[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
 >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
 >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 

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