“Every day we slaughter our finest impulses. That is why we get a heartache when we read the lines written by the hand of a master and recognize them as our own, as the tender shoots which we stifled because we lacked the faith to believe in our own powers, our own criterion of truth and beauty. Every man, when he gets quiet, when he becomes desperately honest with himself, is capable of uttering profound truths.” Miller [1].
3.1 A Journey to the Himalayas
Three things in life that most of us desire are love, money, and an abundance of good food and drink. However, the three things that we usually find are shattered illusions, obesity and death. Nevertheless, some people manage to transcend their peers and stand out for their higher aspirations.
Young Archibald was one of these few: his quest was for Beauty, Truth and Wisdom. To those who warned him that it was unrealistic to hope for all three, he would reply that actually the three were just different facets of the same thing. Beauty is perfection, and perfection cannot be flawed by the capital sin of being false. “Just think of a snowflake,” he said. “The laws of physics and geometry render each flake different, but they are all endowed with the perfection of symmetry. What could be more enchanting than a snow-covered mountain landscape? Wisdom is the ability to recognize this union and be grateful for such a wonderful gift.”
But where can one find beauty and truth? In the cities, the splendour of architecture, art, music and poetry, are offset by pollution, degradation and kitsch. Far better to search in isolated mountain valleys, in villages blessed by the stunning natural beauty of their surrounds, where simpler, but wiser communities prosper.
So Archibald travelled all summer and arrived in late autumn at a secluded village on the slopes of a beautiful mountain in the Himalayas. Since he was searching for wisdom, he was accepted by the Buddhist villagers, and even admitted to the weekly meeting of the ancients. In the first of these, the village Chief ordered that everybody should spend considerably more time collecting firewood for heating, for the predictions were for a very cold winter.
In the second and subsequent meetings, the Chief kept pressing for more and more wood to be collected, until one of the ancients protested and asked how the Chief could be sure that it would be so cold. The Chief did not elaborate how he knew, but said “if you don’t believe me, visit the Wise Old Man in the mountain tomorrow and ask him.”
The next day, Archibald and a delegation of ancients climbed for hours to reach the cave, high in the mountains, where the Wise Old Man was living alone. Archibald was elated: at last he would have a chance to meet the mystic, who in his eyes was the personification of wisdom. When they finally reached the cave, the old man waved them in and demanded their purpose.
As he heard the explanation, he nodded sagely. “Yes, your Chief is right,” he said, “it will be a terribly cold winter.”
But some of the ancients were still unconvinced: “How can you be so sure?”.
The quest for truth and beauty is not confined to adventurers, such as Archibald but, perhaps surprisingly, is shared by scientists, artists, writers and lay people alike. However, not everyone means the same thing when they use these two terms. In this chapter we shall explore further what a scientist means by truth, and how the impact of concepts of beauty has influenced the direction of scientific research.
3.2 A Comparison Between Mathematical and Physical Truth
In Chap. 1, we explored the possibility of multiple truths, and observed that differences may arise between “scientific truth” and “theological truth.” What is perhaps more surprising is that even within the sciences (if one includes mathematics in this category), clear differences of interpretation of what is meant by “truth” are evident. In this Section, we attempt to clarify some of these differences.
It is difficult to imagine any student progressing far in physics without a good grounding in mathematics. The logical nature of physics is closely related to that of mathematics, which undoubtedly explains why mathematics is so entwined with physics. Modern physical theories are formulated in abstract mathematics, which is not the case in most other disciplines. This gives physics its own aura that discourages those with scanty mathematical talent.
To make the correspondence between physics and mathematics a little clearer, let us revisit our high school days and our first introduction to geometry. For most students, this is their first encounter with the logical structure of mathematics. Euclid founded his geometry upon a number of definitions and axioms. The definitions tell us what we mean when we talk of points, lines, circles, etc. The axioms, which are detailed in Appendix 3.1, are sometimes called “self-evident truths”, and are assumed without any attempt to prove them.
All Euclid’s axioms seem to be very reasonable, and few among high school students, or their teachers, question them. From these humble beginnings the whole of Euclidean geometry can be derived by strict application of elementary logic. We have discussed in Chap. 2 the nature of logic, and its origins.
