* Inevitability of ordering products
@ 2001-02-11 20:19 John Duskin
2001-02-13 0:39 ` James Borger
0 siblings, 1 reply; 6+ messages in thread
From: John Duskin @ 2001-02-11 20:19 UTC (permalink / raw)
To: categories
I guess that the point I was trying to make in my post was missed in
my telegraphic submission. I'll try to make what I was trying to say
clearer.
If you take the "representable functor" definition of a product of a
object A with an object B, you say that an object P together with an
ordered pair of arrows (p_A:P--->A,p_B:P-->B) is a product of A with
B if for all objects T , the mapping Hom(T,P)--->Hom(T,A)xHom(T,B),
defined by f |---> (p_A.f,p_B.f) is a bijection. This makes it clear
that P together with (p_A,p_B) represents the product of A with B and
P together with (p_B,p_A) represents the product B with A and that
these are not representations of the same functor but that there is a
natural isomorphism of such an object with itself which "interchanges
the two projections". Note also that this distinction remains even
for a product of A with itself which leads a pr_1 and pr_2 notation
for the two "projections".
Now, as with all such bijections, one can eliminate all reference to
sets of morphisms by a "for all x there exists a unique y such
that..." statement which here becomes: for all ordered pairs of
arrows of the form (x:T--->A,y:T--->B) , there exists a unique arrow
f:T--->P, such that (p_A.x,p_B.y)=(x,y), and the distinction made in
in set theory between AxB and BxA," that they are not equal, but
there is a 'canonical' bijection between them", is perfectly
maintained entirely within category theory... or is it? The use of
the ordered pair term "(x,y)" still appears in the replacement first
order statement, and if the only way that one can use "ordered pair"
is through von Neumann's clever but rather grotesque (x,y)={x,{x,y}},
one has to pull in "Peano's entirely spurious singletons" , as Bill
Lawvere referred to them.
Now if the purpose of an "underlying logic" us to "codify by making
explicit the normal habits of reasoning that all mathematicians will
accept" and that "an object P and arrows p_A:P--->A and p_B:P--->B"
is equivalent to "an object P and arrows p_B:P-->B and p_A:P-->A",
or more starkly, the logical equivalence of,"p_A and p_B" and "p_B
and p_A". Then there would seem to be no way that a purely first
order category theory could make the distinctions that we teach in
every elementary math course about the distinction between (x,y) and
(y,x), unless we appeal to informal aspects of everyday language
where "and" is sometimes commutative and sometimes not, and thus in
this case rely on "everybody's having met (x,y) long before they have
ever heard of a 'category' ".
Apparently this subtlety has surfaced before: In the beginning of
Bourbaki's Theorie des Ensembles,\footnote{which, remember, was
written by "working mathematicians" who did not consider themselves
"professional logicians or professional set-theorists", and may even
have had some contempt for them (among others) if we are to judge
from a certain fronts piece photograph inserted by Andre Weil into
the Fascicule de Resultats.} they introduced as "specific signs", in
addition to those of equality and membership, another sign of weight
2, \couple xy, ultimately written as (x,y) together with the Axiom
(A3) :
(for all x)(for all y)(for all x')(for all y' ) (x,y)=(x',y') implies
x=x' and y=y'. They then define "z is an ordered pair (couple)" by
"(there exists x)(there exists y)(z=(x,y))", which then gives (x,y)
its first and second projections because of the way that "there
exists " is constructed (using their "\tau operator"). The existence
of the cartesian product of the set A with the set B then follows,
as usual, as the set of ordered pairs,since the presence of (x,y)
allows them to describe formal "relations" R|x,y| as properties of
the ordered pair z=(x,y). Only later do they observe, in an
exercise, that the little von Neumann nest of singletons has the
property of the axiom A3, but they use this only to show that
\couple xy together with A3 is relatively consistent with their other
axioms for set theory.
It is clear, however, that \couple xy and A3 could have been
introduced immediately after they had done quantification and long
before any of the axioms for \epsilon were introduced. But, after
all, they were trying to use their "Theory of Sets" as a foundation
for all of mathematics, so most people have considered this whole
business of adding \couple xy and A3 at so fundamental a level an
eccentric and superfluous curiosity, and it has all but been
completely forgotten.
