From: wft@math.canterbury.ac.nz (Bill Taylor)
Subject: Foundations without foundationalism.
Organization: Department of Mathematics, University of Canterbury
Date: Wed, 27 Jan 1993 04:04:30 GMT

"Foundations without foundationalism"; by Stewart Shapiro.

Many thanks to all those who have recommended this book in sci.logic, over the
last few months. (Franzen & Zeleny come to mind, but there have been others too.)

I have finally got around to reading it, and have thoroughly enjoyed it; a
nice mix of technical results and philosophical comment. No doubt large amounts
of it have passed me by, but it leaves me with the feeling that I am better
informed on the matters in question, so to that extent at least it is a success.
It also leaves one with the pleasant feeling that there is more to be discovered
on a re-read, in due course.

I particularly liked seeing defended in print, the "heretical doctrine" that first-order logic is *not necessarily* the be-all and end-all in logic.
It is nice indeed, merely to have the view expressed that it may be a good
idea to "shop around" for the most convenient logic for one's needs.   That
there may not *be* a "one true logic" suitable for all rational thought. Shapiro
stresses the point that this is especially true of mathematics, which might
often be better treated by 2nd-order logic. (There is no actual comment, but a
feeling in the book that there *is* still a "one true math" that is to be
handled by logic, at least as far as arithmetic and analysis go !)

It is nice to see a downplaying of the oft-quoted Lowenheim-Skolem results,
which are treated almost as an artefact of the first-order method, rather than
as significant mathematical results. Indeed it was nice to see stated explicitly,
that one of these, ("downward" L-S), actually needs the use of AC in the
metatheory to derive it:- something that had not been impressed on me before, 
and which may be thought to lessen the significance of the results in themselves.

Contrariwise, it is interesting to see the point thoroughly developed that the
2nd-order axiomatizations of arithmetic & analysis are *categorical*, a concept
quite dear to the typical mathematician.

Dearer, at least, than the (also downplayed) concept of *completeness* which
Shapiro identifies as the chief reason that 1st-order logic has been traditionally
accepted as standard.  It is nice to see completeness removed from its centre
stage position.   I've never been madly impressed with it in a mathematical
context anyway; partly because the Godel-incompleteness of mathematical theories
seems to reduce the importance of logical completeness of 1st-orderness; but
mostly because 1st-order completeness is largely a matter concerning *all*
possible models, whereas in arithmetic and analysis only *one* model is ever of
major concern, (though it is now doubtful if the same could be so confidently
said of set theory, in view of the plethora of independence results).

Anyway, all these matters are handled competently and thoroughly, it seemed to
me, in Shapiro's book. All in all, a good read. Thanks again to all those who
helped bring it to my attention.

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              Bill Taylor              wft@math.canterbury.ac.nz
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                       Kleeneness is next to Godelness.
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