From: Osher Doctorow osher.domain.name.hidden, Tues. Sept. 3, 2002 8:26AM
It also depends on the logic that one chooses (e.g., Lukaciewicz/Rational
Pavelka and Product/Goguen and Godel fuzzy multivalued logics - see P. Hajek
Metamathematics of Fuzzy Logics, Kluwer: Dordrecht 1998 for an excellent
exposition except for his mediocre probability section).. See my
contributions to
http://www.superstringtheory.com/forum, especially to the
String - M Theory - Duality subforum of their forum section (most of which
is archived, but membership is free, and archives are accessible to
members). Or my paper in B. N. Kursunuglu et al (Eds.) Quantum Gravity,
Generalized Theory of Gravitation, and Superstring Theory-Based Unification,
Kluwer Academic: N.Y. 2000, 89-97, which has some further references to my
earlier work.
Analysis including nonsmooth analysis does combine the discrete and the
connected/continuous, but in my opinion it generally regards the discrete as
an approximation to the continuous/connected or piecewise
continuous/piecewise connected (pathwise, etc.).
One confusing point, I think, is the tendency of many mathematical logicians
to identify with algebra and in fact to claim that their field is a branch
or outgrowth of algebra. This was originally claimed by *Clifford
Algebra,* but Clifford himself and many of his wisest descendants/followers
such as Hestenes of Arizona State U. realized than the opposite true -
*Clifford Analysis,* *Spacetime Algebra,* and so on are typical terminology
used by the latter and others to indicate that they are really dealing with
analysis and geometry and related things. Why do so many mathematical
logicians identify with algebra? Largely, in my opinion, because algebra
is much more mainstream-accepted than mathematical logic (and popular, and
respected, etc.), but also because algebra is abstract and mathematical
logic seems to many of its practitioners to be more abstract than concrete.
I have cautioned in various places that even in pure mathematics there needs
to be a balance between abstractness/abstraction and concreteness/physical
application. Analysis historically has had much more of this balance (rough
equality of abstraction and concreteness).
There are also many built-in biases in mathematical and theoretical physics,
and one of them in my opinion is the bias toward dissolution of geometry at
the sub-Planck level. Part of this is the pre-quantum computer bias
toward the discrete and finite or at most countably infinite and the digital
vs analog computer bias (in favor of digital computers). The real line and
real line segments of course are uncountably infinite and connected, and
thee would essentially be no applied mathematics or mathematical physics for
example without it - and not much pure mathematics either. It helps to
occasionally look back in mathematical history, especially to Georg Cantor's
Contribution to the Theory of Transfinite Numbers, which even Birkhoff and
MacLane in their algebra textbooks made sure to include. Of course,
Birkhoff ended up in applied differential equations and hydrodynamics
largely, but MacLane has never been accused of being Analysis-inclined to my
knowledge, and Birkhoff started out at least algebraic.
I hope that we can resist the temptation to go into absolutes. I am glad
to see that you started your reply with a tolerant and compromising tone,
and I will end this posting with a similar tone. The discrete and the
connected are in my opinion different theories or parts of different
theories overall, and they are also parts of different interpretations. My
view is that science progresses by tolerating different theories and
different interpretations for competition and because many supposedly wrong
theories or interpretations end up much later having something useful to
contribute. The majority of scientists (the *Mainstream* as I refer to
them) do not subscribe to this view, but consider that science advances in a
spiral by *killing off* the wrong theories or by only generalizing
(including generalizing in the limit) the partly correct theories - the Law
of the Jungle viewpoint of competition by (intellectual) warfare or
*cannibalistic* absorbtion as opposed to the Competing Teams idea of
competition in which one keeps other teams alive in order to keep competing
and for motivation and ultimately because one respects them and regards them
as like oneself trying to achieve the *Impossible Dream*. You may find my
contributions to math-history (see the Math Forum and epigone sites) to be
interesting in this regard.
Osher Doctorow
----- Original Message -----
From: "Tim May" <tcmay.domain.name.hidden>
To: <everything-list.domain.name.hidden>
Sent: Monday, September 02, 2002 11:07 PM
Subject: Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
>
> On Monday, September 2, 2002, at 09:22 PM, Osher Doctorow wrote:
>
> > From: Osher Doctorow osher.domain.name.hidden, Mon. Sept. 2, 2002 9:29PM
> >
> > It is good to hear from a lattice theorist and algebraist, although I
> > myself
> > prefer continuity and connectedness (Analysis - real, complex,
> > functional,
> > nonsmooth, and their outgrowths probability-statistics and
> > differential and
> > integral and integrodifferential equations; and Geometry).
> > Hopefully, we
> > can live together in peace, although Smolin and Ashtekar have been
> > obtaining
> > results from their approaches which emphasize discreteness (in my
> > opinion
> > built in to their theories) and so there will probably be quite a
> > battle in
> > this respect at least intellectually.
>
>
> I'm not set one way or the other about discreteness, especially as the
> level of quantization is at Planck length scales, presumably. That is,
> 10^-34 cm or so. Maybe even smaller. And the Planck time is on the
> order of 10^-43 second.
>
> One reason discrete space and time isn't ipso facto absurd is that we
> really have no good reason to believe that smooth manifolds are any
> more plausible. We have no evidence at all that either space or time is
> infinitely divisible, infinitely smooth. In fact, such infinities have
> begun to seem stranger to me than some form of loops or lattice points
> at small enough scales.
>
> Why, we should ask, is the continuum abstraction any more plausible
> than discrete sets? Because the sand on a beach looks "smooth"? (Until
> one looks closer.) Because grains of sand have little pieces of quartz
> which are smooth? (Until one looks closer.)
>
> But, more importantly, the causal set (or causal lattice) way of
> looking at things applies at vastly larger scales, having nothing
> whatsoever to do with the ultimate granularity or smoothness of space
> and time. That is, a set of events, occurrences, collisions, clock
> ticks, etc. forms a causal lattice. This is true at the scale of
> microcircuits as well as in human affairs (though there we get the
> usual "interpretational" issues of causality, discussed by Judea Pearl
> at length in his book "Causality").
>
> You say you prefer continuity and connectedness....this all depends on
> the topology one chooses. In the microcircuit case, the natural
> topology of circuit elements and conductors and clock ticks gives us
> our lattice points. In other examples, set containment gives us a
> natural poset, without "points."
>
> (In fact, of course mathematics can be done with open sets, or closed
> sets for that matter, as the "atoms" of the universe, with no reference
> to points, and certainly not to Hausdorff spaces similar to the real
> number continuum.)
>
> The really interesting things, for me, are the points of intersection
> between logic and geometry.
>
>
> --Tim May
> (.sig for Everything list background)
> Corralitos, CA. Born in 1951. Retired from Intel in 1986.
> Current main interest: category and topos theory, math, quantum
> reality, cosmology.
> Background: physics, Intel, crypto, Cypherpunks
>
Received on Tue Sep 03 2002 - 08:59:20 PDT