My point is not to pull category theorists back into the intricacies
of Bourbaki 's treatment of logic, but rather to point out that the
idea of an ordered pair has at least once before been considered a
notion that properly belongs somewhere anterior to set theory and can
be used in category theory without fear of the latter suffering from
any "set-theoretic contamination". In any case, to my eyes, the use
of "lists and addresses" with their attendant ordering seems to be
pretty fundamental in computer science.
Amusingly, Grothendieck "pushed" representability in the forlorn hope
that it would convince working mathematicians that they did not have
to give up their Cantorian Paradise of " set-theory" in order to
make use of the unique insights provided by "category theory", but
then had to (re-)invent "universes" when the old paradoxes of
set-theory and the category of all sets, all groups, etc. carpingly
resurfaced.
Bill Lawvere, in contrast, noticed that when the working axioms of
set-theory were rephrased in purely "category-theoretic" terms, that
they, amazingly, all became "first order" statements , thereby
raising the question of an entirely new way to look at foundational
questions in which the pesky membership paradoxes could not arise nor
even be formally expressible. He, in contrast to Grothendieck,
"pushed" the much more radical move of, effectively, "banning all use
of Hom-sets" and thereby made the divide crystal clear.
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: Inevitability of ordering products
2001-02-11 20:19 Inevitability of ordering products John Duskin
@ 2001-02-13 0:39 ` James Borger
0 siblings, 0 replies; 6+ messages in thread
From: James Borger @ 2001-02-13 0:39 UTC (permalink / raw)
To: categories
On Sun, 11 Feb 2001, John Duskin wrote:
> Bill Lawvere, in contrast, noticed that when the working axioms of
> set-theory were rephrased in purely "category-theoretic" terms, that
> they, amazingly, all became "first order" statements , thereby
> raising the question of an entirely new way to look at foundational
> questions in which the pesky membership paradoxes could not arise nor
> even be formally expressible. He, in contrast to Grothendieck,
> "pushed" the much more radical move of, effectively, "banning all use
> of Hom-sets" and thereby made the divide crystal clear.
Can someone give a reference to an elaboration of this point of view?
Since I am not a real category theorist and have at best a weak
understanding of foundations, something written for the mainstream
mathematician would be even better.
Thanks in advance.
Jim Borger
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: Inevitability of ordering products
2001-02-13 4:54 ` Dusko Pavlovic
@ 2001-02-14 18:40 ` Eduardo Dubuc
0 siblings, 0 replies; 6+ messages in thread
From: Eduardo Dubuc @ 2001-02-14 18:40 UTC (permalink / raw)
To: categories
This is concerning the mail of Dusko in reply to my mail:
>
> Eduardo Dubuc wrote:
>
> > what sense has the concept of unlabeled graph ?
> >
> > try to put an unlabeled graph inside a computer ?
>
> you mean unordered? i would implement it as an ordered graph, with an
> additional involutive map on the edges, ie
>
> Edges <--inv-- Edges ==dom,cod==> Nodes
> dom.inv = cod
> inv.inv = id
>
> --- which, in a way, confirms that
>
yours is not an answer to my question, which I shall explain now (I
thought that it needed no explanations)
by an unlabeled graph i mean the drawing of a graph, in paper, say, or a
graph buildt in space, the skeleton of a building for example. It has
vertices and edges, and everybody knows what it is. Mathematically you
could say a symetric relation on its (finete) set of vertices. But not
quite so ...
If you have n vertices, you have n! different labeling. Each labeling
gives you a different labeled graph.
The minute you have a set (in the mathematical sense) of vertices, you
have a labeling. Namely, the elements of that set are the labels!. So,
with a symetric relation (in the mathematical sense) what you got is a
labeled graph. Not an unlabeled graph !.
And you become well aware of this fact when you want to put a concrete
unlabeled graph (say, the skeleton of a building) inside a computer !!
