FOM: December 1 - December 31, 1999

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FOM: NYC logic conference and panel discussion

In my FOM posting of Sun Dec 05 20:02:08 1999 I said:

 > Alex Heller pointed out that he himself is not an advocate of
 > ``categorical foundations'', but some people such as MacLane and
 > Lawvere have put forth such ideas, and Lawvere has even tried to teach
 > category theory to young children.  
 > [ I was unaware of these pedagogical experiments of Lawvere.  Can
 > anyone give a reference? ]

To this Carsten Butz replied off-line:

 > I am not aware of these experiments, but Lawvere used category theory in a
 > first introduction to mathematics. You probably know the book
 > Lawvere/Schanuel: Conceptual mathematics. A first introduction to
 > categories. (Cambridge University Press, 1997).
 > The book was reviewed by Andreas Blass in MR 99e:18001 (making reference
 > to the earlier version, published by Buffalo University, see
 > MR 93m:18001). 
 > Hope this helps. Best regards,

Yes, these references are indeed helpful.  No, I had not been familiar
with the Lawvere/Schanuel book.  Unfortunately it is not in our
library here at Penn State, but I will try to get it via interlibrary

I looked up the reviews by Blass.  Apparently the Lawvere/Schanuel
book is essentially an edited transcript of a course that Lawvere
taught to American undergraduate math students at SUNY Buffalo,
somewhere around 1990.  The course sounds pretty much like the
standard introduction to rigorous mathematics for undergraduate math
majors, except that everything seems to be phrased in categorical
language, specifically the category of sets.

This seems like an interesting pedagogical experiment.  Was there any
follow-up, to see whether the students learned what they needed for
subsequent rigorous math courses?

Was this the experiment to which Heller was referring?

By the way, the Blass reviews remind me of another book that appeared
recently, in the vein of teaching ``categorical foundations'' as an
alternative to standard set-theoretic foundations.

  Paul Taylor, ``Practical Foundations of Mathematics'', Cambridge
  University Press, 1999, XI + 572 pages.

But Taylor's title made me smile, because the Taylor book seems to
contain a huge amount of advanced category theory, much more than the
Lawvere/Schanuel book.  I question whether it is really ``practical''
to teach category theory to students who do not already know and
appreciate a large amount of advanced, rigorous mathematics,
particularly abstract algebra, at the graduate level.

I also doubt that it is possible to teach topos theory to someone not
already familiar with the elements of set theory.  Has anybody tried
this?  If so, what was the pedagogical outcome?

-- Steve

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