in modern research that again gives rise to the old question in philosophy of science of whether they are mere projections of human theory construction or forms of nature with a "fundamentum in re." By way of the detour of abstract mathematical theory construction and technology built on it, old ideas in the philosophy of nature that proceed from a unitary nature with common basic principles or that let them melt away in a multiplicity of dissolving structures become topical again.

5.31 Symmetry and the Methodology of Scientific Research The analysis of perceptions showed that the observer can spontaneously apprehend the same visual image differently in differing contexts. At first different possibilities are equally valid, so that the context supplies the supplementary information that leads to one pan of the balance spontaneously dipping as opposed the other rises, thus to a breaking of symmetry. Such "fluctuations" occur in scientific discoveries and problem solvings, as well, and are described as "brainstorms" in anecdotes from the history of science and have been greeted by a cry of "eureka " ever since Archimedes. 29 There is the familiar story of the chemist Kekulé and his discovery of the structure of benzene. He relates how after a long, unsuccessful search he fell, exhausted, into a half-sleep and how, in a dream, the atoms and molecules he was examining began to dance, linked up in various formations, and suddenly formed a ring sequence: Eureka - the benzene ring has been found! Sometimes ideas seem to come to people. Plato called this Anamnesis (remembering again). We can prepare ourselves for it in a methodical way. But creativity cannot be forced by method. A pretty example is a letter from C.F. Gauss to W. Olbers, September 3, 1805, in which he has this to report about his work in the Disquisitiones Arithmeticae: "But all my brooding and searching was in vain. I had to lay down my pen sadly again and again. Finally a few days ago it worked - not because of my strenuous efforts, but, I have to say, sheerly by the grace of God. The riddle was solved the way lightning strikes. I wouldn't

29

Cf. Κ. Mainzer, Rationale Heuristik und Problem Solving, in C. Burrichter/R. Inhetveen/R. Kötter (eds.), Technische Rationalität und rationale Heuristik, Paderborn/Munich/Vienna/Zürich, 1985, pp. 83-97.

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have been able to trace the thread leading from what I had already known, which I used to do the last experiments, to what made it succeed. 30

Such depictions recall the spontaneous occurrence of patterns, for example in the laser or in chemical combinations when certain critical values are reached by means of energy supplied from the outside. But that is just an analogy. With innovations as well, psychic processes seem to be a work with mechanisms that are not yet known in detail, but that formally recall the origin of patterns in the breaking of symmetry. Nevertheless, the processes are not to be characterized as "irrational." Closer analysis of them remains a significant task for cognitive psychology since they are of basic significance for innovations in research.31 In the history of science many discoveries and inventions were made because researchers were intuitively oriented to patterns of symmetry as research principles. One principle of research that is related to orientation to symmetry was the requirement that a hypothesis or theory should be simple. For Aristotle simplicity is already a principle of nature and a "sigillum veri." In "De cáelo" he says "At deus et natura nihil prosus faciunt frustra" and "frustra fit per plura quod potest fieri per pauciora." 32 That became a research principle in early modern science. Galileo elucidated the heuristic that led him to the laws of free fall: "Thus when I notice that a stone falling from a significantly high rest position gradually gathers speed, why should I not believe that such increments come about in the simplest way, most plausible to anyone?"33

For him that was the reason for defining the uniformly accelerated movements that were not to be experimentally tested until later. J. Kepler also maintained that "nature loves simplicity and unity."34 The heuristic that guided him in establishing the planetary laws was oriented to the principle of setting forth the simplest curves that corresponded to Tycho Brahe's data. At that time the research precept of simplicity was still associated with the ontological assumption that nature itself behaves in accordance with this 30

31

32 33

34

C.F. Gauß, Brief an Wilhelm Olbers (3. Sept. 1805), ibid., Werke, Ergängzungsreihe IV, Briefwechsel mit H.W.M. Olbers, Vol. I, Hildesheim, 1975, p. 208f. Cf. also H.A. Simon, Models of Discovery and Other Topics in the Methods of Science, Boston, 1977; F. Klix (ed.), Human and Artificial Intelligence, New York/Oxford, 1979; R.E. Mayer, Denken und Problemlösung. Eine Einführung in menschliches Denken und Problemlösung, Berlin/Heidelberg/New York, 1979. Aristotle, Decáelo 1,4, 271a. G. Galilei, Discorsi, Leiden, 1638, German, Unterredungen und mathem. Demonstrationen über zwei neue Wissenzweige, die Mechanik und die Fallgesetze betreffend (ed. and trans. A. v. Oettingen), Leipzig 1890/91, reprinted Darmstadt, 1973, 3rd day. J, Kepler, Opera omnia I (ed, C. Frisch), Frankfurt/Erlangen, 1858, reprinted Hildesheim, 1971, p. 113.

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regulative principle. Copernicus was already motivated by the idea of finding a simpler planetary model than Ptolemy's excessively complicated theory of epicycles. But here too research principles were expected to mirror the actual behavior of nature. Copernicus thought that a simple theory, central symmetry, was the true one, corresponding to reality as well, even at the price of a heliocentric cosmology. Osiander, in his foreword to Copernicus' chief work, was the first to reduce the intentions of the author to mere hypotheses offering advantages for methodology rather than claims about reality.35 Newton's first rule for the study of nature ("regulae philosophandi") was still this dictate: "To admit no more causes of natural things than such as are both true and sufficient to explain their appearances." 36 For Mach simplicity was a question of economy of method, or thrift, for the purpose of using the least possible expenditure of theory to achieve the greatest possible scientific effect. 37 It was in the 19th century that scientists first realized that it was possible to describe reality consistently using differing models. This relativity of alternative theories, which seemed evident, for instance, in the Euclidean and non-Euclidean geometries, posed a sharper question about criteria for choosing methods, such as the requirement of simplicity. Should one proceed on the basis of the effects of natural forces as the source of deviations from Euclidean metrics or go directly from a coordinate curvature, on the basis of non-Euclidean metrics?38 What is intuitively convincing at first, turns out to be rather complicated in scientific practice. That which is psychologically simple and convincing, does not have to be mathematically simple, and what is mathematically simple does not have to be physically true. Simplicity depends on the particular context, and there are criteria that can be precisely defined only in a "local" context that frequently prove not to be very intuitive beyond their narrow realm of application.

35

36

37

38

K. Mainzer/J. Mittelstraß, Kopernikus, in J. Mittelstraß (ed.), Enzyklopädie Philosophie und Wissenschaftstheorie, Vol. 2, Mannheim, 1984, pp. 470-474; F. Krafft, Physikalische Realität oder mathematische Hypothese? Andreas Osiander und die physikalische Erneuerung der antiken Astronomie durch Nicolaus Copernicus, Philos. Nat. 14,243-275 (1973). I. Newton, Philosophiae naturalis principia mathematica, London, 1726, German, Mathematische Prinzipien der Naturlehre (publischer J.P. Wolfers), Berlin, 1872, reprinted, Darmstadt, 1963, p. 380. E. Mach, Erkenntnis und Irrtum. Skizzen zur Psychologie der Forschung, Leipzig, 1905, 1926, reprinted, Darmstadt, 1980, pp. 176f. On conventionalism in geometry, cf. especially H. Poincaré, Wissenschaft und Hypothese, Leipzig, 1914, 51f.; H. Reichenbach, Philosophie der Raum-Zeitlehre, Berlin, 1928, Braunschweig, 1977.

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The dependence on parameters or the degree of a curve may be quite meaningful classifications in certain mathematical realms. 39 Thus, for example, the requirement of linearity in quantum mechanics is surely meaningful from the point of view of computational costs of nonlinear equations and the actual availability of calculation capacity. Nevertheless, alternatives are conceivable that could solve, for example, the difficult problem of measurement in quantum mechanics with the higher degree of complexity of the basic quantum-mechanical equation. Ever since Antiquity, one requirement related to simplicity has been the idea of perfection, which was frequently connected closely with ideas of symmetry and the demands of aesthetics. In Antiquity the sphere and the circle were considered to be perfect geometric forms and were therefore used preferentially in astronomy, while the Platonic bodies, with their perfect polyhedral symmetry, were assumed to be building blocks of matter. In the Platonic tradition simplicity was supposed to symbolize the relationship to the divine Ideas. As has been shown, since Kepler perfection has been associated not only with geometrical bodies, but also with the natural laws themselves, which were interpreted as thoughts of God in the PlatonicAugustinian tradition. These religious and aesthetic aspects, which were said to point to a regular order in nature, also led up to the modern symmetry requirement for mathematical-scientific theories. Leibniz principle of sufficient reason was frequently also interpreted as a principle of symmetry. Moreover, it is not only a justification for what is and becomes (context of justification), but also the "ars inveniendi" and the heuristic for the discovery of new laws (context of discovery). Think of the lever principle which Archimedes already assumed as an α priori principle of symmetry and which was later generalized in multifarious ways. Mach's Mechanics presents this criticism: "One could take the view (according to the so-called principle of sufficient reason) that this is self-understood aside from all experience; that given the symmetry of the whole device there is no reason why the rotation should occur in one direction rather than the other. But in taking this view one would forget that the presupposition includes a set of negative and positive, involuntary, instinctive experiences. For instance, there is the negative experience that dissimilar colors of the arms of a lever, the position of the viewer, an event in the neighborhood, etc., have no influence. And there are the positive experiences... that not only weights, but also distances from the fulcrum determine equilibrium disturbance and that they are determinants of movement. 40 39

40

On the suggestion to measure the simplicity of a function by the number of its freely adjustable parameters, cf. H. Weyl, Philosophie der Mathematik und Naturwissenschaften, Munich/Vienna, 1982, pp. 198f. In that connection also Κ. Popper, Logik der Forschung, Tübingen, 1966, pp. lOOff. E. Mach, Die Mechanik. Historisch-kritisch dargestellt, Leipzig, 1933, reprinted Darmstadt, 1976, pp. lOf.

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Mach criticized Archimedes' famous derivation of the principle of equilibrium to the effect that already the static moment, that is, the product of weight and side length, was hidden in the argument, and thus the reasoning is circular. In fact, Mach criticized important aspects of the justification of the lever law, which prevails only by means of certain experimental test procedures and precise designation of the secondary conditions. However, the principle of reflection symmetry was a guide in the context of discovery, making it possible to start from this simple case to make a priori conjectures about the more complex cases of equilibrium. With a view to the a priori rôle frequently ascribed to symmetry in deriving the principles of modern mechanics, Mach continued: "Even instinctive recognition as logically powerful as the principle of symmetry used by Archimedes can be misleading. Many readers may recall what a mental shock it was when they first heard that a magnetic needle in the magnetic meridian is deflected from the meridian in a particular sense by a conductor indroduced above it in parallel. The instinctive is just as fallible as the lucidly conscious. Above all, it is of value only in a field with which one is very familiar.41

H. Weyl was right when he pointed out later that the principle of symmetry is satisfied: In that connection we assume that by means of reflection in the plane where the current and the magnetic needle are lying, the current indeed transforms into itself, but the magnet does not, instead it reverses the north and south poles. This view is possible because positive and negative magnetism "are inseparable and identical in essence." 42 Therefore assumptions about symmetry, as shown by these examples using reflection symmetry, can be of great heuristic use if they are correctly applied and if their secondary conditions are recognized and not taken to be ontological assertions. The history of science from Antiquity to modern times shows that this assessment of heuristic views is something that had to be learned. An additional heuristic principle related to symmetry is the observation of analogies. One proceeds from an area of graphic and proven problems and makes the conjecture that its relationships are, "by analogy", valid as well in the uncertain problem area. Here are at least a few examples from the history of science: Archimedes discovered the formula for the volume of a sphere by means of a physical-static analogy. He imagined a sphere divided like an apple into infinitesimal circular sections. The circular sections are "weighed out" with

41 42

E. Mach, see Note 40, p. 27. H. Weyl, see Note 39, p. 203.

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corresponding sections of cone and cylinder on a lever.43 Another discovery made by a physical-dynamic analogy is Newton's calculation of fluxions, i.e., his form of differential and integral calculus. Newton's thinking was dynamically graphic. He interpreted curve coordinates as motions. He interpreted fluxions χ and y intuitively as the velocities of a body at a point in time. He determined the derivative of a curve as a tangential velocity vector using the summation theorem of velocities.44 Leibniz, on the other hand, thought in geometrically visual terms. His key idea in the development of infinitesimal calculus was the infinitesimal triangle, which Pascal had already proposed to use for calculation of a circle. Leibniz generalized its application to continuous curves for the calculation of surface areas ("integration"). The skillful introduction of new symbols ("differentials" dx and dy) for the sides of this triangle gave rise to the differential quotient dy/dx to indicate the slope of a curve tangent.45 Newton took the principle of analogy into consideration in his second rule for the study of nature ("regulae philosophandi"): "Therefore to the same natural effects we must, as far as possible, assign the same causes. 46 "

The use of mechanical analogies for discoveries and the formation of concepts is typical in early electrodynamics, with Faraday, Thompson and Maxwell. "What I mean by a physical analogy", Maxwell writes, "is that partial similarity between the laws of one phenomenal realm and those of another, which results in each illustrating the other."47 Finally, H. Hertz writes that we form "internal phantasms" or "symbols of external objects" in such a way "that the conceptually necessary consequences of the images are always also the images of the physically necessary consequences of the envisioned objects." 48 Maxwell's and Hertz' heuristic notions of "analogy" and "image" were preparatory to the modern concept of model and structure, which requires a fully mathematized theory for its precise definition. There will be more

43

44

45

46 47

48

K. Mainzer, Grundlagenprobleme in der Geschichte der exakten Wissenschaften, Konstanz, 1981, pp. 16ff. D.T. Whiteside (ed.), The Mathematical Works of I. Newton I-II, New York/London 1964/1967; idem., The Mathematical Papers of I. Newton I-VIII, Cambridge, 1967-1981. For Pascals manuscript, cf., D.J. Struik, A Sourcebook in Mathematics 1200-1800, Cambridge, 1969, pp. 239-241 ; J.M. Child (ed.), The Early Mathematical Manuscripts of Leibniz, Chicago, 1920, pp. 38-41, likewise, J.E. Hofman, Leibniz in Paris 1672-1676, London/New York, 1974. I. Newton, see Note 36. J.C. Maxwell, The Scientific Papers of J.C.M. I (ed. W.D. Niven), Cambridge, 1890, repr. New York, 1965, p. 156. H. Hertz, Die Principien der Mechanik, in neuem Zusammenhange, Leipzig, 1894, p. 1.

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about that in the next section. First, the heuristic and cognition-guiding function of considerations of analogy must be clarified. In this century the development phase of the early atomic theory of Bohr, Rutherford, et al., was marked by construction of analogies to the development of planetary theory - from the early centrally-symmetrical models (principal quantum numbers) to the elliptical models for more complex atoms (azimuthal quantum numbers, spin numbers). Analogical discovery and invention therefore led to an interaction between logical-conceptual thinking and geometrical intuition. Neurophysiology points to the hardware for this typically human kind of information processing:49 The left half of the brain processes information chiefly in a visual manner, in images, symbols and shapes; the right half does this predominantly in discursive ways, in concepts and calculation processes. Historically, intuition and the language of images appear first, and they are the primary, archaic information processing of early humans. Modern science, to be sure, owes its success chiefly to mastering problems by conceptualizing and calculating. Yet the interplay of intuition and logical-conceptual thinking is a source of creativity even for solving complex problems. Kant's epistemology maintains that concepts without intuition are empty and intuition without concepts is blind. Principles such as simplicity, perfection, analogy, etc., were obviously "regulative ideas" (Kant) that historically proved to be a guide to cognition in solving particular problems for a specific period. Historical analysis has shown that they changed their meaning in new contexts, that they did not stay rigidly firm a priori, but instead sparkled with ambiguity and perhaps for that very reason could offer orientations in the search for new solutions of problems. The modern computer-based heuristics, symmetry assumptions can provide successful or simplifying search strategies for problem solving. For this purpose, however, the contexts and data must be mathematized, so that, for instance, the solution of the problem will be reducible to the solution of particular equations under particular secondary conditions. In this case it could be useful to begin by seeking the symmetrical equilibrium solutions and testing to see whether that solves the problem. Think, for instance, about the tasks of variation theory, optimization theory and game theory which find application in natural science, economics, and the social sciences. In spite of all the successes of modern Artificial Intelligence, our cognitions depend on "brainstorms" which were mentioned above with the result that

49

Cf. also K.R. Popper/J.E. Eccles, The Self and its Brain, Berlin/Heidelberg/London/ New York, 1977, p. 319.

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the saying still holds: The spirit blows as it will, where it will and when it will, even if we understand many of its ways better today than before. 50 In contrast to the "context of discovery", traditional philosophy of science was occupied chiefly with validation and justification of already-stated hypotheses and theories. That reveals a characteristic asymmetry of the methodology. Empirical theories with generalized hypothetical laws that apply to any number of locations, points in time, etc., cannot in principle be verified; they can only b e falsified by contradictory examples. A general logical conclusion cannot be derived from an individual case, but a general conclusion can be contradicted by the syllogism of modus tollens: If a theory results in a logical-mathematical prognosis for an individual case which is contradicted by observation or experiment, then the assumed hypothesis or theory is false for logical reasons. Unfortunately, this compelling logical conclusion is not ipso facto applicable in the empirical sciences. As P. Duhem first systematically showed, a general hypothetical law, such as the Galileo's law of falling bodies, does not at first prove anything about the future behavior of a falling body. For that, we have to make a series of additional assumptions, for instance about the preparation of an appropriate experiment (medium, air resistance, operating forces, etc.), so that only a conjunction of presuppositions is contradicted by the hypothetical law.51 W.V. Quine sharpened Duhem's argument by insisting that in every form of knowledge everything is basically interconnected with everything else and an individual assertion can be considered "locally" only by abstraction in a complex net of dependencies. In philosophy of science this so-called Duhem-Quine thesis has the result that by way of the holism of theories, the scientist gains logical latitude for modifying the presuppositions in order to rescue the theory from contradictions.52 From the point of view of symmetry this methodolical holism à la DuhemQuine is thoroughly justified. Given unrestricted validity of the principle of superposition, we proceed from a holism of quantum systems which is expressed mathematically in the symmetry of its comprehensive automorphism group. The contexts are determined in connection with the EPR50

51

52

K. Mainzer, Der Intelligenzbegriff in wissenschaftstheoretischer und erkenntnistheoretischer Sicht, in B. Reusch/W. Strombach (eds.), Der Intelligenzbegriff in den verschiedenen Wissenschaften, Vienna/Munich, 1985, pp. 41-56. Cf. P. Duhem, La théorie physique, son objet et sa structure, Paris, 1906, German, Ziel und Struktur der physikalischen Theorien, Leipzig, 1908, reprinted (L. Schäfer, ed.) Hamburg, 1978, Chapter 10. W.V.O. Quine, Two Dogmas of Empiricism, in Philos. Rev. 60, 20-43 (1951); G. Wedekind, Duhem, Quine and Grünbaum on Falsification, in Philosophy of Science 36, 375-380 (1969).

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correlations and step-by-step abstractions from them are required in order to constitute objects, facts, etc. From this perspective there are also no "nakedfacts ", such as positivists, empiricists, and common parlance invoke as the ultimate and objective instances of truth. Instead, scientific facts are human "inventions", as L. Fleck emphasized already in 1935, and not "discoveries." 53 However, it should be pointed out right away, of course, that this is not meant to open the floodgates to wild subjectivism. On the contrary: Here the criteria for rationality of scientific objectivity have a meaning that is precisely defined in logicalmathematical terms. The presentation of isolated facts and circumstances presupposes a breaking of symmetry in an originally holistic context. Which contexts are chosen is ultimately a human decision and a question of what research guidelines ("norms") researchers or groups of researchers choose for their orientation. But once the context is chosen, what follows is logically necessary. A logical-mathematical analysis of theory-testing, such as was carried out according to Duhem-Quine and in the light of mathematical symmetry, therefore shows that, along with logical conclusions, methodological norms play a decisive rôle in research. The inquiry into norms to which researchers should be oriented in the choice between interesting contexts is closely connected to the inquiry into criteria of rationality according to which research is to be developed. 54 Accordingly, mere logical tricks and ad hoc hypotheses for immunizing a theory against contradictions, are to be avoided. Instead, modifications of a theory or alternative new theories should contribute to advances in cognition. Consequently Popper, in his later work, set forth objectivity and truth as the goals of research. Science should not just eliminate errors by falsification, but also approach truth. Here truth and objectivity are not intended to be just research regulatives and teleological orientations. Popper attempts semantic definitions in which the truth content of a theory (= the class of its true consequences) and its falsity content (= the class of its false consequences) are presupposed to be unambiguously determinable. Even if a theory's absolute proximity to truth (verisimilitude) cannot be estimated, Popper hopes at least to be able to determine the relative verisimilitude of two theories.55

53 54

55

L. Fleck, see Note 7. The rôle of norms and rationality criteria in the development of science has been discussed, for example, by J. Mittelstraß, Die Möglichkeit von Wissenschaft, Frankfurt, 1974; L. Laudan, Progress and its Problems. Towards a Theory of Scientific Growth, London/Henley, 1977; N. Rescher, Unpopular Essays on Technological Progress, Pittsburgh, 1980. K.R. Popper, Conjectures and Refutations, London, 1963, pp. 231ff.

