Mathematical Aesthetic Principles Nonintegrable Systems

MATHEMATICAL AESTHETIC PAINCIPLES/NONINTEGAHBLE SYSTEMS

This page is intentionally left balnk

MRTHEMRTICHL AESTHETIC PRINCIPLES/NOHINTEGRRBLE SYSTEMS

Murray Muraskin Physics Department University of North Dakota Grand Forks, North Dokota 58202

Wor!d Scientific Singapore »New Jersey London* Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

MATHEMATICAL AESTHETIC PRINCIPLES/NONTNTEGRABLE SYSTEMS Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA.

ISBN 981-02-2200-9

Printed in Singapore by Uto-Print

PREFACE In our research conducted over many years, we have developed and often initiated studies in these basic areas: (1) Mathematical Aesthetics (2) Nonintegrable Systems (3) Construction of Model Universes Using the Computer Articles on these subjects are spread out throughout the literature. In the references we list articles published since 1975. In the fall of 1991 and the fall of 1992 this author gave talks at Iowa State University entitled "Nonintegrable Systems" and "Mathematical Aesthetics and the Simple Sine Curve". Another talk was given at the University of Manitoba in the fall of 1993 on "Mathematical Aesthetics, Sine Curves and Wave Packets" The core of this text is modelled after these mathematical physics and applied mathematics seminars as well as talks given at the 7 th ,8 th , and 9th International Conferences on Mathematical and Computer Modelling. I wish to thank the Iowa State people, in particular C. Hammer, D. Pursey, R. Leacock, D. Ross, B. Young for there hospitality and interest in this project. At Manitoba the author wishes to thank Witold Kinsner for his hospitality and interest. Prof. Avula is thanked for organizing the conferences on mathematical and computer modelling. In the course of our research it was recognized that the basic equations should be nonintegrable. By this we mean that integration from our initial point, called the origin point, to any other point depends on the integration path between the points in general. Thomas E. Phipps Jr. also studied nonintegrable systems in the sense above in his book "Heretical Verities: Mathematical Themes and Physical Description" (see page 326 and following of his book 1 ). From his book and from a private communication it is clear to the author that Dr. Phipps appreciated the importance of nonintegrability in fundamental physics for some time. I would like to personally thank Thomas Phipps Jr. for his insights and encouragement. The 3 basic areas listed in the begining have been blended together in our research. However nonintegrable systems stands on its own. That is, it is not necessary to discuss mathematical aesthetics in order to discuss nonintegrable systems. We use a simple set of equations, called the ABJL equations to introduce the subject of nonintegrable systems in Chapter 2. Even though the ABJL equations are a special case of the mathematical aesthetics program, this point is not essential in the theory of nonintegrable systems. The reader can thus skip to the section on the ABJL equations if he is only interested in nonintegrable systems. How significant is the effect of nonintegrability on the systems we study? We shall find that no integrability opens up whole new "worlds" to study. In fact all the figures in this book (with one exception, Figure 1.3, which involves 6 dimensional space) are associated with nonintegrable systems. We have still not been able to obtain multi maxima and minima solutions to the 4 (or less) dimensional mathematical aesthetic equations when the integrability equations are satisfied. The ABJL system becomes

