The Fungibility of Expressions

1crowThe logician Elliott Mendelson, in his textbook “Introduction to Mathematical Logic”, defines a proof as:

“a sequence of [well-formed formulas] such that, for each , either is an axiom of or is a direct consequence of some of the preceding [well-formed formulas] in the sequence by virtue of one of the rules of inference of .”

In this context a well-formed formula is simply a legal expression in the formal theory (like ), and a rule of inference is a rule that says the presence of one or more expressions means that another expression may be present (like, if and are present then is allowed).

The definition of axioms and rules of inference are the primary work of developing a theory.  And, very often rather than explicitly defining every axiom, mathematicians define axiom schema.  These schema are generalized forms of expressions where irrelevant information is rolled into a one or more variables.  As an example, is a form of the schema where represents and represents .  This technique helps cut the [word similar to crab] so a theorist isn’t burdened with enumerating every possible axiom.

In my career I have often found it useful to think of machinery as proofs.  Each component is a lemma built on axioms and rules of inference derived from the laws of physics.  Each of these lemma then coalesce, under the force of the same rules of inference, to produce the whole machine.  If you wanted, you could try to wrap you head around the concept of an air compressor by only thinking of a probability fog of protons and electrons.  Or you could take a couple steps up the abstraction ladder and define connecting rods, crankshafts, pistons and valves as axioms with fluid dynamics and beam deflection as rules of inference.  In either case, the arrangement of these parts can be used to produce a theorem, namely compressed air.

Recent events have brought into sharp focus how this analogy also applies to organizations.  Here people are the cogs; we enable the theorem.  For the purpose of the organization, an individual is only important to the extent that they fit an axiom schema.  What you see in ‘people as axioms’ that differs from the proofs I’m used to seeing is that a person fits the axiom schema of several theories.  The negation in the antecedent of the example above is irrelevant there, but may well be vital to another theory.

This week I found myself struggling to assert the importance of my particular flavor of axiom, in two combatant organizations.  This futile contest of wills has invaded the time Jan and I normally spend on comics.  Consequently, we could only produce another progress clip.  With jack boots placed where my brothers have instructed, Jan and I should have time this next week to put this comic together.

~Nick…not just an axiom schema


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