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In concrete examples, however, Poppers technical definitions of verisimilitude have proved to have little practicality for establishing a ranking order of theories that are more or less similar to truth.56 Nevertheless, Popper's emphasis on objective truth as the goal of the sciences has been well received by natural scientists who are of the opinion - more from instinct than from philosophy of science - that scientific theories are to be distinguished from philosophical opinions. I. Lakatos of the Popper school proposed pragmatic criteria for growth of knowledge.57 Where there are rival theories, the one with the greatest "heuristic power" should prevail; that is, one that makes new or even unexpected predictions, solves old problems at least as well as the alternative theory, opens new problem areas, etc. Lakatos groups together progressive theoretical developments, such as Antique astronomy from Aristotle to Ptolemy, or classical physics from Galileo to Newton, into research programs that are characterized by research norms (e.g., Newton's "regulae philosophandi"). The great discontinuities in the history of science, such as the shift from Aristotelian to Galilean physics, from Newton's physics to Einstein's and Planck's, are understood to be changes in research programs. Here Lakatos' rationality criteria are comparable to Kuhn's: when old research programs (Kuhn's research paradigms ), are in competition with new ones, the greater heuristic power for solving new and old problems is considered to be the decisive factor.58 Kuhn gives even more emphasis than Lakatos to the significance of the particular scientific-sociological context, i.e., the outlook and interests of the research groups ("scientific community"). To be sure, the interests of the economy, the state, and society, as well as those of individual research groups, set up boundary conditions (contexts) for any possibility of research and for what development will be pursued. But naturally, even the largest capital expenditures cannot force success in problem-solving: ". . . Any science grows slowly, and neither diligence nor administrative actions can establish it."59 Instead, what is central are the "ideas", the "brainstorms" and the "power of innovation" that were discussed at the outset as a special kind of symmetry breaking. They can be facilitated by methodological, economic, administrative and social boundary conditions, but not forced by them. Accord-

56

57

58 59

D. Miller, Popper's Quantitative Theory of Verisimilitude, in Brit. Joum. Philos. Sci. 25, 166-177 (1974); P. Tichy, Verisimilitude Revisited, in Synthese 38, 175-212 (1978). Cf. I. Lakatos, Falsifikation und die Methodologie wissenschaftlicher Forschungsprogramme, in I. Lakatos/A. Musgrave (eds.) Kritik und Erkenntnisfortschritt, Braunschweig (1974), pp. 89-189. Cf. T.S. Kuhn, Die Struktur wissenschaftlicher Revolutionen, Frankfurt, 1976. F.L. Bauer, Was heißt und was ist Informatik? in IBM-Nachrichten 223, 333 (1974)

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ing to Popper, society's openness is a central precondition for facilitating creativity and innovation by means of the competition between alternative theories. P.K. Feyerabend intensified this pluralism of theories uncompromisingly. It is not only the individual scientific theories that must compete with each other, but also, on the meta-level, the various methodologies of philosophy of science. 60 Strictly speaking, it is only at the point of such a pluralism of methodologies that the philosophy of science submits itself to the standard of rationality that it requires of the individual sciences. Accordingly, norms of scientific theory are subordinated to historical changes, so that new situations can call for new norms. One example of that is in the sequence of quantum field theories that has been proposed since quantum electrodynamics: the requirement of renormalization, that is, the requirement that in a rational quantum field theory divergent and infinite magnitudes must be eliminable (cf. Chapter 4.3). This norm, for example, has proven to be a restrictive requirement for the unification theory of gravitation and the other fundamental physical forces. The symmetry requirements imposed on the fundamental theories of modern physics such as the gauge concept, belong to this category also. In relativistic cosmology, as well, symmetry requirements such as the cosmological principle proved to be successful research regulatives. In contrast to the traditional symmetry requirements prevalent from Antiquity, this is not a matter of individual, intuitive ideas that are tinged by philosophical, aesthetic or religious outlooks. Today, rather, behind this norm stands a complex logical-mathematical structure theory that is understood to be the core of mathematical-scientific theory formation. The next section will discuss the framework and network of these symmetry structures, their deviations and breaks. 5.32 Symmetry and the Structures of Scientific Theories Structures are familiar to us from everyday life. In perception we register a figure as a totality. In geometry we decompose it into a set of points; for example, we distinguish straight lines and curves as subsets of the whole point set and particular sections, angles, parallels, etc., as relations between these objects, by establishing their characteristics in axioms and definitions. Such a system of sets, subsets and relations is a simple example of a structure. A population of living organisms can also be grasped as a structure that is determined by a relational system of kinship relationships, functional tasks, 60

P.K. Feyerabend, Wider den Methodenzwang, Skizze einer anarchistischen Erkenntnistheorie, Frankfurt, 1976; idem, Von der beschränkten Gültigkeit methodologischer Regeln, in Neue Hefte Philos. 2-3 (Dialog als Methode ), 124-171 (1972).

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etc. Likewise an ecosystem such as a forest consists of a system of organisms and populations that are structured by a complicated network of relations such as food chains. As we have seen, a molecule or a crystal is described by a structure that consists of a set of elements (atoms) among which relations of sequence, spacing, etc. are defined. Different objects can be examples of the same structure, as is demonstrated by the group structure of molecules. Thus structures provide the possibility of classifying the complex variety of appearances into units and wholes and of making them easy to overview. In the logical set-theoretical language of modern mathematics there is, in principle at least, no difficulty in defining and classifying structures. On the basis of an axiomatic set theory (for example, according to ZermeloFraenkel = ZF), structures are introduced through sets or systems of sets and relations are defined for their elements.61 Relations are themselves sets of ordered pairs or general η-tuples of the basic elements. Thus the 2-tupel relationship "being married" consists of the set of all couples in the assumed set of persons who are married to each other. Likewise the 3-tupel group relationship consists of a set of ordered triples of elements that fulfill the axiomatically defined group characteristics. As has been shown, we can imagine these elements as being actualized in completely different ways, for example, as two rotations in space that are carried out in succession and that together result in a third rotation, but also, for example, as two numbers that are added and provide the result of addition as the third number. On the basis of an axiomatic set theory (e.g., ZF), mathematics as a whole can be understood as the theory of abstract structures. Mathematical theories are concerned with the various kinds of structures that are introduced in set theory and can be classified in a coherent manner. That is related to the fact that set theory, together with a standard logic, also postulates strong non-logical axioms about sets, for instance, that for every set X there exists also the power set Pot(X) as the set of all subsets of X and that there are infinitely many sets. For a set X the Cartesian product X 2 = Χ χ X can be defined as the set of all pairs of elements of X (in general the set X" as the set of all η-tuples of elements of X). In general a structure is a finite system of sets whose type and their species are determined axiomatically. Thus a group (G,g) is a structure with a basis set G (e.g., real numbers) and a 3-tupel relation g on G, with the typification 61

N. Bourbaki, Elements of Mathematics: Theory of Sets, Paris, 1968; H.-D. Ebbinghaus/H. Hermes/F. Hirzebruch/M. Koecher/K. Mainzer/A. Prestel/R. Remmert, Numbers, New York/Berlin/Heidelberg/London/Paris/Tokyo/Hong Kong/Barcelona, 1990.

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(1)

591

g G Pot(X 3 ).

The structural species is defined by the group axiom (2)

α (G, g)

according to which, for example, the operation on G defined by g fulfills the associative rule, the axiom of the inverse element, etc.62 What seems so abstract at first glance provides us with a decisive advantage for the theory of science. Namely, we obtain a single linguistic framework for formulating with logical precision the enormous multiplicity of all thinkable structures, their theories and reciprocal dependencies. This makes available a coherent metatheory of all mathematical theories. If it is also possible to connect these structures by means of appropriate mapping principles with experiments, measurements, etc., i.e., with empirical "reality", then even a general metatheory of the empirical sciences (e.g., physics) would be at hand. At first this program in philosophy of science seems to recall Carnap's logical empiricism.63 But there are underlying differences. For example, the intention of logicism - to reduce mathematical concepts to logic - is not pursued. Instead a standard logic and an informal axiomatic set theory is assumed as in modern mathematics, without first getting into a discussion of mathematical foundations. The question as to how strong the set-theoretic assumptions must or may be, for introducing the mathematical structures of, for example, modern physics, shall be left to a later discussion. Another contrast to Carnap is that there is also no absolute, empirical basis with sensory and measured data, statements of protocol, and language of observation to be preferred, which would be linked to the theory by rules of correspondence. These distinctions made by Carnap in his Der logische Aufbau der Welt (The Logical Structure of the World) have proven to be impossible to carry out in the system of the natural sciences. Instead, the realm of reality and the rules of application of a mathematical theory generally depend on the theory itself. On the level of perceptions, we have already become acquainted with the dependency of knowledge on context. The part of the reality realm that is independent of mathematical theory and rules of application is called the basis domain of theory.64 However, this basis domain is not independent of all physics and experience. 62

63

64

Cf. Ν. Bourbaki, see Note 61, Chapter IV; E. Scheibe Invariance and Covariance, in J. Agassi/R.S. Cohen (ed.), Scientific Philosophy Today, Boston, 1981, pp. 311-331. Meant here is R. Carnap, Der logische Aufbau der Welt, Berlin, 1928; reprinted Hamburg, 1974. G. Ludwig, Die Grundstrukturen einer physikalischen Theorie, Berlin/Heidelberg/New York, 1978, § 2-3.

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Thus current does not belong to the basis domain of electrodynamics, since it was first defined in this theory. But mechanical forces that are introduced in mechanics belong in the basis domain of electrodynamics. For that reason G. Ludwig called mechanics a pre-theory of electrodynamics.65 For the Newtonian theory of gravitation the orbit of a satellite belongs to the basis domain. It can be determined by a pre-theory that includes geometrical optics and terrestrial geometry and is independent of gravitational theory. Methodically, therefore, a pre-theory is pre-given a priori relative to its theory. In this sense it is the task of philosophy of science to reconstruct a methodical order of the relative a priori among individual theories. As has already been mentioned, the way that mathematical theories come about is that, along with standard logic and the axioms of the ZF set theory, additional axioms about species of structures are put forward. In the same way physical theories also deal with particular kinds of structures. As an example, look once more at Newton's theory of gravitation. In Chapter 3.1 the following were assumed as basis sets: Newton's absolute time as a set Τ of points in time, absolute space as set M of points in space, and set Ρ of bodies. These sets are each individually structured, by time metrics and space metrics or functions of masses. In kinematics a connection between points of time, space points, and bodies is produced in such a way that at a specific time a specific body assumes a specific location. Therefore, kinematics can be typed as the structural element (3)

kin € Pot (TxMxP).

Its structural species is determined by the dynamic law of Newton's theory of gravitation, the equations of gravitation. The gravitational equations constitute a system of differential equations for real functions, i.e., the motions of bodies in space and time are mapped onto coordinate systems in real numbers. The place of physical structure is taken by an isomorphic structure of number sets in which the physical relationships are mapped. Therefore the corresponding axiom (4)

a(T,M,P; ... kin ...)

about the structural species would express that there is a real coordinate system f in which the basis sets Τ of time, M of space, Ρ of the set of solids, and the structure elements such as kin are mapped and the corresponding differential equations with secondary conditions apply. Obviously this type

65

G. Ludwig, see Note 64, §§ 10, 12.

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of structure differs from group structure, for instance, only in its greater complexity.66 In general we subdivide a structure (X,s) into basis sets X (abbreviation for Xi, ..., X n ) and structure elements s (abbreviation for si..., s m ). The structural type (5)

s G σ (Χ)

is established by a ladder set on X, i.e., a set that comes from X by iteration of the operation "power set of a Cartesian product." The structural species of (X,s) is established by an axiom (6)

α (X,s),

which determines the structure uniquely with respect to isomorphism, i.e., (7)

(X,s) ~ (X',s')

(a (X,s) «-> a(X',s')).

This requirement imposed on the structural species says that the axiom α does not change its truth value if one replaces the structure (X,s) with an arbitrary structure (X',s') that is isomorphic to it. In the example the group axioms are valid for the rotations of an equilateral triangle as well as for the real numbers. The axioms of the Newtonian theory of gravitation are valid for artificial satellite orbits as well as for planetary orbits of the solar system. Isomorphisms are one-to-one ('bijective') mappings of the basis sets X onto the basis sets X', whereby the typified set s is mapped onto the corresponding set s'. The typification in that case remains unchanged by this, since the corresponding copy is given by the ladder set σ (X). An invariance or symmetry postulate, which we shall characterize in the following as the canonical invariance or symmetry characteristic of a structure therefore enters into the general definition of a structure with (6) and (7).67 It can be shown in detail that the various symmetry characteristics that we elucidated in previous chapters using examples from natural science can be generally derived from the canonical symmetry characteristics of a structure. On that subject, let us again remember F. Klein's successors' characterization of geometry by means of group theory.68 Let M be the space of the 66

67

68

Cf. also E. Scheibe, Struktur und Theorie in der Physik, in J. Audretsch/K. Mainzer (eds.), Philosophie und Physik der Raum-Zeit, Mannheim/Vienna/Ziirich, 1988. In N. Bourbaki (see Note 61), Chapter IV, the requirement (7) is termed "transportability." E. Scheibe, see Note 62, p. 314, speaks of "canonical invariance." Cf. also J. Ehlers, The Nature and Structure of Spacetime, in J. Mehra (ed.), The Physicist's Conception of Nature, Dordrecht, 1973, pp. 71-91; K. Mainzer, Philosophie und

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geometry in question and G a transformation group of the real number space IRn. Then (M,F) is a structure with a typified set F 6 Pot2 (Μ χ IRn)

(8)

of coordinate systems and the structural species (9)

OG (M,F),

wherein the axiom G, which satisfy the Schrödinger equation. The structural species introduced in this way is canonically invariant in the sense of (7). The familiar invariance of the Schrödinger equation

Geschichte von Raum und Zeit, in J. Audretsch/K. Mainzer (eds.), Philosophie und Physik der Raum-Zeit, Mannheim/Vienna/Ziirich, 1988; G. Rosen, Galilean Invariance and the General Covariance of Nonrelativistic Laws, in Amer. J. Phys. 40,683-687 (1972).

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with respect to the unitary transformations U with ψ'(ί) = U ψ(ί) and H' = UHU" 1 (cf., Chapter 4.22) is a special case of it.69 Up to this point only structural types and structural species on a settheoretical foundation have been introduced. In other words, we constructed and classified thought structures. Now it is a matter of rules (instructions) for applying mathematical theory to a domain of reality. Out of the thought structures come physical image structures, out of the structural propositions come physical image propositions. A principle of mapping indicates how observations made in a domain of reality can be notated as elementary laws about a thought structure.70 One example is the results of measurement of planetary orbits according to which a planet is at a particular location at a particular time. The kinematic structural element of the Newtonian theory of gravitation is thus at first only a thought structure that is introduced by axioms of structure as a whole. The elementary propositions about the results of measurement give the structural elements an interpretation that is partly physical but never entirely so. Theoretical totalities such as planetary orbits therefore remain only thought physical models in comparison with the fragments supported by measurement. On the occasion of the 300th anniversary of the death of Johannes Kepler, one of the fathers of modern physics, Einstein maintained: "It seems that human reason has to construct forms on its own before we can distinguish them in things themselves. From Kepler's wonderful lifework we recognize this truth especially well: that knowledge cannot bloom forth from mere empiricism alone, but only from the comparison of what has been thought up and what has been observed."71

In the course of the philosophy of science Newton had already addressed the problem of "incomplete induction" in his "regulae philosophandi." There is also the fact that the results of measurement are inexact and elementary propositions such as "At time t planet χ is at location y" are also idealizations. To be sure the deviations which make up the fuzziness of measurement can also be described by means of particular mathematical structures. Thus in the example of the planetary orbit the inexactitude of the measured location depends on the distance from the observer. The so-called uniform structures make available the sets of fuzziness that are necessary in order to describe the deviations. Here it is a matter of sets of pairs of points in space that are not distinguishable by measurement. Thus for approximate measurements of location χ or time t, uniform surroundings ux und ut are 69 70 71

Cf. also E. Scheibe, see Note 62, p. 317. G. Ludwig, see Note 64, § 5. A. Einstein, Johannes Kepler. Zum 300. Todestag am 9. Nov. 1930, in the Frankfurter Zeitung, reprinted in: A. Einstein, Mein Weltbild (C. Seelig, ed.), Berlin 1964, p. 151.

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available, which would make a body with an inexactitude of ux und u t be at location χ at time t.72 The function of uniform structures is well-known from the measurement practices employed by scientists. Whether it is the astronomer evaluating the photographic plates of distant quasars, or the elementary particle physicist evaluating the photographs of subatomic particle tracks or the physician, radiograms of the human body, in all instances imprecise sets must be taken into consideration as well. We take such imprécisions into account in our everyday perceptions also. To be sure, it is necessary to make certain distinctions among these uncertainties as was explained in the previous chapters. It must be determined whether they are merely of an epistemic kind, i.e., whether they are attributable to the inadequacy of our instruments for measurement and observation while behind these uncertainties there are assumed to be clearly defined systems and causal developments as in the case of classical and relativistic physics; or whether one is dealing with characteristics ("potentialities") that are incompatible in principle which cannot be actualized in clearcut measurements, as in the case of quantum mechanics. Chapter 5.31 showed how scientists tried throughout history to orient the development of research to methodological norms that would guarantee the progress of knowledge from "less good" to "better" theories. Such judgments often appear to be subjective and arbitrary. Indeed, subjective judgments and attitudes of expectation on the part of research groups have playled a decisive rôle during the development of science, as documented by numerous scientific-sociological case studies. On the other hand, a structural analysis reveals the conditions under which one theory provides more information than another about a domain of reality, and thereby also more solutions of problems. These criteria do not depend upon whether a research group finds a theoretical development to be "better" or "worse." Rather it is a matter of exactly defining when a structure and the theory that characterizes it is more information-rich and more comprehensive than another one. In scientific practice a case can definitely occur in which one chooses the theory that is structurally poorer because under certain research constraints it provides adequate and fast problem solutions. One structure is called richer than another if both structures possess the same basis set and the same typification (cf. (5)), but the axioms of the richer structural species (cf. (6)) include those of the poorer structural species.73 A mathematical example is the transition from an ordered set to a lattice structure. Both have the same underlying basis set and order relation as their 72 73

G. Ludwig, see Note 64, § 6. G. Ludwig, see Note 64, § 8.

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structural type. The structural species of the lattice structure requires more axioms for this structural type than the usual axioms of order. One structure is more comprehensive than another if in addition new basis sets, structural types and structural species are added to it. An example of that is the above-mentioned step-by-step development of quantum mechanics, which finally extends from an Abelian group over Hilbert spaces to the self-adjoint operators of the Schrödinger equation. Correspondingly, one theory is called richer in structure than another if the first is determined by a richer structure than the other, but both have the same principal basis domain and the same mapping instructions for the application of the mathematical theory. Thus the more structurally rich theory makes it possible to make more statements and thereby to provide more information, about the same facts than the other theory does. The structurally richer theory is therefore a special case of a more comprehensive theory. Further operations known from mathematical structural analysis can be carried over without modification. Thus one speaks of one structure being "embedded" in another one by means of corresponding mappings. In the same way one structure can be "restricted" to another one by means of corresponding rules. Two theories are called equivalent if they refer to the same principal basis domain and if both can be called reciprocally more comprehensive. Examples from the history of science are at hand. Thus the transition from the Newtonian space-time theory to Einstein's is obviously a transition from a less comprehensive theory to a more comprehensive theory. Frequently the assumption of simultaneity strata in Newtonian space-time (cf. Figure 2, Chapter 3.12) and its negation in Minkowski geometry (cf. Figure 1, Chapter 4.11) is depicted as an unbridgeable contradiction that evokes the impression of erratic theoretical progress. But from the point of view of a mathematical structural analysis this is misleading. In fact, Newtonian space-time is not false (from the point of view of Minkowski geometry). Einstein's theory, namely, can be restricted to a space-time theory with inertial systems that move slowly, compared with the speed of light - to the Newtonian inertial system of the planetary system. Moreover, these subsets of inertial systems are not, in any case, spread out over too much of the cosmos. Thus restricted, Einstein's space-time theory can now be embedded in the Newtonian theory. Besides, there is at least an approximate system of strata of simultaneity in which the sun does not move, or moves only very slowly. The fact that one theory structure is richer or even more comprehensive than another one thus proves to be an objective relationship between theories that is precisely defineable in logical-mathematical terms. Such a theoretical transition is therefore just as cumulative in natural science (such as

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physics) as in mathematics, as far as increase in complexity, information content and capacity for problem solving are concerned. Thus one can talk about "upheaval" and "revolution" in psychological, sociological and ideological contexts only where such structural expansions have historically taken place. This applies to the Copernican change as well as to the historical philosophical discussion that has been going on since Einstein's introduction of relativity theory in the twenties. Indeed, after the first world war many people felt that Einstein's relativistic revision of the Newtonian conception of space-time was the collapse of an old world that had had absolute standards: "Everything is relative" was a popular slogan in an era of disintegrating values and may have furnished the ideology for a greater acceptance of Einstein's theory by some people or increased reservations and rejection by others.74 However, the example also shows that a more comprehensive theory is not necessarily a better one. In many areas of technology - such as automobiles - where we look at slow speeds compared to the speed of light, we are working successfully using classical mechanics. For other areas such as high-energy technology that is no longer true. From the point of view of structural analysis, the change from classical mechanics to quantum mechanics is nothing but the transition from a less comprehensive theory to a more comprehensive one. This structural analysis has already been carried out in Chapter 4.2, where classical mechanics and quantum mechanics were embedded in a generalized algebraic quantum mechanics. In such a structural analysis, many "contradictions", "paradoxes", and "upheavals" that had been associated with the traditional interpretations of quantum mechanics, prove to be obsolete. An example is the confusion that arose historically around the cat paradox that Schrödinger used as an illustration of the process of measurement {cf. Figure 3 in Chapter 4.23). But it arises only when von Neumann's quantum mechanics is assumed, with its unrestricted validity of the principle of superposition (and thereby without classical observables) and with the using of classical measuring instruments. In generalized, algebraic quantum mechanics the principle of superposition is restricted.75 The process of mea74

75

The political, economic and cultural background of quantum mechanics in the 20's can be analyzed analogously. Cf. Forman/von Meyenn, Quantenmechanik und Weimarer Republik, Braunschweig/Wiesbaden, 1984. Cf. also G.G. Emch, Mathematical and Conceptual Foundations of 20th-century Physics, New York/Oxford, 1984, Chapter 9. In this general mathematical framework different epistemological and ontological interpretations of quantum mechanics can be further developed. From the standpoint of dialectical materialism, H. Hörz (Materiestruktur. Dialektischer Materialismus und Elementarteilchenphysik, Berlin, 1971) had emphasized the

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sûrement can be described as an interaction of a classical system (measuring instrument) with a non-classical quantum system which results in separated final states of both systems with specific measured values of the instruments of measurement. The wave and corpuscle descriptions, historically felt to be "contradictions", are also cleared up by structural analysis. The wave description and the corpuscle description are, namely, two different approximation theories of quantum mechanics. The two approaches prove to be equivalent. Bohr's concept of complementarity can therefore be precisely defined in logicalmathematical terms. To be sure, in the case of the theory of electrons the particle description emerged first, historically, and the wave description later, whereas in the case of optics Huygens' wave theory prevailed first and not until later did Newton's idea of the corpuscle prove to be optically useful. But in the case of optics, given appropriate restriction of the wave and corpuscle theories to the domain of reality of definite diffraction experiments, equivalent theories exist. Equivalence also always means the production of redundant copies which can prove to be advantageous, to varying degrees, in the case of theories like biological evolution. Thus a structural analysis shows that Newton's, Lagrange's and Hamilton's versions of classical mechanics are equivalent. For quantum mechanics only Hamilton's version proves to be the appropriate point of departure, whereas the remaining versions can be advantageous for other problem-solving. Naturally, general rules cannot by any means be cited for the transition from less comprehensive to more comprehensive theories. Only for the actually existing structures is it possible to state in retrospect whether they are more comprehensive, structurally richer, etc. Furthermore, the context of discovery or the heuristics of the search for new structures, respectively, is an area of creative innovations. New structures are new possible physical descriptions; they open up new possible domains of reality. Here it appears that the structural forms of the domains of reality do not by any means have to be identical or similar to the structures of our imme"unity of symmetry and asymmetry" (ibid., p. 364), which now occur as the mathematically precisely definable symmetries and breaks of symmetry of classical and non-classical systems. The goal of the Copenhagen philosophy of complementarity was also a unity of physics (C.F. von Weizsäcker, Ν. Bohr, et al. ) by which the interaction of classical and quantum systems is meant. Nevertheless the philosophical distinction remains the primacy of the subject of cognition, which - according to the Copenhagen interpretation - uses the (classical) measuring instruments to create the conditions for cognition. Materialism and realism emphasize the objectivity of physical processes instead, and in these views the interaction of the measuring instrument and the quantum system is only a special example of the interaction of classical and non-classical systems, although by no means an outstanding one.

