Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Bertrand Russell (1872 - 1970)
Source: "The Study of Mathematics", Mysticism and Logic: And Other Essays
Mathematics is to the mind what the heart is to art. Without the latter there is no need for the former, but the former is the ladder to the latter. When the latter and the ladder come together, the former is the maker and the maker is the man... and the woman.
An axiomatic system establishes a reverberating relationship between what a mathematician assumes (the axioms) and what he or she can derive (the theorems). In the best of circumstances, the relationship is clear enough so that the mathematician can submit his or her reasoning to an informal checklist, passing from step to step with the easy confidence the steps are small enough so that he cannot be embarrassed nor she tripped up.
Source: The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer, Pages: 49..50
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.
Source: The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer, Pages: 50
For the most part, it is true, ordinary men and women regard mathematics with energetic distaste, counting its concepts as rhapsodic as cauliflower. This is a mistake--there is no other word. Where else can the restless human mind find means to tie the infinite in a finite bow?
Source: The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer, Pages: 24