The fifth of Euclid’s axioms states that parallel lines never meet. Over the centuries the question was raised as to whether this axiom was really necessary, or indeed, true. From a purely mathematical point of view, mathematicians attempted to deduce the axiom logically from the other four (in which case the fifth would have lost its status as an axiom, and become a theorem), but had no success. We live on a spherical planet, which, because of its size, appears to us to be totally flat locally (except for hills, etc.). Hence, we might expect from our experience that the fifth axiom could be valid locally, but not globally.
Let us, for instance, imagine two-dimensional animals living on a large sphere, analogous to the Earth. If two such individuals set off from different points on the equator on parallel trajectories heading due north, they will meet each other at the North Pole because of the curvature of their spherical domain. In their world, parallel lines do indeed meet. Eventually, nineteenth Century German mathematician, Bernhard Riemann, developed a geometry for such a world. As we shall see in Chap. 7, in Einstein’s General Theory of Relativity, where space–time is not flat, Euclid’s geometry does not apply and Riemannian geometry is essential.
To summarise, mathematics uses logic to deduce complex “truths” from underlying assumed simpler truths. In some branches of mathematics, these basic truths may not be all that intuitive. However, they represent the foundation, on which the towering edifice of any field of mathematics is constructed.
In the case of Euclidean geometry, the axioms reflect the reality of the terrestrial space in which we live our daily lives. However, it is not necessary for the axioms to have any connection with physical reality at all. Many fields of mathematics are highly abstract, and some mathematicians consider only mathematics that has no practical applications to be “pure”. G. H. Hardy, whom we will meet again later in this Chapter, made a distinction between “real” mathematics, “which has permanent aesthetic value”, and the remainder, “the dull and elementary parts of mathematics”, that have practical use [2].
Hardy’s aloofness provoked retorts, such as the following, from those of a more practical bent: A group of physicists and engineers were enjoying a flight in a balloon until sudden gusts of strong wind compelled them to seek a safe landing place. After they had set down, a physicist in the party leaned out of the basket and asked a passing local where they were. “In a balloon”, came the reply. The physicist turned back to his companions. “That man is obviously a mathematician,” he declared. “His answer is totally correct, but absolutely useless.”
Let us now see how the axiomatic structure of mathematics compares with the nature of physics. In physics, we start with definitions of quantities, such as energy, momentum, velocity, mass, length, time, force etc., that are analogous to the definitions of Euclid. Some of these are more fundamental than others: for instance, velocity is defined as the length travelled in a particular direction divided by the travel time. Then, instead of axioms, we have physical “laws”, which are inferred through observations and experiments. For example, Newtonian Mechanics, which describes the motion of objects in the everyday world, is based on three laws (see Appendix 3.2). From them and from the basic definitions, the interactions of bodies, both in the laboratory and in the heavens, can be calculated using the rules of logic, as embodied in mathematics.
Although the analogy between physics and mathematics is clear from the above arguments, there is an important difference. The axioms of mathematics are in a sense a priori, or originating from the mind of mathematicians.1 There may be a correspondence between these axioms and the physical world, but it is by no means necessary. Likewise, Riemann developed his own geometry where parallel lines meet, not to describe travels on the Earth, but as a kind of mathematical game. When Einstein developed his Theory of Relativity, Riemannian geometry was already there, waiting for him to use as a framework for space–time.
The laws of physics, on the other hand, are a property of nature. They represent the physicist’s attempt to understand the working of the cosmos. From these laws, the use of mathematics leads to predictions of the behaviour of interacting bodies that can be tested by experiments carried out in a laboratory, or by observations made through a telescope, or similar instrument. If these predictions do not agree with what is actually observed, it is assumed that the original laws are inaccurate and must be modified in some way, or abandoned altogether.
Strictly speaking we might alternatively say that the rules of logic that we followed are wrong, and some new logic must be devised. However, this alternative is seldom seriously considered. One might say that it is part of the “faith” of the physicist that the logic used by mathematicians is adhered to by nature. We will discuss this proposition further in Sect. 3.3 of this Chapter, and in Part 3 of this book we shall also examine the possibility of non-classical logics.
