FOM: RevMth, Friedman'sPrinciple, Aristotle, ElementaryProof
Robert S Tragesser
RTragesser at compuserve.com
Fri Dec 12 20:22:13 EST 1997
Friedman's Principle for Ordinary Mathematics: When the theorem is
proved from the right axioms, then the axioms can be proved from the
theorem." [Exposed in Simpson;s appendix to Takeuti II]
So it turns out that after all reverse mathematics has an absolute
philosophical sense, and this in connection with the notion of elementary
proof:
In a previous posting I explained the fundamentally Aristotelian
motive at the bottom of the quest for elementary proofs.
In Aristotle one obtains closure, self-containedness, by deducing
the cuases or first pricnples of a theorem (backward) from the theorem.
There is no question now that this is what Aristotle meant by induction in
a proper science [there was a rather silly tradition in which Aristotelian
epagoge -- "induction" -- was taken to be empirical induction].
For Aristotle the first principles/causes are analytically
contained in the theorems and so can be obtained from them by "something
much like deduction"(Aristotle PrrAnalytics].
The powerful suggestion which emerges is the Reverse Mathematics
puts in our hands the means to isolate the "self-contained" or elementarily
provable parts of mathematics, those parts of mathematics which conform to
Aristotelian episteme.
Remarkable!
[N.B. Evert Beth had remarked that after 15 years of struggling to
articulate foundational questions against the background of Kant, he even
bitterly realized that this was mistken -- that one should rather frame
them against the background of the Aristotleian Organon, especially the
Posterior and Prior Analytivcs -- it is interesting that after some
centuries of its being ignored, everyone is putting aside their Kant and
Wittgenstein and reading the Posterior Analytics these days. Hao Wang had
also stressed many times that no discussion of the origin of axioms could
proceed without a close mastery of the Post.Analytics. Of the three
recent books on the Posterior Analytics, I recomend most strongly
P.H.Byrne's ANALYSIS AND SCIENCE IN ARISTOTLE]
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