On the Shoulders of Giants

Lines M E

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351427067
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

On the Shoulders of Giants investigates the relationship between the disciplines of physics and mathematics and shows how many of the most significant advances of 20th-century physics rely on mathematics developed, sometimes much earlier, with no particular physics application in mind. Quoting from mathematicians such as Poincaré and Euclid and physicists such as Newton and Feynman, the links between the two disciplines are explored in the author's entertaining style, providing a fascinating account of the twists and turns in scientific progress through the ages.

1

Introduction

Perhaps Oscar Hammerstein II, in conjunction with composer Richard Rodgers, said it most melodiously (albeit in a most unscientific setting) when he put the words ‘nothing comes from nothing’ into a ballad in the 1950s musical The Sound of Music. Or was it the irrepressible Harvard mathematics professor and songster Tom Lehrer who, in that same decade, expressed it in the more cynical form ‘plagiarize, plagiarize, let no-one else’s work evade your eyes’ in his irreverent jingle concerning the supposed advice of the (actually much respected) nineteenth century Russian mathematician Nikolai Ivanovich Lobachevsky on how best to get ahead in research endeavors? In any event, the sentiment that none of us creates anything truly worthwhile entirely out of a vacuum is not one restricted to musical plays or satire. In a much more serious vein, and in better accord with the theme of this book, this has appeared in many and diverse forms at least as far back as Roman times, and probably much farther.

The most famous quote along these lines is almost certainly the one usually attributed to Sir Isaac Newton who, in a rather uncharacteristically modest fashion, spelled it out in a letter to his distinguished scientific compatriot Robert Hooke in 1675 in the form ‘If I have seen farther [than others] it is by standing upon the shoulders of giants’. However, this oft-quoted remark is not a Newtonian original. One can find essentially the same statement in the historical epic Bellum Civile (the Civil War) by the first century Roman poet and republican patriot Marcus Lucanus in words that literally translate as ‘pigmies placed on the shoulders of giants see more than the giants themselves’. The sentiment is the same, but one can see why Newton chose to paraphase the original.

Although it should come as no surprise to anyone that research in any field of endeavor conventionally builds upon the existing ‘pool of knowledge’ already established for that field—something we like to refer to as ‘pushing back the frontiers’—we shall attempt to establish in the following chapters the existence in the physical sciences of something that often goes, rather mysteriously some might think, significantly beyond this assertion. In these chapters we attempt to demonstrate the manner in which many of the most significant advances of twentieth century physics have found the mathematics which is ideally suited (and sometimes essential) for their theoretical understanding quietly lying-in-wait (almost in the nature of a solution looking for a problem) in the mathematics literature of earlier generations. In other words, the giants whose shoulders are most desperately required by research scientists often turn out to be those of long-ago-expired mathematicians. Mathematicians, on the other hand, would claim that they are rarely, if ever, to be found standing on the shoulders of scientists.

To many it may not seem surprising that mathematics has played an essential role in science, and particularly physics—after all, physics is a quantitative science. Nevertheless, mathematics and physics are by their nature very different disciplines indeed. In fact, mathematics possesses a methodology that is unique among all the sciences, since it is the only science in which deductive logic alone determines truth. It follows that mathematical theorems which were established thousands of years ago are every bit as true today as they were on the day that they were first proved. In many cases they are still taught as part of the essential curriculum in today’s High School mathematics courses. Not for the mathematician are the messy uncertainties concerned with instrumental errors, statistical accuracy of data, or ‘tentative’ assumptions that so often go along with potential advances in other sciences. For this reason, mathematicians tend to believe that the quality of their truths is higher than that of other sciences and are justly proud of their standards of rigor.

