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21 May 2013

The third culture: The power and glory of mathematics

In 1959 C P Snow delivered a celebrated lecture in which he decried the man-made gulf between the arts and the sciences. Yet there is one subject that already spans the divide and is unjustly neglected — mathematics.

By Ian Stewart

In his 1959 Rede Lecture, the scientist and novelist C P Snow memorably deplored what he saw as a gulf between the two cultures of arts and sciences. The lecture provoked the prominent literary critic F R Leavis into responding that there was only one culture: his. Public ignorance of basic science didn’t matter, but ignorance of Shakespeare did. This missed the main point of Snow’s argument, which was that even if that were the prevailing view among the literati, it ought not to be.

Snow’s lecture was based in part on an article he had written for the New Statesman in 1956. He was continuing a tradition that goes right back to the magazine’s first editorial, which adopted a broad cultural stance: “We shall deal with all current political, social, religious, and intellectual questions . . . We shall strive to face and examine social and political issues in the same spirit in which the chemist or the biologist faces and examines her test-tubes or his specimens, ignoring none of the factors, seeking to demonstrate no preconceived proposition, but trying only to find out and spread abroad the truth whatever it may turn out to be.”

Perhaps not wishing to alarm potential readers too much, the editorial expanded on its scientific metaphor: “Social problems may not be – indeed, are not – susceptible of scientific analysis in the popular acceptation of that term, since human beings are not to be weighed in balances nor measured with micrometers . . .” It was a reasonable view then, but times have changed. Today very few social problems are not tackled by measuring aspects of human attitudes, behaviour or bodily form. Consider the current concerns about an obesity epidemic, backed up by extensive statistics in which people are literally weighed in – on balances.

The NS editor clearly had an inkling that such changes were imminent and continued: “. . . unless there can be applied to [social problems] something at least of the detachment of the scientific spirit, they will never be satisfactorily solved. The cultivation of such a spirit and its deliberate application to matters of current controversy is the task which the New Statesman has set for itself.” It was a worthy task, pursued with aplomb and considerable success; it is a task not yet finished, and if anything it is now even more vital than it was a century ago.

The cultural divide between art and science has narrowed perceptibly since Snow delivered his lecture and the issues have been thrashed out extensively, so we now have a better understanding of their nature. However, it might be more accurate to say that the divide has been spanned by a number of bridges, rather than made smaller. Some of these bridges are broad and strong, but many are flimsy and liable to topple at any moment. We still educate would-be scientists in a very different manner from the way we educate those destined for the arts. But at least both streams of intellectual activity now recognise the other’s existence without suggesting that it makes no worthwhile contribution to intellectual life or human society.

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There is, however, a third culture, one that I have occupied for most of my life. It overlaps art and science but is contained in neither, not even in their union. It resembles both, yet is distinctive enough that it is difficult to assign it to either. It is, I think, the only subject in which a university degree may, in some institutions, be awarded either as Bachelor of Arts or Bachelor of Science – for exactly the same work. The subject is mathematics, and the mathematical subculture is gravely misunderstood by most members of the public, many prominent scientists, nearly all artists, the vast majority of politicians and almost every bureaucrat on the planet.

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Leavis wasn’t against science; he just felt that it played no part in culture. You didn’t need to understand it to consider yourself educated, but it had practical value. Mathematics often gets worse treatment, with people parading their ignorance as if it were a virtue. I suspect this is a defence mechanism. When meeting a mathematician, the stock reply “I was never any good at maths at school” acts as a protection against having to engage with the subject. It is often followed by, “You must be very clever,” which is equally misconceived. Mathematics has caused huge changes to our world and its influence underpins virtually every aspect of our society. It is intensely creative, advancing more rapidly now than at any other time in its history. Yet most people are unaware of what it is, what it does or why it is worth doing. Worse, they imagine they know the answers to all of these questions and it never occurs to them that they may be wrong.

