From: Simon Magson (twix_570@hotmail.com)
Date: Wed Nov 17 2004 - 11:10:45 GMT
Scott Roberts wrote
>That's not what Pirsig said. He said "The MOQ denies this. (That Reason
>perceives truths which are incapable of verification in sense-experience."
>Mathematical truths are not verified in sense-experience. They are verified
>through reason, so if you now say that the rational verification process is
>itself experience, in order to keep with the next sentence: "Reason grows
>out of experience and is
>never independent from it.", then you are saying that, in the case of
>mathematics, reason grows out of reason.
I recommend reading ZMM. In it you will find that Pirsig proposes, with the
help of Jules Henri Poincare, that the choice of conventions and axioms from
which mathematical truths are derived is made on the basis of
preintellectual value. Mathematical truths are thus patterns of values.
Value is phenomenal, it is sense experience. Therefore, mathematical truths
are verified by sense experience. Reason itself is a pattern of values.
"To solve the problem of what is mathematical truth, Poincaré said, we
should first ask ourselves what is the nature of geometric axioms. Are they
synthetic a priori judgments, as Kant said? That is, do they exist as a
fixed part of man's consciousness, independently of experience and uncreated
by experience? Poincaré thought not. They would then impose themselves upon
us with such force that we couldn't conceive the contrary proposition, or
build upon it a theoretic edifice. There would be no non-Euclidian geometry.
Should we therefore conclude that the axioms of geometry are experimental
verities? Poincaré didn't think that was so either. If they were, they would
be subject to continual change and revision as new laboratory data came in.
This seemed to be contrary to the whole nature of geometry itself.
Poincaré concluded that the axioms of geometry are conventions, our choice
among all possible conventions is guided by experimental facts, but it
remains free and is limited only by the necessity of avoiding all
contradiction. Thus it is that the postulates can remain rigorously true
even though the experimental laws that have determined their adoption are
only approximative. The axioms of geometry, in other words, are merely
disguised definitions.
<....>
Mathematics, he said, isn't merely a question of applying rules, any more
than science. It doesn't merely make the most combinations possible
according to certain fixed laws. The combinations so obtained would he
exceedingly numerous, useless and cumbersome. The true work of the inventor
consists in choosing among these combinations so as to eliminate the useless
ones, or rather, to avoid the trouble of making them, and the rules that
must guide the choice are extremely fine and delicate. It's almost
impossible to state them precisely; they must be felt rather than
formulated.
Poincaré then hypothesized that this selection is made by what he called the
"subliminal self," an entity that corresponds exactly with what Phĉdrus
called preintellectual awareness. The subliminal self, Poincaré said, looks
at a large number of solutions to a problem, but only the interesting ones
break into the domain of consciousness. Mathematical solutions are selected
by the subliminal self on the basis of "mathematical beauty," of the harmony
of numbers and forms, of geometric elegance. "This is a true esthetic
feeling which all mathematicians know," Poincaré said, "but of which the
profane are so ignorant as often to be tempted to smile." But it is this
harmony, this beauty, that is at the center of it all.
<...>
Poincaré's contemporaries refused to acknowledge that facts are preselected
because they thought that to do so would destroy the validity of scientific
method. They presumed that "preselected facts" meant that truth is "whatever
you like" and called his ideas conventionalism. They vigorously ignored the
truth that their own "principle of objectivity" is not itself an observable
fact...and therefore by their own criteria should be put in a state of
suspended animation.
They felt they had to do this because if they didn't, the entire philosophic
underpinning of science would collapse. Poincaré didn't offer any
resolutions of this quandary. He didn't go far enough into the metaphysical
implications of what he was saying to arrive at the solution. What he
neglected to say was that the selection of facts before you "observe" them
is "whatever you like" only in a dualistic, subject-object metaphysical
system! When Quality enters the picture as a third metaphysical entity, the
preselection of facts is no longer arbitrary. The preselection of facts is
not based on subjective, capricious "whatever you like" but on Quality,
which is reality itself. Thus the quandary vanishes."
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