For most of us outside the subject, mathematics is difficult to understand. Even insiders do not always seem to find it completely straightforward. Bertrand Russell, the philosopher and mathematician, said “mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true”.
Russell’s “definition” probably overstates his own incomprehension but it does not begin to scratch the surface of mine. For me the questions of what mathematicians are talking about, or whether it is true, are a long way down the road. I want to know how they decide what can be said, and whether they could have decided to say different things. Does mathematics consist of truths that were already there, that mathematicians have discovered, or is it a series of ingenious inventions that they have fabricated? Surprisingly, not all mathematicians give the same answer to this question.
Most mathematicians are platonists as far as their subject is concerned. They accept the view of Plato, the ancient greek philosopher, who believed that knowledge exists before men discover it. Initially we get only the faintest glimmering of knowledge – Plato described it as like looking at shadows cast by firelight – but we can achieve a clear understanding by the application of reason.
In support of the platonist view David Fowler of the University of Warwick cites the “unreasonable accuracy” of mathematics. By using mathematical theories of quantum mechanics physicists can predict the magnetic properties of the electron. Physical measurements agree with the prediction to an accuracy of one part in 100 million.
The fact that mathematics makes it possible to describe the world so accurately makes it hard to dismiss it as a mere fabrication. Even so, Fowler admits that there are many tantalising difficulties. Not the least of these is the question of how mathematicians could find out things about the real world by making up puzzles about it, rather than by investigating it directly.
“Serious mathematics often grows out of puzzles and serious mathematicians often indulge in them” Fowler says. The rules of the puzzles are well established, and the games are all aimed at trying to work out what are the consequences that follow from the rules. Fowler concedes that it isn’t really possible to say why mathematics should provide such good descriptions of science and nature. “The rules of mathematics have nothing to do with the real world” he says.
Paul Ernest of the University of Exeter is one of the rarer breed who believes that mathematics is fallible. He emphasizes that it consists of a series of games with agreed rules. Just because following the rules leads to surprising and useful results doesn’t change the fact that the rules were invented in the first place he says. He explains the stability of mathematics by the strong traditions within the subject in terms of the problems to be solved and the methods for solving them.
But the stability of the individual areas does not prevent growth of the subject as a whole. “Mathematics is an amazing subject where you can invent new areas” Ernest says. This has always been the case. Many of the new areas have grown in response to external needs. Arithmetic was invented to support taxation in the ancient world, other branches grew up in response to the needs of astronomers, navigators, military gunners, gamblers, insurance companies, and even scientists. In recent decades information systems, missile guidance and cryptography have been significant driving forces. According to Ernest it is the fact that mathematics usually starts off from real world problems that explains its usefulness in solving them.
Fowler concedes that new branches of mathematics can be opened up. But he talks in terms of discovery rather than invention. “It’s like exploring a landscape and suddenly you find another world that you never dreamed existed.” But the new worlds of mathematics always link up with the old ones. This supports the view that they were waiting to be discovered.
John Coates, of the University of Cambridge works on number theory, a large branch of which deals with the properties of integers (whole numbers) and rational numbers (numbers formed by dividing one integer by another). “Many things that are trivially easy to do with irrational numbers (infinitely recurring decimals) are impossible with rational numbers. Integers turn out to have the most interesting properties of all.” Coates says.
Coates has no doubt that number theorists are “uncovering a structure that’s there already.” Number theory links up with many other branches of mathematics implicating them in the preexisting structure – the proof of Fermat’s last theorem depended on linking together many new and esoteric branches of mathematics even though the theorem itself is over three hundred years old, and only mentions integers.
In the end, Fowler suggests that perhaps the most important reason that mathematics are Platonists is that they are too modest. “They can’t beieve that they could possibly make up something so subtle” he says.