Mathematics is a
"formal" discipline formed around a
human cognitive mechanism which accounts for
perception of
discrete entities (and by extension,
quantitative relationships). It should be noted that discrete entities and quantitative relationships do not actually
exist in
nature per se; rather, we accept the
notion that they do as part of the natural
reductive process of our
consciousness (a sometimes-convenient
compromise). In this context, Mathematics is a (characteristically
exhaustive and
rigorous)
compendium of
metaphors based exclusively on discrete and quantifiable subject matter (i.e.
numbers), where that
description is itself, as much as possible, discrete and quantifiable.
Kurt Godel's Incompleteness Theorems (published 1931) correctly identify that any formal system (i.e. Mathematics) is inherently flawed, specifically "in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system." To restate, Mathematics is effectively a study of the consequences of the particular compromises inherent in human cognition. Caveat emptor.