That is not entirely their fault. For mathematics to be useful in today’s society, it has to be invisible. If anyone using a mobile phone had been required first to take a PhD in mathematics, the device would never have appealed to a large enough market to make it worth anyone’s while to manu­facture. However, without a significant number of engineers who know some very advanced and apparently esoteric mathematics, mobile phones wouldn’t work. Their signals would be so degraded that it would be impossible to deduce what the person on the other end was saying, and the capacity of the network would be so small that we would have to book calls ahead of time – as was the case for international phone calls when Snow gave his lecture.

It is not entirely clear why mathematics occupies such a curious position, teetering dangerously across the gulf between science and art while being neither. The reasons date back at least to the Renaissance, when the seven liberal arts were the trivium (grammar, logic and rhetoric) and the quadrivium (arithmetic, geometry, music and astronomy). Two of these, arithmetic and geometry, are manifestly mathematical, and so was astronomy at the time. Indeed, in 7th-century India, mathematics was a branch of astronomy. In modern times, logic has also become a part of mathematics, and it formed the basis of Euclid’s notion of proof. On the other hand, music is clearly an art, though one with mathematical features such as the structure of musical scales. Grammar and rhetoric point towards literature.

Later, the application of mathematics to astronomy gained a much stronger role with­in the physical sciences, thanks to Johannes Kepler, Galileo, Isaac Newton and others. Newton’s Philosophiæ Naturalis Principia Mathematica revealed the “system of the world” through mathematical laws. Even if mathematics were not itself a science, it quickly became an indispensable tool in the physical sciences. The main reason for not considering it to be a genuine science is that mathematical truths are established by logical proof, not by experiment.

Mathematics also lacks the freedom of expression so central to art, because it is heavily constrained by the need to be logically consistent. Innovative mathematics has the freedom to revolutionise points of view and create concepts; it can change what mathematicians think is important, but it cannot overturn established truths. It can modify them to suit changed contexts: Pythagoras’s theorem is false in non-Euclidean geometry. Even so, it remains true in Euclidean geometry.

Not entirely a science, and even less plausibly an art, mathematics is rightly counted among the sciences for the purposes of government funding and educational administration – BAs and BScs notwithstanding. If you are forced to choose between the two, it has to be a science. The good news is that, as a science, mathematics attracts more funding than it would as an art; the bad news is that, it being the Cinderella science, confined to menial tasks, its funding is pathetic compared to that of particle physics, radio astronomy and genetics.

This is frankly scandalous. A study carried out for the US government found that the mathematical sciences – pure and applied mathematics, statistics and computer science – contributed $37trn to the American economy in the first decade of this century. The 2012 report Measuring the Economic Benefits of Mathematical Science Research in the UK, by Deloitte, concluded that 2.8 million people were employed in mathematical science occupations in 2012, contributing £208bn (gross value added) to the economy that year. The main areas were industrial research and development, computer services, aerospace, pharmaceuticals and architectural activities and consulting – a list that will surprise many. Yet mathematics is the most poorly funded of the sciences.

Agreed, finding the Higgs boson (if that is what the new particle actually is) is good basic science, contributing a missing piece to “fundamental” physics, but the particle is not fundamental in the sense that the remainder of science cannot proceed without it. On the contrary, few areas of science would experience any change, Higgs or no Higgs, because they are based on phenomena that are already firmly established. It may be reassuring to find that these phenomena are consistent with the existence of the Higgs, but if they weren’t, it’s the particle physics that would be wrong.

If 10 per cent of the $9bn spent on Cern’s Large Hadron Collider had been allo­cated to research in the mathematical sciences instead, the benefits to society would have been far greater and would have occurred more rapidly. The development of the next generation of supercomputers, a project estimated to cost about $1bn, is struggling to find funding. The result would be a machine with applications throughout the sciences – for example, in climate change and in the design of new materials and alternative energy sources – and it would dominate future computer technology.

In the long run, the hunt for the Higgs is likely to have important applications, but right now the main spin-off is new technology invented to make particle detectors. Such benefits could have been obtained more quickly through cheaper and more focused projects. This is not to decry par­ticle physics, or even to argue for a reduction in its funding, but governments need to recognise the value of the mathematical sciences and increase funding for them significantly. Mathematics is not as sexy as a machine five miles across, but it is extra­ordinarily versatile and its benefits are extensive and diverse.