As a humorous illustration of this rigor, I am particularly fond of the story concerning three academics supposedly traveling by train from London to Edinburgh to attend a conference. Upon crossing the Scottish border, they casually gazed out of the train window to see two lone black sheep in a field. The first academic, an astronomer by discipline, expressed immediate interest. ‘Just look at that,’ he said, ‘the sheep in Scotland are black.’ The second academic, being a physicist, was jarred by this unwarranted generalization. ‘That is nonsense,’ he responded. ‘All that the evidence before us demonstrates is the fact that at least two of the sheep in Scotland are black!’ After a moment or two of silence, during which time the physicist basked in the glory of his superior powers of reasoning, he found himself equally rebuked by the third member of the group, a mathematician. ‘Your conclusion is equally unfounded,’ the mathematician retorted. ‘The correct interpretation of the observation is that at least two of the sheep in Scotland each have at least one black side!’ Although the story is perhaps a little unkind to astronomers, it does focus on an important point. The physicist, never having seen or heard of the existence of a sheep anywhere on earth with one black side and another side of a different hue, was happy to accept the ‘fact’ that none exists. To the mathematician, in the absence of any proof that no such animal could possibly exist, such an assumption was completely unwarranted.

Other somewhat more serious demonstrations of this difference in attitude between physicists and mathematicians are not difficult to highlight-examples which again focus on the uniqueness of mathematical rigor. For instance, it is often said that of all scientists, the mathematician is the only one who does not know how to pack spheres together to form the densest possible arrangement in three-dimensional space. All physicists, for example, ‘know’ that the answer is just the ordered stacking that you often see for oranges in supermarket displays or for cannonballs at War Memorials. It is called ‘ordered close packing’ and fills up very nearly 75% of all space if continued ad infinitum. The mathematician, looking at the same problem, is quite happy to accept that this is, indeed, the densest ordered (that is, repeating or periodic) arrangement, since a rigorous proof of the fact exists. However, as yet, no mathematical proof has been given that some non-periodic arrangement could not do better. In truth, the physicist also initially had doubts, and so he experimented-first by shaking balls into containers and later, after the advent of the computer, by creating non-periodic algorithms (or sets of rules) to ‘build’ such packings by computer simulation. But try as he would, over years and years of effort, he found that no randomlike packing could ever be assembled with a packing density which came even close to the 75% of space filling achieved by the periodic orange and cannonball arrangements. In fact, after decades of trying, only about 65% of space could be filled by even the cleverest computer-prepared nonperiodic packings, a value not significantly greater than the one achievable by simply shaking balls in any large container. Thus, for the physicist the question is settled ‘beyond a reasonable doubt’. But for the mathematician, even an unreasonable doubt is not sufficient. In the absence of a proof that satisfies the full rigor of mathematical methodology, the question of the most efficient packing of identical spheres in three dimensions remains unanswered.

In a similar vein, the late physics Nobel Laureate Richard Feynman once presented a physicist’s proof of Fermat’s Last Theorem. This most infamous of all (unsolved) mathematical conjectures states simply that the equation xⁿ + yⁿ = zⁿ cannot be satisfied by any positive integer values for x, y, z and n if the exponent n is larger than 2. I place the unsolved adjective in parentheses since, after some 350 years, a rigorous mathematical proof which may possibly withstand the scrutiny of the refereeing panel has been presented during the writing of this book. It was offered by Andrew Wiles, of Princeton University, US, and announced to a stunned audience in a lecture hall at the Isaac Newton Institute in Cambridge, England. All ‘proofs’ until now have failed to hold up, and Wiles’ effort is at present undergoing a ‘fine toothcomb’ examination by experts. However, initial comments—such as ‘the logic is utterly compelling’, ‘it simply has the ring of truth’ and ‘even if there’s a small mistake, it’s likely to be fixable’—seem most encouraging so that, by the time you read this, the question may finally have been settled. But this ‘breaking’ news story has forced me to digress from my main point, the ‘physicists’ proof’.

Even in the days of Feynman’s comment, the conjecture had been verified by computer up to extremely large integers, so that Feynman decided to make use of the random occurrence of numerals in still larger numbers to calculate the statistical probability that the Fermat equation might be satisfied. In this fashion he concluded that, even when integers of all possible digit lengths (beyond those already tested) were included, the probability that any particular set would satisfy the Fermat equation was much smaller than a physicist would be more than happy to accept as conclusive for any physical experiment. Thus he asserted (in characteristically impish fashion) that physicists should be just as happy to accept the validity of his ‘proof’ of Fermat’s Last Theorem as they are to accept any experimental ‘conclusion’ resulting from their discipline.