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diate environment. Instead, it is precisely in the development of relativity theory and quantum mechanics that it becomes clear how the structures of the domains of reality become diverse and more and more dissimilar to our immediate surroundings the further we penetrate into astronomical expanses or atomic depths. An impression emerges of superposed levels of reality of diverse complexity which, however, from the standpoint of philosophy of science, are at first only thought physical models and are only fragmentarily supported by measurement data. 76 Nevertheless we orient our research to such thought wholes without which just as little knowledge would result from finitely many data as the perception of a picture emerges from the individual points of color in a painting. At the conclusion of Chapter 4.44 it became clear how, in the course of their development, the natural sciences have outlined a differentiated hierarchy of complex structures with superposed and interconnected domains of reality. In Figure 11 of Chapter 4.44 there is a schematic representation of the hierarchy that emerges from the stuctures of the subatomic and atomic quantum world through the structural species of molecules and macromolecules up to the macroscopic systems of animate and inanimate nature. The structural species of the quantum world are investigated in the quantum field theories. Corresponding to the states of the quantum systems that have been discussed there are the Hilbert spaces, whose automorphism groups establish the canonical invariance in this structural species. Here is an expression of general symmetry that is assumed in the physical structural picture of the quantum world. The particular space-time proves to be a special structural species (for example the Galileo group) since its representation group is a subgroup of the general group of unitary operators in Hilbert space by which, according to Wigner's famous theorem, the automorphism groups of the quantum systems can be represented. The kinematic symmetry of the quantum world is encompassed by it. The dynamic interactions of the quantum systems; i.e., the physical fundamental forces are described by the structural species of the gauge groups and they establish the dynamic symmetries of the quantum world. Chapters 4.3 and 4.4 described the step-by-step introduction of elementary particles, atoms and molecules through breaks of symmetry to which incisive abstractions from actually existing quantum-theoretical correlations (EPR) correspond. Thus from the standpoint of structural analysis the quantum world is structurally richer and more comprehensive than is assumed, for example, in the atomic and molecular models of molecular quantum chemistry. In that view, molecular quantum chemistry arises through restriction and approximative embedding in the more comprehensive quan76

G. Ludwig, see Note 64, § 10.

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601

tum mechanics. Analogously, Newtonian space-time was described above as the restriction and approximative embedding in the more comprehensive relativistic space-time for the sake of producing an approximative description of the spatial-temporal circumstances of our environment. Thus each case is a matter of reduced descriptions of the areas of reality which are adequate for the particular everyday contexts (as in the case of Newtonian space-time) and for the context of the chemical laboratory (as in the case of quantum chemistry). In these cases we speak of weak theory reductions, since the characteristics of reduced structural species can be expressed only approximatively and not explicitly by the characteristics of the more comprehensive structural species. Here again it is clear that the relationships of the structures among each other do not have to be identical to historical developments. In the case of space-time it did happen historically that the reduced structural species of Newtonian space-time existed first, whereas in the other case, Schrödinger's general quantum mechanics came first historically and then the reduced structural species were developed in quantum chemistry for the purpose of calculating molecular relationships approximately. Thus from the standpoint of structural analysis breaks of symmetry prove to be complexity reductions of more comprehensive structures. But only by means of such complexity reductions can we recognize atoms, molecules, organisms, populations, etc., in the holism of the quantum world. Breaks of symmetry - of logical symmetries as in the case of quantum chemistry, as well as of states of order of macroscopic systems - are therefore linked with the emergence of new patterns and structures that were not yet present at the previous hierarchical level. Whether it is a matter of the recognition of patterns or the emergence of patterns, from the standpoint of structural analysis they are triggered by breaks of symmetry. The "new" emerges for us not only because we proceed to more and more comprehensive and more symmetrical structures, but also because of "symmetry breaking" and "complexity reduction." The double strategy of cognition is called richness (Reichhaltigkeit) and diversity (Vielfalt), as well as unity and symmetry. In the transition from finite to infinite quantum systems there is a characteristic increase of complexity which it is necessary to assume for the microscopic description of macrosystems in chemistry and biology. The philosophy of science requires that thermodynamics with its complex systems of many degrees of freedom must, additionally, first be embedded in a generalized quantum mechanics. In Chapters 4.24 and 4.44 it was mentioned that infinite quantum systems can display irreversible development processes and thereby an arrow of time. In contrast to finite quantum systems a temporal breaking of symmetry emerges here. It is what makes possible the dis-

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tinction between past and future, which applies to chemical and biological evolution just as much as to human history. It is noteworthy that it is not only our image of nature that is described by means of ever more complex structures, but human culture and society as well.77 In the course of modem time institutions, industries, markets, social roles, etc., have achieved such complexity and interconnectedness that the resulting profusion of information can scarcely be mastered any more. Just as reductions in complexity are often what make knowledge about nature possible, our social and political behavior require that we make simplifications and reductions so that the complexity of political, economic and social reality will not render us helpless. Goethe said: To take action one must be without a conscience. To know anything, one must leave out a piece of the truth. Our contemporary image of nature is based on abstract structural species that are linked by a complicated net of pre-theories, observations, instruments of observation and work in laboratories. Structures are set-theoretical wholes, or universals. Thus a question arises as to whether they are merely convenient tools for thought based on axioms for ordering measurement data, or whether they in any way provide information about structures of nature. In the history of philosophy this question as to the status of structures and symmetries is clearly in the tradition of the quarrel of the universals that was carried out on the eve of modern philosophy and has subliminally determined the discussions on the foundations of logic, mathematics and the natural sciences ever since.78 The example of the concept of structure and symmetry shows the scale of possible positions in the quarrel of the universals - from heavily realistic-ontological presuppositions to nominalism and positivism. Platonic ontology makes the most ambitious claim. With regard to the problem of universals one could summarize it in the expression: "Symmetria est ante res. " It holds that symmetrical structures are the real realities, and that we perceive breaks of symmetry as appearances and "shad77

78

The discussion is frequently carried on under the catchwords "generalized system theory" or "structuralism." Cf. also J. Habermas/N. Luhmann, Theorie der Gesellschaft oder Sozialtechnologie - Was leistet die Systemforschung?, Frankfurt, 1971. Luhmann characterizes complexity reduction particularly as an achievement of systems. To be sure, the basic concepts of sociological system theory are still not very precise, and show only vague analogies with the mathematically formulated structure and system concepts. On the quarrel of the universals, cf. I.M. Bochenski, Zum Universalienproblem, in I. M. Bochenski/A. Church/N. Goodman, The Problem of Universals. A Symposium, Notre Dame, Indiana, 1956, pp. 33-57; W.V.O. Quine, Ontological Relativity and Other Essays, New York/London, 1969; W. Stegmüller, Das Universalienproblem einst und jetzt, in Arch. f. Philos. 6, 192-225 (1956), ibid. 7,45-81 (1957).

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ows." Plato presupposes that perfect and ideal regular bodies are the building blocks of matter, not approximative substantial models as they occur in some crystals, for example. In the Christian-Augustinian tradition the Platonic ideas become the thoughts of God, which give nature its laws. The ontological-Platonic conception of natural laws is also found in the early mathematical physicists such as Galileo and Kepler. The physical world is conceived of as a second scripture (book of nature) side by side with the Holy Scriptures; God reveals himself to human beings through both. The book of nature is written in the language of mathematics, so that in consequence the laws of nature can be grasped only by one who masters this language. By contrast, the point of view of the Aristotelian philosophy of nature can be summarized in the expression: "Symmetria est in rebus. " The multifariousness of Being is actualized in the Aristotelian hierarchy of substantial forms. The pure possibility of matter becomes actuality via intermediate stages. According to Aristotle, the distinction between matter and form is only an abstraction we employ to describe the motion of matter. If one conceives of structures as Aristotelian forms, they are "in things", [figuratively speaking]. Thus they do not exist separately from matter. Instead it is in the motions of matter that structures, as potentialities, are actualized. In the age of mechanics the Aristotelian doctrine of forms was often misunderstood and was vehemently attacked as an obstacle on the way to mathematical physics. Leibniz is an exception. He interpreted substantial forms as the new mathematical laws of nature. As has already been mentioned, the late Heisenberg interpreted the incompatible characteristics of quantum mechanics as potentialities and related them to the Aristotelian doctrine of forms. Hermann Weyl offers an epistemological interpretation of symmetry: The invariance of natural laws shows that their validity is independent of the different frames of reference of different observers. In this sense invariance shows the intersubjective validity of natural laws (categories): "Symmetria est in mente. " According to Kant, the forms of natural laws (categories) are already pregiven through our subjective constitution of cognition. Only in this way is it possible for us to formulate natural laws at all. In speaking of natural laws Kant uses a typically political metaphor of the Enlightenment: We human beings do not recognize ontologically alleged natural laws as thoughts of God. Instead, we ourselves are "lawgivers of nature" in the framework of the constitution of our reason.79 Besides making 79

I. Kant, Prolegomena § 36: The proposition itself which has been elaborated in the course of this whole section, namely that it is possible to recognize general natural laws a priori, leads of itself to the principle that nature's supreme law-giving must lie in us ourselves,

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our own laws within the framework of our political constitutions, we also achieve autonomy vis-à-vis nature. Thus structures are products of reason, intuition and imagination and are applied according to categorial schemata for the purpose of giving order to the multifariousness of perceptual phenomena by means of physical "images" (sic Kant!). To be sure, according to Kant these structures are limited a priori to the laws of Euclidean geometry and Newtonian physics, whereas in the previous section they were introduced as set-theoretical totalities on an axiomatic basis in order to do justice to the diversity of modern scientific theory formations. The nominalist view appears in a philosophically sharper form in the conventionalistic and instrumentalistic orientations. In these orientations symmetry assumptions characterized mathematically only by their simple and transparent structure must prove their worth physically in the explanation of measurement data or for purposes of prognosis. Regarded this way, they are at best structural principles of computational formalisms. At first glance the advantage of this position seems to be that mathematical symmetry structures are not associated with symmetrical entities. Therefore symmetry is not bought at a high price of ontological assumptions. To some extent instrumentalism wants to shop for the advantages of symmetry assumptions at an ontological discount: "Symmetria est vox", one might add, in Roscelinus' formulation. Thus the situation has not changed philosophically since the days of the controversy about universals. Yet today the logical-mathematical methods are sharper, the results of measurement more exact. For that reason symmetry can be made mathematically precise as a canonical universal ("invariance property"). This is a matter of automorphism groups, as we have seen from many examples in this book. After that, however, the philosophical discussion begins again. Is this structural species a separate immaterial identity "before [all] things" as is assumed in Platonic tradition? Is it a structure of reality ("in things"), which we must presuppose in order to be able to speak mathematically about symmetry in nature? Should we use Occam's razor to cut off the superfluous Platonic creation of entities and confine ourselves to introducing mathematical structures only as useful and simple instruments for mastering nature? Now physicists do establish relationships between empirical measured data (for instance time and position coordinates) by means of transformations. Consequently the following objection was soon raised to traditional that is, in our reason, and that we must not look for general laws in nature as mediated by experience, but rather, conversely, we should look for nature - in accordance with its general lawfulness - exclusively in the conditions for the possibility of experience which lie in our senses and reason.

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nominalism which claimed only concrete measurements and observations as statements about reality: Individual measured values cannot be thought of without presupposing a "general" one, namely their relationship to other measured values. From the standpoint of mathematics this objection views what is assumed to be "general" as a mathematical function or relation. In quantum mechanics the situation is even more complicated. There the measured quantities ("observables") are already abstract mathematical objects, namely operators over a Hilbert space (thus a function space and not a number space). As was shown in the previous section, the complexity of such structural elements can be typified exactly by set theory. This addresses the discussion about the mathematical foundations in which the old quarrel of the Platonists and nominalists about universale continued in the argument between set theorists and constructivists. For the set theorists the axiomatic basis of, for example, the ZF-set theory proves its worth insofar as it excludes the familiar set-theoretical antinomies of the naive set theory. To be sure, no absolute proof of consistency is, as yet, at hand. 80 Therefore constructivists limit their concept-formations to constructive ("effective") procedures, in order to exclude contradictions in the infinity from the outset. Proceeding from the natural numbers as objects of basic elementary calculations, and using the counting model, one introduces new types of mathematical objects step-by-step - for instance, functions of natural numbers, functions of functions of natural numbers, etc., which can be represented by computational terms of growing complexity. The consistency at elementary arithmetics with natural numbers is thus extended step-bystep to more complex objects which, however, remain denumerably infinite in principle. Then for every structural element that was typified in (5) by the formation of ladder sets, it is required that the presupposed basis sets have been constructively introduced and that only such functions, formations of relations and subsets are permissible which are derivable from definite computational terms - but by no means an arbitrary power-set formation. Today research of mathematical foundations offers a scale of more-or-less complex finite and transfinite construction procedures with which mathematical terms can be introduced. For the constructivist, therefore, it is a "façon de parler" ("vox") to interpret a function as a new abstract entity. In this in-

80

K. Mainzer, Philosophische Grundlagenprobleme und die Entwicklung der Mathematik, in Grazer Philosophische Studien. Intern. Zeitschrift f. analytische Philosophie 1984, pp. 179-197.

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terpretation, all one needs to know for practical computing is the specific computational term.81 Research of mathematical foundations cannot be pursued further at this point. However, it must be pointed out that it is not possible to describe all functions and relations that are investigated in modern mathematics in definite computational terms. The axiom of choice in set theory, in particular, provides adequate examples to the contrary. Thus modern nominalism in its different varieties conceives only certain branches of mathematics, but it has the advantage of being able to furnish them with proof of consistency in principle. The natural sciences now face the question of how high a price we are willing to pay in terms of abstraction of mathematical structure formation and whether the advantages we gain justify this price. 82 First of all it must be emphasized that the use of abstract set-theoretical "ontologies" does not necessarily lead to Platonism. That is to say, they are accepted as axiomatic conceptual constructs, for example, that can make it possible to construct a coherent physical theory, for example. As such, they are thought structures, but they are not associated with the Platonic claim of reflecting an immaterial reality "behind things." We do need to follow Quine's requirement of "thrift" in making settheoretical/ontological assumptions. We pay for them at a high price - by sacrificing experience. For that reason we should be "conservative" (another of Quine's requirements) in holding onto experience.83 The example of the discussion of foundations of quantum mechanics in Chapter 4.2 allows for frankly drawing up directly an "ontological price list" of what the various positions are prepared to pay. There are realistic interpretations with very high ontological costs: The Everett interpretation requires acceptance of myriads of unobservable worlds, that is structural species, in order to be able to explain the process of measurement. 81

82

83

Cf. also P. Lorenzen, Differential und Integral. Eine konstruktive Einführung in die klassische Analysis, Frankfurt, 1965; K. Mainzer, Operative Mathematik, in J. Mittelstraß (ed.), Enzyklopädie Philosophie und Wissenschaftstheorie, Vol. 2, Mannheim/Wien/ Zürich, 1984, pp. 806-809; Κ. Mainzer, Mathematik, in J. Ritter/K. Gründer (eds.), Historisches Wörterbuch der Philsophie, Vol. V, Basel, 1980, pp. 926-935. Η. Field pursues a radical introduction of nominalism into physics in Science without Numbers. A Defense of Nominalism, Princeton, 1980. Field tries to transform some classical field theories into structural species that have been reduced by type-logic. However, the attempted reduction to a logic of the first order has to be paid for by great complication. If such a reduction should succeed, which is by no means certain, then the usual ontological/set-theoretical representation would prove to be an equivalent "façon de parler" which would, however, be methodologically less complicated for the "working physicist." W.V.O. Quine, Die Wurzeln der Referenz, Frankfurt, 1976, § 36.

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At the other end of the price list there are the positivists with their bargain rates, accepting quantum mechanics only as a convenient instrument for prognosis and calculation. Naturally, whoever does not want to pay, doesn't get much. The process of measurement can be explained only under the assumption of ad hoc hypotheses: Why the measuring instrument indicates specific measured values, but the Schrödinger equation provides only static expectation values is still a miracle to the dyed-in-the-wool positivist. Whoever tries to drive ontology out of the door altogether, lets misunderstood "miracles" in at the back door. But sometimes the ontological costs are too high. The "hidden variables" of local-realistic theories in the tradition of Einstein, Rosen and Podolsky have to be excluded as structural elements since they demonstrably lead to contradictions and experimental refutation. The opinion market thus offers a scale of more-or-less realistic interpretations of quantum mechanics at varying prices: Whoever wants to know what hold the world together in its innermost [translator's note identifying the allusion to Faust ] must pay for it.84 In his Critique of Pure Reason Kant lamented the never-ending "wars of the metaphysicians" and tried to bring them under the control of a rational constitution - the "court of pure reason." Kant's bloody battlefields of metaphysicians are perhaps an all-too-martial metaphor coming from his century. We prefer the more peaceful image of an open market of free formation of opinions which nevertheless cannot be arbitrary, as became clear from the example of quantum mechanics. A good market should be regulated by a balance of supply and demand. What ontological interests do scientists have? What structural species do they work with in their theories and laboratories? Everyone believes in a particular kind of realism : In everyday life we believe in the reality of our observations; in the research laboratory we believe in the reality of our instruments of measurement and the results they indicate. It would be strange and contradictory to expect a scientist to believe in the existence of an external world in everyday life and in the research laboratory, but not when he is working on his theories about elementary particles or black holes in the cosmos, for instance. Sometimes the concept of reality in the different disciplines of the natural sciences seem to differ. A high-energy physicist may be satisfied with the statistical interpretation of a ψ-function. But for chemists electrons, atoms, molecules or crystals are individual substantial building blocks that sometimes have remarkable, non-classical properties. 84

K. Mainzer, What Is the Price of Realism in the Quantum World?, in Manuscrito. Revistta de filosofia (Brazil, 1987), pp. 31-52.

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Generalized algebraic quantum mechanics offers a possibility for explaining classical and non-classical finite and infinite systems in one theory85 (cf. Chap. 4.24 and 4.25). In this structurally richer and more comprehensive theory - compared with classical physics and the traditional quantum mechanics of von Neumann - the paradoxes that seemed unavoidable in the simultaneous application of classical and non-classical, finite and infinite physical models of matter, also vanish. The problem of measurement becomes explicable as well. To be sure, the price for these advantages consists of the acceptance of more abstract and more complex structural species. Alternative and incompatible ways of looking at things are possible and explicable in this theoretical framework. By means of this structural analysis the complementarity of the different ways of seeing, which Bohr discussed in an intuitive way, can be made precise. Accordingly, in certain contexts it is logically consistent to ascribe an individual and real existence to electrons, atoms and molecules, as was shown above. What one may not say is that matter consists of electrons, atoms and molecules, as absolute building blocks. Electrons, atoms and molecules are actualizations of structural species in which matter presents itself under certain theoretical and experimental presuppositions. In this generalized theoretical framework the dialectics of standpoints, the diversity of contexts and the breaks of symmetry of various structural species can be rationally discussed. Those are the advantages that we purchase at the price of greater abstraction. 5.33 Symmetry and the Dialectics of Nature The modern discussion of foundations of natural science is obviously in the old traditions of the philosophy of nature, but the latter are no longer familiar to many contemporary logicians and philosophers of science. Therefore it was frequently scientists themselves, as shown by the examples of Heisenberg, Schrödinger, Einstein, among others, who recalled these traditions, while logicians and philosophers of science withdrew to methodological positions that frequently do not do justice to the scientist's own understanding of his work. Here the old philosophical problem of the mediation of concept and reality is flaring up in the discussion of symmetry. Whenever the subject of the complementarity of alternative and incompatible ways of looking at things (for example, the wave-particle dualism) has come up in scientific discussion, it is a matter of philosophical dialectics, where argu85

G.G. Emch, see Note 75; K. Mainzer, Symmetries in Nature, in Chimia 5 1988, p. 161171.

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ment and counter-argument are brought out. But the mathematical structural species are not only concepts and foundations of arguments, but are intended as images of physical reality. Moreover, modern natural science projects the model of hierarchically ordered structural species of subatomic, atomic, molecular and macroscopic matter which do not by any means lie rigidly in place, but instead are in a constant process of development - whether it is a matter of the cosmic development of the elementary particles and their forces of interactions, the emergence of dissipative patterns of nature, the evolution and selforganization of nucleic acids on up to the living beings, the function and ordering of cell processes and organisms or complex ecological systems. In the previous chapters this process of development was described in previous chapters in symmetry breakings and the formation of new substructures. With that, dialectics is addressed not only as a logic of arguments and positions, but also as the logic of structural development as "the way things work", 86 as Hegel wrote. When the twilight of the progress of knowledge begins and the natural sciences cast their long shadows, in which the old open questions of philosophy of nature are hidden, then the owls of Minerva rise once more to seek their answers in the history of philosophical thought. With his concept of dialectics, Hegel placed himself in the tradition of Heraclitus, from whom the motto of this book is borrowed: "There is not one theorem of Heraclitus that I have not included in my logic." 87 In contrast to the Eleatists' static and monist conception of being, Heraclitus proceeded from the empirical world, in a process of continuous change and becoming, which develops by opposites according to the dynamic principle of the Logos. The Logos reconciles the opposites in a hidden harmony. Glistening like all the pronouncements of the great "dark" presocratic, the Logos emerges partly as a fire-like substance endowed with reason, partly as the world law of the natural processes. 88

86

G.W.F. Hegel, Logik I, Sämtliche Werke IV (H. Glockner, ed.), p. 52. In contrast to Hegel's "Dialectics of the Mind," F. Engels developed a "dialectic of nature," which was meant to represent not only the developments of social history (as with K. Marx), but also those of all natural occurrences. Engels formulated three main laws for the naturalistic dialectics, namely: 1) the "shift from quantity to quality" and vice versa, 2) "reciprocal interpénétration of the polar opposites and - when carried to the extreme - turning into each other," 3) "development by means of contradiction or negation of negation." Cf. Marx/Engels, Werke X X , Berlin (East), 1956-1968, p. 307. Η. Hörz gives an interpretation of the physical concept of symmetry from the standpoint of dialectical materialism. See Note 75.