As soon as we mention experiments, we encounter the major difference between physics (or indeed any science) and mathematics. Every experiment, or observation, has associated with it a measurement error. For a well-designed and executed experiment, this error can be relatively small (say 0.1%), but it is always there. The consequence is that we can never verify our underlying physical laws once and for all time, because sometime in the future a more accurate experiment may reveal a small but significant difference between the predictions of our theory and the actual observations.
Thus we have finally arrived at the real difference between “truth” in mathematics and in physics. In mathematics, if a conclusion can be deduced as the logical outcome of a chain of reasoning traced back to the original axioms, it is considered true, and is announced to the world as a “theorem”. In physics, there is no such thing as “truth”: the best one can say is that within the bounds of current measurement error, the predicted result is correct.
3.3 Is Near Enough Good Enough?
Here, each of the three integers 3987, 4365 and 4472 is raised to the twelfth power.2 If we try to check the accuracy of this relationship with our calculators, we will need to be inventive, as the individual terms exceed the largest integers (231–1) that can currently be stored in a normal computer. Let us now pose the question: is this equivalence correct, or not?
To a mathematician, the answer is straightforward: the relationship is wrong because it violates a theorem proposed in the margin of a book in 1637 by Pierre de Fermat, and now known as Fermat’s Last Theorem. It took 358 years for a proof of it to be discovered by Andrew Wiles [3]. The theorem states that integer relationships, such as the one above, are not possible for powers greater than 2. A mathematician would not need to carry out any arithmetic to know that this relationship must be wrong.3
However, an actual numerical evaluation would reveal that the difference between the two sides of the equivalence is exceedingly small (about one part in a hundred billion), and quite negligible compared with the measurement errors that are observed in the most accurate experiments in a physics laboratory. To most physicists, the equivalence is therefore “true”, and they would not hesitate to use it, if any of their theories required it. In this sense, near enough is good enough.
This attitude might seem reprehensible, but the aim of physics is to explain nature, as far as measurements allow it. Physical laws are constantly being refined as new experimental information comes to hand. Mind games are best left to mathematicians.
We have already encountered this different mindset in an example, which we discussed in Chap. 2. Newton’s Law of Gravity cannot be solved exactly for a many-body system, such as our solar system, comprising the Sun, Earth, Moon, other planets and rocket ships. However, extremely accurate approximations to the solutions can be obtained, accurate enough to send space-crafts to distant planets. These approximations may not be exact solutions, but they are near enough.
3.4 Faith in Physics
In a letter to Max Born, dated 29th April, 1924, Einstein wrote: “I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist” [4]. Einstein was expressing his opposition to the random, or probabilistic, nature of Quantum Mechanics. We shall discuss Quantum Mechanics in Chap. 5. However, what is relevant here is that Einstein was essentially expressing an article of faith, rather than a scientific argument. Today few physicists share Einstein’s point of view. As we shall see, the experimental evidence in favour of Quantum Mechanic’s weird predictions is just too all-encompassing to ignore.
If someone of the stature of Einstein could adopt such an “unscientific” attitude, which other articles of faith have become embedded in the science of physics without attracting overmuch critical comment? In the remainder of this Chapter we shall try to unearth a few, with the aim of encouraging our readers to seek out more for themselves.
Before beginning, let us recall for a moment a story from the field of mathematics. British mathematician, G. H. Hardy, was a child prodigy who could write numbers up to millions at the age of two years. As an adult, he spent most of his research life at Cambridge University, where he introduced an increased rigour into British mathematics, which at that time, just prior to World War 1, differed from European mathematics in the extent that it regarded rigour as relevant. By rigour we mean a strict adherence to formal logic, where literally nothing is taken for granted.
An example of rigour carried to the extreme is Principia Mathematica (PM) by Russell and Whitehead [5]. Their objective was to set mathematics on a sound formal basis. Their monumental work is hardly bedside reading, and we suspect is more often cited than read. However, one person who did read it, and was inspired by it, was Kurt Gödel, whose work we will discuss in Chap. 4. Another mathematician who realised its importance was G. H. Hardy.
