Time as a Lattice of Partially-Ordered Causal Events or Moments
On Saturday, August 31, 2002, at 11:31 PM, Brent Meeker wrote: Time is a construct we invented to describe things. Most basically we use it to describe our sequence of experiences and memories. We feel hot and cold, but we needed to quantify hot and cold and give them operational definitions in order make definite predictions about them. So we invented temperature and thermometers. For mechanics we needed a quantified, operational definition of duration - so we invented time and clocks. Besides psychological time,there are at least three different possible definitions of time used in physics What they all have in common is that they assign numbers to different physical states, i.e. they index different states into some order so that this sequence of states can be compared to that sequence of states. I don't have a comprehensive theory of time, but I am very fond of causal time. Picture events as a series of points in a lattice (a graph, but with the properties I talked about a while back in a post on partially-ordered sets). Basically, a lattice of events where there is at most one edge connecting two points. (There are formal properties of lattices, which the Web will produce many good definitions and pictures of.) Lattices capture some important properties of time: * Invariance under Lorentzian transformations...any events A and B where B is in the future light cone of A and A is in the past light cone of B, will be invariantly ordered to all observers. * The modal logic nature of time. Multiple futures are possible, but once they have happened, honest observers will agree about what happened. (Echoing the transformation of a Heyting algebra of possibilities into the Boolean algebra of actuals...this sounds like it parallels quantum theory, and Chris Isham and others think so.) * Personally, I believe the arrow of time comes from more than statistical mechanics. (I believe it comes from the nature of subobject classifiers and the transformation Heyting -- Boolean.) * I am indebted to the books and papers of Lee Smolin, Fotini Markopoulou, Louis Crane, Chris Isham, and several others (Rovelli, Baez, etc.) for this interpretation. None of us knows at this time if time is actually a lattice at Planck- or shorter-time-scale intervals. But discretized at even the normal scales of events (roughly the order of seconds for human-scale events, picoseconds or less for particle physics-scale events), the lattice-algebraic model has much to offer. * I don't see any conflict with Huw Price, Julian Barbour, and others (haven't read Zeh yet), though I don't subscribe to all of their idiosyncratic views. --Tim May
Rucker's Infinity, Tegmark's TOE, and Cantor's Absolute Infinity
I am a new member of this group as of 09/01/02. I read Hal Finney's and others comments regarding Rucker's Infinity and the Mind which I have read several times. I also am aware of Max Tegmark's TOE and I have read much of the material related to Georg Cantor's transfinite numbers. I have a question regarding these individuals' work. I hope someone can present their opinions on this subject. First, Max Tegmark (U. of Penna.) has proposed a TOE that all Mathematical Structures have Mathematical Existence and, in addition, also have Physical Existence in the form of other universes. Max has told me that all the Surreal numbers have concomitant existence with regards to being both a Mathematical Structure and also having Mathematical/Physical existence. My question is this: Is there a one-to-one correspondence between the Mathematical Structure for Absolute Infinity studied by Cantor and mentioned by Rucker and the Physical Existence proposed by Tegmark for his TOE. In other words, does Cantor's Absolute Infinity not only have Mathematical Existence; but, does it also have Physical Existence in terms of the total number of universes? In addition, does the proper class, surreal numbers, also have other proper classes beyond it which have Mathematical Existence and, therefore, Physical Existence. Thank you for your help!!! Dave P.S. This e-mail address ([EMAIL PROTECTED]) belongs to a friend. When I am at home my e-mail address is: [EMAIL PROTECTED] and I'll be using both e-mail addresses.
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 [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 connectednessthis 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