87

G.W.F. Hegel, Vöries. Gesch. Philos., Sämtliche Werke XVII (H. Glockner, ed.), p. 344. B. Snell (ed.), Heraklit, Fragmente (Greek/German), Munich, 1926, Darmstadt, 1979; Κ. Held, Heraklit, Parmenides und der Anfang von Philosophie und Wissenschaft. Eine

88

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As was shown in Chapter 1, the presocratic philosophy of nature first of all reflects an intuitive understanding of nature, which precedes all science. Human beings succeeded at survival, continuance and cultural development only when they acted in consonance with the great cycles, the first consciously experienced symmetries of nature. Early people had to take notice of low tide and high tide, the alternation of the seasons, of day and night, the changes of the constellations, fruitful and unfruitful periods, the periods of the woman, etc. Therefore it is not surprising that the ever-recurring natural cycles with their life-giving and -destroying power were interpreted mythologically and provided the model for the early nature religions. Nature itself appeared as a great organism with human beings tied into its natural processes. The nature mythologies and their rituals therefore served the purpose of allowing human beings to live in harmony with this organic nature. Philosophy of nature superseded the nature religions when questions were raised about the original reasons and causes of the changes in nature. An assumption was made that behind the bewildering diversity, the ongoing changes and the great cycles of nature, there was an unchanging principle of order that human beings can recognize by thinking. Although the mathematical models in the Pythagorean-Platonic tradition and in Democritean atomism seem more familiar to the contemporary scientist, in those times they were too speculative and not very convincing. Why should one interpret even the familiar life processes with abstract, rigid, dead things that one cannot even perceive? For the people then, the converse was more immediate and realistic: interpreting the unknown according to the prototype of the familiar organic life processes. That was the approach taken by Aristotle, whose philosophy of nature prevailed until the beginning of the modern age. 89 In modern terms, Aristotle rejected atomism and the mathematization of nature as speculative. He was a bom botanist, zoologist and physiologist. As we have seen, these discipline therefore decisively shaped his image of nature. Here again we encounter the familiar idea of organic life cycles which was also used in explaining inorganic processes.

89

phänomenologische Besinnung, Berlin/New York, 1980; K. Reinhardt, Heraklits Lehre vom Feuer, in Hermes 77, 1-27 (1942). Cf. also W. Kullmann, Wissenschaft und Methode. Interpretationen zur aristotelischen Theorie der Naturwissenschaft, Berlin/New York, 1974; G.A. Seeck, Die Naturphilosophie des Aristoteles, Darmstadt, 1975; W. Wieland, Die aristotelischen Physik. Untersuchungen über die Grundlegung der Naturwissenschaft und die sprachlichen Bedingungen der Prinzipienforschung bei Aristoteles, Göttingen, 1962, 2 1970; S. Toulmin/J. Goodfield, Materie und Leben, Munich, 1970.

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Whereas we today explain organic and physiological functions such as procreation, conception, heredity, digestion, maturation, etc. by inorganic and molecular processes, Aristotle with his organic paradigm of nature proceeded conversely from the familiar physiological functions and tried to explain all inorganic processes on that basis. Along with early medicine, the physiological paradigm of nature had a great influence on AntiqueMedieval chemistry and alchemy. We encounter the image of nature creating itself and actualizing itself according to its own immanent measure and ends, a "poietic" nature, as Aristotle called it, a "natura naturans" as Medieval Aristotelianism developed it.90 Moreover, faith in self-healing powers in nature is thus a typically Aristotelian attitude. By contrast, modern natural science starts out from nature as an object of cognition, a "natura naturata " that has to bend to the methodical and experimental constraints of the recognizing human being. The mechanistic world view of the early modern age is characteristic of it. As in a great clockwork, one wheel is supposed to grip into another and produce motions. Once God has wound the clock of nature, it functions according to mechanistic laws in pre-established harmony, as Leibniz expressed it. Anyone who has ever heard the many artful clocks ticking in a Baroque castle and has seen the castle garden calculated out with a compass and ruler, can visually recreate this 17th- and 18th-century mechanistic and geometrical image of nature. The physiology of the life processes was also to have a mechanistic explanation, according to which the mechanism of an organism resulted from the position of the organs. According to Descartes, "it does that with the same necessity as the mechanism of a clock follows from the force position and arrangement of its weights and wheels." In physiology and medicine, just as in physics, the organic conception of nature in Aristotle's tradition was to be driven out. In his poem "The Gods of Greece" Friedrich Schiller91 mourns the paradigm change from the Antique vision of an organic nature to the lifeless mechanism of modern science:

90

Natura naturans is a Scholastic conceptual construct derived from Aristotelian distinctions. Cf. Aristotle, Met. Δ 4.1014bl6 ff.; Phys. Β 1.193Ò12-18; de cáelo A 1.268al922; Thomas Aquinas, Summa theologiae I-II qu. 85 art. 6, de div. nom. IV, 21 (C. Pera, ed., Truin/Rome, 1950, p. 206). Cf. also H. A. Lucks, Natura naturans - natura naturata, in The New Scholasticism 9, 1-24 (1935); J. Mittelstraß, Das Wirken der Natur. Materialien zur Geschichte des Naturbegriffs, in F. Rapp (ed.), Naturverständnis und Naturbeherrschung. Philosophiegeschichtliche Entwicklung und gegenwärtiger Kontext, Munich, 1981, pp. 36-69; H. Siebeck, Ueber die Entstehung der Termini natura naturans und natura naturata, in Arch. Gesch. Philos. 3, 370-378 (1890).

91

F. Schiller, Die Götter Griechenlands, in F. Schiller, Werke (in 5 volumes. Nationale Forschungs- und Gedenkstätten d. klass. dt. Lit. in Weimar, ed.), Vol. I, Berlin/Weimar, 14 1976, 82-85.

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"Nature gutted of gods bent, like a thrall, to gravity's ponderous law, the dead beat of the pendulum clock."

Critical voices also arose against the Enlightenment's mechanistically determined image of nature. Thus Goethe derided the analytical method in Newton's optics, in which white light was dispersed by a prism into colored light and then recombined. In this way, said Goethe, the object under investigation is destroyed just as it is by physiologists who kill creatures to dissect them for the purpose of fathoming the mystery of life, or by botanists who tear a plant out of its natural setting in order to preserve its wilted corpse in a glass case. Thus Goethe stood clearly in the tradition of an organic interpretation of nature, according to which one can observe the changing natural forms only as a participant.92 At the very beginning of the 19th century a development began that was heavy with consequences for university politics. As a reaction to the Enlightenment's concept of nature, grounded in physics, Schelling, Hegel, et al., had initiated a romantic philosophy of nature in which it was thought of once more as an organism ensouled by spirit. However, their attempts to derive electricity, magnetism, chemical bonding, etc., from this speculative metaphysics, were sternly rejected by many mathematically and experimentally oriented natural scientists.93 From that point on, "mind" and "nature" were felt to be opposites. The separation of the humanities from the natural sciences was ushered in and was also institutionalized for the first time by the establishment of different faculties. Thus Schelling, referring to the older distinction between "natura naturans" and "natura naturata", distinguished between philosophy of nature, which was to deal with nature as a creative subject or as "productivity", and natural science, which was to deal with nature as a "product" and object of cognition.94 Hegel strives for a "reconciliation " of mind and nature in which nature itself is interpreted as a development leading to mind: "Rational considera92

93 94

Cf. also M. Kleinschneider, Goethes Naturstudien. Wissenschaftstheoretische und geschichtliche Untersuchungen, Bonn, 1971; D. Mahnke, Leibniz und Goethe. Die Harmonie ihrer Weltansichten, Erfurt, 1924. A. Hermann, Schelling und die Naturwissenschaften, in Technikgesch. 44, 47-53 (1977). F.W.J. Schelling, Einleitung zu dem Entwurf eines Systems der Naturphilosophie (1799) §6 II, Sämtliche Werke II (M. Schröter, ed.), p. 284. In the Ideen zu einer Philosophie der Natur als Einleitung in das Studium dieser Wissenschaft ( 1792, 2 1803), (Sämtliche Werke I, 720) he writes: The philosophy of nature follows upon the blind, conceptless way of studying nature that was generally established after the ruination of philosophy by Bacon and of physics by Boyle and Newton. It marks the beginning of a higher Erkenntnis of nature, the creation of a new organ for the intuitive understanding of it.

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tion of nature must regard how nature in itself is this process of becoming mind, of suspending its differentness." 95 Thus Hegel's dialectics of nature and mind criticizes the modern subjectivist attitude with its aim of reducing nature to objects of cognition and technical serviceability on the basis of natural science. Nature is not simply Fichte's "non-ego " alone - the mere negation of the modern self-awareness that wants to dominate, make and create everything - but it is synthesized in a superordinate unity of nature and mind, subject and object. Nature is not only the world that human beings have technically developed and culturally processed, but a Whole to which human beings belong as a mere part. Ecological discussions today try - less speculatively than in Hegel's dialectics - to counter the destructive consequences of an appropriation of nature by human beings. An ecological philosophy of nature is not merely addressing a reality authored under the technical and cultural conditions of industrial society, but is also aiming for a renewal of nature's self-sufficiency which would ultimately serve to preserve human life as a part of the whole. Nature acting self-sufficiently ("poietically " ) would then be comprehensible in the tradition of nature understood in Aristotelian terms. The background for this is not only Romanticism, but also the insight, based on natural science, that the hierarchy of the structural species in nature has by no means been closed off by the ecosystems of the plants and animals. Human society is included in nature's chains of food, raw materials and energy. On the basis of their scientific knowledge, human beings have developed a complicated technical-industrial network by which they are coupled to nature's cycles and can intervene in them directly. This what we today call the technical-scientific world. 96 Although what we are speaking of here is cycles and an understanding of the harmony of nature, the contrast to the organic understanding of nature in the tradition of Aristotle is apparent: This is no longer an appreciative acceptance of an immanent balance of nature but an intention to intervene directively. It is now indisputable that such technical interventions are what assures nutrition, health, life expectancy, prosperity, etc., in fact that modern industrial society could no longer exist without its technicalscientific foundations. However, it is just as clear that these interventions and here lies their greatest danger - can bring about incalculable and irre-

95 96

G.W.F. Hegel, Enc. phil Wiss. II, Sämtliche Werke IX (H. Glockner, ed.), p. 50. Cf. also C.F. von Weizsäcker, Der Garten des Menschlichen. Beiträge zur geschichtlichen Anthropologie, Munich/Vienna, 1977, pp. 47ff.; M. Drieschner, Einführung in die Naturphilosophie, Darmstadt, 1981; H. Rumpf/H. Rempp/M. Wiesinger, Technologische Entwicklung. Vol. I: Allgemeine Entwicklungslinien (Vol. 109,1 der Schriftenreihe der Kommission für wirtschaftlichen und socialen Wandel ), Göttingen, 1976.

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versible changes in the balances of the structural species of nature, in their complicated network.97 This insight leads to an orientation to the world that again deserves to be called "Aristotelian. " It is not to be understood, however, in the sense of an idyll that was historically lost and now needs to be restored. This stylization by later centuries never existed anyway, and it would only reflect another subjectively colored yearning for harmony and again, with Goethe and Rousseau, enter the lists against Newton and Darwin as they were wrongly understood. Instead, the goal would be a common cause with nature leading us at the same time to a normative purpose for natural science and technology, namely technical-scientific knowledge as a means toward the realization of humane purposes and the safeguarding of a humane life-world. In any case, the current discussions about the concept of nature show philosophers that they cannot withdraw into methodology and theory of natural science. On the other hand, their own history teaches them that they have to beware of veering from "enlightenment" to "enthusiasm." Thus the goal is a philosophy of nature and science in which problems of the foundations of the natural sciences remain related to the classical questions of a metaphysics of nature. The discussion of symmetry across the centuries provides impressive examples of that. Thus in quantum mechanics the old discussion about the whole and the part has reached a new significance, one that is immediately related to the concept of symmetry and that lies outside the traditional opposites of "organism", "mechanism", and "atomism." The holism of the quantum world is described by a comprehensive symmetry structure, the quantum systems by mathematical representations of symmetry groups, and their interactions by dynamic symmetries of gauge groups. The classical discussion about potentialities and actualities in nature that has been going on ever since Aristotle, through Thomas, Leibniz and Schelling to Whitehead, likewise finds its expression in the modern discussion of foundations. 98 The incompatible properties of quantum systems can be interpreted as potentiali97

98

Cf. also B. Commoner, The Closing Circle. Nature, Man, and Technology, New York, 1960; J. Mittelstraß, Technik und Vernunft. Orientierungsprobleme in der Industriegesellschaft, in J. Mittelstraß, Wissenschaft als Lebensform, Frankfurt, 1982, pp. 3764; I Prigogine/I. Stengers, Dialog mit der Natur. Neue Wege naturwissenschaftlichen Denkens. Munich/Ziirich, 1981 ; E.F. Schumacher, Die Rückkehr zum menschlichen Maß. Alternativen für Wirtschaft und Technik, Reinbeck, 1977. A.N. Whitehead, Process and Reality. An Essay in Cosmology, New York, 1978; J.B. Cobb, Jr., Whitehead and Natural Philosophy, in H. Holz/E. Wolf-Gazo (ed.), Whitehead und der Prozeßbegriff, Freiburg/Munich, 1984, pp. 137-153; H. Hendrich, Bemerkungen zu einer möglichen Bedeutung der organismischen Philosophie Whiteheads für die theoretische Biologie, ibid., pp. 205-219.

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ties that can be actualized in processes of interactions (for example, in the measurement process), but do not have to be necessary. The notions of a creative-organic nature that emerge in the AristotelianMedieval concept of natura naturans, and extend on through Leibniz and Schelling's philosophy of nature and to Whitehead's "creativity", are also not at all alien to the discussion of foundations carried on in chemistry and biology." Living systems can be understood as structural species interconnected in a complex way in which there are operational principles of selforganization that are very familiar from physics and chemistry. Here too ideas of symmetry assume a key function. At sufficient distance from thermodynamic equilibrium, chemical and biological processes can shift onto a higher organizational plane and form new holistic structures. In this case, symmetry breakings lead to new symmetry structures. This is not to say, by any means, that the old metaphysical concepts of an "autopoiesis" of nature are absorbed by the modern mathematical descriptions of structural self-organization.100 Yet it becomes clear that differentiations made between an organic, holistic, ecological interpretation of nature and a description of nature in the terms of mathematics and natural science, are based on traditional misunderstandings that historically have already had fatal consequences in the polarization of different cultural traditions ( "The Two Cultures " ), and that now provoke downright dangerous biases in attitudes about nature. Here the metaphysics of nature takes over the task of historical and systematic approach . In the area of history, in conjunction with history of science, it provides instruction about earlier concepts in the philosophy of nature and their rôle in scientific research. In systematics it points out deficiencies and limitations, as well as excessive claims, of modern conceptual constructs, and thus keeps the discourse open. A further example is the modern discussion of the status of laws of nature and facts. The "thoughts of God", which early-modern thinkers in the Platonic-Augustinian tradition read into the laws of nature, became, in the course of modern secularization, the "iron" laws of nature and the "naked" facts that the so-called scientific world-views of the 19th century sought to build upon. As was shown in Chapter 5.31, ever since Kuhn at the latest, the modern philosopher of science has been very aware that the so-called 99

100

Cf. also F. Rapp/R. Wiehl (eds.), Whiteheads Metaphysik der Kreativität, Freiburg/Munich, 1986. The fact that Whitehead's philosophy of nature does not necessarily have to be seen as a traditional ontology, but can be read as a system of categories that can be criticized and revised, is shown by H. Posner, Whiteheads Kosmologie als revidierbare Metaphysik, ibid., pp. 105-125. Cf. F.J. Varela, Autopoiesis and Cognition. The Realization of the Living, Dordrecht, 1980. A.L. Plamondon, in Whitehead's Organic Philosophy of Science (Albany, 1979), emphasizes the organic model in Whitehead's philosophy of nature.

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"facts" are established only in the context of theoretical premises and that therefore they are "loaded with theory. " It is less well-known that Schelling laughed at his contemporaries' gullibility about facts: "Without real theory there can be no real observation... Facts as such are nothing." 101 Analogously it becomes clear that so-called "natural laws" originate from predetermined theoretical contexts, namely symmetries and structural species that can be mathematically precisely defined. For example, in physics the chosen space-time symmetry, the dynamic gauge group, etc., are fundamental, and the laws of nature can be derived from them as corresponding equations of force and motion. Thus the iron laws of nature become useful mathematical formulas. Recalling a famous pronouncement by Nietzsche, one could speak of the virtual "death of natural law", or more exactly the "death of belief in natural law ", which is by no means just a figment of intellectual whimsy, but a result of the better insight into the process of scientific theory formation. 102 The choice of structural species and structural elements as the bases of theories is not arbitrary, but it does depend on the specific research context. For example, the chemist's classical laws of motion of quasi-electrons are not identical to the physicist's quantum mechanical equations of electrons, as has been shown. It is the complementarity of alternative and incompatible research contexts that is characteristic of modern research and thoroughly describable in a way that is mathematically free of contradiction. Leibnizian metaphysics does not speak of contexts, but of various "points de vue " which each open up new perspectives onto the whole and comprise a unity only in their multiplicity.103 A philosophy of nature that qualifies in this way mathematically, experimentally, and metaphysically , is in the long tradition of a "philosophia perennis " which has as its study not only the methodological foundations of science, but also the fundaments of our existence. They include human society and its history, the inquiry into which has traditionally been kept distinct from the "dialectics of nature" and the sciences. The mathematical system theory of complex dynamic systems, which was discussed in the previous sections, contains formulations by which the 101

102

103

F.W. J. Schelling, Allgemeine Betrachtungen, in Zeitschrift für spekulative Physik 1.2 1800, 130 (Werke IV, 532). F. Nietzsche, Die fröhliche Wissenschaft, Chemnitz, 1882, 5. Buch: "What our cheerfulness is all about. - The great news - that 'God is dead,' that belief in the Christian God has become unbelievable - is beginning to cast its shadow over Europe." For the relationship between Leibniz' "point de vue" or concept of perspective, and the philosophy-of-science concept of context-dependency, cf. Κ. Mainzer, see Note 17. For Whitehead's concept of perspective, which stands in Leibniz' tradition, cf. S.D. Ross, Perspective in Whitehead's Metaphysics, Albany, 1983.

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investigations of biological systems are broadened to include social-science systems in order to push back the limits of scientific methods. 104 The physical, chemical and biological examples of phase transitions of complex systems are described mathematically by nonlinear equations of evolution or population. Mathematically, the nonlinearity of these models brings out the complex interconnections of these systems: in evaluating them, one very quickly comes up against the computational limits of today's computers. At the same time possibilities open up for constructing mathematical models that go beyond the classical limits of the natural sciences. The foundation for that is a general theory of complex systems whose structure formations are described by nonlinear phase transitions and display characteristic synergistic effects. An initial application to the human being is provided by the psychology of perception, in which the spontaneous recognition of holistic patterns and shapes can be described analogously to the emergence of patterns in the above-named scientific examples. The following example originates from the sociology of population : 105 A study was conducted in the U.S. Department of Transportation in which the evolution of cities and agglomerations of population were simulated as system-theoretical phase transitions. The settlement pattern that emerged showed a heterogeneous distribution with commercial centers, areas of concentration, etc. such as is typical for North American cities. Models of business trends in the economy can be understood as phase transitions of complex systems in which external factors bring about new economic ordered states (for example, a business cycle) after unstable phases. Therefore in a general systems theory of complex dynamic systems the classical dividing lines between the natural and social sciences are relativized and are investigated in analogous models. 106 In this common theoretic framework the criteria for differentiating among natural sciences, social sciences and humanities - as they were proposed in the hermeneutic and neo-Kantian tradition - prove to be no longer binding. Uniqueness, historically unrepeatable developments, and wholeness prove to be phenomena that already appear in the natural sciences and are generally described mathematically in the dynamics of complex systems. Just think of the unique and 104

105

106

Cf. also K. Mainzer, Einheit und Grenzen der Wissenschaften in wissenschaftstheoretischer und wissenschaftshistorischer Sicht, in H. Mäding (ed.) Grenzen der Sozialwissenschaften, Konstanz, 1988. P.M. Allen et al., Dynamic Urban Growth Models. Report No. TSC-1185-3. Cambridge (Mass.): Transportation Systems Center, U.S. Dept. of Transportation, 1977. I. Prigogine, Order through Fluctuation: Self-Organization and Social System, in E. Jantsch/C.H. Waddington (eds.), Evolution and Consciousness: Human Systems in Transition, Reading (Mass.), 1976, pp. 93-133.

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unrepeatable process of cosmic and biological evolution. Complex systems such as an organism, a population, societal systems and their environment cannot be reduced to their parts without permanently disturbing and altering the system. In this sense they form a wholeness whose parts, by means of cooperation, evoke synergistic effects that are constitutive for the whole system. Under these theoretical framework conditions therefore, the hierarchy of disciplines could be continued from physics, quantum chemistry, chemistry, biochemistry, biology, by way of ecology and theory of evolution on to sociology, economics and psychology - whose systems, when they are studied, show increasing degrees of complexity. At first glance this perspective on modern system theory seems to recall A. Comte's positivistic theory of science.107 According to Comte, namely, the history of science yields a logical and historical classification of the individual disciplines from the point of view of the degree of positivity achieved. The hierarchy of the disciplines proceeds in the sequence of mathematics, astronomy, physics, chemistry, biology and finally sociology. Mathematics is not only the oldest science, but has, according to Comte, the highest degree of positivity. According to Comte the other disciplines that are built on it logically and historically can apply mathematical methods only in decreasing degrees and therefore have decreasing degrees of positivity. Comte ascribed the decreasing degree of scientific rigor to the increasing complexity of the object under investigation. Therefore he tried to develop a separate method of historical comparatistics for sociology to describe the historical genesis of institutionalized life forms. Comte's theory of development toward ever new, more complex states of order is, at least formally, not as far removed from Hegel's logic of development - of the subjective mind into objective, and ultimately absolute, Mind as the ultimate totality - as the philosophical-historical differentiation between positivism and idealism suggests. Both of them proceed from a development-logic of rationality, which they conceive to be dynamic no longer static. Both describe a success story of human rationality, which Comte, to be sure, envisions as being actualized only under the conditions of technical-scientific progress. Accordingly, Comte develops a theory of technical-scientific progress which is interpreted, against the background of the French Enlightenment and the success story of modem natural science, as a liberation from speculative-metaphysical world views. According to Comte, a theological or fictitious state in the history of the development of the human spirit is followed by a metaphysical or abstract state, 107

A. Comte, Cours de philosophie positive I - VI, Paris, 1830-1845, critical ed., Paris, 1975; A. Comte, Système de politique positive, ou Traité de sociologie I - IV, Paris 1851 -1854.

