I always saw a close kinship between the needs of “pure” mathematics and a certain hero of Greek mythology, Antaeus. The son of Earth, he had to touch the ground every so often in order to reestablish contact with his Mother; otherwise his strength waned. To strangle him, Hercules simply held him off the ground. Back to mathematics. Separation from any down-to-earth input could safely be complete for long periods — but not forever.
Also: if mathematics in contact only with mathematics becomes “less mathematical” than mathematics in contact with praxis, then how can praxis in contact with mathematics become more practical than praxis out of contact with mathematics?
-Benoit Mandelbrot
While I agree, where could the earth be getting its strength from?
Also: if mathematics in contact only with mathematics becomes “less mathematical” than mathematics in contact with praxis, then how can praxis in contact with mathematics become more practical than praxis out of contact with mathematics?
If you have no mathematical techniques, you don’t know how to think about your empirical evidence.
If you have no empirical evidence, you have nothing to use your mathematical techniques on.
You need both.
Circular reasoning. One chunk pushes against the next, which pushes against the next....until you’re back where you started.