FOM: topos-theoretic "foundations"; logical positivism; Dupre

[Stephen G Simpson]

topos-theoretic "foundations"

1. A correction to my "bankruptcy" posting of 18 Jan 1998 14:39:24.
The McLarty quotes came from his posting of 17 Jan 1998 17:54:03 and
not as stated.

2. Concerning topos-theoretic "foundations" versus set-theoretic
foundations, McLarty 19 Jan 1998 20:09:12 says:

 > > In set theory, the picture is, sets.  In topos theory, the
 > > picture is -- what?  There are too many wildly differing models
 > > for this to be a viable educational and/or foundational option.
 >       
 >         I think this is the central difference.

There are other differences.  One is that set theory is "practically
complete" (Harvey), while topos theory isn't.  Another is that set
theory uses classical logic, while topos theory uses something else.
Another is that the language of topos theory has many primitives,
while set theory gets by with just one, membership.

3. I would like to thank Carsten Butz 19 Jan 1998 21:22:50 for a
straightforward, honest exposition of the senses in which topos theory
supports (intuitionistic) real analysis.

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logical positivism and Dupre's book

I don't want to waste a lot of FOM bandwidth on Dupre's book, because
(i) it's not really a very important book, (ii) it says nothing about
mathematics or f.o.m.  But here are two additional remarks.

1. When I expressed in 13 Jan 1998 13:29:45 a desire to stand up for
the unity of human knowledge, I emphatically wasn't referring to the
Vienna circle.  My view of the logical positivists is that they didn't
contribute anything to unity of knowledge.  Indeed, their most
significant legacy was to popularize a destructive fragmentation, the
fact/value dichotomy.  If Sol Feferman would like to read a truly
devastating indictment of positivism, I recommend Brand Blanshard's
"Reason and Analysis", Open Court, 1962.  I'm also opposed to
"reductionism" in Sol's 18 Jan 1998 19:21:48 sense, i.e. the
positivist belief that subjects become scientific only by reduction to
physics.  I find this unjustifiable and arbitrary.  The kind of unity
that I am talking about is a hierarchy of concepts based on sense
perception and logic and encompassing all of human knowledge, as in my
definition of f.o.m. at
http://www.math.psu.edu/simpson/Hierarchy.html.

2. Dupre is not mainly concerned with logical positivism.  Rather, he
brings forth a series of weak arguments seeking to disparage and
undercut various Aristotelian unifying concepts: natural kinds,
essences, species, causality, etc.  Since the book mentions Aristotle
hardly at all, it quickly degenerates into a series of attacks against
an opponent who is never named and so never gets a chance to defend
himself.  That's why I find this book so uninteresting.

-- Steve