Why Can't It Be Done by Computer?

We often hear it said that the enormous power and capabilities of modern computers are rendering mathematical work increasingly obsolete--at least in practical settings, where real life is more complex than the theoretical ideal. However, when it comes to developing new methods, mathematics is in more demand now than it ever was in the past. Computers can do extraordinarily complex things to the numbers that we give them, but they do not offer us intuition, or a clue to what all the numbers mean. In contrast, mathematical theory provides a way of simplifying the complex, of understanding what it is that the computer is doing, and of deducing structure and orderliness from apparent randomness and irregularity. Each of the practical problems that I shall discuss requires mathematical theory for developing both intuition and methods, and powerful computers for implementation.