Math Aesthetics/Nonintegrability

vi

uninteresting when the integrability equations are satisfied (some of the variables do not change from point to point while other variables are unbounded). Thus, we lose the sinusoidal dependence along any path segment when integrability is satisfied. In addition the 3 component lattice system (Equation 3.18) becomes trivial when we require integrability. Also, the sine within sine system Eq. (5.19) and Eq. (5.25) again becomes trivial when the parameters involved are taken to be Eq. (5.27), the choice of which forces the system to now be integrable. Thus, the study of nonintegrable systems enables us to obtain a variety of interesting situations which would be lost if the equations were required to be integrable. In addition, nonintegrability can be considered more natural than integrability as different paths traverse different environments, so there is no reason to expect results of integration to be independent of path. Thus, we can expect nonintegrable systems to receive greater attention as the years go by. We recall, basically on grounds of expediency, that linear systems eclipsed nonlinear systems, as well, until more recent times. In this book we present new techniques to deal with nonintegrable systems developed by the author over many years. I wish to thank my colleagues here at the University of North Dakota, in particular W. Schwalm, G. Dewar, and L. Jensen for going over my talks with me before presentation. The administration here at North Dakota has also been very helpful in supplying the computer time necessary for the project as well as providing an excellent climate to perform research. In this regard I would like to thank T. Clifford, A. Clark, B. O'Kelly, B. S. Rao, W. Weisser, H. Bale, D. Rice, A. Koch, E. Strinden, K. Dawes, G. Kemper, D. Vetter, and A. W. Johnson. At the computer center I would like to thank D. Bornhoeft, A.Lindem, D. Dusterhoft, and R. Johnson. S. Nemmers and D. Home helped in getting us started using the scientific computer language EXP. I also wish to thank G. Adomian and L. Smalley for their support. B. Ring helped with the computer work prior to 1976. My former students M. Ramanathan and C. Weyenberg also contributed to the research. R. Molmen, M. Brown, and D. Rand also helped with regards to the supercomputer project. Also, I give thanks to my wife Margaret whose emotional support made this work possible. I am also indepted to her for her tireless work on the word processor. S. Krom was instrumental in the preparation of the book. C. Cicha also has helped in the typing aspect. P. Erickson and R. Snortland are responsible for the preparation of the illustrations. I would like to also thank P. Erickson for spotting typographical errors in the manuscript. The text begins with a discussion of mathematically aesthetic principles (Chapter 1), including the reasoning behind such a study. Needless to say there exists mathematical principles that can be classified as being "aesthetic". It never ceases to impress how much is implied by a few simple mathematically aesthetic ideas. That is, with such a small imput we obtain, in a mathematical sense, lattice systems, soliton systems, closed string particles, instantons, chaotic looking solutions, as well as wave packet systems among other things as output. Arguments making use of mathematical aesthetics have long been used in developing physical principles. For example, in obtaining the Dirac Equation of relativistic quantum mechanics, Dirac required that all first partial derivatives be treated in a uniform manner.

vii

Preface

Of course, mathematical aesthetics was not the sole ingredient in obtaining the Dirac Equation. On the other hand, in this book we elevate the study of mathematical aesthetics to a discipline all of its own. There are no "physical" type arguments to be used in conjunction with the mathematical aesthetics. In other words, we study, here, mathematical aesthetics for its own sake. We find that the mathematical aesthetic ideas can be cast into a set of nonlinear equations whose solutions have considerable content as mentioned above. In our early work, as described in reference 2, our results were very limited. It was only after we recognized that the nonlinear system of equations implied by the mathematical aesthetic principles should be nonintegrable as well did we obtain interesting mathematical model universes. Although the studies of mathematical aesthetics, nonintegrable systems and construction of model universes using the computer is mathematical in scope, the expectation is that these studies will contribute to the understanding of some of the unsolved problems in basic physics. A considerably greater commitment of computer resources would be needed in this regard. Even without such resources we shall still be able to obtain certain insights useful to physics. In this book we shall study how these mathematical principles relate to such problems as the arrow of time and the concept of nonlocality. In order for a deterministic theory to account for quantum effects it is necessary that the theory be nonlocal (Bell's theorem). We shall see that the wave packet solution arising from the aesthetic principles leads to a situation that can be described as nonlocal. The traditional way of obtaining basic equations, at present, is to generalize equations that have been shown to be valid in some domain. We recall, even equations such as Maxwell's Equations need to be generalized. This is because Maxwell's Equations are quantized, which means that they are only valid in an average sense. However, there are an infinite number of ways to generalize existing equations. How are we to decide between all the possible generalizations as there are inherent limitations to empiricism? As it is inconceivable to us that the foundation for physics is of a capricious nature, we make the hypothesis that the basis of physics lies in mathematical aesthetics. Only modest steps in such a program have been studied and are recounted here, much more awaits future research. A discussion of outstanding problems yet to be done is found in Chapter 7, section 5. This book is written so that a graduate student or advanced undergraduate student in physics or applied mathematice should be able to master the material. The material is taught as part of my graduate course in Mathematical Methods of Physics. In particular Chapters 1,2, part of 3, 4 and 5 and the appendices are made use of in this course. The appendices are included so as to make the book self contained, and to show how the mathematical aesthetics program relates to standard mathematical subjects. The book, especially Chapters 6 and the first three sections of Chapter 7, also serves as a cohesive record of a research program that obtains sine solutions, lattices of different varieties, soliton behavior, instanton behavior, closed strings, sine within sine curves, wave