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which is ultimately superseded by the scientific or positive state. By contrast, Hegel interpreted the increase in complexity as a further development of metaphysics, and not as the history of metaphysics being eased out and displaced in the course of the development of the positive sciences. Here Comte's dazzling concept of metaphysics should be observed critically. His logic of development also ends in a quasi-metaphysical and theological final state in which his social morality, organized in the form of a positivistic church, is supposed to make all class distinctions disappear. Apparently metaphysics - as we see here again - cannot be displaced; instead it turns up in new places and garbs as secularized wisdom about salvation and a sense of direction as exemplified again, a few decades after Comte, by the Marxist logic of development. The critical difference between mathematical system theory and Comte's positivistic hierarchy of disciplines is that the former does not intend any naturalistic logic of development of natural and human history in which human cultures are predetermined as inevitable continuations of biological evolution. It is instead, a theoretical framework model intended to make interdisciplinary work by natural scientists, social scientists and scholars in the humanities possible, without totally eliminating the dividing lines between their disciplines. These system-theoretical models of highly complex social systems do not make the claim that history must develop in this way, but that under certain boundary conditions it can do so or probably will. Thus for philosophy of science, therefore, the individual case depends on the chosen context. If one takes these conditions into account, then such models of development can assume an important rôle in deliberations about political decisions, although their great complexity poses large conceptual and instrumental problems, even in this age of fast computers. Planning models and ecological, economic, technological and other developments prove to be, to a high degree, systems that are interconnected in a complex way in which the "whole" must be seen in its interrelatedness and simplistic models that have an insulated, biased perspective can become a danger to the development of the whole human life-world. In the system-theoretical formulation, the social sciences seem to be outgrowths of the natural sciences of physics and chemistry through biology, ecology and theory of evolution, and on to population biology. On the other hand, the natural sciences and technology are outgrowths of social and cultural developments, as the discussion of philosophy of science and history of science demonstrated. The discussion in philosophy of science that was triggered by Kuhn, Lakatos, et al., can be interpreted as an actual change of perspective that opened up a social-science perspective on the natural sciences and inaugurated the process of historicizing, psychologizing, and sociologizing the natural sciences. Thus the dividing lines are crossed over

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from the social sciences and the humanities to the natural and technical sciences as well. Science and technology would then be, in Max Weber's sense, the products of purpose-rational actions, in fact, specifically, of scientists and groups of scientists who argue about values like truth, utility, prestige, power, etc., in institutions like the university.108 The hierarchy of disciplines is now becoming a closed ring from the social and cultural sciences to technological and natural sciences and back. Which perspective one should choose at a particular time - whether from the social sciences to the natural sciences or from the natural sciences to the social sciences - depends on the particular context. In any case, the two ways of looking at things are not mutually exclusive but supplementary. In Niels Bohr's sense one could speak of two complementary approaches that are dependent on each other. But the natural and social sciences cannot provide the conceptual orientation needed for choosing values and goals to guide our actions. What should be, is not derivable from what is. But what is, can set limits to what should be and can mark out the potential range of actions and of possibilities for development. 109 At this point the perspective of the humanities on the natural and social sciences is fundamental. To be sure, the humanities would not be able to provide either a complete catalog of values as guidelines for action nor should they be exploited as auxiliary discipline that are supposed to contribute to increased acceptance of technological-scientific progress by using persuasion and window dressing. On the other hand, their rôle is not to be reduced as it is in O. Marquard's sceptical description of it: 110 The rôle of the humanist, he says, is to tell stories and myths and interpret art and literature in order to compensate for the damage caused by modernization by technologicalscientific progress, or at least to make it bearable by offering something "quite different." More than a cheerfully sardonic, relaxed attitude is needed to provide guidance about values and to solve problems in technologicalscientific cultures, however. What is needed is judgment that sees the individual instance of a problem in the total context of the matrix of problems, judgment that knows how to integrate any specific value into the total context of interests and gain a clear view of the whole by a change in perspective. In short, judgment looks at the particular and recognizes the general and the total interrelatedness and 108

M. Weber, Gesammelte Aufsätze zur Wissenschaftslehre (J. Winkelmann, ed.), Tübingen, 3 1968. Weber takes the view that his own position on the philosophy of science that was developed by the social sciences, is a victory over positivism and naturalism. 109 ψ Wild, Naturwissenschaften und Geisteswissenschaften - immer noch zwei getrennte Kulturen? in Universitas 1, 25-36 (1987). 110 O. Marquard, Abschied vom Prinzipiellen, Stuttgart, 1984.

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is therefore a central supplement to the technical-scientific culture of experts with its necessarily partial perspectives. The old trivium consisting of grammar, dialectics and rhetoric as the historical part of the humanities, was directed at the development of such abilities and virtues in order to supplement scientific training with general education. The humanities, therefore, are not genteel accessories to technical-scientific progress. They are also not restricted to compensatory functions of a literary-artistic nature. They are also not absorbed into social sciences or natural sciences in the sense of a methodological theory of convergence . Instead they educate for judgment about altering perspective and in this sense they offer guidelines for action. In that they are, to be sure, dependent on expert knowledge in the natural or social sciences. Therefore neither the humanities nor the natural or social sciences by themselves can assume a leading role for governing action. Today what we know, and what we should do, have become too complex and too dependent on each other in the technological-scientific world to allow for a single discipline to assume the king's rôle, or better, the queen's rôle. Even in science the complexity of the problems has made us into citizens of a republic. Thus the unity of knowledge consists of the diverse perspectives of complementary scientific fields. They do presuppose temporary dividing lines between the disciplines, but these are relativized in an interdisciplinary way in reciprocal complementarity. Therefore we can take up Leibniz' formula of "unity in diversity" in postmodern times as well, when the pluralism of world views is experienced as an unburdening and an expression of freedom, but the interdisciplinary orientation is posed as a shared task.111

5.4 Symmetry in Modern and Post-Modern Art If philosophy, according to Hegel, is "the thoughtful consideration of its times", then art is the visualization of its times. Until the Renaissance there was a mandatory canon of proportions based on geometry which science used for a foundation of the laws of harmony of the world, and art used for a foundation of the harmonious representation of the human body. According to the Antique ideal these proportional relations were to be mirrored in architecture as well, to make them graceful and consonant to the measure of man. In modern times this common basis of science and art broke apart. Mathematical natural science developed an abstract concept of symmetry that went beyond the geometrical theory of proportions. Art was no longer 111

K. Mainzer/H. Sund, Wird die Wissenschaft unübersehbar?, 9. Konstanzer Universitätssymposium, Konstanz, 1988.

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oriented to a strict geometrical canon of proportions according to the Antique model. Yet geometrical views of symmetry played a great role in certain art epochs such as the Baroque. And - though not in the sense of quantitative regularities - Classicism and Romanticism spoke of "inner perfection", "harmony of the soul", etc. According to Schelling, the cosmic harmony was expressed in them. 112 Thus the concepts of symmetry and harmony were further developed independently in science and art and were frequently alienated and broken and then taken up again from another point of view. Modern art's breach with tradition at the turn of the century coincided strikingly with the paradigm shift that scientific theory underwent for the development of modern physics. The concept of symmetry took on a preeminent rôle in this new development of mathematical natural science, as was shown in the previous chapters. In the following section my thesis will be that modern art and architecture are seeking a new center and new structural laws. The fundamental revolution in art took place during the first decade of this century in the rise of abstract art. Its goal was to work out a method of representation that would allow the painter to express his view of the world without having to draw on graphic objects and their superficial appearance. There is an astounding parallel to the problems of abstract quantum mechanical formalism, which was to be developed out of graphic classical mechanics. Artists like Picasso and Braque certainly did not read Planck, Einstein or other physicists. Their cubism had instead followed Cézanne's pronouncement that objects are made of geometrical forms such as spheres, cones and cylinders. In addition there is the recourse to archaic art. The object represented in the picture is broken up into stereometric atoms and then reassembled in a new way for the purpose of making the basic and archetypal forms of the world vivid. In 1912 a theory of cubism was formulated. 113 While modern painting has remained limited to pictorial representation, in the twenties the Bauhaus set about giving artistic form to the technicalindustrial life-world in a synthesis of the arts of architecture, painting, sculpture and functional art. Here as in Antiquity it was a matter of comprehending the human being and his life-world as a unity, but now it was 112

113

Cf., for example, F.W.J. Schelling, Philosophie der Kunst (1859), Darmstadt, 1980, pp. 26ff.; J.G. Herder, Plastik - Einige Wahrnehmungen über Form und Gestalt aus Pygmalions bildendem Traum (1778), Cologne, 1969, p. 56; J.J. Winkelmann, Gedanken über die Nachahmung der griechischen Werke in der Malerei und Bildhauerkunst (L. Uhlig, ed.), Stuttgart, 1969. Cf. J.. Metzinger, Le cubisme état né, Souvenirs, Paris, 1972; G. Apollinaire, Le peintre Cubiste, Oaris, 1913. The Cézanne memorial exhibition of 1907 measurably influenced Picasso and Braque. The theorie of cubism was developed by Metzinger and Gleizes in their 1912 essay "Du Cubisme."

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from the point of view of science, technology and industry. Corresponding to the paradigm shift in science there was thus an artistic upheaval and structural change striving toward a new measure of things and a new center. The standard is a purpose-oriented functionalism that aims to comprehend the human being in his new life-world. The words "new" and "modern" became the fashion in the twenties, which saw the political and social collapse of old world orders. The use of the word "new" ranged from the "Neues Bauen" (new constructions) and "Neues Wohnen" (new dwelling) by way of the "Neue Sachlichkeit" (new practicality) to Huxley's "brave new world", with its disillusionment and irony. Interrupted by war, many projects of modernism were not actualized or further developed until the fifties and sixties. The doubt and disenchantment that set in about the modern industrial culture began to be discussed in architecture under the catchword of postmodernism. This signalled breaks of symmetry in the functionalistic structure of the modern life-world, including breaks of style in architecture, deviations from industrial functionalism, disturbances in the ecological balance, and alternative life forms. Postmodernism raised a question of whether what had occurred was really a "loss of the center" and a loss of ideas and orientation, or whether it was instead an expression of the complementarity of incompatible ways of seeing that are ultimately related to a "hidden harmony" in Heraclitus' sense. 5.41 Symmetry in the Art and Architecture of Modernism At the beginning of this century the Royal Württemberg Provincial Trade Museum in Stuttgart organized a programmatic exhibition.114 Its purpose was to convey an integrated understanding of nature and the structures of technology. The exhibit did not present the concept of nature and symmetry as a narrow geometrical canon of proportions. Instead, under the influence of 19th-century biology, it propagandized a naturalness and classical balance that were thought to be apparent in the wealth of forms in organic nature, especially in the plant world. One of the fathers of modern sculpture, Aristide Maillol, wrote this declaration of a natural state of balance: "Nature always proceeds symmetrically in corresponding forms ... I apply the same principle to the human figure."115 George Kolbe, who became a sculptor under the influence of 114

115

G.E. Pazaurek, Symmetrie und Gleichgewicht. Katalog der Ausstellung im Königl. Württ. Landesgewerbemuseum, Stuttgart, 1906. Quoted according to A. Kuhn, Aristide Maillol, Landschaft, Werke, Gespräche, Leipzig, 1925.

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Maillol and Rodin, produced an outstanding example of this natural balance in his bronze sculpture "Dancer" in 1911. With her arms spread out like a symmetrical scale, this delicate figure of a girl seems to hover, and conveys the impression of perfect psychological and physical harmony. Artists like Paul Klee applied the ideas of symmetry and law to abstract art in order to probe the structure of forms, movements and counter movements: "Every energy requires a complement in order to bring about a state of resting-in-itself, situated above the play of forces." 116 Klee particularly pointed out the parallel with mathematical natural science. In his study "Exact Experiments in the Realm of Art", he wrote: "Art has also been given enough space for exact research, and the gates have been open to it for some time. What was done for music before, until the end of the 18th century, is at least beginning in the field of sculpture. Mathematics and physics are providing the opportunity, in the form of rules for persistence and for alteration. This compulsion to concern oneself first with the functions instead of beginning with the finished form, is a wholesome one. Algebraic, geometric and mechanical tasks provide training toward the essential, the functional, in contrast to the impressive. One learns to see behind the facades and to grasp a thing by its roots. One leams to recognize what is flowing underneath it - the prehistory of the visible - and to dig into the depths and to expose, substantiate and analyze. 117

The essay culminates in the demand to "learn logic, learn the organism." The organism, art and mathematical natural science are no longer regarded as antithetical. Instead they are related to each other. In Kandinsky's book Point and Line to Surface118 the elements of form are elucidated in their tensions and harmonious compositions by examples from mathematics and natural science. Concentric star clusters from astronomy, variational possibilities of physical curves, line formations of lightning, structures of animal tissues, swimming movements of microscopic organisms, etc., alternate with constructions of modern technology. Abstraction is no longer "artificial" and "strange", but corresponds to the newly discovered and created world of forms in nature and technology. Klee and Kandinsky taught in the Bauhaus in the twenties. Another, more significant representative of the Bauhaus was Oskar Schlemmer, who applied the Bauhaus idea of a functionalistic unity of technology, science and the life-world to the representation of the human being. In his paper, "Human Being and Artistic Figure" (1925), he says that "ab116

117

118

P. Klee, Das bildnerische Denken. Schriften zur Form- und Gestaltlehre (J. Spiller, ed.) Basel/Stuttgart 1964, p. 79. P. Klee, Exakte Versuche im Bereich der Kunst, in Bauhaus, Zeitschrift für Bau und Gestaltung 2, No. 2/3, Dessau, 1928. W. Kandinsky, Punkt und Linie zu Fläche. Beitrag zur Analyse der malerischen Elemente, Bauhaus Book No. 9, Munich, 1926; Bem-Bümplitz, 1973.

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straction" is "the sign of our times ... and has the purpose of shaping a new whole in bald outline."119 "The human organism is position on the stage in the cubist, abstract space. Human being and space are both imbued with law. Which kind of law applies? ... The laws of cubist space are the invisible line network of planimetrie and stereometric interconnections. Corresponding to this mathematics there is the mathematics inherent in the human body which creates balance by means of movements that are essentially mechanical and determined by reason. This is the geometry of physical exercises, rhythmics and gymnastics ... The laws of the organic human being, on the other hand lie in the internal, invisible functions of heartbeat, circulation, breathing, and brain and nerve activity. If these are determinative, so is the human center, whose movements and aura create an imaginary space. It follows that, cubistically, abstract space is only the horizontal-vertical framework of this aura." 120 (Figure 1)

Fig. 1

In the Bauhaus Schlemmer had organized a separate course on the theme of "The Human Being." 121 A new human image in an industrial life-world from the point of view of technology, science, philosophy and psychology was to be transmitted to future artists and architects. The Bauhaus conceived of itself as a workshop and a place of apprenticeship that was goal-directed and that coordinated the arts and produced laws of form. In his "Doctrine of the Human Being", Schlemmer quoted from Goethe's Wilhelm Meisters Wanderjahre : 119

120 121

O. Schlemmer, Mensch und Kunstfigur, in Die Bühne am Bauhaus, Bauhaus Book No. 4, Munich, 1925. O. Schlemmer, see Note 119. O. Schlemmer, Der Mensch, Unterricht am Bauhaus, Nachgelassene Aufzeichnungen (H. Kuchling, ed.), Mainz, 1969.

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"As they conducted the guest across the next border, he suddenly saw an entirely different way of building. Here the houses were not dispersed and they were not like huts; instead they proved to be arranged in an orderly way, sturdy and beautiful on the outside, roomy, comfortable and adorned on the inside. Here I saw an uncrampted well-built city, compatible with its surroundings. The fine arts and the crafts related to them are at home here, and a singular tranquillity prevails in the rooms." 122

The battle cry of the Bauhaus people was "Wir (we) modern modern!" It was directed against the antiquated academics, and as such it deserved to be regarded as a paradigm shift. Beyond that, it incorporated the new theory: For the Bauhaus, to be modern was to anchor the fine arts in skilled trades and industry, and, in architecture, to see the center to which the arts were meant to be connected. This architecture calls itself "practical" and "functional" and is interested in modern functional building in an altered technical-industrial life-world. Functional building is understood to be the optimally functional and technical solution under the given conditions, and it needs no decorative sculptures or paintings. Analogous evaluative criteria are put forward that are known to us from philosophy of science. Thus Schlemmer called for "simplicity" for the optimal solution "because there is a power in simplicity in which every real innovation is rooted. Simplicity, understood as the elementary and typical, from which the organic and multifarious peculiar develops, simplicity understood as a tabula rasa and general purification from all electricistic frills of all styles and times, was bound to establish a route called the future!" 123 For Schlemmer the human being was not only an objective motif, but the center of a new life-world that was to be given artistic form. He viewed the human being as a psycho-physical wholeness in a nature that was a technological-scientific life-world, and also in human psychological-social relationships. Accordingly he called for a general education for the artist and architect in science, philosophy and psychology. This would provide the background for exercising artistic capabilities, such as drawing: "For 'The New Life,' which regards itself as a modem world-view and vital consciousness, the recognition of the human as a cosmic being is essential: the conditions of human existence, human connections to the natural and artistic surroundings, human mechanism and organism, human material, spiritual and intellectual manifestation, in short: the human as a corporeal and intellectual-spiritual being, is a subject of instruction that is as necessary as it is significant."124

Drawing is considered to be central to figure studies. It

122

123 124

J.W. Goethe, Wilhelm Meisters Wanderjahre, in Werke (in 14 volumes, Erich Trunz, ed.), Vol. 8, Munich, 1977, p. 249. O. Schlemmer, see Note 121, p. 23. O. Schlemmer, see Note 121, p. 28.

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"deals with the schemata and systems of the linear, the two-dimensional and the corporeally three-dimensional: the standard measures, theory of proportions, Diirer's measuring and the Golden Section. From these are developed the laws of movement and the mechanics and genetics of the body, both in itself and in space, both in natural space and in cultural space (construction). The latter theme is by nature expecially important: The relation of the human being to housing, to its furnishings, to objects. The paths of movement, the choreography of the everyday, lead over to deliberate movement in gymnastics and dance and on to the art form of the stage (q.v.). Analyses of figurative representations in old and new art bring this part to a close. 125

For the scientific part of the instruction Schlemmer envisioned a general presentation that would be easy to understand, beginning with the formation of the cosmos. In the sequence of a natural hierarchy, these subjects follow: "Cells and seeds, birth and growth, life and death, the organization of the skeletal joints, the muscle functions, the internal organs, the heartpump and blood circulation, lungs and breathing, the intestines and metabolism, the sex organs and the sense organs, the brain and nerves, are interpreted from the biomechanical and biochemical standpoints, and in this connection matters of nutrition, hygiene and clothing are touched on. The tactile sense and the sense of sight are by their very nature very significant: Opposite to the factual world of anatomy - instruction about the structure of the body - there is the less unambiguous physiology - instruction about life processes. This leads over to the third kind of approach, the philsophical one."

In the philosophical part, Schlemmer made it his task "to explain the human being as a thinking, feeling being and the world as notions, concepts, ideas..." 126 In search of new laws of form, Schlemmer studied the Egyptian system of measurements and the canon of Polycletus according to Leonardo da Vinci and according to Dürer, but he emphasizes: "Certainly, geometry, the Golden Section, the doctrine of proportions. But they are dead and fruitless if they are not experienced, felt. We must let ourselves be surprised by the miracle of proportions, the glory of numerical ratios and congruences, and we must form laws from results of this kind." 127

Schlemmer sought the new laws of form in a scientific and philosophical contempation of nature. The contemplation begins with the structure of the cosmos and leads via the atom and the molecule to the cell, germ theory, and germ-layer theory, as well as to the ontogenesis of the internal organs. 125 126 127

O. Schlemmer, see Note 121, p. 28. O. Schlemmer, see Note 121, p. 28. O. Schlemmer, see Note 121, p. 55; cf. in that connection also E. Panofsky, Die Entwicklung der Proportionenlehre als Abbild der Stilentwicklung, in idem., Sinn und Deutung in der bildende Kunst, Cologne, 1975, p. 91: "The Italian Renaissance held the theory of proportions in the highest regard, but in a way completely opposite to the Medieval appreciation of it. Once regarded as a technical aid, it was seen now as the realization of a metaphysical postulate."

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Human anatomy includes above all the system of bones, ligaments and muscles and touches upon the vessels, nerves, internal organs and the tools of the senses, emphasizing eye and ear. Formal human physiology takes a broad scope, above all the chemical composition of the body and the problems that arise from movement. In Schlemmer's drawings the human organism is geometrically analyzed and presented in its life lines and lines of movement and function. His wellknown marionettes and wire reliefs (now partly in the Neue Staatsgalerie in Stuttgart) are his three-dimensional transformations of these studies. They are by no means formalized robots pressed into the Procrustean bed of just any contrived canon of proportions. The intention, instead, is to grasp the forms and functions of a complete organism of body, mind and psyche in its new environment and to turn them into drawings. This new environment is to be shaped architectonically to correspond to the new technical-industrial conditions of life. As Gropius wrote, "the challenge is to master organizationally the gigantic tasks of our time - the whole traffic, all human work, material and intellectual".128 Form, function and economic requirements had to be reduced to a common denominator as a task of optimization. Then A. Behne, in 1923, formulated the law of modern functional construction: space must "stay in balance" between the "relative" - the particular concrete accomplishment of purpose - and the "absolute", the will to form. 129 That is the new concept of symmetry which is expressed as law and no longer in external proportions - thus thoroughly comparable to the change in the interpretation of symmetry in the sciences. The architecture of modernism is therefore meant to be economic and functional in order to obtain the greatest efficiency of construction and maximal utility, both with the least expenditure of material. P. Mondrian expressed the new concept of symmetry in this way: that the new design was "before the time of design giving expression to the turning point of human development - of the epoch of the equilibrium of the one and the other."130 This "equilibrium" is what is considered to be the "new harmony" of art and life in the new technical-industrial world. It is characteristic of modernism that the environment and surroundings of human beings were intentionally considered to be "non-natural surroundings", namely the new modern city. (This was later to be a central criticism on the part of postmodern architecture.) With the new organization 128

129 130

W. Gropius, Der stilbildende Wert industrieller Bauformen, in Jahrbuch des Deutschen Werkbundes 1914, p. 29. A. Behne, Der moderne Zweckbau (1926), Berlin/Frankfurt/Vienna, 1964, p. 11. P. Mondrian, De nieuve beelding in de schilderkunst, in De Stijl 1, No. 9 (1918), p. 105.

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and shaping by technology a new culture was expected to come into being, with artistic laws that would be autonomous from nature. In fact it was outright propagandized that culture was total anti-nature in the sense of modern technology. On the theme of machine aesthetics van Doesburg in 1921 exaggerated as follows: "... and we are not far from a chemical and radiomechanical suspension of our last remaining dependence on nature." 131 Here the project of modernism proved to be a radicalization of the Enlightenment, which proclaimed the autonomy of the Subject, not only in political lawgiving, but also - as in Kant's Critique of Pure Reason - vis à vis nature. 132 In Fichte the only way nature appears is as the "non-ego", a construction of human consciousness of self. 133 What had seemed to many a contemporary at the beginning of German idealism to be speculative was now solid and graphic to everyone involved in modern city planning. The goal is the functional city, in which a balance is sought between economic, social and psychological needs. This balance or symmetry cannot be achieved by the architect alone; it presents an interdisciplinary task. The functional city has to be described in a complex structural plan that is a balanced whole, in which public buildings, installations for services and traffic patterns are taken into account just as much as the quality of individual housing. 134 Along with the Bauhaus, there are a series of other representatives of modern architecture that could be pointed out. One was the Dutch group represented in the magazine De Stijl. Theo van Doesburg and Cornells Eesteren emerged as leading theoreticians of this group. In their article "On the Way to a Collective Construction", 1924, they called for structuring the life-world "according to creative laws" that emerge from a "creative principle." 135 They went on to emphasize: "These laws, which are linked to the economic, mathematical, technical, hygienic laws, lead to a new sculptural unity." They are stumbled upon only in joint work, by experience: "The basis for this experience lies in simply knowing about the elementary and universal elements of expression, so that starting from there it will be possible to arrive at a method for ordering those elements into a new harmony. This harmony is based on the knowledge 131 132 133

134

135

Th. van Doesburg, Schilderkunst en piastick, in De Stijl 7, No. 75/76 (1926/27), p. 37. Cf. Note 79. J.G. Fichte, Sämtliche Werke (I.H. Fichte, ed.) I, 305: "In the ego I posit a divisible nonego over against the divisible ego." Thereby Fichte accounts for the content of determinations of objects in nature as originated by the subject. Cf. C. van Eesteren, Tentoonstelling de rationalle woonwijk, in de 8 en Opbouw 13,131132 (1932). Th. van Doesburg/C. van Eesteren, (German) Auf dem Weg zu einer kollektiven Konstruktion, in De Stijl 12, Issue 6/7 (1924); cf. H.L.C. Jaffé, Mondrian und De Stijl, Cotogne, 1967.

