Euler, who created much of the mathematics used today, had never heard of the notion of mathematical rigour: he played happily with unruly divergent sums, was almost never led astray by his magnificent intuition, and made discovery after discovery.
Cantor, who developed a theory of the infinite in order to understand God better, remained unperturbed by the paradoxes he discovered along the way, unlike the other mathematicians of the day. He steadfastly insisted that the essence of mathematics resides in its freedom to imagine and create.
Then there was Ramanujan, who never comprehended the idea of proof. He said that an equation is of value only if it expresses one of God's thoughts, and believed that his ideas were communicated to him by the goddess Namagiri.
So what of our own, unheroic, age? Would Euler, Cantor and Ramanujan be welcome now? Definitely not. As William Byers points out in this courageous book, mathematics today is obsessed with rigour, and this actually suppresses creativity. Take a look at the fictional but humorously accurate portrait that Byers paints of a professor teaching a class. As Byers tells it, the students learn not to ask questions because, when they do, all they get in reply are the same formulae repeated again, more slowly, slightly louder.
What went wrong? Well, it started around the end of the 19th century with David Hilbert's vision of complete formalism, of proofs so thorough a computer could check them. It was a vision widely propagated by the French Bourbaki school of mathematics, which, strangely enough, preferred a rigid, Prussian, vision of maths rather than their own more sensual tradition.
Twentieth-century mathematics decide to eschew words, ideas, diagrams, examples, explanations and applications in favour of formulae. This is a lawyer's vision of maths, where the main goal is the nit-picking avoidance of mistakes. But this, as Byers observes, is also rigour mortis: it is not creative, it leads nowhere. Not surprisingly, fewer and fewer students are now attracted to mathematics. The subject is quietly dying.
To create a new field of mathematics, you have to feel comfortable with paradox, with creative tension, with sloppy and dangerous new ideas, and you have to want to rock the boat, not conform slavishly. As Byers stresses, perfectly formalised ideas are dead, while ambiguous, paradoxical ideas are pregnant with possibilities and lead us in new directions: they guide us to new viewpoints, new truths.
It is time to free mathematical creativity from this prison. We need a radically new mathematics for our postmodern era, a mathematics of complexity, computation and information, a mathematics that applies to complex biological systems. In fact, Byers isn't content merely to rescue maths. His goals are even higher: he dares to oppose the entire zeitgeist. As he correctly points out, our view of maths spills over into our view of ourselves. To paraphrase his dramatic final chapter: if mathematicians think they are machines, they will behave like machines; if mathematicians think they are trivial, then they will be trivial.
We have computers now, so we don't need to have people imitating machines. The 21st century is beginning: time to throw off our chains, and unleash the power of our imagination and creativity. We should be as unlike machines as possible. Bravo, Professor Byers, and my compliments to Princeton University Press for publishing this book. Can this mean that the tide is turning? I hope so.
Gregory Chaitin is a mathematician at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York state. His most recent books are Thinking about Gödel and Turing and Meta Maths
28 July 2007 | NewScientist | 49