Math Aesthetics/Nonintegrability

viii

packets, an agitated vacuum, irregular oscillations, i.e. basic building blocks, with virtually no imput. All that is needed are a few mathematically aesthetic principles. A note in using this book, numbers taken off the computer are truncated rather than rounded off. Our notation used is such as to conform to the notatation in the research articles. We use r j k to refer to the change function. We work in terms of a Euclidean space and use Cartesian coordinates in all our computations. Thus, we emphasize, as we do in the text, that r j k has nothing to do with Christoffel symbols, gij in the text refers to a dynamical field, and is not related to the metric tensor. On the other hand, in Appendix E, we discuss curvilinear coordinates. In this appendix Gy refers to the metric tensor, and Ajk refers to Christoffel symbols, also referred to as the connection. Vectors (tensors), as contrasted with components of these quantities, are written in bold face or with an arrow over the vector. The fourth component is written interchangably as a zero component, as the meaning is to be the same. A summation convention is assumed when upper and lower indices are the same. References 1. T. E. Phipps Jr., Heritical Verities: Mathematical Themes and Physical Description, Classical Nonfiction Library, Urbana Illinois, 1987 2. M. Muraskin, Particle Behavior in Aesthetic Field Theory, Intl. J. of Theor. Physics, 13 303,1975.

ix

Preface

Math Aesthetics/Nonintegrability

x

Figure Caption Wave packet solution to the set of equations based on mathematical aesthetics (see Chapter 1). The origin point data is described in Chapter 5, section 10. Plot is for a representative component, r'n, along the x axis. This is a "big" picture type plot where 24000 points along x are compressed onto a single computer page. The plot is at z=-5, y=15 (units of y and z are 0.005). The grid along x is 0.0005859375 and the spacing between x points is 0.075. The grid along y and z is 0.00005. The system of equations in addition to being nonlinear, is also nonintegrable (see Chapter 2 and 3). In order to obtain this plot, we made use of the path specification approach, and integrated first along z, then y, then x to reach any point. This plot demonstrates the great wealth of information contained in a few mathematically aesthetic ideas. The "vacuum" between wave packet structures also is anything but empty and may have lessons to teach us. A study of the vacuum in such a wave packet solution is found in Chapter 5, section 8. Greater computer resources would be needed to understand the mechanice obeyed by such wave packet structures.

Table of Contents Preface Chapter 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

v

1. Mathematical Aesthetics Why Mathematical Aesthetics All Things are Numbers Flat Space versus Curved Space Some Nonlinear Equations Mathematical Aesthetic Principles The Aesthetic Field Equations (Gamma Equations) Other Relations Obtained from the Basic Principles The Integrability Equations Computer Project Sinusoidal Wave Solutions to the Gamma Equations Multiparticle Solutions

1 1 1 1 2 2 4 6 7 10 11 12

Chapter 2. Nonintegrable Systems 1. The ABJL Equations 2. A Formulation of No Integrability Theory 3. Generalized Derivatives 4. Summation Over Path Method 5. Second Approach to Nonintegrable Systems 6. Computer Results for a Soliton Lattice System

17 17 20 21 23 26 28

Chapter 3. Commutator Method 1. The Commutator Mathod and the ABJL Equations 2. Commutator Method for a Three Component Lattice 3. The Product Method 4. Random Path Approximation

40 40 45 50 51

Chapter 4. Nonintegrability and the Arrow of Time 1. The Arrow of Time

54 54

Chapter 5. The Gamma Equations as a Source of Fundamental Building Blocks 59 1. Introduction 59 2. Sinusoidal Behavior Along any Path Segment Arising from 60 Mathematical Aesthetics Program 3. Linear Sine Within Sine System 63 4. Another Linear Sine Within Sine System 65 5. Nonlinear Sine Within Sine System 66

Math Aesthetics/Nonintegrability

6. 7. 8. 9. 10. 11.

xii

Nonlinear Sine Within Sine Behavior and the Aesthetic Fields Program Wave Packet Solution A Study of Properties of the Vacuum for the Wave Packet Solution Assignment of Origin Point Data A More Viable Wave Packet Solution Features of the More Viable Wave Packet Solution Solution

69 75

79 82 85 87

Chapter 1. 2. 3. 4. 5. 6. 7. 8.