As an example of the lengths to which Russell and Whitehead were prepared to go, we have included a small excerpt from PM in Appendix 3.4. After 362 pages of close mathematical reasoning, the authors have almost reached the stage where they can prove that 1 + 1 = 2. As Whitehead himself remarked: “It requires a very unusual mind to undertake the analysis of the obvious.”
Perhaps the strangest collaboration in the history of mathematics was between G. H. Hardy and Srinivasa Ramanujan. Hardy (left) was a child prodigy, atheist, and a product of the English public school and university system; Ramanujan (right) was a self-taught shipping clerk and a devout Hindu. Images are from Wikimedia Commons (Image 1 by Unknown (Mondadori Publishers) [Public domain], https://commons.wikimedia.org/wiki/File:Godfrey_Hardy_1890s.jpg (accessed 2020/8/31); Image 2 by Konrad Jacobs - Oberwolfach Photo Collection, original location, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=3911526 (accessed 2020/5/12))
Realizing that Ramanujan was no crank, Hardy invited him to Cambridge, and thus began one of the strangest collaborations in mathematics, which Hardy described as “the one romantic incident in my life.” Ramanujan had little interest in the type of formal logic that Hardy regarded as essential. His ideas came to him in flashes of inspiration, or as he claimed, in visions from a Hindu goddess. It is unlikely that this assertion would have impressed Hardy, who was a confirmed atheist.
Not all of Ramanujan’s conjectures4 turned out to be true, but many of them did, and they were ground-breaking in their originality. Hardy and Ramanujan worked together to develop formal proofs for them, an exercise that interested Hardy much more than Ramanujan, who preferred to approach his Goddess for further exciting titbits of information. This unusual collaboration was cut short by the untimely death of Ramanujan from consumption in 1920 at the age of 32. The story of this strange partnership has been narrated by Robert Kanigel [6]. Some of Ramanujan’s highly unorthodox results have inspired much research by later generations of mathematicians, with applications in fields as diverse as String Theory and Crystallography. The reason for recounting this story here is to show how, even in the normally rigorous discipline of mathematics, different points of view can be adopted. Certainly no modern mathematician would feel comfortable with the ad hoc conjectural methodology of Ramanujan, and would not be satisfied until a formal proof of their conjectures had been established. However, how many are willing to go to the lengths of Whitehead and Russell in their search for rigour? This last question becomes even more relevant when we discover, in Chap. 4, that such a quest is doomed from the outset to failure.
3.5 Truth and Beauty
‘Beauty is Truth, Truth Beauty,’ - that is All Ye Know on Earth, and All Ye Need to Know.
The above lines from John Keats’ “Ode to a Grecian Urn” have incited much debate in the world of English literature. Whether the poem can be classed as good or bad is not for us to discuss here. Suffice it to say that on that topic there is a multitude of different opinions.
Terracotta Panathenaic prize amphora ca. 525–500 BCE, attributed to the Kleophrades Painter. On view Metropolitan Museum of Art, New York. Image: Public Domain (https://www.metmuseum.org/art/collection/search/247959 (accessed 2020/9/11))
It may seem incongruous, in a book on physics, to talk about beauty, and yet, as we shall see in the following, beauty and symmetries have played their part in guiding scientific thought, and we shall see that analogies can be drawn between the concepts of what is attractive in the worlds of art and science. In fact, this analogy has been carried so far as to suggest that physics and poetry are but two different paths in the discovery of reality [7].
The first, and possibly the most important, observation is that beauty to one person may well be ugliness to another. Helen of Troy was reputedly the most beautiful woman in the ancient world and her abduction led to the Trojan War. She has been depicted many times over the centuries, and these portraits provide an insight into what the artists (mostly men) at various times considered to be their ideal female form.
Young Woman Braiding Her Hair, 1876 by Auguste Renoir. Image: Open Access, courtesy of NGA, USA (https://www.nga.gov/collection/art-object-page.52207.html)
Facial scarring of men and women is regarded as tragic in the western world. However, in some African tribes, ritual facial scarring is commonplace and considered beautiful in women and showing strong character and loyalty to tradition in men. The practice is now becoming less common as western values become all pervasive.
