The Book of Nothing Page 15

  Einstein’s brilliant success in bringing together all that was known about motion into a simple and mathematically precise theory was the end of the nineteenth-century ether. Einstein’s theory had no need of any ether to convey the properties of light and electricity. His postulate that the speed of light must be the same for all observers had the FitzGerald-Lorentz contraction as a direct consequence, and the non-detection of any light-delay effect in the Michelson-Morley experiment was a key prediction of his theory. Many years later, on 15 January 1931, Einstein made a speech in Pasadena to an audience containing many of the world’s greatest physicists. Michelson was there, making what would turn out to be his last public appearance before his death four months later. Einstein paid tribute to the importance of the experiment that Michelson first performed in guiding physicists to their revolutionary new picture of space, time and motion:42

  “You, my honoured Dr. Michelson, began this work when I was only a little youngster, hardly three feet high. It was you who led the physicists into new paths, and through your marvellous experimental work paved the way for the development of the Theory of Relativity. You uncovered an insidious defect in the ether theory of light, as it then existed, and stimulated the ideas of H.A. Lorentz and FitzGerald, out of which the Special Theory of Relativity developed. Without your work this theory would today be scarcely more than an interesting speculation; it was your verification which first set the theory on a real basis.”

  In fact, Einstein’s career intersected with the ether on many occasions. It only became known after his death that at the age of fifteen he became interested in the stationary elastic ether. He even wrote an article about what happens to the state of the ether when an electric current is turned on, which was not published until 1971.43 Later, he also contemplated carrying out experiments which would be able to verify the existence of an ether. Gradually, he began to doubt its existence. In 1899, he wrote to his girlfriend Mileva Maric of his doubts:

  “I am more and more convinced that the electrodynamics of the bodies in motion, such as it is presented today, does not correspond with reality and that it will be possible to formulate it in a simpler way. The introduction of the word ‘ether’ in the theories of electricity leads to the idea of a medium about the motion of which we speak without the possibility, as I think, to attribute any physical sense to such a speech.”44

  As a student he learned about Lorentz’s theory of electrodynamics, and the role played by the ether, in his course textbooks. When his thinking drove him towards his new theory of motion, he found he had no need of the ether or of a vacuum with any special properties. It was enough to be able to talk about bodies moving in space and through time. That space was empty unless one chose to add further ingredients to it. It was a matter for investigation whether one needed to include a magnetic or an electric field everywhere in the Universe. If such fields of force were ubiquitous, then his theory could handle them but, equally, it could apply itself to the movements of bodies in a completely empty space.45

  The developments in our understanding of matter and motion in the first few years of the twentieth century brought to an end what is sometimes called ‘classical’ physics. Just a few years before there had been serious speculation that the work of physics was all but done. There were refinements to make, further decimal places to establish in experimental accuracy, but all the great physical principles of Nature were thought by some to have been mapped out. The details merely had to be filled in. The discoveries of the quantum theory of matter and the relativity of motion changed everything. New vistas opened up. But they were vistas that did not need a theory of the vacuum, or even a clear notion of what it was. Emphasis switched to the study of how fields and particles influenced one another. Ancient dilemmas like that of the extracosmic void or the nature of absolute space were issues that philosophers still talked about but they were not subjects that promised new insights. Physicists seemed rather relieved to be able to ignore the vacuum for a change, rather than find it like the proverbial tail wagging the dog, in steering the direction of theories of electricity, magnetism and motion.

  This brief era of nothingless physics was soon ended. Within ten years of Einstein’s issue of a redundancy notice to the ether, the issue of the vacuum was back in a central and puzzling place in scientific thinking. The deeper and wider extensions of the special theory of relativity and the quantum picture of matter would reinstate the vacuum in a central position from which it has yet to be dislodged at the start of another century.

  “The fool saith in his heart that there is no empty set. But if that were so, then the set of all such sets would be empty, and hence it would be the empty set.”

  Wesley Salmon

  ABSOLUTE TRUTH – WHERE IS IT TO BE FOUND?

  “As lines, so loves oblique may well

  Themselves in every angle greet

  But ours, so truly parallel,

  Though infinite can never meet.”

  Andrew Marvell, ‘Definition of Love’1

  Like the Grand Old Duke of York, who marched his men to the top of the hill and marched them down again, nineteenth-century physicists had been busily filling the ancient void with ether and emptying it out again. In the meantime, what had been happening to zero, that handy little circle that provided the final piece in the jigsaw of symbols that went to make our modern system of arithmetic?

  During the nineteenth century, mathematics began to move in a new direction and its scope expanded beyond the paths mapped out by the ancients. For them, mathematics provided a way of making precise statements about quantities, lines, angles and points. It was divided into arithmetic, algebra and geometry, and formed a vital part of the ancient curriculum because it offered something that only theology would also dare to claim – a glimpse into the realm of absolute truth. The most important exemplar was geometry. It was the most impressive and powerful instrument wielded by mathematicians. Euclid created a beautiful framework of axioms and deductions that led to truths called ‘theorems’. These truths led to new knowledge of the motions of the planets, new techniques for engineering and art; Newton’s greatest insights were achieved by means of geometry.

