Arithmetic is where the content lies, and not logic; but logic prompts certainty, and not arithmetic.
Arithmetic is where the content lies, and not logic; but logic prompts certainty, and not arithmetic.
The same procedure may be used in the predicate calculus, but it is complicated, tedious, and ugly. It is for this reason--plain laziness, too--that the logiciain repairs to axiom schemata instead of axioms when formalizing the predicate calculus. Axiom schemata do not themselves appear in the formal system. They are part of the logician's own vernacular, expressed in the same language that he or she employs to talk about formulas and predicate symbols. Each axiom schemata specifies the form of a formula, and each axiom of the system itself is obtained from the form as an instance.
The motion of the mind is conveyed along a cloud of meaning.~ There is this paradox that we get to meaning only when we strip the meaning from symbols.
The individual variables x, y, z,... now make a semiformal appearance, performing the function in logic that pronouns perform in ordinary English, the sentence "She is blonde", cognate to the proposition "x is blonde", both she and x specifying something but specifying that thing indeterminately.
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.
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.
Aristotelian logic is massive and marmoreal, but every monument accumulates graffiti.
"Young men wish always to dream of what they have lost."
"And old men?"
"Of what they have not found."
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.
The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system.
