[FOM] Formalization Thesis
messing
messi001 at umn.edu
Sat Dec 29 19:14:18 EST 2007
>In ZFC a category is defined as an
>ordered sextuple: C = (M, O, s, t, c, i) where M and O are sets
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>Kutateladze wrote: In ZFC, yes; but a category may fail to be a set.
If one wants to view a category as an abstract "structure" subject to
some axioms, which are not expressible in ZFC, I would like to see an
assertion concerning categories that can be made and proven which is not
equivalent to one expressible if one takes the definition of category in
ZFC (augmented, if necessary by the Grothendieck axioms on universes).
Certainly as a mathematician who uses category theory every day in my
research (see, for example, my paper with Larry Breen The Gifferential
Geometry of Gerbes, math.AG/0106083), I know of no such example. If
Kutateladze means that a category may fail to be a set because one wants
to speak of the category of all groups, then this is fine in NBG, as it
exists as a class. As I already pointed out in my post concerning FLT,
in rerpsonse to the "anonymous number theorist", the problem from this
point of view, as noted explicitly by Grothendieck in his Algiers
lectures of November 1965, is that if one allows in NBG categories to be
classes, then, if C and D are such "large categories", there is no
category HOM(C,D) whose objects are the functors F:C --> D (because
functions whose source is a class, do not themselves form a class. This
is precisely why Grothendieck introduced universes, as it is essential
in algebraic geometry to have such HOM categories, e.g. categories of
presheaves, sheaves for a chosen Grothendieck topology, e.g. etale,
fppf, fpqc, crystalline, syntomic ....
>Messing wrote:... hieroglyphs... nothing but an historical
>circumstance.
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>Kutateladze wrote:This is not so. There is an analogy between the
>natural languages(Maya and Latin, for instance) and first order
>theories (ZFC and CM) inexpressible in one another. My tacit hope is
>that this analogy might help to acknowledge the inadequacy of the FT.
>Historical circumstances are much more than nothing---that's my
>another point.
I do not understand. I never maintained that the analogy between
natural languages, whether Mayan or Russian or English or pig Latin,
could be expressed in ZFC, or conversely. "Analogy" is not a logical or
mathematical notion and certainly what is being said here is, it seems
to me, comparable to saying "drinking vodka is not the same as drinking
chocolate milk." As Bourbaki says on page 10 of his introduction to the
volume Set Theory: "Nous ne discuterons pas de la possibilite
d'enseigner les principes du language formalise a des etres dont le
developpement intellectuel n'irait pas jusqu' a savoir lire, ecrire et
compter." As at least one member of the editorial board of FOM has
written that his French is not up to understanding this quote from
Bourbaki, allow me to translate: "We will not discuss the possibility of
teaching the principles of a formalized language to beings whose
intellectual development has not reached the level of knowing how to
read, write and count."
I certainly concur that historical circumstances are much more than
nothing. But this is, I think, irrelevant, in a discussion of the
Formalization Thesis. The mutiny on the Potemkin in 1905 was caused by
historical circumstances and this is relevant in discussing events
during the Autumn of 1917. One does not have to be Wittgenstein to note
this.
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