  Geometry was not seen as merely an approximation to the true nature of things, it was part of the absolute truth about the Universe. Like part of some holy writ, the great theorems of Euclid were studied in their original language for thousands of years. They were true, perfectly so, and they provided human beings with a glimpse of absolute truths. God was many things but he was undoubtedly also a geometer.

  We begin to see why mathematics was of such importance to theologians and philosophers. With no knowledge of mathematics you might have been persuaded that the search for absolute truth was a hopeless quest. How could we fathom its bottomless complexity given the approximate and incomplete nature of our understanding of everything else in the world around us? How could a theologian claim to know anything about the nature of God or the nature of the Universe in the way that medieval philosophers seemed to do so confidently in their pronouncements about the vacuum and the void? Their justification was in the success of Euclid’s geometry. It was the prime example of our success in understanding a part of the ultimate truth of things. And if we could succeed there, why not elsewhere as well? Euclid’s geometry was not just a mathematician’s game, a rough approximation to things, or a piece of ‘pure’ mathematics devoid of contact with reality. It was the way the world was. A similar exalted status was afforded the system of logic that Aristotle introduced as the means by which the truth or falsity of deductions made from premises could be ascertained. Aristotle’s logic was accepted as being true and perfectly representative of the working of the human mind. It was the one and only way of reasoning infallibly.2

  Euclid’s geometry is a logical system that defines a number of concepts, makes a number of initial assumptions, sets down what rules of reasoning are to be allowed, and then allows an edifice of geometrical truths to be deduced by applyi
ng the rules of reasoning to the concepts and axioms. It is rather like a game of chess. There are pieces and rules governing their movement together with a starting position for all the pieces on the board. Applying the rules to the pieces produces a sequence of positions for the pieces on the board. Each possible configuration of pieces that can be reached from the starting position could be regarded as a ‘theorem’ of chess. Sometimes one encounters inverse chess puzzles that challenge you to decide whether or not a given board position could have been the result of a real game or not.

  Euclid’s geometry described points, lines and angles on flat surfaces. It is now sometimes called ‘plane geometry’. He set out definitions of twenty-three necessary concepts and five postulates. To get the flavour of how pedantically precise Euclid was, and how little he took for granted, here are a selection of his definitions:3

  Definition 1: A point is that which has no part.

  Definition 2: A line is a breadthless length.

  Definition 4: A straight line is a line that lies evenly with the points on itself.

  Definition 23: Parallel straight lines are straight lines that, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

  Euclid’s aim was to avoid using pictures or practical experience. All truths of plane geometry must be deduced by using these definitions and five other axioms or ‘postulates’ from which everything follows by logical reasoning alone. This arena for plane geometry was circumscribed most potently by one of its axioms, the fifth, which stated that parallel lines never meet.4 Usually it is known as the ‘parallel postulate’. There had always been special interest in this axiom because some mathematicians suspected that it might be an unnecessary stipulation: they believed it could be deduced as a logical consequence of Euclid’s other axioms. Many claims were made at different times to have proved the parallel postulate from the other axioms, but all were found to have cheated in some way, subtly assuming precisely what was to be proved along the way.

  The great success of Euclidean geometry had done more than merely help architects and astronomers. It had established a style of reasoning, wherein truths were deduced by the application of definite rules of reasoning from a collection of self-evident axioms. Theology and philosophy had used this ‘axiomatic method’, and most forms of philosophical argument followed its general pattern. In extreme cases, as in the works of the Dutch philosopher Spinoza, philosophical propositions were laid out like the definitions, axioms, theorems and proofs to be found in Euclid’s works.5

  This confidence was suddenly undermined. Mathematicians discovered that Euclid’s geometry of flat surfaces was not the one and only logically consistent geometry. Carl Friedrich Gauss (1777–1855), Nikolai Lobachevski (1793–1856) and Janos Bolyai (1802–1860) all contributed to the revolutionary idea of giving up the quest to prove Euclid’s parallel postulate from his other axioms and, instead, see what happens if one assumes that it is false.6 This revealed that the fifth axiom was by no means a consequence of the other axioms. In fact, it could be replaced by another axiom and the system would still be self-consistent.7 It would still describe a geometry but not one that exists on a flat surface.

  There exist other, non-Euclidean, geometries that describe the logical interrelationships between points and lines on curved surfaces (see Figure 5.1). Such geometries are not merely of academic interest. Indeed, one of them describes the geometry on the Earth’s surface over large distances when we assume the Earth to be perfectly spherical. Euclid’s geometry of flat surfaces happens to be a very good approximation locally only because the Earth is so large that its curvature will not be noticed when surveying small distances. Thus, a stonemason can use Euclidean geometry, so can a tourist travelling about town, but an ocean-going yachtsman cannot.

