Explaining the unreasonable effectiveness of the natural sciences in mathematicsby Marcus Chown / September 1, 2019 / Leave a comment
Galileo was one of the first to realise a profound truth about the Universe: mathematics expresses perfectly the behaviour of the physical world. “Philosophy is written in the grand book (I mean the Universe) which stands continually open to our gaze,” he wrote. “But it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. This book is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it.”
Since the 17th century, mathematics has time and time again demonstrated that Galileo was right—it is indeed the unique language of the Universe. Among its spectacular successes have been the predictions of the existence of radio waves, black holes, antimatter, the Higgs boson and gravitational waves. In 1960, the Austrian physicist Eugene Wigner articulated what many had been thinking since the time of Galileo when he remarked on “the unreasonable effectiveness of mathematics in the natural sciences.”
So why is mathematics so effective in distilling the essence of the world? How is it possible that someone sitting at a desk can write down an arcane mathematical formula that predicts the existence of something previously unsuspected—something that people later discover actually exists in the real world? To put it more bluntly: why does the Universe have a mathematical twin that appears to mimic it in every way? “The astonishing effectiveness of mathematics in physics has enthralled me since I was a schoolboy,” says the former scientist and science writer Graham Farmelo. And now it has prompted him to explore the connection in his book The Universe Speaks in Numbers.
Perhaps the most striking example of mathematics predicting things in the real world was provided by the British physicist Paul Dirac, the subject of Farmelo’s bestselling biography The Strangest Man (2009). The thought processes of most great physicists are essentially like those of normal people (though souped-up). They seek everyday mechanical analogues of the Universe, which they can then describe mathematically.
Famously, this was the method of James Clerk Maxwell, a Scot who in the 1860s created a model of electricity and magnetism in which the Universe was filled with turning, toothed, invisible cogs. Once mathematised, the model became “Maxwell’s equations,” which predicted the existence of a vast array of invisible “electromagnetic waves” that have made possible the ultra-connected world of the 21st century.
Dirac’s thought processes, however, were not like Maxwell’s, or indeed like those of any normal human being. He believed, at least metaphorically, that “God is a mathematician of a very high order,” and that the best way to describe the Universe was to try and concoct a beautiful mathematical equation.
The big problem in the late 1920s was uniting the two towering achievements of early 20th-century physics: Einstein’s 1905 theory of special relativity, which describes things moving close to the speed of light; and quantum theory, which describes the world of atoms and their constituents. In the lightest atom, hydrogen, the solitary electron orbits a proton at only about one per cent of the speed of light. However, in the heaviest atoms, the electric pull from large numbers of protons whirls an electron around at close to the speed of light, making a relativistic description essential. And it was just such a description that Dirac plucked from thin air while sitting at his desk in his spartan rooms in St John’s College, Cambridge, in November 1927.
The Dirac equation is inscribed on a flagstone near the memorial to Isaac Newton in Westmister Abbey. It perfectly describes everything that was then known about the electron. But when Dirac wrote it down he noticed, to his dismay, that the machinery of the equation was duplicated. It appeared to describe not only a negatively charged electron but a particle with the same mass as the electron and a positive electric charge. In August 1932, 6,000 miles away in Pasadena, California, Carl Anderson, completely unaware of Dirac’s prediction, stumbled on the “positron.” It was the first particle of antimatter discovered.
Dirac’s formula, concocted for its self-consistency and beauty, had predicted the existence of a previously unsuspected Universe in which every subatomic particle has an oppositely charged doppelgänger. The Dirac equation is arguably the most productive equation ever devised, with the most wide-ranging and startling consequences.
Einstein was actually a pioneer of Dirac’s mathematical approach. His main motivation for both the special theory of relativity and the general theory of relativity of 1915—also known as Einstein’s theory of gravity—was not any experimental result or observation, but the requirement of mathematical consistency. In particular, he realised that the laws of physics are uniquely determined by the fact that they must appear the same to all observers moving at uniform speed with respect to each other (special relativity) and to all observers no matter what their point of view, or “coordinate system” (general relativity).
The central importance of “mathematical symmetries” in the Universe was recognised in 1918 by the German mathematician Emmy Noether. She discovered that simple “global” symmetries of space and time are actually the wellspring of the great “conservation” laws of physics such as the one that dictates that energy can neither be created nor destroyed.
