Axiomatics in its Element

[Further thoughts along the lines of the thought-themes treated in the essay that begins here.]

John Locke, An Essay Concerning Human Understanding (1690), §IV.ii.8:

The necessity of this intuitive knowledge, in each step of scientifical or demonstrative reasoning, gave occasion, I imagine, to that mistaken axiom, that all reasoning was ex praecognitis et praeconcessis;  which  how far it is mistaken, I shall have occasion to show  more at large, where I come to consider proposititions, and particularly those propositions, which are called maxims;  and to show that ‘tis by a mistake, that they are supposed to be the foundations of all our knowledge and reasonings.

Quite otherwise is the role of axiomatics in Set Theory.  Here, the axioms are once again not simply arbitrary -- they always have some prior empirical motivation.  But the consequences of assuming any new axiomatization  are initially quite obscure, and have turned out in many cases  to be enormous (or even fatal).  Hence  taking on a new axiom, or retwiddling an old one, is more akin to creating a new universe for exploration, than tidying up our formal description of the old universe.

In physics, if your axiomatization results in predictions refuted by experience -- so much the worse for that axiomatization.  Its role is not truly foundational, it’s mostly just there for show.

In logic and set theory, the case is much more complex, requiring qualitatively deeper criteria for analysis.  You wind up with logically distinct and often incompatible theories of these utterly fundamental subjects.   (Or, such we had taken them to be.  If there exists no canonical description, is their foundational character impugned?)

The proliferation of spaces (finite-dimensional and otherwise) and of surfaces or, more generally, topological objects within them, is all good clean fun -- like discovering new fauna in newly-explored islands.   But to have incompatible characterizations of logic is more like being suddenly troubled by what actually counts as a life-form.  In our own time, the biologist has had actual glimpses of such worries, in the shape of empirical discoveries not known to earlier centuries, such as viruses (alive? not alive?), and of a continuum of creaturely objects of uncertain individuation:  ant colonies; slime molds; clonal species of certain trees and fungi; and, on another dimension, organisms that divide by mitosis, or that clone themselves;  not to mention the newer view that individual genes are the fundamental vital entity, organisms being just their carrying-cases.   Now add further discoveries, such as Quantum Cats -- separate and independent twins, yet joined in an Einstein-Podolsky-Rosen way, so that they are actually not independent at all.   Or the planet in Solaris, a single organism;  and  distant cousin, the self-aware Oort Cloud.  Or:  certain subsets of integers, like the fateful one in the TV series “Lost”, with malevolent propensities of their own, and measurable biological effects.  -- The consequences for our conception of mathematical reality are quite as drastic as that.

The result is highly uncomfortable for a Platonist.  It seems as though we are coming after all  to the sterile game of creating arbitrary universes at will -- lifeless and pointless entities, with no connection to the Real (or the divine) -- a nightmare of sterile atheism.  Yet here we see one who did not flinch.  John Dawson (Logical Dilemmas, p. 175) quotes Gödel 1946 on the subject:

He asserted that, even if some new axiom ‘had no intrinsic necessity at all”, its truth might come to be accepted inductively  [dbj: Note, actual “truth”, not mere “usefulness”], on the basis of its “verifiable’ consequences” (those demonstrable without the new axiom, whose proofs by means of the new axiom are considerably simpler and easier to discover”).  Indeed, he declared, “There might exist axioms  so abundant in their verifiable consequences, shedding so much light upon the whole discipline, and furnishing such powerful methods for solving given problems… that they would have to be assumed… in the same sense as any well-established physical theory.”

The style of reasoning here strikes me as theological, quite in the tradition of those views that saw the universe, in all its parts, as having been created just so, for the well-being of God’s favorite creature, Man.  To find such an axiom as that, must indeed strike one as finding a kind of key (like that proverbial watch encountered on the empty strand, left there by the Watchmaker…)

            For:  The invention of the vacuum-cleaner certainly added spring to the housewife’s step;  but we do not therefrom conclude as to the nature of The Vacuum.    By contrast, the mathematical lawfulness of physical phenomena, in a myriad of ways, has indeed impressed observers as saying something about the Universe itself.  In similar fashion, should some axiom (with its associated methods) prove to open up, at a stroke, whole vistas of the already otherwise-intuited invisible world, we would feel it had been left there for us:  that it is as real as rocks.