6. A Study of Some Additional Solutions to the Gamma Equations A Lattice of Closed String Solitons Ladder Symmetry Loop Lattice with a Doublet Basis Concept of Imperfect Lattice "Chaotic" Looking System Instanton Solution A Dynamical Lattice Irregular Oscillations along any Path Segment

112 112 116 119 120 125 126 133 141

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

7. Mathematical Aesthetics: Additional Topics Component to Represent the Particle System Lorentz In variance of the Gamma Equations Higher Dimensions A Brief Summary Outstanding Problems Mathematical Aesthetics and Epistemology

147 147 150 150 157 159 160

Appendix A. Elements of the Calculus

164

Appendix B. Theorem of the Calculus Requiring Path Independence for Integrable Systems

167

Appendix C. Elements of Tensors 1. Tensors and Coordinate Transformations 2. The Importance of Tensors 3. Rotations of a Cartesian System 4. Raising and Lowering of Indices and Mathematical Aesthetics

171 171 173 176 178

Appendix D. Elements of Determinant Theory 1. Transformation Law for a Determinant of a Second Rank Tensor 2. Expansion of a Determinant in Terms of Cofactors 3. The Equations for the Change of g 4. Relationship with the Aesthetic Fields Program

181 181 184 186 187

xiii

Table of Contents

Appendix E. Curvilinear Coordinates 1. Parallel Transport and the Connection 2.Mathematical Aesthetics and General Coordinates 3. A Non Dynamical Tensor 4. A Restatement of the Mathematical Aesthetic Principles

189 189 194 196 206

References to More Recent Research Articles by the author

208

CHAPTER NO. 1 MATHEMATICAL AESTHETICS 1. Why Mathematical Aesthetics The underlying hypothesis we make is that the foundation of physics lies in mathematical aesthetics. Otherwise, how can we justify one equation rather than another in view of the limits of empiricism. Although our hypothesis may appear eminently reasonable, the implication of such a program is too broad to handle all at once, so we will focus, at least at the outset, on a few simple questions which can be readily adressed: (1) Do there exist principles which can be classified as mathematically "aesthetic" that can be written down? (2) If so, can these principles be cast into a set of nonlinear equations describing change? (3) If so, what sort of solutions can be obtained from these equations? Can we get multiparticle solutions? 2. All Things are Numbers To Einstein is attributed the far reaching idea that all physics is geometry. A more conservative principle would be embodied in the statement that all physics is mathematics. Or perhaps, getting down to basics "all things are numbers" This latter belief has been attributed to the Pythagoreans. Within the framework of field theory such an idea can be expressed by the absence of a need for units associated with the set of numbers assigned to each point defining the field. Units arise once one has particles with certain attributes. Rods to measure length are to be constructed from aggregates of these particles. Clocks to measure time exploit the periodic motion of particles. Mass and charge are properties assigned to the particles. Thus, we can say that units reflect an underlying particle substructure. In a basic theory the expectation would be that particles are constructs from the basic fields. If so, the need for units at the fundamental level would be done away with. Units represent entities that are not reducible in terms of more basic quantities. Thus, to admit a need for units can be looked on as an additional complication for the theory. Why ask for a more involved theory than would absolutely be necessary? 3. Flat Space Versus Curved Space The notion "all things are numbers" is more general than the notion "all physics is geometry". In addition we shall observe later that nonintegrability is more natural than a theory based on integrability. Nonintegrability is not intrinsically a geometric concept. This offers an example of the restrictive character of the geometric point of view. Furthermore geometric theories have been pursued by Einstein, Schrodinger, Weyl, Eddington and many others without compelling results. Additional arguments can be given favoring a flat space: (1) Flat space is simpler than curved space. One does not usually consider a more complicated situation without fully investigating the simpler alternative. (2) Microscopic physics does not call for curved space, at least not yet. (3) Gupta1 has formulated gravitational-fheoiy rn flat space.