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of contrasts, complexes of contrasts, dissonance, etc., a knowledge that is necessary to make everyting around us visible. The multiplicity of contrasts results in powerful tensions which provide a balance and a state of rest as a result of their reciprocal suspension. This balance of tensions constitutes the quintessence of the new constructive unity. Therefore we call for the application or concrete demonstration of this constructive unity in practice."

Thus the art and architecture of modernism considered symmetry and harmony to be the central structural characteristics that come to expression in particular work rules. De Stijl, again, characteristically speaks of the "new spirit of modern life" which turns against the dominance of nature and against artistic window-dressing and the prescriptions for it. It must be noted here, to be sure, that the upheaval into modernism was also considered to be a threshhold of a new epoch or - in philosophy of science - a paradigm shift that opposed, for example, 19th-century academic drawing and copying of nature. The modernists tried to find bases for the "sculptural unity of all the arts" pursued by the group "De Stijl." Investigations were called for about "the laws of space and their endless variations (space contrasts, space dissonances, space complements, etc.)" that could be "mastered as well-balanced unity." The laws of color in space and time were to be analyzed in the same way, so that "the equilibrium relationships of these elements will ultimately lead to a new, positive graphic art." By 1918 "De Stijl" had published a manifesto that proclaimed: "There is an old and a new time consciousness. The old one is oriented to the individual, the new one to the universal ... The new art has brought to light what is contained in the new time consciousness: an equal relation of the universal and the individual." 136 Again it is characteristic that this "reform of art" dissociate itself from the "natural form." Erich Mendelsohn, widely known as the builder of the Einstein Tower in Berlin, in 1923 asserted the logocentrism of modernism: "Seldom - it seems to me - has the order of the world been revealed so distinctly, only seldom has the Logos of Being opened wider than in this time of supposed chaos." This pronouncement has an effect like Heraclitus' reference to the "hidden harmony" in chaos. Mendelsohn set forth criteria for evaluating architectonic problem solutions that are already well known to us from philosophy of science: "Accordingly, we face a greater task of countering agitation with sense, exaggeration with simplicity, uncertainty with clear law; of rediscovering the elements of energy out of the

136

De Stijl, Manifest I, in U. Conrads (ed.), Programme und Manifeste zur Architektur des 20. Jahrhunderts, Bauwelt Fundamente Vol. 1, Berlin, 1964.

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smashing of energy, of forming a new whole from the elements. - Take hold, construct, recalculate the earth." 137

One of the central representatives of modernism was, without question, Le Corbusier, who not only achieved magnificent edifices, but also came onto the scene as a theorist. In his guiding principles, "Toward an Architecture" (1920), he says this about the engineer's aesthetics and architecture: 138 "The engineer's aesthetics, architecture: at the deepest level the same, one deriving from the other, the one full-blown, the other secretly developing. The engineer, guided by the law of economy and led by calculations, transposes us into accord with the laws of the universe. He attains harmony. By means of his handling of forms, the architect brings into reality an order that is the pure creation of his spirit: by means of the forms he stirs our senses and awakens our feeling for form. The interconnections that he produces give rise to a deep echo in us: He shows us the standard for an order that we feel to be in accord with the world order. He brings about manifold motions of our mind and out heart: thus beauty becomes experience for us."

In the euphoria of the mood of upheaval, Le Corbusier used formulations like this: "A great age is breaking in. There is a new spirit in the world... Industry, tempestuous like a river that is striving toward its destination, is bringing us the new resources suited to our epoch filled with the new spirit."

The outlook on architectural and social problems is striking. Symmetry in construction is required to correspond to the "balance of the social order. " In this context Le Corbusier developed the concept of "mass-produced houses" for which "spiritual preconditions must be created." In his later guiding principles for city planning, "Urbanism"139 (1925), there is a dissociation from nature: "A city! This is man's confiscation of nature. It is a human deed done against nature, an organism made by man for protection and for work. It is his creation... The poetry of nature is, strictly speaking, nothing but a construction of the mind."

A passage follows which elevates Le Corbusier outright to Platonist of modern architecture: "Geometry is the means we ourselves have created for outselves so that we can overcome our surroundings and express ourselves. Geometry is the foundation. It is at the same time the material bearer of the symbols that signify perfection and the divine. It bestows on us the sublime satisfactions of mathematics. The machine proceeds 137

138 139

E. Mendelsohn, Die internationale Übereinstimmung des neuen Baugedankens oder Dynamik und Funktion (Lecture, Nov. 6, 1923), in Architectura, weekblad van het Genootschap, Architectura et Amicitia 28, Issue 2, Amsterdam, 1924. Le Corbusier, Vers une Architecture, in U. Conrads (ed.), see Note 136. Le Corbusier, Urbanisme, in U. Conrads (ed.), see Note 136.

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from geometry; its dreams set out to find the joys of geometry. The modem arts and modern thinking, after a century of analysis, seek their salvation beyond accidental facts, and geometry conducts them to a mathematical order, to a more and more generalized posture... Such passion ensouls deeds, brings forth actions, drenches them in its color, gives them direction. The name of this passion today is: exactitude. An exactitude carried to great length and elevated to the ideal: the striving for perfection...

However, the Platonic emphasis on geometry and harmony in the architecture of modernism is by no means associated with elitist-aristocratic tendencies. In a paper by F. Schumacher about "Social City Planning", 140 (1919), he wrote: "A contemporary metropolis can become harmonious only when its structures, at defining points, conform to certain rhythms that run through the whole. Their sizes and their arrangement must be regulated to express a sense of unity. Inside this elastically planned framework the particular and individual can unfold all the more freely then, undisturbed by any contingencies."

Social and artistic harmony are seen in a unity in which social and cultural politics and economics are to be coordinated with each other: "One can see that bringing about social and artistic harmony requires many kinds of laws and therefore all these questions lead directly into politics. Not that it would be possible to actualize a social or artistic idea just with laws - the creative act is required; laws clear a path for it."

5.42 Symmetry and Symmetry Breakings in Postmodernism One could almost speak of a "logos of modernism" in which symmetry and functional unity would occur as structural characteristics. Art, architecture and science would then be only different forms of expression and projections of one and the same temporal epoch, forms in which human beings are confronted with certain problems of their technical-cultural stage of development. For philosophy, analogously, one could name Wittgenstein's "Tractatus Logico-Philosophicus", Carnap's "Logical Structure of the World", or Neurath's "Unified Science", which express the logocentrism of the era. Meanwhile the combined functionalism and structuralism of modernism has gotten on in years and is the worse for wear. These "breaks of symmetry" are most noticable in architecture. Functionalism became debased, in part, into an "international style" of desolate and unimaginative structures that concreted shut the metropolises of the first, second and third worlds alike, veered into the opposite of its original intentions and single-handedly reduced the cities into uninhabitable exhaust- and noise-plagued knots of 140

F. Schumacher, Sozialer Städtebau, in Kulturpolitik, Jena, 1919.

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traffic, municipal administration, banks and business. The idea of the city as an antiform to nature, as modernism presented it again and again, had taken on a life of its own and had turned against human nature.141 In architecture the critique of modernism has used the catchword postmodern for years, and has expanded into a general cultural criticism using catchwords like "post-industrial society", "post-structuralism", "postmodern scientific theory", etc. 142 A thread running through this sometimes brilliant discussion is that it does not complain of the "loss of the center" 143 and "modernism's symmetry breaking" as a cultural decline, but, rather, as an achievement that offers new opportunities. Now, the ironic treatment of "unity" and "Logos" in modernism from the beginning of this century is not new: In Dadaism, modernism was, to a degree, keeping its own court fool, one that had substituted the principle of "accident" for the "Logos" and that in artists' happenings constantly reminded the protagonists of modernism of the consequences of a "falling away from the center" - just like once the court fools in the Medieval world of divine order. By contrast, postmodern cultural criticism doubts the possibility of a "unity" and a "center" in principle and criticizes them as excessive demands on human reason. Reason is taking dangerous paths in centralization, rationalization and bureaucratization and can shift into totalitarianism, as proven by the most recent historical examples in the East and the West. T. Adorno's critique of positivism and rationalism should be understood against the background of such a "dialectics of Enlightenment. " Nevertheless, Adorno emphasizes that critique does not cause "categories such as unity and self-harmony" to disappear without a trace. 144 Rather, "the principle of harmony remains in play, transformed beyond recognition." Besides, according to Adorno, "the logicity of art" consists of "the balance of the coordinated, of that homeostasis in the concept of which aesthetic harmony

141

142

143

144

Cf. also K. Frampton, Die Architectur der Moderne, Stuttgart, 1983; C. Jencks, Die Sprache der postmodernen Architectur, Stuttgart, 1978. Cf. J.-F. Lyotard, Das postmoderne Wissen, Bremen, 1982; Tod der Moderne. Eine Diskussion (Konkursbuch), Tübingen, 1983; in that connection also, A. Wellmer, Zur Dialektik von Moderne und Postmoderne. Vernunftkritik nach Adorno, Frankfurt, 1985; J. Habermas, Die Moderne - ein unvollendetes Projekt, Theodor-W.-Adorno-Preis 1980 (Ed. Dezernat Kultur und Freizeit der Stadt Frankfurt a.M.), Frankfurt, 1981 ; idem, Moderne und postmoderne Architektur, in: Die andere Tradition, Katalog zur Austeilung Nr. 3 der Reihe Erkundungen, Munich, 1981. The title goes back to H. Sedlmayer, Verlust der Mitte - Die Bildende Kunst des 19. und 20. Jahrhunderts als Symptom und Symbol der Zeit, Berlin, 1955. T.W. Adorno, Äesthetische Theorie, Frankfurt, 1970, p. 235.

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is sublimated as the ultimate." 145 Again asymmetry is comprehensible only against the background of a hidden symmetry. Adorno writes: "It speaks well for the survival of the concept of harmony that art works that strive against the mathematical ideal of harmony and the dictate of symmetrical relationships and that strive for absolute asymmetry do not get rid of all symmetry. Asymmetry, in accordance with its valeurs of artificial language can be understood only in relation to symmetry."146

Here it is important for modernism to emphasize again that its "center" and "symmetry" are not to be confused with external simple symmetry characteristics such as reflection symmetry or axial symmetry. In modern natural science as well, the external geometrical symmetry characteristics of individual bodies emphasized since Antiquity, play a rather subordinate rôle. What is decisive are the uniform, comprehensive (but abstract) symmetry characteristics that are expressed in the mathematical structures of scientific theories. Analogously, the concept of symmetry that is intended in the architecture of modernism should be seen as a characteristic of a uniform structuralism and functionalism. In this sense the architecture of postmodernism comes to "breaks of symmetry." Uniform functionalism is broken up. Ornament and decoration are permitted again, and we find in one and the same postmodern building diverse historical styles quoted in partly ironic alienation - from the small oriel with its Medieval effect, to the Baroque stucco work above the window to the Ionic pillar by the entrance. For many a contemporary it is here that the paucity of ideas of an epigonal era becomes visible, characterized by eclecticism and historicism. Others refer to the ironic breaking of history that allows for playfully fishing in old boxes of building blocks for the styles of former eras. Throughout postmodernism, reflection symmetry and the Golden Section appear as historical quotations of external symmetry which nevertheless are included as set pieces and do not determine the uniform structure of the building - neither in the sense of Antiquity nor of modernism. To be sure, the historical tendency of postmodernism makes itself partly independent then in a macabre way, namely when single-family houses affect to be Greek temples or the historical self-assurance of a small town is paraded in restorations that are flashy instead of original. Such Potemkinlike villages lack the important element of self-irony, without which postmodern architecture can easily become unbearable. Of course, let it be undisputed that in phases of historical awareness important restorations of historically valuable buildings are achieved.

145 146

T.W. Adorno, see Note 144, pp. 235ff. T.W. Adorno, see Note 144, p. 237.

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Where successful postmodern architecture appears, it is characterized by a loosening up of the strict purism, openness and "pointillism " of the styles that seem to come together by chance, but at a second glance, at the latest, prove to be a successful and original ensemble of styles. The emphasis on the "selective", the "casual" and the "individual" versus "unity" and "generality" mirrors a postmodern outlook on life that - on the basis of relevant experiences - reacts rather skeptically and at best ironically to the claims of being the only true technical reason and the belief that modern rationalism will result in universal feasibility and solution to problems. 147 In that connection the concept of nature of classical modernism comes in for special criticism, according to which the modern industrial city - as was expressed in the previous chapter - is understood as an "anti-form" to nature (in the sense of Fichte's non-ego). With a new ecological awareness of the environment, philosophy of nature comes into play as well: The living space that a city offers is really liveable only when it is in ecological balance with nature as its environment. The requirements of a natural environment extend from nearby recreational areas to oxygen providers ("green lungs"). On closer examination, however, that does not constitute a rift with modernism, whose functionalism was directly intended to produce a liveable environment. But an industrial monstrosity in the landscape does not exactly fulfill these functional requirements. In retrospect, modernism's dissociation from nature is only historically comprehensible. Modernism as a creative and original interpretation of art tried to free itself from the academic copying of nature in naturalism, the previous era. In short: the demand to heed ecological balance is not in opposition to modernism, but would only have been the consistent further development of its practical and functional orientation to the life-world of modern man. It gets dangerous only if isolated tendencies become independent as a complex system of tensions. An example of that is the historical recourse to Late-Romantic images of nature. Nature is suddenly made to be an "oppressed" Subject that is not only supposed to exist separately and for its own sake, but also to be, in Rousseau's sense, "good" and "unspoiled" and is said to have been alientated from "true" nature as a result of "destructive" human intervention in it, and turned into Fichte's anti-ego. This view completely loses sight of the fact that nature can also be destructive and misanthropic from viruses, bacteria and malaria mosquitoes to natural catastrophes that are unleashed without man's intervention and that can only be prevented or at least checked by means of his technical intervention. Modernism with its 147

In that connection see O. Marquard, Abschied vom Prinzipiellen. Philosophische Studien, Stuttgart, 1981; H. Blumenberg, Lebenszeit und Weltzeit, Frankfurt, 1985.

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technological-scientific conditions becomes, to an extent, the guarantor of a human environment and the preservation of nature, even if it is conceived of as the creation and a "natura naturans" or - secularized - as an independent agent in front of man.

Fig. 2

The philosophy of nature and philosophy of science sketched in Chapter 5.3, could be called "postmodern" in the sense that breaks of symmetry are included in it. The plurality of "points de vue" or contexts in which nature is viewed, would also be, in this sense, postmodern. That, however, does not mean just noncommittally playing with points of view. What is emphasized instead is that even incompatible points of view in philosophy of nature are related to each other in a complementary way and that in their multiplicity they are what makes it possible to recognize unity. Here the postmodernism of philosophy of science would be overtaken by a "postpostmodernism" in which - just as in the most recent architecture a retroactive awareness of strictness of form and unity in function can be observed once more. In contrast with an art and literature business which easily turns crazy at the fashion stock market of new announcements of trends, philosophy has the capacity to remind one of its own "philosophia perennis " and to point to the insight, uttered repeatedly ever since Heraclitus, that even the violent breaks of symmetry are still related to a frequently hidden symmetry.

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Author Index Abel, N.H. 174, 181 Adorno, T. 632 Alberti, L.B. 128 Alembert, J.l.R.d' 171, 198, 243, 255, 287 Al-Hâzinî 104 Ampère, A.M. 268,273 Anaxagoras 73 Anaximander 66-68, 72, 114, 119 Anaximenes 69 Antisthenes 69 Aphrodite of Cyrene 120 Apollonius of Perga 56, 59 Archimedes 29, 30, 77, 107, 582 Archytas of Tarentum 52, 56 Aristophanes 119 Aristotle 2, 3, 29, 48, 64, 66, 71, 76, 82, 85, 87, 99,106, 117, 135, 233, 242, 252, 292, 343, 377, 384, 561, 567-569, 573, 579, 587, 602, 609, 612 Artin, E. 300 Aspect, A. 400 Augustine 125, 301, 317, 319, 344, 581, 614 Bach, J.S. 169, 172 Baer, R. 906 Bargmann, V. 393 Barlow, P. 153 Bayer, A. von 482 Becker, O. 229 Beeckman, I. 4, 243 Beethoven, L. van 169, 173 Behne, A. 627 Bellemans, A. 330 Belov, N.V. 163 Beltrami, E. 209 Bergson, H.L. 319 Berkeley, G. 226, 239, 561 Bernoulli, D. 255, 308 Bernoulli, J. 200, 302, 308 Berzelius, J.J. 478 Biot, J.B. 267

Böhm, D. 397 Bohr, H. 60 Bohr, N. 6, 374, 376-378, 380, 382, 393, 409, 413, 559, 583, 619 Boltzmann, L. 6, 276, 316, 321-327, 329, 333-336, 338 Bondi, H. 364 Bonnet, C. 47, 521 Bonnet, O. 206 Born, M. 383,483,485 Boscovich, R. 254, 266, 285, 367, 473 Bourbaki, N. 589, 592 Bradley, J. 344 Bradwardine, T. 33 Brahe, T. 89, 564, 579 Braque, G. 621 Bravais, E. 148, 156,479 Broglie, L. de 378 Bruno, G. 97, 360, 364 Bunsen, R.W. 375 Callipus 56 Campanus, J. 33 Cardano, G. 174 Carnap, R. 577,590,631 Cartan, E. 6, 202, 210, 341, 360, 452 Cayley, A. 197, 214, 299 Cézannes, P. 621 Chhang-Fang, Ch. 88 Chhung-Chih, Tsu 110 Chhung, W. 88 Christoffel, E.B. 353 Chün, M. 109 Clarke, S. 239 Clausius, R. 321 Comte, A. 617 Copernicus, N. 2, 57, 63, 89, 94, 579 Coulomb, C.A. 260 Crick, F.C. 508 Curie, P. 494,511,528 Dalton, J.

375, 479

662 Darwin, Ch. 6, 68, 224, 316, 332, 336, 512,541,613 Del Ferro, S. 174 Democritus 64, 75, 76, 102, 344, 397, 479, 540, 568, 609 Desargues, G. 187 Descartes, R. 4, 31,198,199, 243, 292, 369, 567, 569, 610 Dirac, P.A.M. 8,11, 342,414, 416,424, 428, 560 Dirichlet, G.P. 147 Doesburg, Th. van 628 Duhem, P. 323, 585 Dürer, A. 131,187,626 Dyson, F. 547 Eddington, A.S. 351,374 Eesteren, C. 628 Ehrenfest, P. 327 Ehrenfest, T. 327 Eigen, M. 327, 541, 542, 544-548 Einstein, Α. 6, 258, 280, 286, 324, 343, 351-355, 357, 362, 364, 367-369, 372, 375, 395, 398-400,425, 471, 575, 594, 596, 606, 621 Empedocles 71-73, 77, 87 Engels, F. 608 Eötvös, L. 352 Escher, M.C. 164, 193, 195 Eudemus 51 Eudoxus of Knidos 2, 47, 51, 52, 54, 377 Euclid 26,47,50,135,181,186 Euler, L. 187, 223, 242, 255, 259, 287, 310,312-314, 343, 366 Eupalinus 109 Everett, H. 412 Faraday, M. 270-272,510,583 Fatou, P. 553 Fay, C.F. du 259 Fedorov, E.S. 140, 153,163 Fermât, P. de 301 Fermi, E. 436 Feyerabend, P.K. 412, 587 Feynmann, R.P. 8, 430 Fibonacci 45 Fichte, J.G. 612, 628, 634 Field, H. 605 Fischer, E. 482,512 Fleck, L. 586

Author Index Fludd, R. 131 Fock, V. 357 Fraenkel, A.A. 589 Francesca, P. della 44 Frank, F.C. 515 Fraunhofer, J. 360 Fredholm, I. 215 Frege, G. 577 Fricke, R. 140 Friedmann, Α. 362, 365 Frederik the Great 224, 306, 309, 311 Frobenius, G. 219, 374 FuHsüan 110 Galen 120 Galileo, G. 1,4, 76, 223, 239, 243, 294, 352, 357, 464, 579, 587, 602 Galois, Ε. 174, 177-181, 183, 198 Gassendi, P. 375 Gauss, C.F. 5, 32, 147, 174, 181, 203-208, 248, 256, 261, 263, 268, 273, 341, 355, 578 Gell-Mann, M. 452,455-457 Gerisch, G. 537 Gilbert, W. 94, 259,264 Glashow, S. 9, 342,447 Gmelin, L. 479 Godei, Κ. 365 Goethe, J.W. von 5,47, 128, 200, 521, 555,601,611,613,624 Gordan, P. 299 Gordon, W. 415 Grassmann, H. 214 Greenberg, O.W. 459 Gropius, W. 12, 627 Grossmann, M. 353 Grünbaum, A. 326 Habermas, J. 632 Haeckel, E. 526 Haitam, ibn al- 30 Haken, H. 529, 532 Haller, A.v. 332 Hamilton, W.R. 287, 290, 315, 366 Hanson, Ν. 563 Haü, R.-J. 481 Hayyän, G. ibn 102 Hegel, G.W.F. 4, 69, 528, 608, 612, 617, 620 Heidegger, M. 69

Author Index Heisenberg, W. 3, 8, 69, 81, 374, 382, 384, 425,427, 430, 441, 444, 450,463, 502, 602, 607 Heitier, W. 482 Hecataeus of Miletus 68 Helmholtz, H. von 6, 135, 158, 208-210, 341, 561 Hepp, K. 413 Heraclitus 13, 64, 69-71, 75, 83, 292, 528, 608, 622, 629, 635 Herbart, J.F. 157, 214 Hermann, J. 303, 308 Hermes, J. 33 Hermite, Ch. 147 Hero of Alexandria 108 Herschel, J. 480 Hertz, H. 5, 206, 275, 279, 583 Hilbert, D. 6, 215, 299, 343, 366-369, 471 Hipparchus 61 Hippasus of Metapontum 42 Hippodemus 119 Hoffmann, D. 374 Hogarth, W. 131 Homer 70 Honnecourt, V. de 116,127 Hörz, H. 597 Hubble, E.P. 360 Huggins, W. 360 Hui, L. 24 Hume, D. 131, 226, 293, 569, 574 Husserl, E. 229 Huxley, A. 622 Huygens, C. 4, 238, 253, 293, 320, 369, 598 I-Hsing