Music is another branch of art where tastes in beauty vary widely. Classical music may not be appreciated in a household brought up on popular genres, such as rock, country, and jazz. Even within the classical repertoire, there are disparities of style that do not appeal to all. The atonal music of Arnold Schoenberg leaves many repulsed by the strong discords, and it may take considerable exposure before one begins to appreciate the subtleties that lie beneath. Schoenberg is an “acquired taste”, an attribute that he shares with many other “beautiful” things. Spicy food, the bitterness of beer and the dryness of wine are not tastes that the newly weaned infant is likely to enjoy. Rather, that appreciation is brought about by familiarity, born of experience.
What then characterises a beautiful piece of physics, and why should we care? After all, also for scientists “beauty is in the eyes of the beholder”, while science should be by definition objective. Sabine Hossenfelder in her provocative book, “Lost in Math: How Beauty Leads Physics Astray” has polled many modern physicists over what they consider a beautiful theory. She writes: “When asked to judge the promise of a newly invented but untested theory, physicists draw upon the concepts of naturalness, simplicity or elegance, and beauty. These hidden rules are ubiquitous in the foundations of physics. They are invaluable. And in utter conflict with the scientific mandate of objectivity [8]”.
“Naturalness” implies that the free parameters in a theory are not fine-tuned to achieve agreement with experiments or observations. It also includes an aversion to theories which contain very large or very small values for the dimensionless constants (see Sect. 3.7). “Simplicity” will be discussed in the next Section. “Beauty” usually involves underlying symmetries in the mathematical equations. Symmetries allow the prediction of more results from fewer basic assumptions.
The pursuit of beauty is not just a recent interest in physics. The circle, being the most symmetric plane figure, requires only the specification of a central point and a radius to completely identify all of its properties. The early Greek astronomers took it as given that the motion of the planets, Sun and Moon were described by circles, which had for them almost divine properties. Ptolemy wrote: “Our problem is to demonstrate in the case of the five planets, as in the case of the Sun and Moon, all their apparent irregularities as produced by means of regular and circular motions (for these are proper to the nature of divine things which are strangers to disparities and disorders). [9]”. We shall describe the Ptolemaic model of the solar system in more detail in Chap. 10.
This prejudice in favour of the circle persisted even after Kepler had shown that the planetary orbits were, in reality, ellipses. Kepler himself struggled to reconcile the observations made by Tycho Brahe with circular heliocentric orbits, before giving up and accepting that the orbits had to be elliptical. Galileo took no interest in Kepler’s elliptical orbits: he maintained that “only circular motion can naturally suit bodies which are integral parts of the Universe as constituted in the best arrangement” [10].
“If nature were not beautiful, it would not be worth knowing, and life would not be worth living.” – Henri Poincaré;
“I have deep faith that the principles of the Universe will be beautiful and simple.” – Albert Einstein;
“A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data.”– Paul Dirac;
“My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.” – Hermann Weyl.
As we shall see in Part 2 of this book, the pursuit of beauty in physics has led to some spectacular successes. Dirac predicted the existence of a hitherto undiscovered particle, the positron,5 just from the symmetry of his equations. Symmetries in the Standard Model of Fundamental Particles (which we shall discuss in Chap. 9), led to predictions of the existence of two new particles,6 that were subsequently discovered.
As we have seen in the case of Ptolemy’s planetary orbits, being beautiful does not necessarily mean that a theory is correct. Over the years since Kepler, astronomers have by necessity come to appreciate the appeal of ellipses, which once were regarded as not perfect enough to be part of the divine plan. Astronomical observations forced the theorists, in the eighteenth century, to give up their prejudices.
In the fields of Cosmology and Fundamental Particle Physics today, observations and experiments that can seriously test modern theories are difficult and very expensive. As a consequence, there is a lack of observational data to direct today’s physicists away from the theories that they consider beautiful. Without the impetus given by observation and experiment, tomorrow’s physicists may not be encouraged to overcome current prejudices and develop their own concepts of beauty. Progress in these fields may then slow. We shall discuss this issue further in Part 3 of the book.
