‌Chapter 1

The Maths Behind the World

Even stranger things have happened; and perhaps the strangest of all is the marvel that mathematics should be possible to a race akin to the apes.

– Eric T. Bell, The Development of Mathematics

Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.

– Bertrand Russell

In terms of intellectual ability, Homo sapiens hasn’t changed much, if at all, over the past 100,000 years. Put children from the time when woolly rhinos and mastodons still roamed the Earth into a present-day school and they would develop just as well as typical twenty-first century youngsters. Their brains would assimilate arithmetic, geometry, and algebra. And, if they were so inclined, there’d be nothing to stop them delving deeper into the subject and someday perhaps becoming professors of maths at Cambridge or Harvard.

Our neural apparatus evolved the potential to do advanced calculations, and understand such things as set theory and differential geometry, long before it was ever applied in this way. In fact, it seems a bit of a mystery why we have this innate talent for higher mathematics when it has no obvious survival value. At the same time, the reason our species emerged and endured is because it had an edge over its rivals in terms of intelligence and an ability to think logically, plan ahead, and ask ‘what if?’ Lacking other survival skills, such as speed and strength, our ancestors were forced to rely on their cunning and foresight. A capacity for logical thought became our one great super-power, and from that, in time, flowed our ability to communicate in a complex way, to symbolise, and to make rational sense of the world around us.

Like all animals, we effectively do a lot of difficult maths on the fly. The simple act of catching a ball (or avoiding predators or hunting a prey) involves solving multiple equations simultaneously at high speed. Try programming a robot to do the same thing and the complexity of calculations involved becomes clear. But the great strength of humans was their ability to move from the concrete to the abstract – to analyse situations, to ask if/then questions, to plan ahead.

The dawn of agriculture brought the need to track the seasons accurately, and the coming of trade and settled communities meant that transactions had to be carried out and accounts kept. For both these practical purposes, calendars and business transactions, some kind of reckoning had to be developed, and so elementary maths made its first appearance. One of the regions where it sprang up was the Middle East. Archaeologists have unearthed Sumerian clay trading tokens dating back to about 8,000 bc, which show that these people dealt with representations of number. But it seems that, at this early period, they didn’t treat the concept as being separate from the thing being counted. For example, they had different shaped tokens for different items, such as sheep or jars of oil. When a lot of tokens had to be exchanged between parties, the tokens were sealed inside containers called bullae, which had to be broken open to check the contents. Over time, markings began to appear on the bullae to indicate how many tokens there were within. The symbolic representations then evolved into a written number system, while tokens became generalised for counting any kind of object and eventually morphed into an early form of coinage. Along the way, the concept of number became abstracted from the type of object being counted, so that, for example, five was five whether it referred to five goats or five loaves of bread.

The Egyptians had a good understanding of practical mathematics and put this to effect in the construction of the Pyramid of Khafre at Giza, shown here together with the Sphinx.

The connection between maths and everyday reality seems strong at this stage. Counting and record keeping are practical tools of the farmer and the merchant, and if these methods do the job who cares about the philosophy behind it all? Simple arithmetic looks well rooted in the world ‘out there’: one sheep plus one sheep is two sheep, two sheep plus two sheep is four sheep. Nothing could be more straightforward. But look more closely and we see that already something a bit strange has happened. In saying ‘one sheep and one sheep’ there’s the assumption that the sheep are identical or, at least, for the purposes of counting, that any differences don’t matter. But no two sheep are alike. What we’ve done is to abstract a perceived quality to do with the sheep – their ‘oneness’, or apartness – and then operate on this quality with another abstraction, which we call addition. That’s a big step. In practice, adding one sheep and one sheep might mean putting them together in the same field. But, also in practice, the sheep are different and, digging a little deeper, what we call ‘sheep’ – like anything else in the world – isn’t really separate from the rest of the universe. On top of this, there’s the slightly disturbing fact that what we take to be objects (such as sheep) ‘out there’ are constructions in our brains built up from signals that enter our senses. Even if we grant that a sheep has some external reality, physics tells us that it’s a hugely complicated, temporary assemblage of subatomic particles that’s in constant flux. Yet, somehow, in counting sheep we’re able to ignore this monumental complexity or, rather, in everyday life, not even be aware of it.

Of all subjects, mathematics is the most precise and immutable. Science and other fields of human endeavour are, at best, approximations to some ideal, and are always changing and evolving over time. As the German mathematician Hermann Hankel pointed out: ‘In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure.’ From the outset, this difference between maths and every other discipline is inevitable because maths starts with the mind extracting what it recognises as being most fundamental and constant among the messages it receives via the senses. This leads to the concepts of natural numbers, as a way of measuring quantity, and of addition and subtraction as basic ways of combining quantities. Oneness, twoness, threeness, and so on, are seen as common features of collections of things, whatever those things happen to be and however different individuals of the same type of thing happen to be. So, the fact that maths has this eternal, adamantine quality to it is assured from the start – and is its greatest strength.

