On Monday, September 2, 2002, at 09:22 PM, Osher Doctorow wrote:
> From: Osher Doctorow [EMAIL PROTECTED], 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