114

Jacobi, C.G.J. 5, 297 Jordan, C. 140,152 Jordan, P. 427 Joyce, J. 457 Julia, G. 553 Jung, C.G. 106 Kaluza, T. 374 Kandinsky, W. 623 Kant, I. 4, 9, 74, 157, 214, 225, 229, 235, 243, 255, 263, 266, 285, 287, 293, 298-300, 313, 332, 335, 342, 351, 434,

663 473, 511, 522, 561, 567, 569, 574, 584, 602, 606, 628 Kekulé, A. 481,482,498,578 Kepler, J. 1, 33, 38,46, 56, 59, 90, 94, 138, 155, 223, 242, 253, 264, 312, 343, 441,464, 563, 579, 581, 594, 602 Kirchhoff, R. 375 Kirchner, A. 34,144 Klee, P. 623 Klein, F. 5,183, 186, 192, 223, 297, 299, 369, 374,415, 592 Koenig, S. 303,308,311 Koffka, K. 562 Köhler, W. 562 Kolbe, G. 622 Kossei, W. 482 Krüger, L. 327 Kuhn, T.S. 563, 587, 614, 618 Kuhn, W. 512 Lagrange, J.L. 174,198, 214, 223, 248, 256, 263, 287, 314 Lakatos, I. 587,618 Lange, L. 4, 223, 243 Laplace, P.S.M. de 74, 224, 258, 263, 269, 283,332 Larmore, J. 315 Laue, M. von 154 Lavoisier, A.L. 105, 293 Le Bel, J. Α. 481 Le Corbusier 630 Le Verrier, U.J.J. 258 Lee, T.D. 435 Legendre, A.M. 481 Leibniz, G.W. 4,7, 9, 21, 34, 51,134,136, 155, 176, 179, 223, 237, 240-242, 244, 250, 254, 260, 285, 287, 293-295, 300-311, 315, 320, 342, 351, 366, 369, 375,409,413,434, 511, 519, 546, 549, 556, 568, 573, 581, 583, 602, 610, 614, 620 Leonardo da Vinci 44, 128, 131, 137, 187, 626 Leonardo of Pisa 45 Levi-Civita, T. 353 Lewis, G.N. 482 Lie, S. 6, 183, 198, 201-203, 209, 223, 299, 341 Locke, J. 76, 135, 293, 567 Lockhart, C.M. 408,413

664 Loinger, A. 393 London, F. 425,482 Lorentz, H.A. 266, 276-278, 282, 349, 357 Lorenz, E.N. 548 Loschmidt, J. 323, 337 Lotka, A.J. 535 Lucretius 76 Ludwig, G. 590 Luhmann, Ν. 601 Lullus, R. 21 Lyell, C. 333 Mach, E. 223,225,229,323,512,561, 580-582 MacLaurin, C. 242 Maillol, A. 622 Mandelbrot, B.B. 552, 555 Mann, Th. 10 Maricourt, P. de 264 Marquard, O. 619 Maupertius, P.M. de 224, 306, 308, 312-314, 336, 343 Maxwell, J.C. 1, 5, 8, 223, 248, 256, 259, 265, 271, 273-279, 284, 332, 336-339, 348, 464, 583 Mayer, J.R. 334 Meixner, J. 328 Mendelsohn, E. 629 Mersenne, M. 199 Michelson, Α. 281,345 Mie, G. 366,369,471 Mills, R.L. 439 Minkowski, H. 147, 282, 286, 345 Misra, B. 408,413 Mondrian, P. 627 Monod, J. 337,510,544 Morley, E.W. 281,345 Musschenbroek, P. van 259 Ne'emann, Y. 452 Neumann, J. von 7, 339, 394, 411, 607 Neurath, O. 631 Newton, I. 4, 8,90, 94, 135, 223, 234, 237, 239-243, 245, 248-250, 252, 254, 256, 259, 264, 276, 286, 291, 295, 309, 316, 319, 332, 347, 356, 375, 382,473, 580, 587, 591, 596, 598, 600, 611, 613 Nicolas of Cusa 104 Nietzsche, F.W. 6, 69, 343, 346, 615 Noether, E. 5, 297, 339, 350

Author Index Noether, M.

299

Occam, W. 413, 603 Oerstedt, H.C. 5, 264, 266, 270 Olbers, W. 578 Oppenheimer, R. 483,485 Orban, J. 330 Oresme, N. 116 Oslander, A. 580 Ostwald, W. 334 Pacioli, L. 37,44, 130, 155 Pappus 37, 187 Parler, H. 125 Parmenides 71,75,83,292 Pascal, B. 187, 583 Pasteur, L. 153, 342, 479,494,496, 510, 516 Paterno, E. 482 Pauli, W. 369, 427, 436 Pauling, L. 505 Peregrinus, P. 94,116 Pericles 119 Perutz, M. 506 Picasso, P. 621 Pines, A. 330 Planck, M. 315,375,377,587,621 Plato 3, 25, 29, 50, 57, 64, 71, 79-82, 292, 301, 341, 343,414, 441, 477,482, 528, 538, 561, 568, 578, 581, 601, 603, 614 Pliicker, J. 191 Podolsky, B. 399, 606 Poincaré, H. 193, 202, 324, 549 Poisson, D. 248, 256, 260, 263, 270 Polyà, G. 140 Polycletus 119,626 Poncelet, J.V. 188 Popper, K.R. 326, 469, 577, 586 Prigogine, I. 330, 338,408 Primas, H. 403, 484 Proklos 28 Protagoras 120 Ptolemy 32, 55, 59, 62, 86, 89, 196, 242 Pythagoras 41,69,71,232 Quarles, F. 317 Quine, W.V.O. 585, 605 Qurra, T. ibn 29 Rameau, J.P.

169

665

Author Index Reichenbach, H. 326, 577 Rhim, W.K. 330 Ricci, G. 353 Riccioli, G.G. 90 Richelot, F.J. 33 Riemann, Β. 6, 203, 207, 341 Ritter, J.W. 5,264 Robertson, H.P. 361 Robinson, A. 202 Rodin, A. 623 Roemer, O. 344 Rosen, N. 399, 606 Rousseau, J.J. 613,634 Russell, B. 69 Rutherford, E. 376, 583 Salam, A. 9, 342, 446 Savart, F. 267 Scheibe, E. 590, 592 Schelling, F.W. 4, 255, 259, 271, 528, 611, 613-615,621 Schiller, F. von 610 Schläfli, L. 33, 158 Schlemmer, O. 623-627 Schmidt, E. 215 Schönberg, A. 169,171-173 Schrödinger, E. 326, 338, 374, 378, 383, 397, 401,411, 425,487, 600, 607 Schumacher, F. 631 Schur, I. 219,374 Schuster, P. 542 Schwarzschild, K. 315 Schwinger, J.S. 8,431 Seeliger, H.H. von 258 Shubnikow, A.V. 163 Simon, R. 47 Snow, C.P. 1, 165 Socrates, 69 Sohncke, L, 140 Sommerfeld, A. 378 Speiser, A. 136, 141 Spencer, H. 333, 336 Spengler, O. 319 Spinoza, B. de 567 Ssu-Hsün, Ch. 114 Staudinger, H. 502,513 Staudt, V.G.C, von 188 Steiner, J. 188 Stockhausen, K. 169 Stokes, G. 262, 272

Ströker, E. 229 Su Sung 114 Sund, H. 505 Sylvester, J.J. 299 Tartaglia, Ν. 174 Tê-Yen, Wu 110 Thaies of Miletus 27, 66, 68 Theaetetus 35, 78, 150 Theo of Alexandria 112 Thom, R. 534, 556 Thomas of Aquin 83, 610, 613 Thompson, D'Arcy 526, 539, 555 Thompson, W. 274, 323, 336 Torricelli, E. 4 Turing, A.M. 537 Utiyama, R.

357, 440

van't Hoff, J.H. 153,481,482 Vasari, G. 128 Veblen, O. 374 Verhulst, P.F. 534 Vitruvius 108, 120, 122, 128 Voigt, W. 281 Voltaire, F.-M. 5,312 Volterra, V. 535 Waerden, B.L. van der 222, 300, 374 Wafä, Abü-al- 32, 38 Walker, H.G. 361 Wallis, J. 232 Ward, C. 445 Watson, J.D. 508 Waugh, J.S. 330 Weber, E.H. 228 Weber, M. 619 Weinberg, S. 9, 342, 445, 447 Weizsäcker, C.F. von 385, 570, 597 Wertheimer, M. 562 Weyl, H. 1,5,7,9,210,222,238,241, 244, 262, 276, 341, 343, 366, 369, 370, 371-374, 389, 402,424,425, 431, 471, 582, 602 Whitehead, A.N. 528, 613 Wick, G.C. 405 Wightman, A.S. 405 Wigner, E.P. 7, 222, 277, 341, 343, 374, 388, 390,402, 405, 573, 599 Wilson, Ch.Th.R. 378

666

Author Index

Wittgenstein, L. Wolff, C. Wright, S. Wu, C.S.

631

Yang, C.N.

435, 439

308,311 543 435

Zarathustra 41 Zermelo, E. 6, 324, 589 Zweig, G. 455

Subject Index acceptance 619 action at a distance 248, 269, 279 action, quantum of s. Planck's quantum of action action theory 619 actuality s. potentiality actualizing 384, 412, 567, 602, 607 actus 252,384 addition theorem classical 280,348 relativistic 348 ad hoc hypothesis 56, 606 advances in cognition 585 aesthetic 588 air 6 9 , 7 1 , 7 7 , 7 9 , 8 4 , 9 9 alchemy 3, 37, 64, 72, 97f, 99, 102, 105 algebra 3, 5, 147, 175 C*- 375 Lie 202,452 observables 394,402 algebraic quantum mechanics s. quantum mechanics alpha helix 505 amino acid 510 Ampère's law 274 analogy 274,581 analytical geometry s. geometry angles, theorem of the sum of the 197 angle, trisecting of an 29, 182 anisotropy 107 antimatter 476,510 antinomy 604 antiquark 455,457,463 antisymmetry 259, 505, 508 antiparticle 418 architecture 1, 3, 117, 621, 627, 630, 633 Buddhist 123 Greek 119 Islamic 123 of the Renaissance 128 s. modem times s. post-modern times

area-preserving 204 arithmetic 2, 25,40, 168 arrow of time s. time's arrow art 1, 3, 11, 12, 117, 168, 620-632, 623 abstract 623 of architecture s. the same of modem times s. the same of post-modem times s. the same of the Renaissance 131 artificial intelligence 11, 560, 583 astrolabe 112 Astronomia nova 90 astronomy 2, 23, 25, 37, 57, 168 Antique-Medieval 61 Babylonian 49,52 Chinese 50 Egyptian 49,52 Greek 50 of the Maya 50 s. cosmogony s. cosmology asymmetry 315,476,478,495,501 cerebral 565 of time s. time's arrow atom 75-77,81,375,567,621 atomic physics 3, 375 atomism 64, 72, 75, 76, 82, 85, 102, 255, 398, 559, 567, 609, 613 atom model 6, 375, 378 attractor 552 auditory field 227 autocatalysis 540, 542, 546 automorphismus 10,135, 176 automorphism group s. group autonomy 602,628 autopoiesis 614 axiomatics 367 axiom of choice 606 Aztecs 18 background radiation 364 balance s. equilibrium

668 baryon 449,457 basis set 590 Bauhaus 12,623,628 Bell's inequality 400 Bénard effect 536 benzene 498 ß-decay 9, 342, 433,435 big bang 363, 365, 475 Bilateralia 522 biochemistry 10,502,507,512,557-559 homochiral 512 biology 1, 10, 134, 329, 333, 343, 477, 528, 557 biomolecule 514,516,541 biotechnology 543 Biot-Savart's law 267, 269 black hole 358 bodies, regular s. Platonic bodies Platonic bodies s. the same Bohm-Aharanow effect 425 Born-Oppenheimer method 486 Book of Nature 301 Bose-Einstein statistics 389 boson 404,469 boson-fermion field 471 botanies 609 brain 565,583 Bravais lattices 148,152-156 Brownian motion 324, 333 bucket experiment 235 Buddhism 21 C*-algebra s. algebra cabbalistic 20 canon s. canon of proportion carbon atom 481 catalyst 100 s. autocatalysis catastrophe theory 11, 534 s. chaos theory category 11, 298, 335,401, 560, 566, 568, 573 cat experiment s. Schrödinger's cat causa efficiens 573 causa finalis 573 causa formalis 573 causa materialis 573 causality 11, 237, 238, 241, 244, 291, 299, 304, 307, 313, 346, 358,401, 433, 511, 560, 566, 572-575

Subject Index cause s. causa s. causality cell differentiation 482, 537 center (algebraical) 403 central symmetry s. symmetry change (μεταβολή) 83 chaos theory 10, 528, 551-553, 555-557 characteristics, accidental 570 essential 570 charge 259, 263, 265, 273, 278, 436 charge conjugation 418,436 charge conservation 273 charge law, Coulomb's 260 charge multiplets s. particle multiplets chemical bonding 101,478,484,498 chemistry 1, 11, 72, 79, 97, 134, 270, 293, 339,403,406,477, 528, 539, 557 macromolecular 502 racemate 513 chirality 434,494,496, 501, 512, 514, 516 Chladni's figures 380 circular motion 85,89,116 collapse of the wave packet 409, 411 color symmetry s. symmetry commensurable 42,44, 47 commutation relation s. Heisenberg's commutative 7,391,402 compass 95 compass s. ruler compatible 384 compensation 620 complementarity 11,409,410,413,559, 598, 607, 615, 619 completeness 368 complex dynamic systems s. systems complexity 11, 316, 333, 528, 557, 604, 617 reductions 600 increase of 600 computer program 565 configuration, molecular 503 ferromagnetic 168 configuration space 291 conformation, molecular 503 congruence 135,209 conic section 29, 30 conservation, law of 5,10, 66,71, 273,287, 292, 297-300, 350, 370, 386, 416,427, 569 of energy 297 of linear momentum 296

Subject Index of the angular momentum 296 of the center of gravity 298 conservation of energy s. law of conservation conservation quantities 295, 297 constancy s. speed of light constructibility 36, 133 construction, means of 29 problem of 32,180 construction of regular η-sided polygons 181 constructivism 604 context 509, 562, 577, 579, 583-585, 589, 596, 607, 615, 619, 635 context of discovery 576, 580, 586 context of justification 576, 580 conventionalism 579 Copemican model 89 Copemican Revolution 351 correctness 368 correspondence principle 374, 380, 391, 393, 415, 425 cosmic metric s. Robertson-Walker metric cosmogony 67, 71,74, 366,474,479 cosmological principle 359-362, 587 cosmology Chinese 23,96 Greek 82 Newtonian 258 relativistic 359, 363, 366,448,477, 587 Coulomb force 260 Coulomb's convention 269 Coulomb's law 267, 269, 274 coupling constants 476 covariance 248, 285, 354, 420 covariance, principle of 354, 356, 370 creativity 583,586 cross ratio 190, 198 crystal classes 152 crystallography 3, 133, 140, 148, 152, 171,479,480,506 cube 35, 78,150, 159 cube, doubling the s. Delian problem cubism 165,621 Curie's symmetry principle 551 current 266 curvature 206-209, 355, 357, 372 s. tensor analysis Dadaism

632

669 Delian problem 29,182 demon, Laplace's 224, 398, 549 Loschmidt's 330,331 Maxwell's 332, 336, 339, 407, 532 determinism 400,573 dialectical materialism 597 dialectics 13, 607, 612, 620 of Enlightenment 632 of nature 607 differential equation 202, 249, 255, 262, 590 s. equation of motion s. Laplace's equation s. non-linearity s. Poisson's equation differential geometry s. geometry diopter 109 Dirac equation 415,420,425 Dirac field 427,429 Dirac sea 417 directional indicator 109 displacement current 274 dissipative structures 338, 540, 558 dissymmetry 10,478,494,501,510 DNA helix 508 DNA molecule 10,510 dodecahedron 35, 78, 150, 159, 162, 527 Doppler effect 360 double helix 508 D-tartaric acid 481 duality principle 191 Duhem-Quine thesis 584 dynamics peripatetic 86 non-linear 549 earth 7 1 , 7 7 , 8 4 , 9 9 eccentric circles 61 ecliptic 52, 68, 74 ecology 536,612 economy 579 eigenvalue problem 217, 385 eight-fold way 453 Einstein-Minkowski program 286 Eleatism 71,608 electricity 5, 8, 255, 259, 264, 266, 270 electrochemistry 264 electrodynamics 5, 8, 223, 248, 270, 285, 287, 315, 319, 341, 343, 366, 371, 414, 420, 425, 589

670 electromagnetic field s. field electromagnetism 266 electron field, local symmetry of the 423 electron wave 423 electrostatics 64, 223, 256, 259, 260-263, 270, 276, 285 elementary particle physics s. physics elements (philosophy of nature) 84 embedding 599 empiricism 568, 584, 589 enantiomery 494,496,516 s. chirality s. dissymmetry s. parity energy 70 conservation of s. conservation kinetic 251,288,294 potential 251,257,288,294 energy difference, parity-violating 514-516 engineer's aesthetics 630 entangled systems 397,401 s.EPR entelechy 384,536 entropy 316, 319, 321-323, 325, 327, 330, 337,407 entropy operator s. operator epicycle-deferent technique 56, 58, 61 epigenetics 332 epistemology 11, 455,486, 576, 584, 625, 635 post-modem 632 EPR correlation 400,404,406,410,413, 484,487, 585 EPR experiment 397, 399 EPR holism s. holism, quantum mechanical s. superposition principle equant point 62 equation, non-linear s. non-linearity equation of motion 243, 286 classical 249 Dirac's s. Dirac equation Hamilton's 292 Lagrange's 288, 291, 297, 320 mechanical 316 Newton's 287 relativistic 352 Schrödinger's s. Schrödinger equation equation theory 173

Subject Index equilibrium 103, 104, 224, 321, 327, 537, 580,612, 622, 627 ecological 1, 534, 634 sociological 630 thermodynamic s. thermodynamics equivalence principle 258, 352, 357 Erlanger program 5, 184, 223, 246, 297, 299 ether 280 ethics 48,619 Euclidean geometry s. geometry Euclid's algorithm 44 event 233, 237, 346 Everett interpretation 412, 605 everyday 626 evolution 10, 12, 332, 337, 339, 342, 515, 528, 547, 552, 558 biological 316,342,601,618 chemical 342,601 cosmical s. cosmogony pre-biotic 472 evolution equation 541-543 s. non-linearity s. phase transition evolution, rates of 542 evolution reactor 541 evolution, theory of 6, 67, 73,477, 541, 547 expectation value, statistical 385, 390 extension 568 extremal principle 5, 224, 287, 300-302, 306, 546 fact 584,614 factorability 396 s. separability falling bodies, law of (Aristotelian) 85 fallibility 566 falsification 440, 567, 584 Fermat's prime numbers 32 Fermat's principle 301 Fermi-Dirac statistics 389, 459 fermion 404 ferromagnet 165, 168, 442, 529 Feynmann diagrams 446 Fibonacci sequence 47, 123, 521 field 64, 88, 93, 116, 123, 248, 255, 273, 284,420 color 461 electrical 276,289

Subject Index electromagnetic 5, 424, 429 magnetic 271,276,289 matter 429 field (algebraic) 175 field concept (philosophy of nature) 94 field equation 270, 276, 285, 354, 369, 426 field operator 428 field tensor 285 field theory 255,258,275,426 field vector 261,265 final cause 304, 307 fire 6 9 , 7 1 , 7 7 , 7 9 , 8 3 , 9 9 flavor 464 fluctuation 561,577 fluctuation hypothesis 325, 338 force (Newton) 294 (Leibniz) s. vis viva force field s. field force function 248, 254 s. reciprocal effect force, law of (Newton's) 245-249 form(s) (μορφή) 82, 101 geometrical 134 quadratic 147 substantial 567,602 forms, hierarchy of 572 form invariance 354, 361 foundations, discussion about the mathematical 604 four-coordinates 282,427 four-current 283,415 four-momentum 286 four-potential 283,429 four-vectors 282 fractal 11,553-558 Frank mechanism 515 free mobility 209 fulguration 562 functional analysis 215 functionalism 622,633 functional space s. space future 237,601 fuzziness 595 s. structure, uniforme Galilean metric 245 Galileo group s. group Galileo invariance 4, 248, 250, 280, 287, 341, 391 Galileo transformation 247, 349, 393,402

671 Galois group s. group Galois theory 4, 174, 176, 179,182, 202 galvanism 266 gauge field 5,424, 441, 444-447, 576 gauge group s. group gauge invariance 341,425 gauge symmetry s. symmetry gauge theory 223,414, 440, 460, 464 of supersymmetry 471 SU(3) 462,464 SU(5) 466 U(l) 463,464 gauge transformation 289, 297, 371,421, 424, 575 Gaussian curvature 205 Gaussian curve 328 Gauss' theorem (electrostatics) 262, 268, 273 Gedankenexperiment 397, 411 Gell-Mann/Nishijima formula 450 gene coding 10 general theory of relativity s. relativity theory geocentrism 50, 86, 89 geometry 2, 5, 26, 38, 168, 231, 591 absolute 187 affine 187,244 algebraic 299 analytical 59, 184 differential 197, 203, 355, 360, 372 elliptical 195 Euclidean 186,205 hyperbolic 192 Minkowski 574,595 non-Euclidean 560 projective 131, 187, 348, 560 spherical 195 Gestalt s. shape gestalt psychology 561, 564 glucose 510 gluon 460 God 301,305-308,310,320,546,549, 602 Golden Rectangle 44 Golden Section 27, 41, 44,47, 118,120, 123,125, 521,523,626 Golden Spiral 45, 200 Gothic 125 grand unification s. unification

672 gravitational equation 6, 364, 368, 590, 592 classical 252, 257, 590, 592 relativistic 6, 356, 364 gravitational force 6,9, 94, 254, 259, 286, 289, 366, 474 gravitation field 257,351-358,471 gravitation, theory of classical 94, 223, 252, 256-258, 263, 270, 276, 591 relativistic 343, 368, 425 super- s. supergravitation group(s) 133,135,176, 588 affine 592 arithmetical 157 automorphism 3, 7, 136,402, 584,599, 603 character table of 497 color group s. color symmetry continuous 198,200,209,218,221, 223, 374 crystallographic 162 cyclic 136 dihedral 137,522 discrete 136, 140, 147,152, 218 elementary 344 frieze 137,503 Galileo 7, 246, 297, 391-394,405,408, 484, 569, 571, 574, 592, 599 Galois 176-178 gauge 277, 290, 300, 374, 439, 560, 576, 599, 613, 615 kinematic 4, 241, 344, 592 Leibniz 241,246,344 Lie 4, 198, 201, 203, 238, 247, 299 Lorentz 5, 349, 406,426, 484, 571 movement 153, 162 Newtonian 238, 244, 246, 344, 592 of rotations 150 ornament 141, 155, 162 permutation 174,178, 389,404 Poincaré 349,592 point 136,142, 150,154,162 principal 186 representation 493,599 s. representation theory rotation 151 similarity 186,232 space 156,162 space, biological 506 SU(2) 9,438,440

