FOM: Topos book

Colin McLarty cxm7 at
Mon Jan 26 15:26:20 EST 1998

Harvey Friedman wrote:

        I mentioned my book:

>This is an irresponsible way to make your points on the fom. People are
>busy, your book may not be easily accessible, and nobody thinks that the
>best way for you to defend this stuff is necessarily in a six year old
>book, since you now have the benefit of direct criticism from so many
>genuine f.o.m. professionals. E.g., your book is taken out of the library
>here, so I have never seen it.

        It would be irresponsible to refer to the book instead of posting my
points, but I referred to it along with posting my points and a summary of
the relevant book passages.

        Various people here demand "concessions" from me, and accuse me of
shifting my ground. And I reply by pointing out that the things I say on the
list I have been saying in print for years. 

>"Springing from many disparate examples" does not entail "no coherent
>conception of the mathematical universe that unlerlies categorical
>foundations." You still have not conceded that there is no coherent
>conception of the mathematical universe that underlies categorical

        Your observations are correct, and closely related. I do not concede
that "there is no coherent conception of the mathematical universe that
underlies categorical foundations". I happily "concede" that the coherence
is not of the type you and Steve Simpson want: It is not a matter of
formalizing one fairly well defined example of mathematical structure (as ZF
formalizes the example of iterated collection). It is precisely a matter of
finding the unity behind many disparate examples. I am proud of how well
category theory has unified disparate structures in working mathematics (as
I said before, of course it is not of my doing) and the uniformity of method
it gives for foundations: e.g. of set theory, differential geometry, and
recursive analysis. I call this "foundations" in the sense that you have
axioms which (without depending on any prior conception of sets or whatever)
express basic intuitions of these subjects and give the theorems.

        I do not believe, as John Mayberry thinks, that merely interpreting
the theorems gives a foundation.

        The lesson I draw from John misunderstanding me, is that these are
topics where a little difference of opinion can easily lead to a large lack
of understanding. John and I have talked about these things at length, and
corresponded. For him to misunderstand me on a point which is itself so
clear to him, and where I have learned so much from him, shows this is
subtle stuff. Anyone who likes is free to decide it is just my fault, I'm
terribly obscure. But it might be more productive to wonder how many other
apparently wild disagreements are misunderstandings. 

>Grothendieck as well as almost everyone else right now "uses perfectly
>usual set theoretic foundations." Why don't you emulate him, instead of
>trying to use his name in order to promote categorical/topos "foundations?"

        Well, most mathematicians "use" ZF set theory in a pretty loose
sense. They can't give the axioms. They are indifferent to the differences
between ZF and categorical set theory. 

        I don't emulate Grothendieck because we are no longer in 1956. We
know things he did not, until he discovered them.  

>Were you a five year colleague of Lawvere and Schanuel? I know them. And I
>have talked to MacLane a fair amount over the years.

        I am a 25 year friend of Lawvere (not that I like to think its been
that long). And I know that things you say about "category theorists" are
not true of him, or Schanuel or Mac Lane.

        In fact, I have been amazed at some of what people on this list
think of category theorists--various ideas that categorists want to conceal
logic, hate sets, and so on. (On the other hand, though Vaughan Pratt has
lately found a major objection to my vision of category theory, I always
like his observations on category theorists.)   

>Lawvere told me that the existential quantifier is
>not only best understood as some sort of map or functor (or other more
>complex categorical object) - but can only be properly understood that way.

        And did you suppose he meant noone should say things like "there's a
car coming" or "there is an empty set" until they have studied adjunctions
and pullback functors?

        Perhaps you did. It wouldn't be much beyond some of the crossed
communication we've had lately. 

>>pages 177-78 discuss arithmetized real analysis in any topos with a
>>natural number object. The usual definitions are used, but in the weaker
>>topos logic they do not give all the classical results. For a start,
>>the Cauchy and Dedekind reals do not agree.
>You seem to have gone out of your way not to make this point clear in your
>response to my 9:06 PM 1/17/98. Your response was 10:59AM 1/18/98.

        I am sure it did seem so to you, since you say it did. And I am
trying to understand how it could. In your post you asked for analogues to
statements you would use in a calculus course. I gave analogues statement by
statement, as you had more than emphatically requested. Why would you expect
those specific analogues to make this general point about toposes clear?


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