Is mathematics the root of reality?

• New Scientist 27 February 2013 by Brian Greene
• Magazine issue 2906.

http://www.newscientist.com/article/mg21729062.400-physics-crunch-is-mathematics-the-root-of-reality.html?full=true

What makes us so sure that mathematics can reveal nature's deepest workings?

Read more: "Crunch time for physics: What's next?"

In the late 1800s, when James Clerk Maxwell realised that light was an electromagnetic wave, his equations showed that light's speed should be about 300,000 kilometres per second. This was close to the value experimenters had measured, but Maxwell's equations left a nagging loose end: 300,000 kilometres per second relative to what? At first, scientists pursued the makeshift resolution that an invisible substance permeating space, the "aether", provided this unseen standard of rest.

It was Einstein who in the early 20th century argued that scientists needed to take Maxwell's equations more seriously. If Maxwell's equations did not refer to a standard of rest, then there was no need for a standard of rest. Light's speed, Einstein forcefully declared, is 300,000 kilometres per second relative to anything. The details are of historical interest, but I'm describing this episode for a larger point: everyone had access to Maxwell's mathematics, but it took the genius of Einstein to embrace it fully. His assumption of light's absolute speed allowed him to break through first to the special theory of relativity – overturning centuries of thought regarding space, time, matter and energy – and eventually to the general theory of relativity, the theory of gravity that is still the basis for our working model of the cosmos.

The story is a prime example of what the Nobel laureate Steven Weinbergmeant when he wrote: "Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough." Weinberg was referring to another great breakthrough in cosmology, the prediction by Ralph Alpher, Robert Herman and George Gamow of the existence of the cosmic microwave background radiation, the afterglow of the big bang. This prediction is a direct consequence of general relativity combined with basic thermodynamics. But it rose to prominence only after being discovered theoretically twice, a dozen years apart, and then being observed through a benevolent act of serendipity.

To be sure, Weinberg's remark has to be applied with care. Although his desk has played host to an inordinate amount of mathematics that has proved relevant to the real world, far from every equation with which we theorists tinker rises to that level. In the absence of compelling experimental results, deciding what mathematics should be taken seriously is as much art as it is science.

Einstein was a master of that art. In the decade after his formulation of special relativity in 1905, he became familiar with vast areas of mathematics that most physicists knew little or nothing about. As he groped towards general relativity's final equations, Einstein displayed a rare skill in moulding these mathematical constructs with the firm hand of physical intuition. When he received the news that observations of the 1919 solar eclipse confirmed general relativity's prediction that star light should travel along curved paths, he noted that had the results been different, he "would have been sorry for the dear Lord, since the theory is correct".

I'm sure that convincing data contravening general relativity would have changed Einstein's tune, but the remark captures well how a set of mathematical equations, through their sleek internal logic, their intrinsic beauty and their potential for wide-ranging applicability, can seemingly radiate reality. Centuries of discovery have made abundantly evident the capacity of mathematics to reveal secreted truths about the workings of the world; monumental upheavals in physics have emerged time and again from vigorously following the lead of mathematics.

Nevertheless, there was a limit to how far Einstein was willing to follow his own mathematics. He did not take the general theory of relativity "seriously enough" to believe its prediction of black holes, or of an expanding universe. Others embraced Einstein's equations more fully than he, and their achievements have set the course of cosmological understanding for nearly a century. Einstein instead in the last 20 years or so of his life threw himself into mathematical investigations, passionately striving for the prized achievement of a unified theory of physics. Looking back, one cannot help but conclude that during these years he was too heavily guided – some might say blinded – by the thicket of equations with which he was constantly surrounded. Even Einstein sometimes made the wrong decision regarding which equations to take seriously and which to not.

Quantum mechanics provides another case study of this dilemma. For decades after Erwin Schrödinger wrote down his equation for how quantum waves evolve in 1926, it was viewed as relevant only to the domain of small things: molecules, atoms and particles. But in 1957, Hugh Everett echoed Einstein's charge of a half century earlier: take the mathematics seriously. Everett argued that Schrödinger's equation should apply to everything because all things material, regardless of size, are made from molecules, atoms and subatomic particles that evolve according to probabilistic rules. Applying this logic revealed that it is not just experiments that evolve in this way, but experimenters, too. This led Everett to his idea of a quantum "multiverse" in which all possible outcomes are realised in a vast array of parallel worlds.

More than 50 years later, we still do not know if his approach is right. But by taking the mathematics of quantum theory seriously – fully seriously – he may have had one of the most profound revelations of scientific exploration. The multiverse in various forms has since become a pervasive feature of much mathematics that purports to offer us a deeper understanding of reality. In its furthermost incarnation, the "ultimate multiverse", every possible universe allowed by mathematics corresponds to a real universe. Taken to this extreme, mathematics is reality.

If some or all of the mathematics that has compelled us to think about parallel worlds proves relevant to reality, Einstein's famous query – whether the universe has the properties it does simply because no other universe is possible – would have a definitive answer: no. Our universe is not the only one possible. Its properties could have been different, and indeed the properties of other member universes may well be different. If so, seeking a fundamental explanation for why certain things are the way they are would be pointless. Statistical likelihood or plain happenstance would be firmly inserted in our understanding of a cosmos that would be profoundly vast.

I don't know if this is how things will turn out. No one does. But it is only through fearless engagement that we can learn our limits. Only through rational pursuit of theories, even those that whisk us into strange and unfamiliar domains – by taking the mathematics seriously – do we stand a chance of revealing the hidden expanses of reality.

This article appeared in print under the headline "Roots of reality"

Brian Greene is a theoretical physicist at Columbia University in New York. This article is adapted from his book The Hidden Reality (Allen Lane, 2011)