Briefly:
The curious and hardly intelligible current conflation of
consistency and freedom-from-contradiction is an excellent example of:
"Science does not always progress, but sometimes regresses under the
illusion of progress." The positive interpretation of this, Chairman
Mao's three steps forward, two steps backward, should only suggest
that, in the identification of consistency and contradiction-free, we
have taken two (regrouping, simplifying steps backwards), but now
ought to recognize those as backward steps, and move forward by setting
the problem of fully understanding "consistency" as a major of our
concerns.
We have here a case of a resonant and cosmically rich notion --
"consistency"-- being egregiously suppressed, displaced, by
"contradiction-free".
For example, it is absurd to say that contradiction-free-ness
entails being, but it is not so absurd to say that of "consistency".
[understood in the fully resonant and rather complex sense].
"Consistency" carries with it a family of notions [and, as
always when one word is chosen for a complex of inter-related notions,
no one of which is central, it cannot be anything like simply defined].
It carries with it the associations: coherent, of a piece (a unified
if not complete whole), homogenous, standing-together, standing-firm,
senseful, staying true to something, staying true...
The firmest trait of beings, entities, even if like all the
natural entities of Plato or Aristotle did, they have ragged edges, or
are dynamical (like an organism as a life-cycle), is consistency. And
indeed: It is pretty hard to find a trait that qualifies some "thing" as
existing other than consistency!
The upper-half plane model for hyperbolic geometry, Hilbert's
polynomial interpretation of non-Archimedian geometry, the geometrical
interpretation(s) of _i_, etc., establish consistency or we might
better say relative consistency, for they show that the theories or
conceptions in questions can be _seen as_ (aspects of) some more
strikingly consistent (= coherent, together, self-standing,
sense-ful, of a piece, ... mathematics.
The simplest illustration of the issue is that "dispute" between
Locke and Leibniz on the status of the law of non-contradiction.
For Leibniz, this was an innate principle -- the servicing girl who
blushes at a lie is vaguely grasping that she is violating this law.
More to the point:
Are there square circles?
LOCKE: No, for one only has to try to understand "square-circle"
and it immediately emerges that being and square and being a circle
cannot be embodied by anything (in the intended sense), they are in
that sense inconsistent with one another/incompatible. There is not
appeal whatsoever here to contradictoriness.
LEIBNIZ: he required that "square-circle" be resolved into a
contradiction, and the law of noncontradiction implied.
Note: Experimental and theoretical physics at the beginning of the
century was inconsistent (wave/particle conflict); and this was not
good quite independently of the inference of an outright contradiction.
IT WAS ONLY BY A CLOSE STUDY OF HOW IT WAS INCONSISTENT, THAT A DEEPER
UNDERSTANDING EMERGES. (Whereas a study of how it yielded a
contradiction would have been of very little use indeed.)
Note: Tait shows us how PRA is consistent.
Note: the cumulative hierarchy conception powerfully suggests the
consistency of ZF... but the CONCEPTUAL gaps among set theoretical
practice, the sense of the cumulative hierarchy, and various conceptual
gaps among these and the axioms suggests, still, that set theory is
not fully consistent.
The under the aspect of eternity important thing about "proofs
of consistency" of T relative to seemingly finitary or concrete or
concretely intuitive mathematics is not that they assure
contradiction-freeness, but that those limited bits of mathematics are
strikingly consistent (coherent, together, not senseless, of a piece,
etc.), and one can believe by virtue of the relation the "consistent
proof of T" establishes with them that T is at least not-inconsistent,
even if T has not more directly won through to that kind of assurance
that carries with it the stamp(s) of consistency in the
mathematician's/physicist's/(and I should think?)metaphysician's/
epistemologist's important sense.
Robert Tragesser
West(Running)Brook, Connecticut.