FOM: Re: three challenges for McLarty

Colin McLarty cxm7 at
Mon Jan 26 11:44:55 EST 1998

Stephen G Simpson wrote :

>    Challenge 1. McLarty needs to explicitly state fully formal axioms of
>    topos theory plus natural number object etc.  These axioms are to be
>    compared side-by-side with the very concise yet fully formal axioms of
>    set theory which were posted by Harvey in 23 Jan 1998 00:54:20.  (The
>    purpose of the comparison is to see that the set theory axioms are
>    much, much simpler than the topos axioms.)

        I'll come to this in another post, on another day.        

>    Challenge 2. McLarty needs to forthrightly concede that topos with
>    natural number object is not sufficient for undergraduate real
>    analysis, specifically the standard undergraduate material on
>    calculus, ordinary differential equations, partial differential
>    equations, power series, and Fourier series.

        I have "conceded" repeatedly that classical analysis requires
more than a topos with natural number object. I "conceded" it in a book 
6 years ago, before I knew it was concession, and I have said it repeatedly
on this list. When you can say what you mean by "the standard undergraduate
material" then we can talk about that.

        Prof. Simpson's research concerns various subsystems of formal
second order arithmetic. Most of these give less than "the standard
undergraduate material" mentioned above. Whether ANY of them give it all,
depends on just what "the standard undergraduate material" is. Yet he speaks
of doing "real analysis" in these subsystems. That is the sense in which I
say you can do "real analysis" in any topos with natural number object.    

>    Challenge 3. McLarty needs to forthrightly concede that, as Harvey put it,
>     > there is no coherent conception of the mathematical universe that
>     > underlies categorical foundations in your sense.

        I freely concede categorical foundations don't rest on what you and
Harvey think of as a "coherent conception of the mathematical universe". It
rests on a conception I find coherent.

        My post of Mon, 19 Jan 1998 20:09:12 said what I still consider true:
>>        I think this is the central difference. Set theoretic
>>foundations aim at one rather narrowly described picture (though of
>>course there are variants), categorical foundations have many quite
>>different variants. I think the range of variability, the organic 
>>unity of that range, and the different relations of different variants 
>>to mathematical practice, are advantages. Simpson thinks it makes
>>categories foundationally inviable. We may well be at bedrock here.

        The reference to you concerns your post of Sat, 17 Jan 1998 21:06:04.

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