FOM: Re: Tragesser on set-theoretic reduction
neilt at mercutio.cohums.ohio-state.edu
Mon Feb 23 17:35:21 EST 1998
Could Robert Tragesser please explain the metaphor "it could be that set
theory "works" because it is so rich in symmetry (so close to paradox)
that most any bit of conceptual mathematics can break the symmetry"?
Could he also provide an *impossibility proof* to back up the following:
"as an exercise just try to so reduce the proofs of the
first 20 theorems say of Hilbert's FoG as they occur there -- you are
in serious trouble right off because a number of them -- or if one is very
sensitive, none of them -- go over into a predicate calculus derivation."
Concerning the reducibility of mathematics to set theory he also says
"the fact that NT finds something to be amazed about suggests also that live,
practiced, unreduced mathematics has about it a TRACTABLE content that is
lost in set theory."
I disagree. Indeed, there is the tractable content claimed; a content which
varies greatly from one branch of mathematics to another. Thinking about
continuous functions and homeomorphisms seems to draw on different intuitive
and conceptual resources than thinking about discrete systems such as
the natural numbers, or directed graphs. What is amazing is that, despite
the variety of intuitive and conceptual flavours in the unreduced mathematics,
it can all be reduced to the theory of a two-place relation (membership)
among pure sets. I disagree, however, with the claim that the tractable
content is *lost* upon reduction to set theory. It is the job of the
"definer-reducer" to choose the right definitions so that the content is
*preserved* while yet being re-capitulated within set-theoretic terms.
Just one example might help to make this clear. Take the set-theoretic reduction of the continuum, according to which it "is" (i.e. can be taken to be) the
power set P(w) of the set w of all finite von Neumann ordinals. This might
*seem* to destroy the "tractable content" of that nice continuous, more
geometrically graspable stripe that we like to call the continuum. But a
moment's reflection dispels that mistaken impression. For any subset r of w
can be thought of as a binary expansion (1 in the n-th place if n is in r,
0 otherwise), yielding r as a real in a much more familiar guise. Pray tell,
Robert: what has been "lost" upon thus re-conceiving the set R of real
numbers as [in 1-1 correspondence with] P(w)?
With proper care, we should be able to devise "notations that think for
themselves" even in the context of a set-theoretic reduction. Some reductions
are better than others. So we are free to impose further adequacy constraints
on our adoption of definitions, so as to ensure that mathematics can make a
smooth transition to its set-theoretic foster home.
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