A Quote by David Berlinski on axiom, mathematics, symbol, rules, and intuition

An axiomatic system comprises axioms and theorems and requires a certain amount of hand-eye coordination before it works.  A formal system comprises an explicit list of symbols, an explicit set of rules governing their cohabitation, an explicit list of axioms, and, above all, an explicit list of rules explicitly governing the steps that the mathematician may take in going from assumptions to conclusions.  No appeal to meaning nor to intuition.  Symbols lose their referential powers; inferences become mechanical.