The Magicians

by Marcus Chown

  In the 1930s, Eugene Wigner wrote a famous essay entitled ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. ‘The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious,’ he wrote. ‘And there is no rational explanation for it.’4

  Einstein echoed Wigner’s observation. ‘How can it be’, he asked, ‘that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?’5 Famously, he also remarked that ‘The most incomprehensible thing about the world is that it is comprehensible.’ And by ‘comprehensible’, he implicitly meant comprehensible by mathematics.

  ‘Our work is a delightful game,’ said Murray Gell-Mann, who won the Nobel Prize for proposing the existence of ‘quarks’, the ultimate building blocks of matter. ‘I am frequently astonished that it so often results in correct predictions of experimental results. How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature?’6

  So why is mathematics so unreasonably effective in the natural sciences? Why is the universe mathematical? The first thing to say is that not everyone believes these are valid questions. According to Stephen Wolfram, the billionaire creator of the symbolic computer language Mathematica, the universe is not mathematical; it simply looks that way.

  Wolfram points out that most of what is happening in the universe, such as the turbulence in the atmosphere and biology, is far too complex to be encapsulated by mathematical physics. Most physicists would argue that this is because, at present, we lack mathematical tools of sufficient sophistication, but that this is only a temporary situation and that one day we will obtain such tools. Wolfram begs to differ. He thinks the reason we cannot describe complex phenomena like turbulence and biology with mathematics is because it is impossible.

  According to Wolfram, we are in the position of a drunk man hunting for his dropped car keys on a street at midnight. He looks in the pool of light under a street light, for no other reason than that is the only place he can reasonably look. Similarly, claims Wolfram, we use mathematics to describe the only part of the universe that it is describable by mathematics.

  Bertrand Russell, the twentieth-century British mathematician and philosopher, would have agreed with Wolfram. ‘Physics is mathematical’, he said, ‘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.’ Percy Bridgman, an American physicist, said something similar: ‘It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention.’7 Arthur Eddington put it this way: ‘The mathematics is not there till we put it there.’

  If, as Wolfram claims, the universe is not fundamentally mathematical, it is without doubt doing something far from random. There is a regularity. There are rules more basic than mathematical equations. They are encapsulated in simple computer programs, and it is such programs that Wolfram thinks are orchestrating everything we see in the universe. These programs are ‘recursive’, which means their output is continually fed back in as their input, like a snake eating its own tail. In the early 1980s, Wolfram had played around with such simple programs on one of the first personal computers and discovered that, occasionally, they can generate infinite complexity and novelty. It was such a striking finding that it had caused him to wonder whether this might be the secret of how nature creates a rose, a newborn baby or a galaxy.

  Generally, the only way to discover the consequences of such a program is to run it and find out. According to Wolfram, this is true of most of what is going on in the universe, which he terms ‘computationally irreducible’. However, for a small subset of programs, it is possible to discover their outcomes in advance of running them. Wolfram terms these ‘computationally reducible’. The special shortcut that enables the prediction of their outcomes is none other than mathematical physics.

  Most physicists disagree with Wolfram and believe that the universe is indeed mathematical. So the question remains: Why is mathematics so effective in the natural sciences? Why does the central magic of science work? There have been many attempts over the years to answer these questions. A remarkable one has come from the Swedish–American physicist Max Tegmark, and it involves multiple universes.

  In recent years, evidence has been mounting from many different directions that our universe is not the only one. For instance, the fact that the universe was born 13.82 billion years ago means that we can see only those galaxies whose light has taken less than 13.82 billion years to reach the Earth. The ‘observable’ universe is therefore bounded by a ‘horizon’ rather like the membrane of a soap bubble; beyond the horizon are galaxies whose light has not yet got here. In other words, there are other domains – possibly an infinite number of them – like our observable universe, but with different stars and galaxies. The ensemble of such universes is called the ‘multiverse’.

  In addition to this rather trivial multiverse, physicists have reason to believe there may be other universes with different numbers of dimensions, different laws of physics, and so on. No one knows yet how all these multiverse ideas fit together. It is an emerging ‘paradigm’.

  Tegmark takes this idea to its logical conclusion and suggests that there may be an ‘ultimate ensemble’ of universes, in which every piece of mathematics is actualised. So, for instance, there is a universe that contains only flat space, or ‘Euclidean’, geometry, another that contains only arithmetic, and so on. Nothing happens in most of these universes because the rules are too simple to create anything of interest. Only in universes with mathematics as complex as the ‘theory of everything’ that is believed to orchestrate our universe is it possible for the emergence of interesting things, like stars and planets and life. The reason we find ourselves in such a universe, according to the topsy-turvy logic of the ‘anthropic principle’, is that in any simpler universe we would not have arisen to notice the fact.

  Tegmark is claiming that mathematics is unreasonably effective in the physical sciences for a completely trivial reason: because mathematics is physics. ‘Our successful theories aren’t mathematics approximating physics, but mathematics approximating mathematics,’ he says.8 Heinrich Hertz had a similar thought a century and a half earlier: ‘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.’