3.6 A Monk with a Good Razor
Jan Cornelisz Sylvius, the Preacher, circa 1644–1645 by Rembrandt van Rijn, circa 1627–1628. Pen and brown ink on laid paper, National Gallery of America, Washington DC. Image: open access (https://www.nga.gov/collection/art-object-page.45998.html (accessed 2020/9/11))
As assumptions may always be questioned, the fewer assumptions, the more confidence we can have in the conclusions drawn from them. This simple assertion is usually attributed to William of Ockham (c. 1287–c 1347), an English friar and theologian. However, the same, or a similar principle was already used by the ancient Greeks, e.g. Aristotle and Ptolemy. Many other thinkers followed, before and after Ockham, with similar assertions, but either luck, or the empathy of some later philosopher, has granted Ockham the honour of having his name associated with the criterion, which is generally known as Occam’s (or Ockham’s) Razor. It is a rule of thumb, rather than a scientific law. The basic idea is that from among competing hypotheses that explain an observed result, the one that requires fewest assumptions should be selected.
In science, Occam’s razor is used as a practical guide in the development of new theoretical models, but not as a criterion to establish their validity. In fact, it may happen that, later on, new observational or experimental data require us to modify or abandon these models in favour of new ones. It is important to remark that in science there is no place for dogmas, and that every new result must be seen only as one more step in a long journey, rather than as the final word.
Simplicity has of course obvious practical advantages, and also a powerful aesthetic appeal. The main criterion for the acceptability of a given theory is its falsifiability, i.e. the possibility to prove it wrong (see Appendix 3.5). This concept, introduced by Philosopher, Karl Popper, may be difficult to understand and even counterintuitive. However, if it is impossible to prove that a certain theory is wrong, we are left only with the alternative of believing it, or rejecting it altogether. Our decision becomes an act of faith, instead of being based on experimental or observational evidence, in open contradiction with the scientific methodology. Moreover, with fewer assumptions, the burden of falsifying them is lighter.
As an example, let us examine the Bishop of Ussher’s7 assertion that the earth was created at 6 p.m. on Saturday, October 22nd, 4004 BCE in the proleptic Julian calendar,8 near the autumnal equinox. Which time zone this referred to is unknown. Presumably it wasn’t daylight-saving time, as in Genesis the Sun wasn’t created until the fourth day. When presented with evidence of fossils that can be dated to much earlier times than this, some fundamentalists claim that the fossils were also created, along with everything else, in 4004 BCE.
It is impossible to disprove in absolute terms such a thesis. If the fossils, along with all other prehistoric relics, cave paintings and the like, had been created in some kind of celestial scam in order to trick naïve scientists and archaeologists, then the scam cannot be falsified. Ussher’s assertion, by its nature, can always be tailored to explain the observed data. However, according to Popper, such theories are not science.
It is, of course, easy to use Occam’s Razor to counter the old bishop. Scientific theories are much simpler than Ussher’s explanation, involving, as they do, only one Big Bang, compared with the requirement of countless individual acts of creation for each fossil and ancient artefact.
Leprechaun, with his pot of gold. Does every act of magic he performs count as a single assumption when applying Occam’s Razor, or does belief in the leprechaun count as only one assumption? Image from Wikimedia Commons (Image by Croker, T. C. (1862) Fairy Legends and Traditions of the South of Ireland. https://commons.wikimedia.org/wiki/File:Leprechaun_or_Clurichaun.png (accessed 2020/5/12))
We will in the course of this book come across many applications of Occam’s Razor. It lies behind the fundamental desire to reduce physics to as few laws as possible. Perhaps the greatest achievements of 19th Century physics was the unification by James Clerk Maxwell of two separate phenomenologies, electricity and magnetism, into just one: electromagnetism. As a consequence of Maxwell’s work, the number of physical laws required to explain a wide range of physical phenomena was significantly reduced. This success inspired physicists to attempt the unification of the other fundamental forces (weak nuclear, strong nuclear and gravitational) into a Theory of Everything (TOE). As we will see in later Chapters, they were only partially successful.
