The two meanings of mathematical terms

Talking of mathematics as a bondage-and-discipline precise science always seemed to me as misleading. This is not how mathematics is done, even if the result looks so.

As to PG's definition of mathematics as study of terms that have a precise meaning, this is wrong too, as exemplified by problems of set theory and its logic, from which mathematicians escaped into more loosy setting of category theory and categorical logic.

I think article "When is one thing equal to another" by Barry Mazur somewhat fits into the theme of this blog note http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

He's just distinguishing between pure and applied maths.

Pure math -> "level one". Start with axioms and apply logic.

Applied math -> "level two". Take some pure math, and find an analogous system in the real world.

The question in applied math is whether (or better yet when) the analogy is a good one.