  Many physicists would argue that a multiverse containing a possible infinity of universes, most of which are devoid of anything interesting, is a high price to pay for an explanation of Wigner’s remark, but many would accept that, for some mysterious reason, the universe appears to be a manifestation of an underlying mathematical structure. Powerful evidence that this is the case is outlined by science writer Graham Farmelo in his book, The Universe Speaks in Numbers. Not only does mathematics provide insights into physics, says Farmelo, but physics provides insights into mathematics; it is a two-way street. The most striking example of this is ‘string theory’, which views the fundamental building blocks of matter not as point-like particles but as strings of mass-energy vibrating in ten-dimensional space–time. Although the theory is yet to contribute any testable predictions for physics, it has already opened up whole new vistas of enquiry for pure mathematics.

  Tegmark and Wolfram’s explanations for the unreasonable effectiveness of mathematics are not the only ones. The late American physicist Victor Stenger was fond of pointing out that the physics we have discovered is actually no more than the physics of ‘nothing’.

  Recall that in 1918, Emmy Noether showed that the great conservation laws of physics are merely a consequence of deep symmetries. So, for instance, the law of conservation of energy, which states that energy can neither be created nor destroyed, is a consequence of
time-translation symmetry – the fact that the outcome of an experiment does not depend on when it is carried out. The conservation of momentum is the consequence of space-translation symmetry – the fact that an experimental result does not depend on its location in space, whether the experiment is done in London or in New York. ‘If mathematics is the language of nature, symmetry is its syntax,’ says Gian Francesco Giudice.9 What is striking about these symmetries, Stenger pointed out, is that they are also the symmetries of an entirely empty universe. In a featureless void, after all, every time is exactly like every other time and every location is precisely like every other location.

  In addition to such global symmetries, our universe maintains local symmetries. And as explained earlier, the fundamental forces of nature exist merely to ensure that such local ‘gauge invariance’ is enforced everywhere in space and time.10 These local symmetries, like the universe’s global symmetries, are also the symmetries of empty space.

  There are, of course, other fundamental laws of nature besides those that are merely the consequence of deep symmetries. However, the British chemist Peter Atkins points out that these, too, also arise from nothing – or at least, they are not the substantive prescriptions that at first sight they appear to be.

  Take the law which dictates the path taken by a ray of light through a medium such as glass, better known as the law of refraction. It turns out that there is another, entirely equivalent way of determining the trajectory taken by light: it follows the path that takes the least time. A little thought reveals that the only way a light beam can possibly do this is by trying all possible routes through, say, a piece of glass, to determine the quickest path. It may seem mad, but this is pretty much what light does.

  The critical thing you need to know is that light is a wave. Now, imagine that the light travelling through a block of glass takes all possible paths between points A and B. The wavelength of light is small, which means that light rays following neighbouring paths differ substantially in the location of their peaks and troughs. In fact, for each path there is a neighbouring one for which the peaks of one wave coincide with the troughs of the other, and vice versa; consequently, they cancel each other out. The only path that does not suffer such ‘destructive interference’ is the path of shortest time.

  In a sense, the real law of refraction is that there is no law; the light follows every possible route and the passive phenomenon of interference culls all but the shortest-time path. According to Atkins, nature’s law of refraction is nothing but a law of laziness, or ‘indolence’.

  The path followed by a ray of light may not seem to have any wider significance, but it does. The reason is that in quantum theory – our very best description of the microscopic world of atoms and their constituents – the behaviour of the building blocks of matter is described by a wave function. According to the Schrödinger equation, this spreads through space, and where the wave is big, or has a large amplitude, there is a high probability of finding a particle such as an electron, and where it is small there is a low chance.

  In this ‘many histories’ interpretation of quantum theory, which was devised by Richard Feynman, a particle travelling between points A and B tries every conceivable path. And just as in the light example, one trajectory is picked out by interference. Rather than being the one that takes the least time, it is the one that takes the least ‘action’, but the principle is the same.* And just as in the example of the light ray, nature’s law is nothing but a law of indolence.

  So not only are many of the laws of physics the same as the laws of nothing, as Stenger maintained; those that remain are also nothing – or born of indolence – as Atkins claims. ‘Nothing is extraordinarily fruitful,’ says Atkins. ‘Within the infinite compass of nothing lies potentially everything, but it is an everything lurking wholly unrealised.’11

  Even though we live in a universe whose laws are arguably the same as they would be if the universe was an empty void, there remains one sticky question: Why, rather than living in a universe of nothing, do we live in a universe of organised nothing? The answer, of course, is that nobody knows.

  As for the central magic of science, what progress have we made in understanding it? Recall that Paul Murdin and Louise Webster discovered Cygnus X-1, the first black hole candidate in the Milky Way, in 1971. The existence of such an entity had been predicted in 1916 by Karl Schwarzschild, while he was on the Eastern Front, dying of an auto-immune disease that covered his skin in ugly, painful blisters. Murdin, like every other scientist who has ever confirmed a scientific prediction, expressed amazement at his discovery. ‘The surprising thing is that black holes turn out to be real objects,’ says Murdin. ‘Incredibly, they actually exist!’ The fact remains that physicists are just as gobsmacked as they were in Urban Le Verrier’s day about the unreasonable effectiveness of mathematics and the incredible predictive power of science. The central magic remains as magical, and as inexplicable, as ever.

