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.

Leibniz endeavored to provide an account of inference and judgment involving the mechanical play of symbols and very little else. The checklists that result are the first of humanity's intellectual artifacts. They express, they explain, and so they ratify a power of the mind. And, of course, they are artifacts in the process of becoming algorithms.

More than sixty years ago, mathematical logicians, by defining precisely the concept of an algorithm, gave content to the ancient human idea of an effective calculation. Their definitions led to the creation of the digital computer, an interesting example of thought bending matter to its ends.

Commentators who today talk of 'The Dark Ages' when faith instead of reason was said to ruthlessly rule, have for their animadversions only the excuse of perfect ignorance. Both Aquinas' intellectual gifts and his religious nature were of a kind that is no longer commonly seen in the Western world.

The calculus is the story this [the Western] world first told itself as it became the modern world.

Bystanders wandered in and out of the merchant's stall, passing the time, talking of dreams they might purchase. Workers and slaves stooped from labor asked timidly for dreams of wine and ease. Women asked for dreams of love, and men for dreams of women.

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.