REMEMBER I rise the question on unlabeled graph related to the question
that we were discussing:
INEVITABILITY OF NAMING (IN MATHEMATICS)
(naming is not the same as labeling ?)
>> well, unlabeled graph has to be a quotient by an equivalent relation
...
I said that. It seems possible. I explain now the ...
Given two graphs R, S (symetric relations) on a finite set X (of
vertices), consider the natural action of the symetric group of X on the
power set of X x X. Then, R =~ S iff they are in the same orbit. The
elements of the quotient set are the unlabeled graphs.
e.d.
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: Inevitability of ordering products
2001-02-12 20:27 ` Eduardo Dubuc
@ 2001-02-13 4:54 ` Dusko Pavlovic
2001-02-14 18:40 ` Eduardo Dubuc
0 siblings, 1 reply; 6+ messages in thread
From: Dusko Pavlovic @ 2001-02-13 4:54 UTC (permalink / raw)
To: categories
Eduardo Dubuc wrote:
> what sense has the concept of unlabeled graph ?
>
> try to put an unlabeled graph inside a computer ?
you mean unordered? i would implement it as an ordered graph, with an
additional involutive map on the edges, ie
Edges <--inv-- Edges ==dom,cod==> Nodes
dom.inv = cod
inv.inv = id
--- which, in a way, confirms that
> well, unlabeled graph has to be a quotient by an equivalent relation ...
isn't the "ordering" of the components of a product AxB (by the names,
colors A and B), in a similar way, "factored out" by the canonical
isomorphism with BxA? isn't coherence theory the way we can always factor
out such arbitrary annotations on objects?
(SORRY i am posting too much.)
-- dusko
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: Inevitability of ordering products
2001-02-08 18:48 Charles Wells
@ 2001-02-12 20:27 ` Eduardo Dubuc
2001-02-13 4:54 ` Dusko Pavlovic
0 siblings, 1 reply; 6+ messages in thread
From: Eduardo Dubuc @ 2001-02-12 20:27 UTC (permalink / raw)
To: categories
ordering is not inevitable
naming is inevitable
how do you refer to a proyection withot naming it ?
of course naming a proyection by the object it proyects is the mother of
all evil
have you delt with actions of the symetric group, say, on polynomials on
several variables ?
There is a related (if not the same) confusion here.
what sense has the concept of unlabeled graph ?
try to put an unlabeled graph inside a computer ?
well, unlabeled graph has to be a quotient by an equivalent relation ...
and so on ...
^ permalink raw reply [flat|nested] 6+ messages in thread
* Inevitability of ordering products
@ 2001-02-08 18:48 Charles Wells
2001-02-12 20:27 ` Eduardo Dubuc
0 siblings, 1 reply; 6+ messages in thread
From: Charles Wells @ 2001-02-08 18:48 UTC (permalink / raw)
To: categories
Vaughn Pratt made some valid points about my earlier remarks on the
inevitably of binary product projections being ordered. For the most part,
I agree with him (but see below), since my (unclearly made) point was that
it is inevitable in our current mathematical culture, not that it was
mathematically inevitable.
However, I am stuck on one point: Sometimes one needs to refer to one of
the projections, and that involves giving the projections names. I
mentioned "red" and "blue" as examples of names that do not introduce a
spurious ordering. But in practice, we must occasionally give names. This
is not only for computation, either. For example, one sometimes needs to
assume that an n-ary operation factors through one of the projections, and
then deduce consequences from that (Peter Johnstone did something like that
in his study of varieties that are ccc's). In the proof one must give a
name to the projection it factors through.
So I argue that naming the projections is sometimes a practical necessity,
and given current mathematical habits the names are likely to have some
intrinsic (culturally intrinsic!) ordering. But we could use red and blue.
Or vanilla and chocolate.
Charles Wells, 105 South Cedar St., Oberlin, Ohio 44074, USA.
email: charles@freude.com.
home phone: 440 774 1926.
professional website: http://www.cwru.edu/artsci/math/wells/home.html
personal website: http://www.oberlin.net/~cwells/index.html
NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm
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