On the Progress of Mathematics

Abstract

Mathematics remains perhaps the only science for which in the context of philosophy of science, we can still talk about progress. Despite the claims of some critics led by purely philosophical prejudice, this is yet another point where mathematics proves to be fundamentally different from all other natural sciences with which it is traditionally associated. This is an historical tradition. Although this association with other sciences is well justified from the practical standpoint, logically and essentially mathematics is still a rara avis among its “natural” counterparts. In the first place, mathematical objects do not seem to be natural in the same way as living organisms are in biology or minerals are in mineralogy. We shall not go further into this point, for it has long been discussed, though to little effect. We shall simply point out that due to the very peculiar nature of the subject of mathematics, its new results — in contrast with those of other natural sciences — do not cancel the old ones. The theorems of Euclid’s Elements are as valid nowadays as they were in antiquity. As time passes, mathematics only acquires new results; it grows, although it is a very specific type of growth. There is not the mere addition of a new result to the sum of results already known, but an intertwining of this new result into the complex, hierarchical structure of mathematics, e.g., of a new theorem into the Euclidean geometrical system. Clearly, the understanding of mathematics is determined historically. It evolves and sometimes this evolution results in great transformations of the whole structure of mathematical knowledge.