A Comparison Of Two Cultural Approaches To Mathematics

Many people would like to know where scientific ideas come from and how they arise. In the case of mathematics, new ideas often come in the form of new “mathematical objects”: groups, vector spaces, sets, etc. Some people think these new objects are invented, others that they are discovered. By exploring the birth of descriptive set theory in France and Russia in the period 1890–1930 we show that the leading French mathematicians worked within a rational, secular worldview that made them doubt the legitimacy of infinite sets, particularly nondenumerable ones; on the other hand, the creators of the famous Moscow school of mathematics, particularly those who subscribed to a religious doctrine known as “name‐worshipping,” believed that humans had absolute freedom to invent mathematical objects. Partly as a result of their different cultural environments, the French and the Russians took different approaches to the same problem. In the end the Russians created a new field, descriptive set theory, at a time when the French remained hesitant