FOM: Golden Age/FOM plans
Vaughan R. Pratt
pratt at cs.Stanford.EDU
Thu Oct 23 20:01:18 EDT 1997
I haven't responded to any of Harvey's several postings because I am in
general agreement with much of what Harvey says. The one little point
that I would differ on is the place of large cardinals in foundations,
Harvey's ingeniously constructed defenses of them notwithstanding (and
Adrian Mathias' too, I should add).
Large cardinals have been partying seismically in the mathematical
basement ever since Cantor, and Harvey is encouraging them at a time
when we should be chasing them back into their mountain eyries.
The origin of large cardinals is the uninhibited discarding of the
structure most mathematical objects go around decently clothed in, or
in more straightforward language, taking 2^X to be the *set* of all
subsets of X. Just as sets strut round naked as a jaybird, 2^X is all
dolled up like Sigourney Weaver slugging it out with aliens, with more
equipment than any other object on the mathematical planet, namely the
full regalia of a complete atomic Boolean algebra (CABA). (3^X is also
a CABA, having no additional structure; its axes are more finely
divided but it transforms equivalently to 2^X nonetheless.)
Forming 2^(2^X) a la Cantor entails first discarding the structure of
2^X. This happens automatically in the category of sets because no set
has structure, and the underlying set of 2^X (which is a lot smaller
than that of 3^X) is the optimal approximation in Set to the
requirement that 2^X consist of the maps from X to 2. It consists of
them all right, but only discretely so, i.e. minus all their
structure.
People should *pay* to remove that structure. You have no appreciation
for what it costs. You (or the HMO's) blithely accept the
radiologist's stiff fee for an X-ray of your chest, and the surgeon's
far greater fee to open you up, yet you treat stripping off the skin
and muscle of a CABA as if it cost nothing! And so you do it over and
over, and even past \Aleph_\omega you still haven't paid a dime, like
LA stealing all of California's water!
Ok, you ask, so what difference would it make if a fee were levied and
you accordingly left the structure in place except where it was
*really* worth it to you to strip it off, just every now and again
instead of over and over?
Well, when you *don't* take the structure off a CABA, 2^(2^X) is
isomorphic to *the set* X. The CABA homomorphisms from the CABA 2^X to
2 (qua 2^1), treated schizophrenically as the CABA 2, are the monotone
functions that send exactly one atom (and hence one "back wall" or
complete ultrafilter of the CABA) to 1; equivalently (by duality), the
functions from the set 1 (the exponent in 2^1) to the set X, i.e. the
elements of X. Instead of ascending the ladder to Cantor's paradise,
you simply bounce right off the structure ceiling and are returned to
your original place. This happens within the space of two
exponentiations.
To punch through the structure ceiling should cost, just like an X-ray
or a plane ride.
(The rule about schizophrenic objects like 2 is that by default they
have the maximum possible structure, but when they crash a party whose
guests are, shall we say, a little less structured, they are allowed to
shed just enough structure to let them mingle without attracting
attention, like Chelsea's detail.)
Although this perfect return to X might appear critically dependent on
X having no structure, in fact it is completely independent of the
structure in X. In the classical 2-valued-logic world, 2^(2^X) is
isomorphic to X no matter what structure X has. In 3-valued logic
(independently of what structure you choose to equip your 3 values
with, all the way up to the limit), 3^(3^X) is likewise isomorphic to
X. And so on for yet higher values of 3.
A more detailed account of this situation can obtained from my LICS'95
paper, "The Stone Gamut: A Coordinatization of Mathematics", expressing
this view of foundations. It can be found in the LICS'95 proceedings,
or on the web as
http://boole.stanford.edu/pub/gamut.ps.gz
It is a theory of the transformable objects of mathematics, which I lay
out in two dimensions.
The horizontal dimension is the discrete-coherent "gamut," represented
numerically as the real interval [-1,1]. The completely discrete
objects, namely sets, have coherence -1, and the completely coherent
objects, namely CABA's, have coherence 1. Self-dual "mens sana in
corpore sano" objects like finite (more generally locally compact)
abelian groups, finite-dimensional (more generally topological) vector
spaces, complete semilattices, and finite chains with bottom, all have
coherence 0, being a perfectly symmetric blend of discreteness and
coherence. I call that horizontal dimension the Stone gamut
("spectrum" already has another meaning in the theory of Stone
spaces).
The vertical dimension is linguistic complexity, represented as the
ordinals, with 2 being where interesting Stone duality begins. To
match up this notion of linguistic complexity with FOL's, a structure
of total arity k (add up the arities of the relations, and add 1 for
every sort *if* it is a heterogeneous structure---so 1 sort is free, 2
sorts cost 2, 3 sorts cost 3, etc.) corresponds weakly to a linguistic
complexity of 2^k. (I say "weakly" because k-ary relational structures
live at level 2^k but populate that level exceedingly sparsely.)
The basis for this two-dimensional organization of transformational
mathematics is Chu spaces, named for a master's student of Mike Barr
who worked on the V-enriched version of them just long enough to get
his master's thesis in the 1970's, which appeared as the appendix of
Barr's LNM monograph on *-autonomous categories. The case V=Set,
ordinary Chu spaces, was first treated by Laont and Streicher in
LICS'91, my interest in them only began in 1992. A guide to what some
of us here have been writing since then about ordinary Chu spaces can
be found at
http://boole.stanford.edu/chuguide.html
A Chu space over a set K is simply an AxX matrix over K that transforms
via "continuous" functions suitably defined. (The V-enriched case does
all this in the internal logic of the symmetric monoidal closed
category V, and came first historically just like integration came
before differentiation.) A referee was kind enough to describe one of
my more recent attempts at an introduction to Chu spaces as
"accessible", namely
http://boole.stanford.edu/pub/parikh.ps.gz
so this would be one place to start if you can't find it in your heart
or mind to attach "accessible" to the older Gamut paper. :)
Vaughan
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