A survey is given of several key themes that have characterised mathematics in the 20th century. The impact of physics is also discussed, and some speculations are made about possible developments in the 21st century.
This article is based on a transcript of a lecture given at the Fields Institute in Toronto in June 2000.
Author(s): Michael Atiyah
Publisher: London Mathematical Society
Year: 2002
Thank you for inviting me here to take part in this program. Of course, 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, partly because it is 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, G¨odel 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, but perhaps
you will hear more about those in some of the other lectures taking place during
this meeting. 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 those terms. In fact,
if you make a really important discovery in mathematics, you will then 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
the same way.
