when Albert Einstein finally completed his general theory of relativity in 1916, he looked down at the equations and discovered an unexpected message: the universe is expanding.
Einstein didn't believe the physical universe could shrink or grow, so he ignored what the equations were telling him. Thirteen years later, Edwin Hubble found clear evidence of the universe's expansion. Einstein had missed the opportunity to make the most dramatic scientific prediction in history.
How did Einstein's equations "know" that the universe was expanding when he did not? If mathematics is nothing more than a language we use to describe the world, an invention of the human brain, how can it possibly churn out anything beyond what we put in? "It is difficult to avoid the impression that a miracle confronts us here," wrote physicist Eugene Wigner in his classic 1960 paper "The unreasonable effectiveness of mathematics in the natural sciences"(Communications on Pure and Applied Mathematics, vol 13, p 1).
The prescience of mathematics seems no less miraculous today. At the Large Hadron Collider at CERN, near Geneva, Switzerland, physicists recently observed the fingerprints of a particle that was arguably discovered 48 years ago lurking in the equations of particle physics.
How is it possible that mathematics "knows" about Higgs particles or any other feature of physical reality? "Maybe it's because math is reality," says physicist Brian Greene of Columbia University, New York. Perhaps if we dig deep enough, we would find that physical objects like tables and chairs are ultimately not made of particles or strings, but of numbers.
"These are very difficult issues," says philosopher of science James Ladyman of the University of Bristol, UK, "but it might be less misleading to say that the universe is made of maths than to say it is made of matter."
Difficult indeed. What does it mean to say that the universe is "made of mathematics"? An obvious starting point is to ask what mathematics is made of. The late physicist John Wheeler said that the "basis of all mathematics is 0 = 0". All mathematical structures can be derived from something called "the empty set", the set that contains no elements. Say this set corresponds to zero; you can then define the number 1 as the set that contains only the empty set, 2 as the set containing the sets corresponding to 0 and 1, and so on. Keep nesting the nothingness like invisible Russian dolls and eventually all of mathematics appears. Mathematician Ian Stewart of the University of Warwick, UK, calls this "the dreadful secret of mathematics: it's all based on nothing" (New Scientist, 19 November 2011, p 44). Reality may come down to mathematics, but mathematics comes down to nothing at all.
That may be the ultimate clue to existence - after all, a universe made of nothing doesn't require an explanation. Indeed, mathematical structures don't seem to require a physical origin at all. "A dodecahedron was never created," says Max Tegmark of the Massachusetts Institute of Technology. "To be created, something first has to not exist in space or time and then exist." A dodecahedron doesn't exist in space or time at all, he says - it exists independently of them. "Space and time themselves are contained within larger mathematical structures," he adds. These structures just exist; they can't be created or destroyed.
That raises a big question: why is the universe only made of some of the available mathematics? "There's a lot of math out there," Greene says. "Today only a tiny sliver of it has a realisation in the physical world. Pull any math book off the shelf and most of the equations in it don't correspond to any physical object or physical process."
It is true that seemingly arcane and unphysical mathematics does, sometimes, turn out to correspond to the real world. Imaginary numbers, for instance, were once considered totally deserving of their name, but are now used to describe the behaviour of elementary particles; non-Euclidean geometry eventually showed up as gravity. Even so, these phenomena represent a tiny slice of all the mathematics out there.
Not so fast, says Tegmark. "I believe that physical existence and mathematical existence are the same, so any structure that exists mathematically is also real," he says.
So what about the mathematics our universe doesn't use? "Other mathematical structures correspond to other universes," Tegmark says. He calls this the "level 4 multiverse", and it is far stranger than the multiverses that cosmologists often discuss. Their common-or-garden multiverses are governed by the same basic mathematical rules as our universe, but Tegmark's level 4 multiverse operates with completely different mathematics.
All of this sounds bizarre, but the hypothesis that physical reality is fundamentally mathematical has passed every test. "If physics hits a roadblock at which point it turns out that it's impossible to proceed, we might find that nature can't be captured mathematically," Tegmark says. "But it's really remarkable that that hasn't happened. Galileo said that the book of nature was written in the language of mathematics - and that was 400 years ago."
If reality isn't, at bottom, mathematics, what is it? "Maybe someday we'll encounter an alien civilisation and we'll show them what we've discovered about the universe," Greene says. "They'll say, 'Ah, math. We tried that. It only takes you so far. Here's the real thing.' What would that be? It's hard to imagine. Our understanding of fundamental reality is at an early stage."