Mathematicians would obviously be aghast at such an approach to the problem. Assertions concerning the randomness of digits in large numbers are dubious in the extreme. Problems similar to Fermat’s, which have been unquestionably solved and are known to have an infinite number of possible solutions, are also known to pass the Feynman test for insolubility with precision comparable to the finding for the Fermat equation itself. Evidently the sciences of physics and mathematics possess fundamental differences in their approaches to the ‘search for truth’. In a very real sense, mathematics is divorced from experiment, while physics appeals to experiment as the sole arbiter of truth. This difference is illustrated by an oft-told (though possibly apocryphal) meeting between the youthful Albert Einstein and an aging Jules Henri Poincaré, the latter of whom was one of the foremost mathematicians of the nineteenth century. Einstein supposedly said to Poincaré, ‘I considered taking up mathematics but decided against it because of its lack of connection with the real world and the impossibility of telling what is important.’ To which Poincaré replied, ‘In my youth I was seriously tempted to become a physicist, but I decided against it because in physics it is impossible to tell what is true.’ Clearly, if the two subjects are to reach compatibility then some trade-off has to be made between the high standards of truth and the requirements of applicability.

A young Albert Einstein at his desk in the Patent Office in Berne ca 1905. Courtesy of The Hebrew University of Jerusalem, Einstein Archives. Reproduced by permission of AIP Emilio Segrè Visual Archives.

Jules Henri Poincaré, 1854–1912. Reproduced by permission of Mary Evans Picture Library.

As a very simple demonstration of this trade-off let me discuss for a moment the subject of the science of crystallography. By various experimental techniques involving the scattering of x-rays or neutrons from crystals, it is nowadays a common procedure to (as physicists put it) ‘determine’ the ‘structure’ of a particular crystal. By this they mean identify the shape and size of the smallest atomic unit which repeats itself endlessly (or at least of order 10²³ times) in a regular pattern to build up the macroscopic sample. As a simple example we can imagine 10²³ cubes packed together with atoms at the cube comers and, perhaps also, the cube centers or cube face centers. Many of the most common metals in the world have structures of this kind, and here we would appear to have a crystallographic truth that is perfectly representable by mathematics. But we have deceived ourselves. The laws of thermal physics dictate that no such perfect ‘mathematical’ crystal can possibly exist at any non-zero temperature—and the absolute zero of temperature can never be attained. In all actual crystals there must, in addition to atomic vibrations about their ideal sites, be defects—that is, some atoms absent from the sites where they should be, and others in displaced sites where they ideally should not be. Clearly the real crystal is not exactly represented by the mathematical lattice (and can never be, even in ideal physical circumstances). And yet, since the vast majority of atoms are indeed where they ‘should be’, the mathematical description is of great value. In other words, a valuable trade-off between mathematics and physics has been made. In like manner there is much more cooperation than confrontation between the physical sciences and mathematics than their different visions of truth might suggest.

In truth, if one factor has remained constant throughout the twists and turns of the history of progress in the physical sciences, it is the decisive role played by mathematics. It is the story of this role in many of its guises that the chapters of this book will attempt to portray. In many instances, a case can be made for the statement that it is the mathematics that drives the physics. Indeed, the power of existing mathematics to mirror the behavior of the universe has ever been a source of wonder to physicists, and has even attracted the concern of philosophers. The usual procedure is for the physicist to carry out a translation of the physical reality into the abstract language of mathematics by locating the most ‘relevant’ mathematics available for the purpose. Surprisingly often, this turns out to be some previously unapplied corner of purest mathematical abstract enterprise. Application of this existing mathematical framework to its new context then not only assists enormously in clarifying the thinking surrounding the physical phenomenon, but also frequently points the way to fruitful directions in which the physical research in that area can be extended.

