A mathematical universe is born

Solving a decades-old puzzle meant tearing up and rebuilding the most basic ideas in mathematics

Jacob Aron

Whole numbers, addition and multiplication are among the first things schoolchildren learn, but a new mathematical proof shows that even the world's best minds have plenty more to learn about these seemingly simple concepts.

Shinichi Mochizuki of Kyoto University in Japan has torn up these most fundamental of mathematical ideas and reconstructed them as never before. The result is a fiendishly complicated proof for the 27-year-old "ABC conjecture" -- and an alternative mathematical universe that should prise open many other outstanding enigmas.

Not only that, Mochizuki's work also offers an alternative way to prove Fermat's last theorem, a long-standing problem that became one of the most famous results in the history of mathematics when it was proved in 1993 (see "Fermat's last theorem made easy", below).

The ABC conjecture starts with the most basic equation in algebra, adding two whole numbers, or integers, to get another: a + b = c. First posed in 1985 by Joseph Oesterlé and David Masser, it places constraints on the interactions of the prime factors of these numbers, primes being the indivisible building blocks that can be multiplied together to produce all integers, Take 81 + 64 = 145, which breaks down into the prime building blocks 3x3x3x3 + 2x2x2x2x2x2 = 5x29. Simplified, the conjecture says that the large amount of smaller primes on the equation's left-hand side is always balanced by a small amount of larger primes on the right -- the addition restricts the multiplication, and vice versa.

"To learn something new about addition and multiplication at this late date is quite startling."

"The ABC conjecture in some sense exposes the relationship between addition and multiplication," says Jordan Ellenberg of the University of Wisconsin-Madison. "To learn something really new about them at this late date is quite startling."

Though rumours of Mochizuki's proof started spreading on mathematics blogs earlier this year, it was only last week that he posted a series of papers on his website detailing what he calls "inter-universal geometry", one of which claims to prove the ABC conjecture. Now mathematicians are attempting to decipher its dense logic, which spreads over 500 pages.

So far the responses are cautious, but positive. "It will be fabulously exciting if it pans out, though experience suggests that that's quite a big if," wrote University of Cambridge mathematician Timothy Gowers on Google+.

"It is going to be a while before people have a clear idea of what Mochizuki has done," Ellenberg told New Scientist. "Looking at it. you feel a bit like you might be reading a paper from the future, or from outer space," he wrote on his blog.

Mochizuki's reasoning is alien even to other mathematicians because it probes deep philosophical questions about the foundations of mathematics, such as what we really mean by a number, says Minhyong Kim at the University of Oxford. The early 20th century saw a crisis emerge as mathematicians realised they actually had no formal way to define a number -- we can talk about "three apples" or "three squares", but what exactly is the mathematical object we call "3"? No one could say.

Eventually numbers were redefined in terms of sets, rigorously specified collections of objects. Mathematicians now know that the true essence of the number zero is a set which contains no objects -- the empty set -- while the number 1 is a set which contains one empty set. From there, it is possible to derive the rest of the integers.

But this was not the end of the story, says Kim. "People are aware that many natural mathematical constructions might not really fall into the universe of sets."

Rather than using sets, Mochizuki has figured out how to translate fundamental mathematical ideas into objects that only exist in new, conceptual universes. This allowed hi to "deform" basic whole numbers and push their innate relationships -- such as multiplication and addition -- to the limit. "He is literally taking apart coventional objects in terrible ways and reconstructing them in new universes." says Kim.

These insights led him to a proof of the ABC conjecture. "How he manages to come back to the usual universe in a way that yields concrete consequences for number theory, I really have no idea as yet," says Kim.

"A verified proof would set off a chain reaction, in one swoop proving many other open problems"

Because of its fundamental nature, a verified proof of the ABC conjecture would set of a chain reaction, in one swoop proving many other open problems that relate to it, and deepening our understanding of the relationships between integers, fractions, decimals, primes and more.

Ellenberg compares proving the conjecture to discovering the Higgs boson, which it is hoped will reveal a path to novel physics. But while the Higgs emerged from the particle detritus of a machine specifically designed to find it, Mochizuki's methods are completely unexpected, providing new tools for mathematical exploration.

Verifying the proof will itself be an endeavour, as mathematicians must comb through Mochizuki's work line by line to check that the logic of each step holds true. Ellenberg expects the process to take at least a year, though any potential mistakes may be discovered sooner. That is what happened with Andrew Wiles's proof of Fermat's last theorem in 1993. He rectified the error and published the true, proof in 1995.

Crank claims of solving long-standing problems with esoteric methods are common in mathematics, but Mochizuki, who could not be reached for comment, has a credible history. "He has a terrific track record," says Andrew Granville at the University of Montreal in Canada Kim agrees: "This is what makes good mathematicians take his claims very seriously, in spite of the unusual nature of the machinery he has developed."

If Mochizuki's proof is borne out by peer review, it would transform the fundamental mathematics he spent decades studying -- and his career. "My guess is that he would become a megastar," wrote Gowers on Google+. "Mochizuki's method, if it is found acceptable to the mathematical community, is likely to yield a completely new way of thinking about numbers," says Kim.

Fermat's Last Theorem Made Easy a + b = c. This basic equation sits at the very heart of the fiendish ABC conjecture -- now potentially solved (see main story). It links the conjecture to many other mathematical problems, including Fermat's last theorem. In the 17th century, Pierre de Fermat declared there were no possible solutions to the related equation, an + bn = cn, if n is 3 or more. Maddeningly, he did not write down a proof. It was not until 1993 that Andrew Wiles provided one using modern mathematics that Fermat could not possibly have known. Though many doubt Fermat had a credible proof to back up his statement, the ABC conjecture provides an alternative route to the theorem, and could even help illuminate Fermat's line of thought. The two puzzles are linked because if the ABC conjecture is true, it implies that there are no solutions to an + bn = cn, if n is sufficiently large. That does not solve Fermat's theorem outright but it vastly shortens the task: it turns the infinite problem of checking every n, in order to prove Fermat true, into a finite one. Depending on the exact formulation of the ABC conjecture, it could be that only n = 3, 4 and 5 must be checked. "Fermat's last theorem is that easy!" jokes Andrew Granville of the University of Montreal in Canada. There seems no way that Fermat could have proved ABC -- no one even posed the conjecture formally until 1985 -- but perhaps he worked out the "sufficiently large n" part, says Minhyong Kim at the University of Oxford. This could have led him to put forward his own theorem, albeit an unproved one. Others are uncomfortable with speculation about the ABC conjecture and Fermat. "There is zero chance that it has anything to do with what Fermat had in mind," says Jordan Ellenberg at the University of Wisconsin-Madison.