Something doesn’t add up (Image: Steve Allen/Brand X Pictures/Getty)

Mathematicians are facing a stark choice – embrace monstrous infinite entities or admit the basic rules of arithmetic are broken

IF YOU were forced to learn long division at school, you might have had cause to curse whoever invented arithmetic. A wearisome whirl of divisors and dividends, of bringing the next digit down and multiplying by the number you first thought of, it almost always went wrong somewhere. And all the while you were plagued by that subversive thought- provided you were at school when such things existed- that any sensible person would just use a calculator.

Well, here’s an even more subversive thought: are the rules of arithmetic, the basic logical premises underlying things like long division, unsound? Implausible, you might think. After all, human error aside, our number system delivers pretty reliable results. Yet the closer mathematicians peer beneath the hood of arithmetic, the more they are becoming convinced that something about numbers doesn’t quite add up. The motor might be still running, but some essential parts seem to be missing- and we’re not sure where to find the spares.

From the 11-dimensional geometry of superstrings to the subtleties of game theory, mathematicians investigate many strange and exotic things. But the system of natural numbers- 0, 1, 2, 3, 4 and so on ad infinitum- and the arithmetical rules used to manipulate them retain an exalted status as mathematics’ oldest and most fundamental tool.

Thinkers such as Euclid around 300 BC and Diophantus of Alexandria in the 3rd century AD were already probing the deeper reaches of number theory. It …