A century of controversy over the foundations of mathematics

T hanks very much, Manuel! It’s a great pleasure to be here! We’re in a state of euphoria now in the computer business because things are going so well: the Web, e-commerce. It’s all paying for our salaries, and it’s a nice moment to be around, when things are going so well. But I’d like to make the outrageous claim, that has a little bit of truth, that actually all of this that’s happening now with the computer taking over the world, the digitalization of our society, of information in human society, you could say in a way is the result of a philosophical question that was raised by David Hilbert at the beginning of the century. It’s not a complete lie to say that Turing invented the computer in order to shed light on a philosophical question about the foundations of mathematics that was asked by Hilbert. And in a funny way that led to the creation of the computer business. It’s not completely true, but there is some truth in it. You know, most historical statements are a lie, so this one isn’t that much worse than most others! So I’d like to explain the philosophical history of the computer. In a way what happened, and I’ll tell you more, is that Hilbert said we should formalize all of mathematics, mathematical reasoning. And this failed: It took Gödel and Turing to show that it couldn’t be done. It failed in that precise technical sense. But in fact it succeeded magnificently because, formalization of reasoning no, but formalization of algorithms has been the great technological success of our time— computer programming languages! GREGORY J. CHAITIN