Math Aesthetics/Nonintegrability

2

(4) But perhaps most importantly, one can always imbed a curved space in a flat space of sufficiently high dimension. Thus, there is no loss of generality by working in flat space. We shall work, at least at the outset, in a Euclidean space using a Cartesian system of coordinates. Such a coordinate system has the attraction of simplicity and we do not want the coordinate system to simulate any dynamics as would be the case if we adopted a curvilinear system of coordinates. It is implied that the equations should have the same structure regardless of the Cartesian system under discussion (invariance of the equations under 4 dimensional rotations). We will work in four dimensions unless circumstances suggest a greater number of dimensions would be useful. Sometimes we will work with a smaller number of dimensions for calculational ease. We call the x° direction the time axis. 4. Some Nonlinear Equations Consider the equations dx _ • ax -bxy dt ~ dy_ -cy + dxy. dt

(1.1)

These are the Volterra Predator-Pre)' relations. For a,b,c:,d > 0 we get oscillatory behavior for x and y. Consider the equations dx = -X -2xy dt " dy_ = -y. 2 , 2 ■x + y dt

(1.2)

These are the Henon-Heiles equations2 For particular values of the origin point data we get random looking behavior in phase space. What do these equations have in common? They are first order coupled nonlinear equations with quadratic terms on the right hand sides. The values of x and y are given in terms of xo and yo, where xo and yo are the values of x and y at t = 0. Already we see that equations of this type have solutions with an interesting and varied behavior. We mention these equations since they bear some resemblance to the type of equations we will be studying. Equations such as Eqs. (1.1) and (1.2) are not suitable as field equations since they have no terms involving space derivatives of the variables. 5. Mathematically Aesthetic Principles What we seek is particle solutions of nonlinear equations. The first question we may ask is which equations should we choose to study. In formulating field equations one

3

Mathematical Aesthetics

normally starts with a generalization of an existing set of equations which are valid in some domain. Empirically an electron is considered a point. How then does one generalize a point? Thus, there are problems in potential generalizations: (1) There are an infinite number of possible generalizations. Without empirical justification how do we decide between these generalizations? One could think of all these possible equations each written on a separate piece of paper and then placed in a hat. Then one could say, "God does not play the lottery". (2) The generalizations are nonlinear. Thus, in many instances it is not clear what the generalizations really imply. However, the difficulties go deeper than this. Consider the Einstein-Schrodinger generalizations of the Einstein gravitation equations obtained from an action principle: « 5 / £ v / ^ d 4 x = 0.

(1.3)

What is done is to arbitrarily exclude higher derivatives from the Lagrangian density so as to make the equations for gy no higher than second derivative. But there is nothing "wrong" with higher derivatives. One could say that an almost too cavalier approach has been taken with respect to higher derivatives. There may be some local justification for omitting higher derivatives, but it is hard to look at this omission as anything but ad hoc, at least at this stage. (3) Thus the difficulty is one of mathematical inelegance. Consider the equations dip , i?i -rr =[ca ■ P + 0mc2]ip. (1.4) at This is the Dirac equation. Here all first derivatives are treated in a uniform way. There are two natural ways if we wish to treat all higher derivatives in a uniform way: (1) We can have all derivatives appearing in a single equation (or in a few equations). (2) We can have an infinite number of equations. We will consider the latter of these possibilities in what follows. Next we recognize that the basic equations that are currently in use always involve low rank tensors. When was the last time you had to deal with, say, a 6,142,371 rank tensor? Granted, low rank tensors are simpler to deal with compared to high rank tensors. However, this is not sufficient reason to promote a low rank tensor over a high rank tensor on conceptional grounds. The question then is whether it is possible to formulate a set of field equations according to mathematically aesthetic principles. What might these principles be? We list some of the principles we have been working with in our original formulation4. The list of aesthetic principles is meant to be kept flexible at this stage. (1) All derivatives of tensors are treated in a uniform way (with respect to change).

Math Aesthetics/Nonintegrability

4

(2) All dynamical tensors are treated in a uniform way (with respect to change). A dynamical tensor in a Cartesian space is one that changes from point to point. Examples of a nondynamic vector are the Cartesian unit vectors i,j,k. (3) The field is continuous and singularity free and possesses a Taylor series. (4) We require self-consistency. The equations are obtained using Aristotelian logic. (5) We seek a theory without arbitrary functions. In this way we exclude hyperbolic theory. In hyperbolic theory the field and its time derivative are arbitrary on, say, a t = 0 hypersurface. But this would mean that particle structure is arbitrary on this hypersurface. If the aim of the theory is to obtain particle structure then arbitrariness on a hypersurface can be looked at as an incompleteness of the theory. Also, in view of possible conservation laws, how are we to determine the number of particles on a hypersurface if we have arbitrariness there. The rejection of hyperbolic equations with arbitrariness on a hypersurface does not imply that wave solutions do not appear in the theory as we shall see. From a different point of view, we wish to minimize arb­ itrariness in the theory. For these reasons, we shall assume arbitrariness at but a single point. This is similar to the Volterra and Henon-Heiles equations introduced before. (6) As all our computations will be done in a Cartesian system, we require here that the equations have the same structure when we change from one Cartesian system to another. The equations should then be tensor equations. In Appendix C we discuss elements of tensor calculus. We extend the coordinate system to curvilinear coordinates in Appendix E. 6. The Aesthetic Field Equations (Gamma Equations) As most theories admit a vector field we shall start off by assuming the existence of a vector Aj (tacitly we have supposed that tensor fields exist from the second mathematically aesthetic principle). We write for the change of Aj between neighboring points dAi = T\k Aj dxk

