Quotes about Symbol

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.

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.

Within the categorical syllogism, ordinary language represents the ordinary flow of inference. Two premises are given; there is a plash of insight, and one step undertaken. The mind hops right along, not quite knowing where it is going but getting there nonetheless. On the right, a checklist does its work. The logician's clamp retains its force of old, but the inferential steps involve no more than the substitution of symbols for symbols, with the anchor of inference embedded in identities. Inference now proceeds from one identity to the next; no plash of insight is involved, only the solid satisfying ratcheting sound of symbols being substituted for symbols.