Many physicists over the years have expressed their amazement at the potency of this ‘marriage’. The nineteenth century German physicist Heinrich Hertz, after using differential equations in pursuit of an understanding of radio waves, wrote that ‘One cannot escape the feeling that these mathematical formulae have an independent existence and intelligence of their own; that they are wiser than we are, wiser even than their discoverers, and that somehow we get more out of them than was originally put in.’ Later, in the twentieth century, the same feeling of wonder was expressed by Eugene Wigner, the Hungarian-born American physics Nobel Laureate, when he contemplated the successes of various mathematical concepts in the interpretation of quantum mechanics, the physics of the weird and wonderful world of the atomically small. In reference to this mystical and almost supernatural association between the newly discovered physical world and the previously unapplied abstract mathematical formulations he wrote ‘We are in a position similar to that of a man who was provided with a bunch of keys and who, having opened several doors in succession, always hit on the right key with the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.’ Not to suggest, of course, that all physicists (whether experimentalists or theorists) have been quick to recognize the mathematical ‘jewels’ offered to them. The British physicist and astronomer Sir James Jeans, while Professor of Applied Mathematics at Princeton in 1910 (and before his elevation to knighthood in 1928) is purported to have advised the curriculum committee ‘We may as well cut out group theory. That is a subject which will never be of any use in physics.’ He could not have been more mistaken (as we shall see later in the book) and this same group theory, a product of the purest mathematical thought, is now not only the essential language of the physics of crystalline solids, but also completely dominates the thinking of all who are trying to understand the fundamental particles and forces of nature at the subatomic level.

So how do we explain this uncanny ability of formal and abstract mathematics to come to the aid of physical scientists in their efforts to understand the world around them? One recent suggestion by a Harvard mathematician goes something like this: ‘Mathematics is the science of patterns; patterns in number, in space, and in imagination. Theories in mathematics then pursue the relations among these patterns, binding one to the other to yield structures. Applications of mathematics to the physical sciences use these patterns to “explain” natural phenomena that fit the patterns. In recent years, particularly with the advent of computers (which are to computational mathematics what microscopes are to science) the portfolio of these patterns has been increased immensely. Could it therefore be that the reason why mathematics has the uncanny ability to provide the right patterns to explain natural phenomena is simply that the number of patterns now recognized by mathematicians is fast approaching all the patterns that there are?’ Although persuasive, perhaps, as an argument as to why mathematics may continue to fulfill this role in the future, I personally am not impressed by it as reason for its uncanny success in the pre-computer world of the distant, and not so distant, past. At present I am content to let it remain an enigma, and proceed to relate the individual stories that constitute the evidence.

2

From Aristotle to the Structure of Glass

The question of how to fill space, without leaving any gaps, by fitting together identically shaped building blocks is one of the most ancient of all problems about space. In one sense the answer is easy (the bricklayer, for example, has obviously discovered one solution) but in another it is extremely difficult. Which shapes of block are able to fill space in this manner and which are not? In its more general form, perhaps somewhat surprisingly, the question remains unanswered to this day, computer graphics notwithstanding. But interest in pursuit of solutions has produced an enormous amount of effort over the centuries which has, as we shall see, finally left the realm of the academic and become firmly enmeshed in the study of glass structure; glasses being those solids that seem to have trouble in finding a way to pack space without becoming scrambled and distorting their atomic ‘building blocks’.

Questions concerning the filling of space with identical blocks appear to have first arisen ne...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• 1 Introduction
• 2 From Aristotle to the structure of glass
• 3 From Euclid to general relativity
• 4 From plucking strings to electrons in solids
• 5 The perception of number: from integers to quaternions
• 6 From tiling floors to quasicrystals
• 7 Determinism: from Newton to quantum chaos
• 8 Symmetry: from Galois to superstrings
• 9 From coin tossing to entropy
• 10 Topology: from the bridges of Königsberg to polymers
• 11 From parabolas to fractons
• 12 Motion: from Zeno to Schrödinger
• Appendices
• Index