In this digital age, hardly anything functions without mathematical input. To take one example out of hundreds, the global positioning system, used for car satnavs, relies on several entirely different areas of mathematics – as do engineering, materials science and business acumen. Number theory is used to generate sequences of pseudo-random numbers, which underpin the method used to compare the times that radio signals take to travel to your car from different orbiting satellites.

Trigonometry is needed to work out, from those times, the location of your car. Mathematical physics, in the form of special and general relativity, is needed to stop the earth’s motion and its gravitational field from causing the calculated position to drift alarmingly from the real one. Newton’s laws of gravity and motion are used to compute the launch trajectories of the satellites. Mathematics is also now becoming essen­tial in the life sciences. Recent biological research has shown that two long-standing mathematical models of development – the growth of embryos and the form of living organisms – are substantially correct, and that the orthodox theory of “positional information”, determined by a chemical gradient, is probably wrong. One of these mathematical theories goes back to 1952, when Alan Turing wrote down equations to model the formation of animal markings such as spots and stripes. The other goes back to 1976, when Jonathan Cooke and Christopher Zeeman proposed a “clock and wavefront” model for the formation of segments and the spinal column.

Mathematical models are also playing a significant role in the study of cancer, supplementing the conventional approach through genetics and biochemistry by paying attention to the physical mechanisms by which cancer cells grow and spread, and allowing doctors to compare various therapeutic strategies.

The routine uses of mathematics, all vital to modern society, run into the thousands. They include oil prospecting, criminology, plant-breeding, climate modelling, weather forecasting, stock-market trading, aircraft design, fuel efficiency, sustainable energy, CDs, DVDs, graphic effects in films, Google’s search engine, digital cameras, smartphones, three-dimensional TV, computer chips, software verification, foot-and-mouth outbreaks, HIV, heart disease, pacemaker design, machine translation of foreign languages and Hawk-Eye’s ability to predict LBW verdicts in cricket.

Artists of all persuasions are taking inspiration from science and mathematics. The sculptor Peter Randall-Page makes explicit use of geometry and the mathematics of natural form. The architect Charles Jencks has introduced an artform (or revived an old one): large-scale earthworks. He pioneered this idea in his Garden of Cosmic Speculation, motivated by strange attractors in chaos theory. Jencks’s recent Northumberlandia is another, spectacular example, inspired by mythology and the human form.

However, the nature and importance of mathematics do not rest solely on its practical uses or its artistic merit. It has an intrinsic intellectual interest and an idiosyncratic beauty. What makes it so hard to grasp the subject’s nature is that it combines many disparate elements in one ever-growing, ever-changing body of knowledge. It is practical, arcane, precise, obscure, abstruse, rooted in nature, rooted in the human imagination; its history spans 4,000 years, and what was discovered long ago is often just as valid and important today. It is absolutely huge, increasing by a conservative one million pages a year, and it is all one thing. Its internal connections link entirely different sub-disciplines in an intricate, dynamically changing web, so that yesterday’s dead end may suddenly become today’s essential technique.

It is mathematics that has brought social problems within the purview of science. In 1835 the Belgian Adolphe Quetelet published his analyses of data on crime, the divorce rate, suicide, births, deaths, human height and weight – variables that no one expected to conform to any mathematical law, because their underlying causes were complex and involved human choices or accidents of nature. It seemed ridiculous to think that the emotional torment that drives someone to suicide could be reduced to a formula. And that is true if you want to predict exactly who will kill him or herself. But when Quetelet concentrated on statistical questions, such as the proportion of suicides in various groups of people, he saw a common pattern, the bell curve. His data demonstrated that people en masse behave more predictably than do individuals.

Statistics is just one way in which mathematics affects society and human culture. Its most important application is probably to medicine: it is a rare medical study that does not have its significance assessed using statistics. The global financial system also requires a heavy dose of statistics – often abused, mind you, as we saw in 2008 when the banking crisis struck. However, re-engineering the world’s financial sector to make it more stable will undoubtedly require a greatly improved understanding of how the system works, and that will rest on new mathematics.