  Notes

  1 The First Three Minutes: A Modern View of the Origin of the Universe by Steven Weinberg (Basic Books, New York, 1993).

  2 The Assayer (Il Saggiatore) by Galileo Galilei (1623).

  3 The Cosmic Code: Quantum Physics as the Language of Nature by Heinz Pagels (Dover Publications, New York, 2012).

  4 ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’ by Eugene Wigner, in The Collected Works of Eugene Wigner, Volume VI, edited by Jagdish Mehra (Springer Verlag, Berlin, 1995).

  5 ‘Geometry and Experience’, an expanded form of an address by Albert Einstein to the Prussian Academy of Sciences in Berlin, 27 January 1921.

  6 ‘Symmetry and Currents in Particle Physics’ by Murray Gell-Mann (Nobel Lecture, 11 December 1969: https://www.nobelprize.org/prizes/physics/1969/ceremony-speech/).

  7 Mathematics is built from building blocks which mathematicians call ‘formal systems’. There are a large number of such systems, such as ‘Boolean algebra’ and ‘group theory’. A formal system consists of a set of givens, or ‘axioms’, and the consequences, or ‘theorems’, that can be deduced from them by applying the rules of logic. For instance, the axioms of Euclidean geometry include the statement that ‘parallel lines never meet’, while the theorems that can be deduced from the axioms include such statements as ‘the internal angles of a triangle always add up to 180 degrees’.

  8 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality by Max Tegmark (Penguin, London, 2015).

  9 A Zeptospace Odyssey: A Journey into the Physics of the LHC by Gian Francesco Giudice (Oxford University Press, Oxford, 2010).

  10 The Logic of Modern Physics by Percy Bridgman (Macmillan, New York, 1927).

  11 Conjuring the Universe: The Origins of the Laws of Nature by Peter Atkins (Oxford University Press, Oxford, 2018).

  * In physics, action is a quantity that involves both the potential energy and the energy of motion of a particle.

  Further Reading

  1: Map of the invisible world

  The Neptune File: A Story of Astronomical Rivalry and the Pioneers of Planet Hunting by Tom Standage (Penguin, London, 2000).

  The Hunt for Vulcan: How Albert Einstein Destroyed a Planet and Deciphered the Universe by Thomas Levenson (Random House, London, 2015).

  The Hunt for Planet X: New Worlds and the Fate of Pluto by Govert Schilling (Copernicus, Berlin, 2008).

  The Ascent of Gravity: The Quest to Understand the Force that Explains Everything by Marcus Chown (Weidenfeld & Nicolson, London, 2017).

  2: Voices in the sky

  Faraday, Maxwell and the Electromagnetic Field: How Two Men Revolutionized Physics by Nancy Forbes and Basil Mahon (Prometheus Books, New York, 2014).

  The Man Who Changed Everything: The Life of James Clerk Maxwell by Basil Mahon (John Wiley, Chichester, 2003).

  Electric Universe: How Electricity Switched on the Modern World by David Bodanis (Abacus, London, 2005).

  Faraday: The Life by James Hamilton (Har
perCollins, London, 2002).

  3: Mirror, mirror on the wall

  The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius by Graham Farmelo (Faber & Faber, London, 2010).

  Antimatter by Frank Close (Oxford University Press, Oxford, 2007).

  It Must Be Beautiful: The Great Equations of Modern Science, edited by Graham Farmelo (Granta Books, London, 2002).

  4: Goldilocks universe

  Home Is Where the Wind Blows: Chapters from a Cosmologist’s Life by Fred Hoyle (University Science Books, California, 1994).

  Fred Hoyle: A Life in Science by Simon Mitton (Aurum, London, 2005).

  The Magic Furnace: The Search for the Origins of Atoms by Marcus Chown (Vintage, London, 2000).

  5: Ghost busters

  No Time to Be Brief: A Scientific Biography of Wolfgang Pauli by Charles Enz (Oxford University Press, Oxford, 2010).

  Inward Bound: Of Matter and Forces in the Physical World by Abraham Pais (Oxford University Press, Oxford, 1988).

  Neutrino by Frank Close (Oxford University Press, Oxford, 2010).

  8: The god of small things

  Smashing Physics: Inside the World’s Biggest Experiment by Jon Butterworth (Headline, London, 2014).

  The Infinity Puzzle: The Personalities, Politics, and Extraordinary Science Behind the Higgs Boson by Frank Close (Oxford University Press, Oxford, 2013).

  A Zeptospace Odyssey: A Journey into the Physics of the LHC by Gian Francesco Giudice (Oxford University Press, Oxford, 2010).

  In Search of the Ultimate Building Blocks by Gerard’t Hooft (Cambridge University Press, Cambridge, 1998).

  Massive: The Hunt for the God Particle by Ian Sample (Virgin Books, London, 2010).