3.7 O Heaven, Were Man but Constant, He Were Perfect! [11]
Another consequence of Occam’s Razor is the belief that the laws of physics and the fundamental constants they involve are unchanging throughout the breadth of the Universe, that they have remained unchanged throughout its history, and presumably will remain unchanged forever. A word of explanation is necessary here. The laws of physics often include a physical constant, the value of which must be obtained from measurement. An example is Newton’s Law of Gravity, where the strength of the gravitational field is given by G, the so-called Gravitational Constant. Likewise, the speed of light in vacuum, denoted by c, appears in many formulas in any text book of physics.
Both G and c have physical dimensions, which means that their value depends on the system of physical units used to measure them. This is quite arbitrary. For instance, c is normally expressed in metres per second, but could just as well have been given in feet per second, or furlongs per fortnight. Experimentalists in nuclear physics often use the approximate value of one foot per nanosecond for c as a guide to help them lay out detection equipment in their time-of-flight experiments.
However, as we saw in Chap. 2, it is always possible to construct various products and ratios of physical quantities to produce dimensionless parameters. We can carry out this procedure with the fundamental physical constants. For instance, if we consider the ratio of the electron mass to the proton mass, we obtain a dimensionless number that is independent of any arbitrariness arising from the choice of units. This ratio will always have the same value, irrespective of whether we measure the electron and proton masses in kilograms, pounds, or any other units of mass. The value of this ratio has therefore a more fundamental physical meaning than the individual masses.
The very name of these quantities, i.e. constants, reflects the implicit belief that their values do not change throughout the past and future, or over the length and breadth, of the Universe. Likewise, the laws of physics are also believed to be immutable. The laws of Conservation of Momentum and Energy are found to be valid from sub-atomic to cosmological scales. This immutability of physical laws and constants is a credo for which Occam’s Razor provides some sort of justification, i.e. that unchanging laws and constants provide a simpler scenario.“Immediately you would like to know where this number for a coupling comes from: is it related to π or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and we don't know how He pushed his pencil. We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!” [12].
“Since the laws of nature depend on the individual characters of the things constituting nature, as the things change, then consequently the laws will change. Thus the modern evolutionary view of the physical Universe should conceive of the laws of nature as evolving concurrently with the things constituting the environment. Thus the conception of the Universe as evolving subject to fixed eternal laws should be abandoned.”
The speed of light c is ubiquitous in physics; it appears in innumerable formulas in physics textbooks. Over the period 1928 to 1945, the measured speed of light fell steadily by an amount that was outside the bounds of quoted experimental errors. Then in the late nineteen-forties new measurements were made that were in better agreement with the 1928 measurements. Had there perhaps been a cyclic change in c over this time period? Most physicists believe not, and ascribe the result to a self-censorship process by experimenters. Measurements too far from the currently accepted value tend to be treated as statistical flukes and discarded. However, the possibility of a variable speed of light has been raised in alternative models of the early history of the Universe. We shall discuss this issue further in Chap. 11.
The long-term variability of other physical constants, e.g. Planck’s Constant (see Chap. 5) and the electronic charge, has also been suggested. The question of whether the Fine Structure Constant α is varying now, or has changed in the past, is an active research field (see Appendix 3.6). Clearly these investigations have important implications for cosmological theories. However, it will take very convincing observational data to persuade physicists to abandon the assumption of constancy in the physical laws and their associated fundamental constants, partly because of Occam’s Razor, but also because once the laws and constants are allowed to change, anything goes. It is very difficult to put observational constraints on what occurs in another part of the Universe or in the very distant past and future.
If Occam’s Razor is invoked in some areas of physics in a search for simplicity, in others it appears to have been completely disregarded by the specialists. So we have the scenario of multiverses, where at every instant the future timeline of the Universe splits into an infinite number of possibilities. Another theory requires that space–time comprise eleven dimensions, while in Cosmology the presence of Dark Matter and Dark Energy would imply that 95% of the Universe is unknown. As we progress through this book, we will return to these proposals in more detail. One can only imagine what Einstein’s reaction would have been to some of them.