Subject: Intuitionism's Limits

From: categories <[email-masked]>
Date: Sun, 2 Mar 1997 15:18:54 -0400 (AST)

Date: Sun, 2 Mar 1997 15:07:12 +1030 (CST)
From: William James <[email-masked]>

Intuitionism's Limits: if C is a category sufficiently complex to
demonstrate that some C-arrow f:a-->b is monic and B is a subcategory
of C containing just f (and the requisite identity arrows), do we
still know that f is monic? Should we? (Or, in other words, which
view *should* dominate: Intuitionism, Realism, the category
theoretic...?) What if C is something (semi?)fundamental like a
category of all sets and functions, or a category of categories?

I suppose the answer is that monicity is relative to a category,
but what supports this as a claim? And doesn't it seem to contradict
the reasonable realist claim that we can somehow know f in B to be
monic? (Or am I missing something straightforward: that properties
can be granted to f by its relationship to C via an inclusion functor?)

This goes to the issue of the adequacy of category theory as a foundation
in more than the simply technical sense.

(I could be using the term "realism" incorrectly too: I take it to be
a positon, in maths at least, that mathematical entities can
have collections of properties beyond the constraints of a given
theoretical context.)

William James

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From: categories <[email-masked]>
Date: Mon, 3 Mar 1997 10:35:21 -0400 (AST)

Date: Sun, 2 Mar 1997 15:27:28 -0500 (EST)
From: John Baez <[email-masked]>

William James <[email-masked]> writes:

> but what supports this as a claim? 

It seems to me that category theory takes the sensible viewpoint that
mathematical entities (e.g. objects and morphisms) only become
interesting through their relationship with other entities.  Every
arrow looks just like every other arrow if we consider it in
isolation.  Every arrow in a category C is an image of the what James
Dolan calls the "walking arrow" --- the nonidentity morphism in the
free category C0 on a single morphism --- under some functor F: C0 ->
C.  Studying an arrow in isolation is just like studying the walking
arrow, which is completely dull.  The fun begins only when we have a
bunch of arrows and start composing them.

This is one reason why I think n-category theory should be useful in
physics problems like quantum gravity, where it only makes sense to
speak of where or when an event occurs relative to other events, not
with respect to some spacetime manifold of fixed geometry.  For
some of the technical apsects of how this might go, see:

John Baez and James Dolan, Higher-dimensional algebra and topological
quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

Louis Crane, Clock and category: is quantum gravity algebraic?,
J. Math. Phys. 36 (1995), 6180-6195.  

These both appeared in a special issue on diffeomorphism-invariant 
physics.

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From: categories <[email-masked]>
Date: Mon, 3 Mar 1997 10:36:01 -0400 (AST)

Date: Sun, 2 Mar 1997 15:52:17 -0500 (EST)
From: Peter Freyd <[email-masked]>

To William James,

  You must be using a non-standard (philosophical?) definition of
"monic", since it is obvious using the standard (mathematical)
definition that a monic remains a monic in any subcategory containing
it (to get unnecessarily technical, because it's given by a
universally quantified Horn sentence). Could you tell us your
definition?

  (For the record: f: A -> B  is monic iff for all  x,x':X -> A  it 

                    x    f          x'   f
is the case that  X -> A -> B  =  X -> A -> B  implies  x = x'.)

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From: categories <[email-masked]>
Date: Mon, 3 Mar 1997 10:36:46 -0400 (AST)

Date: Sun, 2 Mar 97 21:09 EST
From: Fred E J Linton <[email-masked]>

If  a  and  b  are *two* objects, then, in the category consisting solely of
those two objects, their respective identity maps, and one further map from
 a  to  b  (and nothing more), that map is both monic and epic.  Once embedded
in another category, however, that map may easily fail to remain monic, may
easily fail to remain epic, may remain one but not the other -- there's no
telling.  And if  a = b  instead, and  f  and the identity on  a  are the
only *two* maps there are, then clearly  f  *may* be idempotent, hence neither
monic nor epic; then again,  f  *may* be involutory, hence a true isomorphism.

I think true realism requires that one pay strict attention to the definitions,
refraining from free-associations with the vibrations of the terms defined.

Cheers,

-- Fred

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From: categories <[email-masked]>
Date: Mon, 3 Mar 1997 10:37:39 -0400 (AST)

Date: Mon, 3 Mar 1997 18:40:30 +1030 (CST)
From: William James <[email-masked]>

> category of all sets and functions, or a category of categories?

Whoops! The question is trivialised by using monicity as the relevant
property. Reconsider it in terms of say f as an isomorphism, or of
f holding some property in C that B lacks the resources to demonstrate.

I'm thinking aloud on this question: constructive maths should say that
of f in B there is no demonstration forthcoming, so judgment will be
withheld on whether or not f has the property; a category theorist might
say that category theory does not dwell on elements and that, in context,
B is no different from any isomorph of 2, so there positively is no
further property of f to be had other than that which can be
demonstrated in any isomorph of 2. This is more than Intuitionism will
allow.

Might I, then, go on to say that the philosophies of constructive
mathematics and category theory really are different?

William James (if I'm digging a hole, I want it to be big)

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From: categories <[email-masked]>
Date: Thu, 6 Mar 1997 13:29:47 -0400 (AST)

Date: Wed, 05 Mar 1997 22:36:19 -0800
From: Vaughan Pratt <[email-masked]>

	Date: Tue, 4 Mar 1997 18:19:51 +1030 (CST)
	From: William James <[email-masked]>
	(I grant you the original question would have been more
	recognisable given better use of language: "...philosophies of
	constructive mathematics and *of* category theory...")

Your original question was "Which view should dominate?", where "the
category theoretic view" was one of your options.  (You had several
questions but this one seemed the most central.)

If you are asking whether the primary expression of structure should be
in terms of relations between elements or transformations of objects,
then I would answer this as follows.

The analogous question for physics is whether energy and matter consist
of particles or waves.  The consensus in physics today is that both
energy and matter can be viewed more or less equally accurately, if not
equally insightfully, as either particles or waves.  Which offers more
insight depends on the circumstances.

The corresponding position for mathematics would be that structure can
be expressed more or less equally well in elementary or
transformational terms, and that which approach gives more insight
depends on the circumstances.

The extent to which this is not the consensus in mathematics today is
less a reflection on either approach than on the conceptual health of
mathematics relative to that of physics.

Vaughan Pratt