This replacement is typical of the historical development of
mathematics.
It replaces a vague notion (Tragesser's "= coherent, together,
self-standing, sense-ful, of a piece, ...") with a precise one that can
be proved, disproved, or shown to be undecidable. This is similar to
the evolution of continuity from "a curve freely drawn by the hand" to
the various precise definitions recently much discussed in this list.
They are not equivalent to each other (as various examples show), but
they are mathematics, in that they can be conclusively reasoned about.
They are inequivalent in a much stronger sense to freely hand-drawn,
since conclusions about that are another cosmos of powerfully
significant confusions and phenomena.
Any other mathematization of "consistent" (have there been any?) would
be equally apalling to those who prefer it understood in a "fully
resonant and rather complex sense" unspecified by that sense's
proponent.
Goethe remarked that "Mathematicians are like Frenchmen; they translate
anything you say into their own language, whereupon it means something
completely different," and this is their proper business. Consistency,
continuity, symmetry, infinity all mean something different to
mathematicians than they mean to artists and theologians. There is full
resonance to "God is a circle whose centre is everywhere and whose
circumference is nowhere," which is rather complex too, but is it to be
admitted as a mathematical statement? No more should wide-open senses
of "consistent" be.
It is enough justification of freedom-from-contradiction as consistency
that (a) if it is false of some A, the "resonant" meanings had better be
false of A too, and (b) the analysis of it has led to powerful,
beautiful mathematics with applications all over the map. What more is
needed? Jam on it?
Tim
___________________________________________________________
| |
| Tim Poston Chief Scientist |
| Centre for Information-enhanced Medicine |
| National University of Singapore |
| www.ciemed.nus.edu.sg |
|___________________________________________________________|