Mathematics exists. Of that there’s no doubt. Pythagoras’ theorem, for instance, is somehow part of our reality. But where does it exist when it’s not being used or being instantiated in some material form, and where did it exist many thousands of years ago, before anyone had thought about it? Platonists believe that mathematical objects, such as numbers, geometric shapes, and the relationships between them, exist independently of us, and our thoughts and language, and the physical universe. Quite what sort of ethereal realm they inhabit isn’t specified, but it’s a common assumption that they’re somehow ‘out there’. Most mathematicians, it’s probably fair to say, subscribe to this school of thought and therefore also to the belief that maths is discovered rather than invented. Most, too, probably don’t care very much for philosophising and are happy just to get on with doing maths, in the same way that the majority of physicists, working in the lab or solving theoretical problems, don’t worry a lot about metaphysics. Still, the ultimate nature of things – in this case, of mathematical things – is interesting, even if we never arrive at a final answer. The Prussian mathematician and logician Leopold Kronecker thought that only whole numbers were given, or in his words: ‘God made the integers, all the rest is the work of man.’ The English astrophysicist Arthur Eddington went further and said: ‘The mathematics is not there till we put it there.’ The debate about whether mathematics is invented or discovered, or is perhaps some combination of both, arising from a synergy of mind and matter, will no doubt rumble on and, in the end, may have no simple answer.

One fact is clear: if a piece of maths has been proven to be true, it will remain true for all time. There’s no matter of opinion about it, or subjective influence. ‘I like mathematics,’ remarked Bertrand Russell, ‘because it is not human and has nothing particular to do with this planet or with the whole accidental universe.’ David Hilbert voiced something similar: ‘Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.’ This impersonal, universal quality of maths is its greatest strength, yet doesn’t, to the trained eye, detract from its aesthetic appeal. ‘Beauty is the first test: there is no permanent place in the world for ugly mathematics,’ remarked the English mathematician, G. H. Hardy. The same sentiment, but from the field of theoretical physics was expressed by Paul Dirac: ‘It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power.’

The flip side to the universality of maths, however, is that it can seem cold and sterile, devoid of passion and feeling. As a result we may find that although intelligent beings on other worlds share the same mathematics as us, it isn’t the best way to communicate with them about a lot of the things that matter to us. ‘Many people suggest using mathematics to talk to the aliens,’ commented SETI (Search for ExtraTerrestrial Intelligence) researcher Seth Shostak. In fact the Dutch mathematician Hans Freudenthal developed an entire language (Lincos) based on this idea. ‘But,’ said Shostak, ‘my personal opinion is that mathematics may be a hard way to describe ideas like love or democracy.’

The ultimate goal of scientists, certainly of physicists, is to reduce what they observe in the world to a mathematical description. Cosmologists, particle physicists, and the like are never happier than when they’ve measured and quantified things and then found a relationship between the quantities. The idea that the universe is mathematical at its core has ancient roots, stretching back at least as far as the Pythagoreans. Galileo saw the world as a ‘grand book’ written in the language of mathematics, and, much more recently, in 1960, the Hungarian-American physicist and mathematician Eugene Wigner wrote a paper called ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’.

We don’t see numbers directly in the real world, so it isn’t immediately obvious that maths is all around us. But we do see shapes – the near-spherical shape of planets and stars, the curved path of objects when thrown or in orbit, the symmetry of snowflakes, and so on – and these can be described by relationships between numbers. Other patterns, translatable into maths, emerge from the way electricity or magnetism behaves, galaxies rotate, and electrons operate within the confines of atoms. These patterns, and the equations describing them, underpin individual events and seem to represent deep, timeless truths underlying the changing complexity in which we find ourselves. The German physicist Heinrich Hertz, who first conclusively proved the existence of electromagnetic waves, remarked: ‘One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.’

It’s unquestionably true that the bedrock of modern science is mathematical in nature. But that doesn’t necessarily mean that reality itself is fundamentally mathematical. Ever since the time of Galileo, science has separated the subjective from the objective, or measurable, and focused on the latter. It’s done its best to evict anything to do with the observer and pay attention only to what it assumes lies beyond the interfering influence of the brain and senses. The way modern science has developed almost guarantees that it will be mathematical in nature. But this leaves much that science has trouble dealing with – most obviously, consciousness. It may be that someday we’ll have a good, comprehensive model of how the brain works, in terms of memory, visual processing, and so forth. But why we also have an inner experience, a feeling of ‘what it is like to be’, remains – and may always remain – outside the field of conventional science and, by extension, of mathematics.