Subject Index SU(3) 452,463 SU(4) 463 SU(5) 9, 342,465 symmetrical 178 symmetry 8, 223, 232, 386, 569-572, 613 s. symmetry topological 592 translation 164,233 unitary 402,452, 558 hadron 9,342,449,463 Hamiltonian function 7,291, 296, 298, 315,387,427 Hamiltonian representation s. mechanics Hamilton operator 382, 386,404,408, 415,436, 500 Hamilton's principle 290, 314, 343, 366-369 handedness s. chirality Harmonice mundi 38,138 harmony 1, 6, 17, 25, 39,42, 47, 51, 64, 67,69,97, 103,117, 119, 125-128,131, 168, 224, 262, 368, 539, 612, 620,627, 628-631,632 Hartree-Fock method 488, 516 Heisenberg's commutation relation 382, 427 Heisenberg's uncertainty relation 430 heliocentrism 62 hemoglobin 507 heuristics 578, 580, 583, 586, 598 hidden variables 400, 606 hierarchy 600 epistemological 617 ontological 601,602 structural theoretical 599, 608 Higgs field 447 Higgs mechanism 443-445 Hilbert space s. space Hinduism 116, 123 hippopede 53, 54, 55 history of philosophy s. philosophy, history of holism ecological 556 epistemological 584 quantum mechanical 394,400,600, 613 homeostasis 547,632

673

Subject Index homogeneity 51, 67, 109, 208, 210, 225, 230, 240, 297, 350, 359 hózhó 17 H-theorem 323,326 Hubble constant 364 Hückel model 499 humanities 620 Hydra 537 hydrodynamics 255 hypercharge 452,456 hypercube 157-159 hypercycle 542 I Ching 22 icosahedron 35,45, 78, 79, 150, 159, 162, 527 idealism 628 idea, regulative (Kant) 583 identity 491 impact, laws of 294 impetus, theory of 86 incommensurability 42,47, 179 incompatible 385 incompleteness 398 indefiniteness 395,400 indistinguishibility, principle of 7,134, 177, 389 individualization 571-573,576 induction 400 induction, law of (electrodynamical) 270, 272 industrial society 12, 619, 624, 633 inertia, law of 207, 242, 244, 245 inertial system 243, 245, 248, 352, 595 infinitely distant lines 191 infinitely distant points 190, 196 infinitesimal calculus 294, 582 s. non-standard analysis inner product 215 innovation 578, 586, 598 instrumentalism 603 integral equation 262 s. differential equation interaction 79, 254, 567, 573 electrodynamical 9, 277, 366,406, 414, 433,445,464,474, 478 electromagnetic s. electrodynamical gravitational s. gravitational force strong 9, 366, 414, 449, 464, 468,474 superweak 436

weak 9, 366, 414,433,436,464,468, 474 interference term 395 intuition 561,584 forms of s. perception intuitive space 158 invariance 4, 6, 133, 174, 176, 183, 189, 197, 204, 243, 245, 277, 280, 289, 296, 297, 349, 369, 386, 390, 567, 603 canonical 592,599 s. group s. symmetry invariants, theory of 299, 353 inversion 184,491 irreversibility 316, 317, 323, 324, 326, 338, 407, 600 Islam 102 isometry 135 isomorphism 591 isospin quantum number 450 isospin-symmetry s. symmetry isotropy 6, 10, 51, 71, 75, 107, 124, 208, 211, 213, 225, 231, 281, 350, 360-362 Jina 19 judgment

619

kaleidoscope 34, 144,155, 342 Klein-Gordon equation 415 ladder set 593 Lagrange equation 291,367,426 Lagrange operator 443 Langrangian density 426 Langrangian function 288, 290-292, 297, 314,426 Laplace equation 258, 263, 269 Laplace operator 263, 269, 283 Laplace's demon s. demon laser 531-534,541,548,558 left-right difference 153, 235 s. chirality s. dissymmetry s. enantiometry s. parity lemniscate 54 length-preserving 204 lepton 9, 433, 467 lever, law of 86,580,581 Lie algebra s. algebra

674 life 326 s. biology s. evolution life cycle 82, 84, 609 life-world 624 technical-industrial 621 light 272,301 light cone 346 s. Minkowski space light, speed of 279, 281, 344, 575 limiting value, thermodynamic 406 linearity 558,580 s. non-linearity Liouville operator 408 living force s. vis viva Lobatchevski space s. space local-realistic theory 389, 607 lock and key hypothesis 512 logicism 589 logocentrism 629,632 Logos 12, 40, 70, 120, 608, 631, 632 Lorentz function 276, 278 convention 278,284 force 266, 277, 286, 289 group s. group invariance 6, 248, 282-286, 341, 343, 357, 3 9 1 , 4 2 0 , 4 2 7 , 4 3 3 , 4 3 6 , 576 transformation 282, 349 Loschmidt's demon s. demon Lotka-Volterra equation 535 L C A O method 4 9 8 , 5 0 0 L-tartaric acid 481 machines, simple 108 macrocosm 473,551 macromolecule 505, 506, 557 magnetic field 264, 267, 274 magnetism 5, 8, 6 4 , 9 4 , 1 1 6 , 255, 264, 266, 272 magnetostatics 96, 223, 256, 264, 268, 270, 273, 276 Mandelbrot set 555 manifold 217 homogeneous 210 pseudo-Riemannian 354 Riemannian 207 many worlds view 413 mapping principle 590, 594 mass 244, 245, 248, 252, 263 distribution of 257

Subject Index rest 286 materia prima (πρώτη υλη) 10, 99, 342 mathematics 106, 168, 601 Chinese 24 Egyptian 25 Greek 2 5 , 4 8 Indian 25 modern s. constructivism s. foundations s. set theory s. structure matrix representation 220 matter (υλη) 83 matter 3 6 9 , 4 7 6 , 5 1 0 s. field Maxwell field 4 2 7 , 4 2 9 Maxwell's equation 5, 274, 278, 284-368 Maya 50 mean, arithmetic 4 1 , 4 9 geometric 41 harmonic 41 measurement process 411,412, 593, 596, 615 quantum mechanical 403,408, 575 measuring instrument 7 mechanics 589 classical 242, 374, 596 Hamilton's 287,291,382,570,598 Lagrange's 287, 315, 598 Newton's 287, 382, 598 quantum s. quantum mechanics mechanism 613 medicine 64, 72, 610 megalith 117 meson 450 meso-tartaric acid 4 8 1 , 4 8 2 metabolism 407, 479, 516, 529, 542, 548 metamorphosis 5 , 5 2 1 metaphysics 570, 573, 606, 613, 615, 617 metatheory 589 meteorology 530 methodology 577, 579, 586 Michelson-Morley experiment 345 microcosm 374,473, 551, 555 mind 612 Minkowski cone 351,433 Minkowski metric 282, 347, 351, 354 Minkowski space s. space model, concept of 582 model, conformai 193-196,204,348 physical 593,599

675

Subject Index projective 193-196, 204, 348 modernism 12, 620, 622, 625, 627, 630 molecular structure 494 molecules, non-stationary 496 stationary 496 momentum, conservation of s. conservation monadology 155,156, 254, 311 morphogenetics 407, 537, 558, 566 morphology 547,552 motion, absolute 238, 241 equation of s. the same movement (κίνησις) 83 multiplets s. particle multiplets multiplicity 635 s. unity music 2,11,25,40,134,168 mutation 517,542 Mysterium cosmographicum 90 myth 15 mythology 1,20 naturalism 308,336 natural science 440, 611, 620, 623 natura naturans 610, 611, 614, 635 natura naturata 611,612 nature (φύσις) 83,611 poietic 610,613 nature, law of 224, 246, 300, 568, 602, 614,615 nature, philosophy of 3-5, 12, 22, 37, 61-64, 68, 72, 73, 79, 223, 252, 255, 288, 443, 465, 510, 528, 539, 540, 546, 578, 634, 635 Antique-Medieval 64 Aristotelian 82, 88, 96, 610, 615 Chinese 88, 259,401 Greek 259 modern 341,398,576,607,613 of modem times 223 Plato's 77 pre-socratic 65, 292, 609 romantic 266,270,611 Stoics' 88, 96, 105, 123, 276 Taoist 87, 96, 105,123, 264, 276, 397 Navajo Indians 15-19 Neoplatonism 3 neurology 335 Noether's theorem 297, 298 nominalism 601,604-606 non-commutative 374, 385, 402, 439

non-dissymmetry 512 non-equilibrium 557 s. thermodynamics non-linearity 533, 549, 555, 558, 580, 616 non-standard analysis 202 norm 585, 587, 613 nuclear structure 478, 486 nucleic acid 508,541,548 nucleón 437 number theory 33, 47, 147 observable 7,341,374,384,390,391, 394, 571,605 classical 7, 384, 392,402, 404,486 incompatible 339 non-classical 392 observables, algebra of s. algebra occasionalism 307 octahedron 35, 38,45, 78, 150, 159, 527 ontology 368, 413, 573, 578, 602, 605, 606 operator 605 adjoint 216 entropy 407,558 Hamilton 593 Hermitian (self-adjoint) 217 linear 216 self-adjoint 384 self-adjoint linear 593 time-reversal 390 unitary 217,392,402 optical acitivity 153,479,496,510 optics 5, 8, 223, 279, 301, 315, 344,481 optimizing 304, 308, 543 orbital 380,487,498 order 336 organism 87, 516, 610, 613, 623, 627, 631 organism, model of the 64 origin of life 547 ornamentation 134, 164 ornament group s. group ornament symmetry s. symmetry painting 130,621 pair annihilation 429 pair generation 429 paleontology 547 pantheism 239 paradigm 586,610 paradigm, change of 629

676 paradoxes, cosmological 258 parallel postulate 192, 231 parity 9, 223, 236, 342,434,435, 506, 511 parity violation 506,510,513,516 particle 64 particle multiplets 450-458 past 237,602 patterns, emergence of 565, 578, 600 recognition of 563, 565, 600 Pauli principle 389, 417, 459, 471, 487 PC parity 445 PCT theorem 436 pentagram 41 perception 11, 225, 560, 564 forms of 298 perfection 580, 584, 621 permutation 235,373 permutation symmetry s. symmetry perpetuum mobile 116, 127, 293, 337 perspective 130,619 s. context perspective, change of 618 pharmacology 502 phase shift 425 phase space 292 phase transition 407,532, 616 s. non-linearity philosophia perennis 223, 313, 615, 635 philosophy, history of 608 philosophy of nature s. nature, philosophy of philosophy of science 12, 455, 486, 576, 584, 625, 635 post-modem 632 photochemistry 512 photon experiment 396 photosynthesis 334 phyllotaxy 521 physicalism 308 physics 1,4,11, 61, 82,105, 134, 333, 339, 528 Aristotelian 89 Chinese 64 classical 223, 248, 569, 574, 602, 607 elementary particle 8, 79, 133, 155, 179, 358,414, 575 Plato's 79 quantum s. quantum mechanics solid-state 165 physiology 335, 583, 610, 626 Planck's quantum of action 375 plane ornaments 138,144

Subject Index plants 520 Platonic bodies 36, 37, 124, 150, 157, 159,162, 198, 527, 581 Platonism, mathematical 604 pluralism 620 Pneuma 285,540 Poincaré transformation 471 point de vue 409,486, 615 pointillism 560,634 Poisson equation 257, 263, 269, 356 polygons, regular 26-34,44, 144 semi-regular 37 polyhedron, regular 35 semi-regular 37 polymerization 502,514 population 534, 535, 548 population equation 534, 541, 616 s. non-linearity position, relationship of 239 positivism 441,584,601,605,606,618 positron 8,418 postmodernism 13, 621, 623, 632, 634, 636 potentia 252 potential 257, 277, 284, 289, 406, 420, 474 electromagnetic 367,373 gravitational 367 potentiality 83 (δύναμίς), 384, 397, 398, 401,413,567, 594,602,614 potential theory 248, 250, 256, 258, 260, 263, 269, 276 pre-established harmony 5, 287, 300, 313, 315, 320, 351,512,610 pre-formation theory 332 presence 237 pre-socratics 3, 64, 77, 609 pre-theory 591 principle, chemical 105 of least action 300, 302, 306, 308 of sufficient reason 51, 240, 243, 304, 580 of the general covariance s. principle of covariance principle (Kant) s. category of persistence 293, 298, 569 s. conservation principle s. substance of time sequence 574 s. causality of simultaneity 574

Subject Index s. interaction probability 323,383 progress, technological-scientific 619 projection central 188 parallel 188 proper time 285, 353 proportion 24, 31, 41, 47, 56, 81, 117, 120-123 proportions, canon of 119, 122, 169, 524,

620 proportions, doctrine of 48, 119, 133, 627 protein 505,507 proton, decay of the 468 proximity effect 262, 269, 271 psychology 335,619 Pythagoreans 1, 25, 35, 39,42, 50,168, 368 Quadrivium 2, 49,168 qualities 102 primary 76 secondary 76 quantization, first 427 second 426,427 quantum chemistry 478, 483,486, 558,

600 quantum chromodynamics 9, 449,459, 463 quantum electrodynamics 8,414,431, 449,463, 587 quantum field theory 8, 414, 426,432, 437,449, 558, 576, 587 quantum logic 7, 403 quantum mechanics 4, 6, 218, 224, 287, 299, 341, 370, 373, 393, 566, 571, 576, 592, 596, 602, 606 C* 7 Galileo-invariant 405,414 generalized algebraic 7, 339, 403,406, 413, 484f, 597, 608 Lorentz-invariant 414 molecular 485 von Neumann's 7, 339-341, 375, 608 quantum optics 558 quantum system s. systems quark 9, 163, 455,457, 460, 463, 467 quark-antiquark pair 463 quasi-electron 487

677 Radiolaria 526 rationalism 409,633 realism 8, 375, 398,477, 592, 602, 604, 606, 607 reality (ένέργεια) 83 s. hierarchy, ontological s. ontology s. realism reality, realm of (epistemological) 590 reciprocal effects s. interaction recurrence paradox 324 redshift 360 reductionism 11,484, 488 reflection 4 , 8 , 1 8 4 , 3 5 0 regulae philosophandi 582, 586 regular polygon s. polygon relativity, principle of 4, 344, 369 Galilean 297 Einstein's (special) 281, 344 (general) s. equivalence principle s. covariance principle kinematic 242 relativity theory 4,7, 224, 299, 343, 370, 432, 566, 596 general 6, 8, 341 special 6,341,343,351,574 religion 1, 124, 234, 310, 587, 640 renormalization 432,464, 587 representation s. mechanics representation theory 213, 374, 392,402, 405,491,570, 599 reproduction 508 res cogitans 410 extensa 410 research program 587 rest mass s. mass restriction 599 "saving the phenomena" 51,58 reversibility 319, 324, 338 reversibility paradox 323 Ricci calculus 204 Riemann-Christoffel tensor 355 Riemannian manifold s. manifold ring structure 498 Robertson-Walker metric 361 rotary inversion 491 rotation 4, 8, 246 rotation symmetry s. symmetry ruler 26,28,32

678 sandpainting (Navajo) 16 scalar product 214 scale 104,623 scholasticism 125 Schrödinger equation 7, 380, 384, 386, 390,408, 411,415, 558, 572, 592 Schrödinger's cat 339, 397, 409, 411, 597 scientific community 587 selection 343, 512, 517, 525, 541, 547 selection process 543 selection rule 404, 500, 501 selection value 542, 544 self-optimization 544 self-organization 12, 540, 546, 560, 614 self-replication 548 self-reproduction 10, 517, 518, 541 s. autopoiesis self-similarity 552,553 Sensorium Dei 234, 253, 332 separability 398,400 sequence space s. space set theory 588, 589, 604 shape 489,562 shape formation s. morphogenesis similarity 134,205 simplicity 11, 52, 56, 63, 242, 276, 343, 368,441,578,583, 625 simultaneity 237, 241, 574 Slater determinant 486 slime mold 537 snow crystal 529 social sciences 620 sociology 619 solid-state physics s. physics space 239,314,566 absolute 4,234,235,239 functional 221 Hilbert 7,213-216,218,341,383,392, 401,599,604 Lobatchevski 193,348 maximally symmetric 211 Minkowski 6, 343, 346-351,420, 576 musical 171 oriented 229 relative 234,240 Riemannian 211 s. manifold sequence 543,547 symmetrical 203, 208, 210, 360 vector 213, 217, 219 velocity 348

Subject Index space-time 225, 233, 246, 277, 281,402, 595 Galilean 298 Leibniz' 238 Minkowski 345 Newtonian 234 space-time-symmetry s. symmetry special relativity theory s. relativity theory spectroscopy 133,140, 375, 377, 404 spheres 28,50 spheres, harmony of the 51 sphere, symmetrical s. symmetry spin 382 spiral 199,521 squaring the circle 29 standardization s. unification standard logic 588 standard model (cosmological) 362 star polygon 33, 37 state (quantum mechanical) 383 pure 394,402 s. system stationary position, absolute 233, 237, 241, 244 steady state model 364 stereochemistry 3, 10,478,482, 501, 513 Stoicism 248, 255, 285, 540 Stoke's theorem 251, 262, 272 strangeness 457 stripe ornament 137,173 structural element 590, 604 structural formula 486 structural species 588-591, 592, 598-601, 605, 607, 612, 615 structural type 591,592 structure 587 comprehensive 594,598 rich 594 uniform 593 structure, concept of 582 structure, embedding of 595, 599 structure, restriction of 595, 599 substance 11, 287, 298, 314, 370, 401, 432, 560, 566-570, 574 supergravitation 10, 342, 464, 471-473, 475 Super-Higgs mechanism 472 superposition, principle of 7, 409, 411,484, 572 classical 250,260 quantum mechanical 394-397, 401-404

Subject Index superselection rule 7, 402-404, 572 superstrings 473 supersymmetry 10, 464, 469 gauge theory of 471 global 471 local 471 spontaneous breaking of 407,472,515, 533 transformation 470 surface metric 203 symmetrical group s. group symmetry algebraic 177 bilateral 11 breaking s. symmetry breaking central 127-129 class 497,501 color 162,459,469, 528 dynamic 277,357 external 8, 10, 277 functional 518 gauge 287, 300, 358, 366, 370, 407 geometric 177,357 global 5,341,344,351,357,366, 420-423, 443, 460, 576 helical 521 internal 8, 277, 290 isospin 8, 437, 445,452,469 global 438 local 438 local 5,341,343,351,357,366, 420-424, 464, 577 SU(2) 439 logical 572 molecular 489 morphological 10 of music 162 orbital 498 ornament 142 particle-antiparticle 418 permutation 7 reflection 17, 86, 107, 128, 136, 169, 172, 316, 374, 388, 435,481,491, 520, 522, 555, 561 right-left 4, 9, 135 rotary reflection 491 rotation 17, 108, 136, 165, 298, 374, 520 spaces, symmetrical s. space space-time 4, 7, 8, 241, 280, 284, 287, 295, 374, 394, 471, 560

679 spin-rotation 407 spiral 527 statistical 519 structural 518, 522, 527 SU(2) 440, 443, 445,476 SU(2)xU(1) 9,446-449, 465, 476 SU(3) 9, 342, 453, 455, 460, 476 SU(4) 458 SU(5) 342, 466, 467, 469, 476 supersymmetry s. the same time 6,321,390,436 transformation 491 translation 137, 172, 298, 407, 520 U(l) 8, 445, 476 symmetry breaking 2, 10, 19, 120-122, 124, 173, 339,448, 474, 478, 496, 517, 521, 528-531, 536, 541, 547, 557, 559, 561, 563, 572, 577, 585, 586, 597, 600, 615, 632, 636 spontaneous 441,447 spontaneous of supersymmetry s. supersymmetry SU(2)xU(l) 516 SU(5) 476 temporal 225,407, 557 symmetry group s. group symmetry operation 7,218,520 symmetry principle s. Curie's synergetics 532 systems, classical 8, 408,413, 598 complex dynamic 528, 553, 555, 557, 615 infinite thermodynamic 407, 600 open 530, 531, 548 quantum (non-classical) 7, 385,408, 413, 573, 598 system theory 562,615 tactile field 228 Taoism 64, 87, 94, 96, 105, 116, 123, 248, 255 technology 3,12,107, 613 teleology 302,312,559,585 tensor analysis 353 tensor product Hilbert space 395 tetrahedron 35, 38, 78,150,159,482 Théodicée 5, 224, 305, 308, 313 theology 238, 300, 307-309, 617 theoretical progress 595

680 theories, mathematical 588 pluralism of 586 theories, equivalence of 598 theory comprehensive 595 local-realistic s. the same richer in structure 595 theory, loadedness with 615 theory reduction strong 486 weak 486,600 thermodynamics 6, 224, 315, 327, 334, 336, 339,528, 557, 600 of equilibrium 327-329,338 of non-equilibrium 329, 338,408, 529, 614 phenomenological 321 second law of 316, 319, 334-338, 528 statistical 321 three-bodies-problem 549 throwing, law of (Aristotelian) 85 s. theory of impetus time 240,314,566 absolute 4,234, 347 arrow 224,316,317,326,408,600 asymmetry of 316 direction 319 metric 232 s. recurrence paradox relative 234 reversal of 9, 320, 330, 350 -reversal operator s. operator s. reversibility paradox s. space-time symmetry s. symmetry topological 232 translation 247 topology 197 transformation charge s. charge conjugation s. Galileo s. gauge s. group s. Lorentz parity s. parity s. Poincaré reversal of time s. symmetry similarity s. similarity s. symmetry translation symmetry s. symmetry translation velocity 244, 298

Subject Index transmutation 97, 100 Twin paradox 347 typification 588 uncertainty relation 408, 557 s. Heisenberg's unification 5, 6, 8, 183, 223, 254, 285, 342, 366, 370, 414, 464,469 grand 465 s. supergravitation s. supersymmetry unitary s. group unity 12, 127, 132, 165, 266, 272, 278, 285, 341, 343, 366, 407,433,464, 600, 620, 623, 628, 632, 635 unity in diversity 620 universals, quarrel of the 601 universe closed 363 s. cosmogony s. cosmology finite 364 open 363 s. standard model unlimited 363 vacuum 85,428,443 vacuum polarization 462 value (s. ethics) 619 variation theory 290, 300-304, 367,427, 583 vector space s. space velocity space s. space Verhulst population equation 549 verisimilitude 585 viruses 517 visual field 226 vis viva 250, 293 vortex theory (philosophy of nature) 74 Wallis' criterion 232 water 66, 71, 77, 79, 84, 99 wave equation 278-280, 283, 378 s. Dirac equation s. Schrödinger equation wave-function, Schrödinger's 378, 383 wave-particle dualism 383, 420, 427, 429 waves (philosophy of nature) 88 Weinberg angle 447 Weinberg-Salam theory 448

681

Subject Index wholeness 98, 397, 401, 487, 517, 559, 566f, 602, 616, 619, 626, 632 s. holism Woodward-Hoffman rule 500 work 250 world line 244 world radius 362

Yang-Mills SU(2) theory 439-441, 443 Yang-Yin dualism 22, 72, 87, 94, 259, 264, 397, 510 Zhabotinsky reaction zoology

609

539