Most of us view the third culture as something we were dragged through at school, didn’t understand, didn’t like and never use. It exists only to produce the next generation of mathematics teachers. It is pointless and uncreative and nothing has happened in it for hundreds of years because basically it has all been done already. By way of proof, every question in the maths textbook has an answer at the back. However, this is nonsense. The myths survive because few people realise just how deeply mathematics has penetrated the foundations of society – just how much we rely on the subject for almost everything we do.

I often think it’s a pity that the word “mathematics” is used to describe what is taught at school level. Doing so leads people to assume that mathematics is what they have experienced, which largely consists of repetitive drill on artificial problems. Why learn how to add or multiply numbers when you can grab a calculator? That would be a reasonable argument if the sole point of teaching mathematics were to teach us how to add and multiply, but that is not quite the point. We are taught how to add or multiply numbers so that we understand what those operations mean. Unless we internalise them so that they become a reflex, we will not be able to proceed to higher levels of the subject.

Now, many of us won’t do that. So why are children who dislike mathematics required to study it anyway? Why not let them go off and play football? Let me turn the question round. Suppose I were to approach a parent and tell them: “Your child is clearly no good at mathematics, and trying to teach her will be a waste of time. So we’re not going to. That, of course, does mean that she will never be able to become a computer programmer, an airline pilot, a surgeon, an ophthalmic nurse, an electrician, a currency trader, an engineer, a medical statistician, or a plumber. But as we already know, without even letting her try, that she isn’t capable of succeeding, why bother?” I doubt that the parent would agree. They would probably be outraged. I’m not suggesting that mathematics is needed for every worthwhile area of human activity. But it is a vital part of every citizen’s mental toolkit and it would be grossly unfair to rule out half of the interesting jobs on the planet for the child before she reaches the age of ten.

I say that deliberately, because the belief that the only job you can get with a degree in mathematics is that of a mathematics teacher is a myth, and a very silly one. If it were true, governments worldwide would have taken mathematics out of the curriculum long ago as a waste of money. The truth is almost exactly the opposite: far too few mathematics graduates go into teaching, because many better-paid jobs with better conditions are available to anyone with a mathematics degree. (This may be one reason why children dislike the subject: too many of its teachers don’t understand it, and they communicate their unease to the class. This is not to criticise the teachers, who are doing their best.) The range of professions available to a mathematics graduate is vast because mathematics is so broadly applicable. Surveys have shown that students with mathematics degrees earn more, on average, than those who have studied almost any other subject.

There is, naturally, more to life than money, but if money is what you want, mathematics is a good way to go. If you’re not that mercenary, mathematics offers advantages of a more intellectual kind. The idea that the subject was mined out centuries ago is little better than a sick joke; the belief that because the answers are in the book everything in the subject must already be known makes no sense. No one thinks that their school physics text, or biology text, or even English literature text, contains everything that has ever been discovered. A textbook is an introduction to the simplest parts of the topic, not a comprehensive encyclopaedia. So, why do so many people think that mathematics texts contain the entire subject? Dame Kathleen Ollerenshaw, a celebrated mathematician, politician and educator, hints at a possible reason in her autobiography, To Talk of Many Things. She relates that when at school, she informed a classmate that she wanted to study mathematics at university in order to create more of it. “Why bother?” replied the friend. “There’s too much of it already.”

The young Kathleen understood something that never occurs to most of us: that mathematics, far from being dull, boring and repetitious, is creative. It may even be one of the most creative areas of human activity. But the creative side emerges only when you look at how new mathematics comes into being. Most school mathematics is like practising musical scales. Mathematical research is like composing new tunes, sonatas, even symphonies. However, you can listen to great music without being a composer. It is much harder to appreciate the beauties of mathematics without doing any. Mathematics is not a spectator sport – or at least, it wasn’t until recently.