Why has the human brain evolved to be so extraordinarily good at a subject – mathematics – that it doesn’t need for survival?

On the one hand, Platonists believe maths to be a land that already exists, awaiting our exploration of it. On the other, there are those who insist that we invent mathematics as we go along to suit our purposes. Both positions have weaknesses. Platonists struggle to explain where things like pi might be outside the physical universe or the intelligent mind. Non-Platonists have a hard time denying the fact that, for example, planets would continue to orbit the Sun in ellipses whether we do the maths or not. A third school of mathematical philosophy occupies a middle ground between the other two, by pointing out that, in describing the real world, maths is not as successful as it’s sometimes made out to be. Yes, equations are useful in telling us how to navigate a spacecraft to the Moon or Mars, or design a new aircraft, or predict the weather several days in advance. But these equations are mere approximations to the reality of what they’re intended to describe, and, moreover, they apply to just a small portion of all the things going on around us. In touting the success of mathematics, the realist would say, we downplay the vast majority of phenomena that are too complex or poorly understood to capture in mathematical form, or that, by their very nature are irreducible to this kind of analysis.

Is it possible that the universe isn’t, in reality, mathematical? After all, space and the objects it contains don’t directly present anything mathematical to us. We humans rationalise and make approximations in order to model aspects of the universe. In doing so we find mathematics extremely useful in enabling us to understand it. That doesn’t necessarily imply that maths is anything other than a convenience of our making. But if mathematics isn’t present in the universe to start with, how is it that we’re able to invent it in order to put it to such use?

Mathematics is broadly divided into two areas: pure and applied. Pure maths is maths for maths’ sake. Applied mathematicians put their subject to work on real-world problems. But often, developments in pure maths, with seemingly no bearing on anything tangible, have turned out later to be surprisingly useful to scientists and engineers. In 1843, the Irish mathematician William Hamilton hatched the idea of quaternions – four-dimensional generalisations of ordinary numbers of no practical interest at the time but which, more than a century later, have turned out to be an effective tool in robotics and computer graphics and games. A question first tackled by Johannes Kepler in 1611, about the most efficient way to pack spheres in three-dimensional space, has been applied to the efficient transmission of information over noisy channels. The purest mathematical discipline, number theory, much of which was thought to have little practical value, has led to important breakthroughs in the development of secure ciphers. And the new geometry pioneered by Bernhard Riemann that dealt with curved surfaces proved ideal for the formulation of Einstein’s general theory of relativity – a new theory of gravity – more than 50 years later.

In July 1915, one of the greatest scientists of all time met one of the greatest mathematicians of the age, when Einstein paid a visit to David Hilbert at the University of Göttingen. The following December, both published, almost simultaneously, the equations that described the gravitational field of Einstein’s general theory. But whereas the equations themselves were the goal for Einstein, they were what Hilbert hoped would be a stepping-stone towards an even grander scheme. Hilbert’s passion, the driving force behind much of his work, was a search for fundamental principles, or axioms, that might underlie all of mathematics. Part of this quest, as he saw it, was to find a minimum set of axioms from which he could deduce not only the equations of Einstein’s general theory but any other theory in physics as well. Kurt Gödel, with his incompleteness theorems, undermined faith in the notion that maths might have the answers to all questions. But we remain uncertain as to what extent the world in which we live is truly mathematical or just mathematical in appearance.

Whole swathes of mathematics may never be put to use, other than to help open up yet more avenues of pure research. On the other hand, for all we know, it may be that much of pure maths is enacted, in unexpected ways, in the physical universe – or, if not this universe, then in others that might exist throughout what cosmologists suspect is a multiverse of incomprehensible scale. Perhaps everything that is mathematically true and valid is represented somewhere, sometime, somehow in the reality in which we are embedded. For now there is the journey to keep us occupied: the weird and wonderful adventure of the human mind as it explores further the frontiers of number, space, and reason.

In the chapters that follow we’ll take a deep dive into subjects that are both bizarre and astonishing and yet, at the same time, have very real connections with the world we know. True, some of the maths may seem esoteric, fanciful, and even pointless, like some strange and convoluted game of the imagination. But, at its core, mathematics is a practical affair, rooted in commerce, agriculture, and architecture. Although it’s developed in ways that our ancestors could never have dreamed about, still those links with our everyday lives remain at its heart.