FOM: reply to the "list 2" crowd
Vaughan Pratt
pratt at cs.Stanford.EDU
Thu Jan 22 15:11:44 EST 1998
From: Charles Silver <csilver at sophia.smith.edu>
>I don't mean to quibble, but isn't the *basic* or *foundational*
>concept that of a "collection" rather than of a "set"?
Judging from the proceedings I'd say this was true for more than 95%
of the readership. I want to take this opportunity to describe at this
very basic level how the category theoretic perception of the starting
point differs from that of the set theoretic one.
As I've emphasized several times before, I am for neither one nor the
other. However I am extremely interested in understanding the points
of similarity and difference between the two.
To avoid misunderstanding it should be said that ZF itself does not start
with a preconceived notion of collection and go from there. ZF assumes
only the language of first order logic with equality and an uninterpreted
binary relation (membership). In fact there is no a priori assumption
at all about the universe being described by the axioms, other than the
traditional (but unnecessary) assumption that it is nonempty. ZF weaves
the entire notion of set from the whole cloth of first order logic.
How one *reads* ZF is another matter. It is plausible that one starts
with some preconceived notion of "collection" (which may or may not be
weaker than the notion of "set"), and either verifies that the axioms are
sound for that notion, or if not then refines the notion (e.g. by adding
commutativity and idempotence to the properties of the union operation)
to make them sound.
Now, how do category theorists read Colin McLarty's book, which develops
topos theory axiomatically. In particular do they begin with a notion
of collection, or with something else?
Here's my own understanding of the tabula rasa that category theorists
picture themselves starting from when developing category theory
axiomatically, or at least should if the axioms are to come across as
common-sensical. I'd be very interested in Colin's response, as well
as that of Carsten Butz (whose insights on category theory have to my
taste been as on-target as any in this forum).
Whereas set theory starts from the notion of a discrete collection
and aims for the notion of continuum without ever quite reaching it
satisfactorily, category theory does the exact opposite. It starts
from the notion of "continuum," conceived just as fundamentally as
"collection", and aims for the notion of collection without ever quite
reaching it satisfactorily.
Ordinary category theory is one-dimensional, its atomic continuum being
the morphism with its two boundaries. (As an aside, unlike Colin I
believe in starting axiomatically with 2-categories at a minimum, i.e. a
planar continuum. In the category section of my universal algebra class,
I treat categories, 2-categories, and n-categories before treating natural
transformations and adjunctions, on the ground that both the underlying
intuition for and axiomatization of n-categories are simpler than those
of either natural transformations or adjunctions.)
Whether you work in one, two, or n dimensions, category theory is
fundamentally continuum-oriented. Instead of conceiving of the continuum
as a collection containing enough elements for all practical purposes,
it is a basic starting notion.
A single continuum, qua morphism, is simply not enough--you would be
unable to bisect it, for example. To escape triviality, a category must
have more than one morphism.
But it won't do for these continua to be disconnected the way set
theorists conceive of the elements of a set. At a minimum one needs a
graph, which is what the boundaries or faces are for: to provide entities
that morphisms, as graph edges, can share in order to become connected.
But mere connection is still not enough to recognize when an edge has
been bisected by its constituents. One further needs some relationship
between those constituents and the single edge they constitute. This need
is met by composition, which assigns to every consecutive pair of edges
the edge that is that path. The collection of all finite paths in a graph
(including one zero-length path at each object) is the free category on
that graph under path concatenation. The free category has no diagrams
that commute nontrivially; conversely every category is a quotient of
the free category on its underlying graph, namely that defined by its
commuting diagrams.
(Harvey, note that composition is *total* when its domain is taken to be
consecutive pairs of edges, the type C^3 where 3 denotes the category
*->*->* having two consecutivity nonidentity morphisms, the *arity*
of that type. It is natural for a set theorist to want to take pairs
of disconnected paths but this is not the natural domain for a category
theorist. Discreteness is the cradle of the set theorist but the holy
grail of the category theorist. The category theorist's cradle is the
continuum, as reflected in its choice of representation of numbers as
the connected categories 0, *, *->*, *->*->*, ... in preference to the
discrete categories 0, *, * *, * * *, ... .)
Higher dimensional edges (not needed for ordinary one-dimensional
category theory) are necessarily simply connected (no holes),and
for these the domain of n-dimensional composition must be a simply
connected arrangement of n-cells called a pasting. Although the notion of
n-category is quite elementary, that of pasting has turned out not to be.
Definitional problems enter at dimension three, with pasting diagrams of
a mere dozen or so cells already presenting problems, and things really
bog down at dimension four. This is characteristic of the difficulty of
passing from a continuum conceived purely to one conceived as being made
up of parts. In this regard Colin is doing the right thing staying away
from 3-categories, but he has no such excuse for not pressing 2-categories
into service.
>From this perspective set theory and category theory are equally
fundamental *provided* you have not done so much of one that the other has
passed from the realm of self-evident to that of highly problematical.
I maintain that before one has been corrupted by either camp, both
notions, collection and continuum, are equally good starting points for
a foundation of mathematics. Many (most?) category theorists recognize
this---as the underdogs today they are in a better position than set
theorists to do so---and are fond of reflecting wistfully over a beer
what might have been had only category theory been invented before 1870.
The cradles and holy grails of both set theory and category theory
are clearly apparent to those who work in these respective fields.
They are effectively mirror images of each other.
Occam's razor seeks the most attractive path from cradle to grail.
>From what I've seen of the two camps, it is clear that the choice of
path optimal for Occam's razor is highly dependent on its orientation.
Vaughan Pratt
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