FOM: algebraic simplicity? the obliteration of logic

Stephen G Simpson simpson at
Sat Mar 14 15:44:59 EST 1998

Steve Awodey 12 Mar 1998 17:58:43 writes:
 > >The purpose of such a dialogue is to get to the bottom
 > >of exactly why the claims of categorical dis-foundations are mistaken,
 > >and in the process hopefully to illuminate some genuine f.o.m. issues.
 > Now that this prejudice is out in the open, it's clear that there is no
 > (good) reason for categorical logicians to try to have a discussion with
 > you.

Why are you unwilling to have a discussion with me?  Are you only
willing to talk to other category theorists?  However much we
disagree, we are both professional scientists.  My professional
opinion is that the claims of "categorical foundations" are
exaggerated and wrong.  I have given solid reasons for my opinion.
When scientists disagree, it's appropriate to engage in dialogue to
find out who is mistaken, or at least to illuminate the issues in
question.  I am raising some legitimate and important scientific
issues.  Why don't you want to discuss them?  Is it because you have
nothing to say?

One specific issue that I wanted to discuss with you is your claim
that the topos axioms are in some sense simpler than the ZFC axioms.
Now that the topos axioms and the ZFC axioms have been spelled out
explicitly here on the FOM list, I'd like you to back up your claim.
The topos axioms have about 5 times as many bytes and 13 times as many
primitives as the ZFC axioms.  Why do you think they are simpler?  In
what sense do you think they are simpler?  Do they present a more
coherent picture?  What is the picture?  The picture behind the ZFC
axioms is that of sets and the cumulative hierarchy.  Is there any
similarly compelling picture behind the topos axioms?  McLarty seems
to be saying that there isn't.  What do you say?  Do you think there
is no need for such a picture?

In your posting of 26 Jan 1998 01:09:16 you said:

 > the topos axioms are essentially algebraic, a condition that can
 > arguably be interpreted as a kind of logical simplicity (and is so
 > regarded by category theorists).  By contrast, ZFC does not have
 > this property (not even close).

Why do you think that being "essentially algebraic" is to be
interpreted as "logical simplicity"?  This seems strange on the face
of it.  Traditionally, algebra and logic are regarded as two different
subjects, each presumably with its own criterion of simplicity.
(Recall the earlier discussion of the fallacy of metabasis.)  Do you
disagree with this traditional view?  You seem to think that logic as
a subject is obsolete and is to be replaced by algebra.  Why?  I don't
think that obliteration of logic would be conducive to the unity of
science.  Do you think it would be?

-- Steve Simpson

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