[ggm]

Fair. I like Polyani so I shall put a Polyani quote here which I feel goes to your point: Mathematics as a purely formal system of symbols without a human being possessing the know-how for dealing with the symbols is impossible (1969) the wiki page I found it on is the Brouwer-Hilbert controversy (constructivism vs formalism, which seems a reasonable laymans take on a component of your point) and says: Despite the last-half-twentieth century's continued abstraction of mathematics, the issue has not entirely gone away. which is I feel, what I first said. It's a continuing difference, and as a non mathematician I take heart in that. All of these observations stem from looking at the law of the excluded Middle btw. There was a causal chain of weblinks to get there, I didn't just grab at random.