FOM: Toposy-turvey

Colin Mclarty cxm7 at
Tue Jan 20 09:15:07 EST 1998

Reply to message from kanovei at of Mon, 19 Jan
>It seems that the Harvey Friedman's critics will 
>not make a big problem for Colin McLarty, because 
>the latter has silently distanced his "course" 
>from the categorical mainstream. 

	No, I haven't. And on the other hand I doubt Friedman
will agree with you that his criticisms are no problem for me. 

>Indeed he does not start his "course" with 
>a prayer like 
>"there are no Categories except for Categories 
>and the Topos is their Prophet". 

	Does this make you suspect that categorists are not a 
religious cult? Or do you only conclude that we are a secretive

>He starts and continues with the Bourbaki set 
>theory. (He did not specify this clearly, indeed.)

	This is a good point. You missed some technically important
but practically insignificant distinctions: Bourbaki can form the 
intersection of any two sets, I can only intersect subsets of some 
single set. And I treat elements as a special case of functions. 
These both came into clause 1. But neither stood out to you.

	You focussed on the most important point. Even on the level
of semi-formal rigor that Friedman proposes, categorical set theory
is not very different from the element-based set theories. The 
differences appear in the style of axiomatization when you are quite 
formal about it, in considerations of the whole universe at once,
and in the relation to working methods of advanced algebra and 
geometry--none of which are often issues in a calculus course.

best, Colin

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