~

In Modern Philosophy (1994; p.99) the philosopher and theist Roger Scruton  lists some familiar foundational principles of Truth Theory -- things like “If the sentence ‘p’ is true, then so is the sentence ‘ “p” is true’; and vice versa -- but calls these platitudes.  Very nice.  Putting axiomatics in its place.

Another way of putting them in their place  is to observe that no determinate character of ‘axiomaticity’ inheres in any one of them -- a salient illustration of Quine’s point in “Two Dogmas of Empiricism”.  Thus, from a very straightforward and standard textbook, with no philosophical (let alone relativist)  axe to grind:

What is axiom and what is theorem  is often an arbitrary choice, since one is often able to derive each logically from the other.

Sometimes one can go a step further, and make all the axioms of a mathematical system into theorems  by basing the system entirely upon some other system.  Thus, analytical geometry makes it possible to base all of plane geometry upon the properties of the real numbers.  This is also possible with R itself, throwing it back onto modern set theory.

-- Creighton Buck, Advanced Calculus (1956, 3^rd edn. 1978), p. 57

You sketch my hand,  I'll sketch yours

~

Andrew Gleason, by contrast, acutely proposes a distinction between postulates and axioms,  restoring lustre to the latter.  The distinction only makes sense  on a Realist account of mathematical truth :

When we leave the domain of abstract configurations, what we call postulates  take on a different significance.  In the abstract domain  we are in charge; we can frame postulates as we please  and simply exclude from consideration  configurations which fail to satisfy them.  But we cannot take this attitude toward concepts which have any sort of independent existence.  What are appropriately called postulates in the context of configurations  become axioms when we deal with independent conceptions.  They are axioms because they are accepted as true, or at least granted for purposes of argument, for intutitive reasons.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Such a conceptual armature  in effect brings mathematics into comparison with both theology and physics, while increasing the contrast with both postmodernism and finger-painting.

Gleason is not here reporting settled usage, but putting forth a semantic proposal, one of répartition or desynonymization.  As he states earlier, anent the defining properties for an ordered set:

The conditions appearing in the definition are called the axioms or postulates for an ordered set.  The word axiom has long carried the connotation of being self-evident, but it is hard to find a sense in which these conditions are self-evident.  The word postulate (from Latin postulare, to demand) seems more appropriate.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 59

It is, incidentally, refreshing, to hear a mathematician who helped settle one of the Hilbert Problems  freely concede that a (not specially complex) set of condititions  is not, in fact, self-evident.
(In the classroom, he was very much like that:  No trace, either of arrogance or of false humility.)

~

The establishing of first principles  is not a matter for math and science only;  the widest field for its application is the law.  And as we saw in the case of mathematics and of physics, so too in law, the axioms do not historically arise first:

Primitive law is made up of simple, precise, detailed rules  for definite narrowly-defined situations.  It has no general principles.

-- Roscoe Pound, An Introduction to the Philosophy of Law (1922, 1954)

In time, such atomistic empricism is systematized.  (For a fable along these lines, consult our parable of the mathematizing woodchuck.)

In place of detailed rules, precisely determining what shall take place  upon a precisely detailed state of facts,  reliance is had upon general premises, for judicial and juristic reasoning.  These legal principles, as we call them, are made use of to supply new rules [and] to interpret old ones.

-- id.

Spinoza

Once a structured body of legal principles has arisen, judgments may be arrived at more geometrico --  though there will always be a residue of hard cases:

Judicial treatment of a controversy  is a measuring of it by a rule  in order to reach a universal solution for a class of causes, of which the cause in hand is but an example.

Administrative treatment of a situation  is a disposition of it  as a unique occurrence.

-- id.