Quotes about Axiom

Finally, we entered Chetaube County, my imaginary birthplace, where the names of the little winding roads and minuscule mountain communities never failed to inspire me: Yardscrabble, Big Log, Upper, Middle and Lower Pigsty, Chicken Scratch, Cooterville, Felchville, Dust Rag, Dough Bag, Uranus Ridge, Big Bottom, Hooter Holler, Quickskillet, Buck Wallow, Possum Strut … We always say a picture speaks a thousand words, but isn’t the opposite equally true?

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.

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.

The facts and arguments in this book are intended to signpost new directions of thought and action for a better rather than a worse future - but are the goals they point to achieveable ?  Certainly they will involve change,  and pretty radical change at that,  and as past experience shows,  it will not happen without strict observance of the axioms.