Mathematics in the 20th Century

If you talk about the end of one century and the beginning of the next you have two choices, both of them difficult. One is to survey the mathematics over the past hundred years; the other is to predict the mathematics of the next hundred years. I have chosen the more difficult task. Everybody can predict and we will not be around to find out whether we were wrong. But giving an impression of the past is something that everybody can disagree with. All I can do is give you a personal view. It is impossible to cover everything, and in particular I will leave out significant parts of the story, partly because I am not an expert, and partly because they are covered elsewhere. I will say nothing, for example, about the great events in the area between logic and computing associated with the names of people like Hilbert, G6del, and Turing. Nor will I say much about the applications of mathematics, except in fundamental physics, because they are so numerous and they need such special treatment. Each would require a lecture to itself. Moreover, there is no point in trying to give just a list of theorems or even a list of famous mathematicians over the last hundred years. That would be rather a dull exercise. So instead I am going to try and pick out some themes that I think run across the board in many ways and underline what has happened. Let me first make a general remark. Centuries are crude numbers. We do not really believe that after a hundred years something suddenly stops and starts again. So when I describe the mathematics of the 20th century, I am going to be rather cavalier about dates. If something started in the 1890s and moved into the 1900s, I shall ignore such detail. I will behave like an astronomer and work in rather approximate numbers. In fact, many things started in the 19th century and only came to fruition in the 20th century. One of the difficulties of this exercise is that it is very hard to put oneself back in the position of what it was like in 1900 to be a mathematician, because so much of the mathematics of the last century has been absorbed by our culture, by us. It is very hard to imagine a time when people did not think in our terms. In fact, if you make a really important discovery in mathematics you will get omitted altogether! You simply get absorbed into the background. So going back, you have to try to imagine what it was like in a different era when people did not think in our way.