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. 57You 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.)

(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.

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