Painting a picture brings an immediate feeling of having created something. I, too, paint, when I find time, mainly landscapes, and when I’ve finished I sometimes hang the result on the wall: Machu Picchu, a Galapagos dolphin leaping out of the water, moai on Easter Island. But it is a slightly trivial form of creativity. If I used the sig­nificance of the work as a criterion, I’d hang up my forthcoming paper on neural network models of visual illusions instead. It was created within a set of very limiting technical constraints, and instead of being whatever I imagined, or derived from whatever image I happened to take as my starting point, it represents an attempt to solve a real-world puzzle by thinking about it very hard. If the idea is correct – which is moot – it will long outlive my painting of the dolphin. Great art is another matter, nothing trivial about that. But most people who say that painting is creative but mathematics is not are not referring to great art.

Bertrand Russell, writing in the NS in 1913, believed that society’s tenden­cy to disparage science was “not the fault of science itself, but the fault of the spirit in which science is taught. If its full possibilities were realised by those who teach it, I believe that its capacity of producing those habits of mind which constitute the highest mental excellence would be at least as great as that of literature . . .” This is even more so for mathematics, a subject to which he contributed in his epic analysis of its logical foundations with Alfred North Whitehead in Principia Mathematica, a conscious echo of Newton. This experience may have led Russell to his celebrated characterisation of mathemat­ical beauty: “Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” I think he overstated his case a little.

I said earlier that mathematics is not a spectator sport, but I must now qualify that: the development of high-quality computer graphics has made some aspects of mathematical beauty visible for all to see. This is one reason why Benoît Mandelbrot’s fractals – multicoloured, psychedelic structures – now decorate walls all over the planet. It is also a reason why his ideas were decried by some mathematicians as trivial hype; however, the topic has now proved its worth in numerous areas of applied science and in mathematics itself, so that judgement turns out to have been shallow. At home next to my landscapes are two elegant images of symmetric fractal attractors, a contribution to chaos theory made by my colleagues Michael Field and Marty Golubitsky. I have hung their research on the wall. Its logical beauty goes deeper, and is more akin to Russell’s austerity, but even a superficial glance shows a remarkable mix of chaos and order. The images contain clues to the hidden mathematical structure that makes them possible.

Public perception notwithstanding, we are living in a golden age of mathematics. Discoveries are being made at an accelerating rate; ancient problems are crumbling under the onslaught. Computers sometimes help, but the crucial work is done by those “little grey cells”, as Hercule Poirot would say. The subject’s inclination towards abstraction in the 1960s, which outsiders often criticised as pointless navel-gazing, has turned out to be exactly what its practitioners claimed at the time: an essential move towards generality and clarity of thought that would open up new vistas for the entire subject.

Many of the greatest open problems – Fermat’s Last Theorem in number theory, Kepler’s conjecture about the most efficient way to pack spheres together and Poincaré’s characterisation of a basic aspect of three-dimensional topology – have been solved in recent years by applying the abstract point of view. Outsiders may find that way of thinking uncomfortable, but it has proved its worth many times over. In the 1960s, physics departments instructed their students to avoid taking mathematics courses because they would ruin their minds. Today, the UK’s Biotechnology and Biological Sciences Research Council is about to require all PhD students in biology to take a specially designed course in mathematics.

In his NS article, Russell also wrote: “A life devoted to science is . . . a happy life, and its happiness is derived from the very best sources that are open to dwellers on this troubled and passionate planet.” The same goes for a life devoted to the arts, and one to mathematics. The diversity of human thinking is to be celebrated, not restricted to just one of the three cultures on specious grounds of alleged superiority. As resources dwindle, populations explode and our impact on the planet becomes increasingly dire, we will need every trick we can think of to survive. The oft-neglected third culture of mathematics will be vital here. It is time to release it from its supporting role and allow it to take centre stage.

Which leads me to sign off with another quotation from that first editorial in the New Statesman: “We have put our cards on the table. We have said enough to show our readers where we stand . . .”

Ian Stewart is an emeritus professor of mathematics at Warwick University and the author of “The Great Mathematical Problems” (Profile Books, £15.99)