(1.5)

Tjk are a set of coefficients called the change function as they determine the change of Aj. We have dropped terms of higher order in dxk. For a second vector field we write dBj = rj k Bj dxk.

(1.6)

Thus, we are assuming that r j k is a universal change function that determines the change of all vectors in a uniform way. That is, one set of numbers corresponding to a vector should not be treated any differently from any other vector set of numbers. Said in another way, one vector does not have "something painted on it" that says it is any more significant than any other vector. We required that Aj appear on the right in Eq. (1.5) so that the sum Aj+B; would behave like any other vector with respect to change. From the product AjBj we have from Eq. (1.5) and Eq. (1.6) d(A( Bj) = (T'k A, Bj + Tjk Aj Bt) dx k .

(1.7)

5

Mathematical Aesthetics

Now AjBj is an example of a second rank tensor. According to our principles, we then require that r ] k determine the change of all second rank tensors in a uniform way. For a second rank tensor gy we then get (replacing A,Bj by gtj and AjBt by gjt) d

gij = ( r !kgtj + r] k gi,)dx k

(1.8)

Going one step further, an nth rank tensor is taken to behave like a product of n vectors. From Aj, gy, we define A' by means of the equation (tensor calculus allows the introduction of upper indices even for Cartesian systems as we emphasize in Appendices C and E) Ai=gljAJ.

(1.9)

Then from Eq. (1.5) and Eq. (1.8), provided g;j has an inverse, we get cLV = -rj k Aj dx k .

(1.10)

In a Cartesian system the difference between two vectors, dAj, even at different points is still a vector. Thus, from Eq. (1.5) I \ is a third rank tensor, and thus behaves like A'BjCic- Therefore the change of T'k is given by (using the same rules as before A'BmCk is replaced by TJ^, etc.)

drjkKrur^ + ^rs-r^r^dx 1 . Then, since T':k are assumed well behaved (aesthetic principle (3)) we have dT' arj,i d x dI

* = id

'

(i.ii)

FromEq. (1.11) and Eq. (1.12) we then get dT'

^

= rjnkr]]1 + rj n ,rs-r]rri n ,

(1.13)

These are 256 equations for the change function alone. We note from Eq. (1.13) that derivatives of gamma are given by products of gamma. We may call these equations the Aesthetic Field Equations, or more simply the gamma equations. We note that we can obtain Eq. (1.10) by requiring that r[ k behaves like D k . Thus, it is not necessary that gy has an inverse to obtain Eq. (1.10). We summarize what we have done. We have introduced a change function that determines the change of all functions. But the change function is itself a function. Thus, it must determine its own change by Aristotelian logic (aesthetic principle 5). The functions we are dealing with are Cartesian tensors and the manner in which T]k

Math Aesthetics/Nonintegrability

6

determines change is such that all Cartesian tensors are treated in a uniform way with respect to change. We have thus obtained a set of partial differential equations from a small amount of input, figuratively we can say we have obtained "something out of nothing" or at least almost nothing. Equation (1.13) arises from what Schwalm5 calls logical self-implication in his article "The Lure of Mathematical Science". We note that Eq. (1.13) is a set of first order coupled nonlinear equations with quadratic type terms appearing on the right hand side. Fjk is given in terms of rj k at a single point. Thus Eq. (1.13) has similarities with the equations such as the Volterra Predator-Prey equations and the Henon-Heiles equations mentioned earlier, which we know have considerable content. Other Relations Obtained from the Basic Principles From Eq. (1.8) we get dgjj