FOM: category theory, cohomology, group theory, and f.o.m.

[Till Mossakowski]

Steve Simpson Tue, 22 Feb 2000 00:21:24 -0500 (EST) wrote:

>As a non-category-theorist and a human being, I of course find this
>way of viewing quantifiers somewhat unnatural.  But putting that
>aside, don't you agree with me that this alleged definition of
>quantifiers in terms of adjoint functors is circular?  Quantification
>has to be understood *before* you can even define what you mean be a
>category, let alone a functor and a left adjoint.  After all, a
>category is defined as a certain kind of algebraic structure where
>*every* pair of morphisms of a certain kind has a composition, and
>*for every* object *there exists* an identify morphism, etc etc etc.
>This illustrates why logic is more fundamental than category theory.

You cannot define quantifiers without circularity.
The formal definition of quantifiers already assumes that you
informally know what they are.

There always is a circularity in the foundations of mathematic:

  
  Logic <----> Set theory


This circularity is illustrated nicely e.g. by the footnote on
p.9 of Joseph Shoenfield's "Mathematical logic":

   This paragraph is not intended as an introduction to elementary
   set theory, with which we assume the reader is familiar. Our
   object here is merely to establish the terminology and notation.
   An axiomatic treatment is given in Chapter 9, but only very elementary
   results will be needed before then.
   
Actually, on p.47 Shoenfield uses the Teichmueller-Tukey-Lemma (which is
proved using AC in Chapter 9), and on p.78, he uses cardinal arithmetic,
in his meta theory of mathematical logic.

Now category theory, like group theory, is based on both logic
and set theory:

    -------------------------------
    |                             |
    |                            \/
  Logic <--->  Set theory --> Category theory
  
Thus, category theory is *outside* the "foundational circle".
  
I would understand the question of the foundational significance of 
category theory as the question if there are "foundational arrows"
also going from category theory, like in:

    -------------------------------
    |                             |
    |                            \/
  Logic <--->  Set theory <--> Category theory
    /\                            |
    |                             |
    -------------------------------  

Indeed, explaining quantitfiers using adjoint functors is 
an "arrow" from category theory to logic, because a concept
of logic is explained in terms of category theory (circularity
is not the point here, since the whole picture is circular).
However, the question is if this arrow from category theory to logic
is really of the same foundational significance as the double
arrow between logic and set theory. If the arrow going
from  category theory to logic is just the explantation
of quantitfiers using adjoint functors, the answer is definitely: no.


Till

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Till Mossakowski                Phone +49-421-218-4683, monday: +49-4252-1859
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