FOM: Kreisel on the elementary, the purity of method
Robert S Tragesser
RTragesser at compuserve.com
Sun Dec 14 05:54:14 EST 1997
THIS COMPLETES MY WORDY [AND I CAN'T IMAGINE READ] POSTINGS ON
ELEMENTARY PROOF, ETC. Here I manage to get the perspective on the
matter which I have been straining toward. (There is a geography which I
must now go on to map out in detail.)
Anders Goeransson pointed me to Kreisel's RS memoir on Goedel,
where Kreisel places the issue of elementary proof, purity of method, at
the center.
PURITY OF METHOD AS THE CENTRAL PROBLEM OF THE PHILOSOPHY OF MATHEMATICS:
Here is exactly the thought I have been straining toward. "The
LOGICAL question is to settle to what extent purity of methods can be
achieved--in all of mathematics, parts of mathematics, in fact, in logic
or metamathematics itself. But this leaves open the PHILOSOPHICAL
question whether purity of method is at all basic, in the sense of
fundamental, to mathematical knowledge, the sort of thing one cannot know
to much about."
ARISTOTLE RATHER THAN KANT: The pernicious influence of Kant.
Kant has entranced us with the vision of an epistemological
hierarchy [for us, from the ultra-infinitary up through the transfinite --
to speak very roughly] bearing within it some cut-off level, where below
that reasoning is "elementary" and reasoning occuring above that level is
non-elementary and must be interpreted and evalauated in the lower
elementary part of the hierarchy [N.B. the important part of the Critique
of Pure Reason is not the wholly unconvincing transcendental aesthetic,
but Chapter II of Book II, the chapter "System of All Principles of the
Understanding" where, roughly speaking, Kant takes principles from
traditional metaphysics/epistemology and translates or interprets them with
respect to the domain of possible experience; second in importance are
then the chapters on the logic of illusion where Kant resolves traditional
paradoxes by interpreting the metaphysical theses involved in the light of
those principles which were relativized to the domain of possible
experience]. Reductive proof theory has in fact made a mess of that
hierarchy -- mathematics cannot be organized by the epistemological
hierarchy. We now wonder what the point of trying to do so was? (Now
that we are no longer captivated by Kant.)
For Aristotle, for every genos of being (self-sufficiency is a
boundary for a genos) there is reasoning which is native -- which is to say
basic or elementary -- to it, and reasoning which is metabasic,
transcendental, a foreign import. For every genos of being there is a
method purely for it, hence one can and must strive for purity of method.
Thus in pure, transfinite set theory, we can speak of elementary proofs
and mean by this proofs secured by pure set-theoretic reasoning. Thus CH?
can be regarded as a question about the cardinality of the power set of a
countably infinite set, capable of being settled only by purely set
theoretical reasoning. Or the CH? can be regarded as an impure question
about the cardinality of spatiogeometric continua, with the accompanying
impure thought that it might be that the full power set misrepresents such
continua. We then ask, what set of subsets fairly and squarely represents
such continua, and revise the statement of CH, making it a question about
the cardinality of that set of subsets. This is applied (impure) set
theory.
That the methods of pure transfinite set theory might be called
elementary strikes us as crazy is due to the pernicious influence of Kant.
As the Aristotelian Husserl would put it: to every genos of being there
is correlated a genos of reasoning proper to it. Against Kant, there is
not some privileged genos of reasoning.
I understand Kreisel's problem (quoted above) in terms of this
Aristotelian concept of purity of method.
[N.B. there is a literature associated with Husserl that argues
that Husserl would have an antirealist philosophy of mathematics. But
this literature does not notice that Husserl in Ideen I says that he will
explain how -- what we call -- anti-realism is a consequence of an
incomplete view of the intentional acts of consciousness; he gives us
that full explanation in the curiously seldom read, but absolutely
fundamental for any good understanding of Husserl, Ideen II.]
The question is: does mathematics indeed fall into self-contained,
distinct subject matters? or is mathematics deeply unified?
It is clear why (as Kreisel reports) Goedel might have progressed
in his readings from Kant, through the Aristotelian Husserl, to
Schelling's Naturphilosophie. Naturphilosophie, recall, had a powerful
effect, inspiring the likes of Oersted/Faraday and Riemann that truth is
to be found trans-genos, in unities.
REMARKS ON THE CHOICES OF AXIOMS.
There are two kinds of choices of axioms that are particularly
worthy of being kept in mind. One, in the spirit of Emily Noether's
popularization of the axiomatic method -- choosing axiom for a subject
matter which embody, and so give one, immediate penetrating mathematical
insight into that subject matter. Another: axioms chosen for
metamathematical purposes (such as Tarkski's axioms for elementary
geometry) which yield some sense of how the mathematical subject for which
the axioms are given is situated with respect to other (parts of)
mathematics.
Noetherian axiomatization is then more in the spirit of the
Aristotelian.
ReverseMathematics is clearly a way of axiomatizing in the second
sense -- of finding axioms which locate the power of a theorem, but not in
any sense that correlates with Cartesian-Kantian epistemology (right?).
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