Other physicists, following her lead in the second half of the century, discovered that nature enforces a more restrictive symmetry, known as “local gauge symmetry.” This led directly to the Standard Model, which describes all of the fundamental building blocks of matter plus the non-gravitational forces that glue them together, and which is widely regarded as the crowning achievement of 350 years of physics.
Farmelo covers all of this with his characteristic clarity, having carried out comprehensive research and personally interviewed many of today’s front-rank physicists. They include Edward Witten, often compared with Einstein, and Nima Arkani-Hamed, one of the world’s most energetic and imaginative physicists. The result is a book which, with its bird’s-eye view of the panorama of modern physics, is as authoritative as it is fascinating.
Einstein, though a pioneer of the idea that mathematics might illuminate the deepest laws of physics, made a fatal wrong turn, refusing to incorporate quantum theory into his worldview with the dismissal that “God does not play dice with the Universe.” Einstein’s failure inaugurated a period when physics and mathematics, according to Farmelo, went their separate ways. Now, thankfully, he says, they are back together again.
One thing that has been largely responsible for the rapprochement has been “string theory.” String theory is the only mathematical description of the world of fundamental physics that unifies both quantum theory and special relativity and which, as a spin-off, predicts the existence of gravity. One drawback, which critics rightly point to, is that the theory, which views the ultimate building blocks of matter not as tiny point-like billiard balls but tiny vibrating “strings” of so-called mass energy, works only in 10 dimensions of space-time. The problem is that we appear to live in a Universe with only three dimensions of space and one of time. Oh, and there is the little matter that string theory has yet to make any testable prediction about the real world. The jury is still out on whether string theory is a mirage or a tantalising glimpse of the fabled “theory of everything” that Stephen Hawking boldly predicted.
String theory is the ultimate embodiment of Galileo’s idea that mathematics is the language of nature. However, not everyone believes that it is. Stephen Wolfram, the creator of the computer language Mathematica, thinks that Wigner’s “unreasonable effectiveness” is an illusion. Imagine, Wolfram says, a drunk man looking for his dropped car keys on a city street at midnight. The man searches under a street light not because that is the best place to look, but because it is the only place illuminated. Similarly, Wolfram thinks that mathematics illuminates only the bits of the Universe that mathematics is able to illuminate.
Physicists are perfectly happy to accept that they cannot describe many things in the Universe. However, they would say this is simply because our current mathematical tools are inadequate and, in the future, maybe in the 22nd century, we will obtain better tools. Wolfram, however, believes we will never obtain such tools because most of what the Universe is doing is not mathematical (he believes it is computing things—shades of The Hitchhiker’s Guide to the Galaxy).
Farmelo does not cover Wolfram’s views. However, he points out an extraordinary and under-appreciated thing that appears to bolster the view that the connection between mathematics and physics is more than an illusion. The extraordinary thing is that mathematics is not only effective in physics but that physics is also effective in mathematics. When, in the 1970s, the British mathematician Michael Atiyah saw the mathematical potential of local gauge theories, says Farmelo, he inspired a generation of mathematicians to work with physicists on a joint enterprise that has been dubbed “physical mathematics.” Since then, whole new fields of mathematics are being opened up, with gauge theory and string theory spawning insights into topology, the study of geometrical shapes.
The fact that the mathematics-physics connection is a two-way street is remarkable and something I have not seen emphasised in any other popular science book. In a way, it compounds the puzzle of the unreasonable effectiveness of mathematics in the natural sciences. Now we also have to explain the unreasonable effectiveness of the natural sciences in mathematics. The mystery is even bigger and even more intriguing than Wigner suspected.
I, for one, am not disappointed that the central mystery of the unreasonable effectiveness of mathematics remains a mystery—and, in fact, is now an even deeper mystery. What excites me about science is not that we know so much but that there is still so much more to find out. But although the mystery of the mathematics-physics connection is unsolved, Farmelo reminds us that the immense practical consequences of this symbiosis should not be overlooked. “Physicists have not one but two ways of improving their fundamental understanding of how nature works by collecting data from experiments and by discovering the mathematics that best describes the underlying order of the cosmos,” he says. “The Universe is whispering secrets to us, in stereo.”
Graham Farmelo’s The Universe Speaks in Numbers (£20) is published by Faber & Faber.