  This simple mathematical discovery revealed Euclidean geometry to be but one of many possible logically self-consistent systems of geometry. All but one of these possibilities was non-Euclidean. None had the status of absolute truth. Each was appropriate for describing measurements on a different type of surface, which may or may not exist in reality. With this, the philosophical status of Euclidean geometry was undermined. It could no longer be exhibited as an example of our grasp of absolute truth. Mathematical relativism was born.

  Figure 5.1 A vase whose surface displays regions of positive, negative and zero curvature. These three geometries are defined by the sum of the interior angles of a triangle formed by the shortest distances between three points. The sum is 180 degrees for a flat ‘Euclidean’ space, less than 180 degrees for a negatively curved ‘hyperbolic’ space and more than 180 degrees for a positively curved ‘spherical’ space.

  From this discovery would spring a variety of forms of relativism about our understanding of the world.8 There would be talk of non-Euclidean models of government, of economics, and of anthropology. ‘Non-Euclidean’ became a byword for non-absolute knowledge. It also served to illustrate most vividly the gap between mathematics and the natural world. Mathematics was much bigger than physical reality. There were mathematical systems that described aspects of Nature, but there were others that did not. Later, mathematicians would use these discoveries about geometry to discover that there were other logics as well. Aristotle’s system was, like Euclid’s, just one of many possibilities. Even the concept of truth was not absolute. What is false in one logical system can be true in another. In Euclid’s geometry of flat surfaces, parallel lines never meet, but on curved surfaces they can (see Figure 5.2).

  These discoveries revealed the difference between mathematics and science. Mathematics was something bigger than science, requiring only self-consistency to be valid. It contained all possible patterns of logic. Some of those patterns were followed by parts of Nature; others were not. Mathematics was open-ended, uncompleteable, infinite; the physical Universe was smaller.

  Figure 5.2 Lines on flat and curved surfaces, where ‘lines’ are always defined by the shortest distance between two points. On a flat surface only parallel lines never meet; on the spherical surface all lines meet whilst on a hyperbolic space many lines never meet.

  MANY ZEROS

  “The ultimate goal of mathematics is to eliminate all need for intelligent thought.”

  Ronald Graham, Donald Knuth & Owen Patashnik9

  The discovery that there can exist logically self-consistent geometries which are different from Euclid’s was a landmark.10 It showed that mathematics was an infinite subject. There was no end to the number of different logical systems that could be invented. Some of those logical systems would have direct counterparts in the natural world, but others would not. Only a fraction of the possible patterns of mathematics are used in Nature.11 From now on, some new choices would have to be made. What mathematical system is appropriate for the problem under study? If we wish to survey distances we need to use the right geometry. Euclid is no good for determining distances on the Earth’s surface which are great enough for its curvature to be important.

  The proliferation of mathematical systems (see Figure 5.3) led to the notion of what is now called ‘mathematical modelling’. Particular pieces of mathematics help us describe aerodynamic motion but if we want to understand risk and chance we may have to use other mathematics. On the purer side of mathematics, it was recognised that there exist different mathematical structures, each defined by the objects (for example, numbers, angles or shapes) they contain and the rules for their manipulation (like addition or multiplication). These structures have different names according to the richness of the rules that are allowed.

  One of the most important families of mathematical structures of this sort is that of a group. It is a precise prescription for a collection of objects that are related in some way. A group contains members, or ‘elements’, which can be combined by a transformation rule. This rule must possess three properties:

  closure: if two elements are combined by the transformation rule, it must pro
duce another element of the group.

  identity: there must be an element (called the identity element)12 which leaves unchanged any transformation it is combined with.

  inversion: every transformation has an inverse transformation which undoes its effect on an element.

  These three simple rules are based on properties that are possessed by many simple and interesting procedures. Let’s consider a couple of examples. First, suppose that the group elements are all the positive and negative numbers (… −3, −2, −1, 0, 1, 2, 3, …). The group transformation rule will be addition (+). We see that this defines a group because the closure condition is obeyed: the sum of any two numbers is always another number. The identity condition is obeyed. The identity element is zero, 0, and if we add it to any element it is left unchanged by +. The inversion property also holds: the inverse of the number N is −N so that if we combine any number with its inverse we always get the identity, zero; for example 2 + (−2) = 2 − 2 = 0.

  Note that if we had taken our elements to be the same natural numbers but the transformation combining them to be multiplication rather than addition and the identity element to be 1, then the resulting structure is not a group. This is because the inversion property fails for all numbers other than +1 and −1. The quantity that we need to multiply, say, the number 3 by to give the identity, 1, is ⅓, which is not a whole number and so is not another element of the group. If we allow the elements to be fractions, then we do have a group with transformation defined by multiplication.

  We notice that in these two examples the identity operation which leaves an element of the group unchanged is a null operation. In the first example of adding numbers it corresponds to the usual zero of arithmetic. Its status as the identity element of our group is guaranteed by the simple property that N + 0 = N for any number N. In the second example the identity element is not the usual zero at all. The null operation for multiplication is provided by the number 1 (or, as a fraction 1/1, which is the same thing). The usual zero is not a member of the second group.13