When I was studying at Cambridge, the Department of Pure Mathematics and Mathematical Statistics was in a dingy converted warehouse, off a side street. The new mathematics building at Cambridge University is designed to make everyday discussion and exchange of ideas easier (see pic above). The architects write:
“We start by gathering the offices on galleries round stairs and a lift… Three floors of galleries go to make a pavilion… and on the ground floor, somewhat visible from all three floors, is a coffee room/seminar/meeting room for larger, pavilion-sized meetings, arranged and un-arranged. The meeting room of each pavilion overlooks a large shared space where you will come across people from the other pavilions and where you eat, relax, have meetings, work things out together, or organise big gatherings.
“We like to think of the pavilion meeting rooms overlooking the large shared space as classic American front porches”.
The entrance to the new building leads straight into the cafeteria. A lot of the time, even when the cafe bit is closed, the cafeteria area is abuzz, each table surrounded by mathematicians talking, puzzling, or arguing about maths.
Mathematics is, contrary to its popular image, the most social of sciences. It is almost impossible to learn mathematics by yourself. To make progress you need exchange of ideas, swapping of insights, access to areas which others know better than you do.
History confirms this. In many fields, you can bypass organised and collective learning and reach the forefront by individual study. The economist John Maynard Keynes and the philosopher Bertrand Russell had their university education in maths, not economics or philosophy. The philosopher Ludwig Wittgenstein had his in engineering; the historians Leopold von Ranke and E H Carr both had theirs in classics. The organised education from which Karl Marx went on to develop his social theory was in law and philosophy.
Good writing, art, or music can be done without an organised education in literature, in fine art, or in music.
There have been few productive mathematicians who did not have an organised education in mathematics. The popular image of the mathematician tends to be of a loner. It’s the opposite of the truth.
The Bourbaki group of French mathematicians at a meeting, 1951