Re: The Time Deniers and the idea of time as a dimension
Hal Finney wrote: Physicist Max Tegmark has an interesting discussion on the physics of a universe with more than one time dimension at http://space.mit.edu/home/tegmark/dimensions.html , specifically http://space.mit.edu/home/tegmark/dimensions.pdf . Wouldn't it be true that in the manyworld, every quantum branchings that is decoupled from other quantum branchings would in effect define its own time dimension? The number of decoupled branchings contained by the observable universe is very large. Linear time is only an illusion due to our limited perspective of the branching/merging network that our consciousness traverses. While our consciousness may spread over (experience) several OMs or nodes in that network, it can only perceive a single path through the network. George
Re: The Time Deniers and the idea of time as a dimension
Le 21-juil.-05, à 08:33, George Levy a écrit : Hal Finney wrote: Physicist Max Tegmark has an interesting discussion on the physics of a universe with more than one time dimension at http://space.mit.edu/home/tegmark/dimensions.html , specifically http://space.mit.edu/home/tegmark/dimensions.pdf . Wouldn't it be true that in the manyworld, every quantum branchings that is decoupled from other quantum branchings would in effect define its own time dimension? The number of decoupled branchings contained by the observable universe is very large. Linear time is only an illusion due to our limited perspective of the branching/merging network that our consciousness traverses. I think so. And Tegmark paper is indeed interesting. While our consciousness may spread over (experience) several OMs or nodes in that network, it can only perceive a single path through the network. Comp entails by itself that we should be able to perceive, in some indirect way, the presence of the many bifurcating or differentiating orthogonal path by looking sufficiently close to our probable neighborhood. But then this is confirmed by the very existence of the MW interpretation of the quantum theory. Are there reason to believe that (physical, or local) time could have a scale invariant fractal dimension (between 1 and 2, bigger?) ? Does it make sense ? I guess we must wait progress in string theory, or loop gravity, or even comp (!) to solve a so difficult question ... Bruno http://iridia.ulb.ac.be/~marchal/
Duality (was NEAR DEATH LOGIC)
Le 20-juil.-05, à 14:50, I wrote: It would help me to proceed if you tell me, you Stathis or any reader of the list, if you have understand (or not) that the fact that [Bf, Bt, actually any Bsomething hold in the dead-end world] is as obvious as the fact that [for ALL numbers x, if x is bigger than 2 then x is bigger than 1.] And I was expecting some answer or comment, but I can perhaps help a little bit more. OK, I will give you a completely different explanation why, for any proposition p, Bp is true in any cul-de-sac world. But first I explain a fundamental duality in modal logics. In modal logic there is an important duality, which has been discovered by Aristotle (the founder of modal logic), and which is encapsulated in the Aristotelian Square: Read Bp as p is true in all (accessible) world. What I will say holds both for the Leibniz multiverse (= a collection of observer-moments or worlds), and Kripke multiverse (the same but with a accessibility relation among observer-moments). To make it concrete read B by everywhere in Belgium, and p by it rains. For someone (in Belgium or not) we can distinguish the following four nuances, which I put in the Aristotelian square: 1. Bp 2. ~Bp 3. B~p4. ~B~p 1. (Bp) means p is true in all (accessible) worlds (here: it rains everywhere in Belgium) 2. (~Bp) means it is false that p is true in all (acc.) worlds (here: it is false that it rains everywhere in Belgium). That means that there is a (acc.) world where p is not true. Here:there is a place in Belgium where it does NOT rain. 3. (B~p) means ~p is true in all (accessible) worlds, which is the same as p is false in all (accessible) world, or there is no (acc.) worlds in which p is true. Here: there is no place in Belgium where it rains, or nowhere does it rain in Belgium, or good wether everywhere in Belgium 4. (~B~p) means it is false that ~p is true in all accessible worlds. This is the same as saying that there is an accessible world where p is true. In our example: it is false that [it does not rain] everywhere in Belgium, or there is a place in Belgium where it rains. ~B~p is called the dual of Bp. Bp means p true in all accessible world, more precisely (Kripke) Bp is true in some world \alpha if p is true in all world \beta such that \alpha reaches \beta (which we could write \a R \b). Thus, ~B~p true in \a, means it is false that ~p is true in all world accessible from \a, but this means (by classical logic again) that there is a world accessible from \a where p is true. Let us write Dp for ~B~p. D can be seen as a new modality defined from B. Not that ~D~p, the dual of D, is the same as (by definition) ~(~(B~(~p))), but that is Bp (in classical logic p is equivalent with ~~p). So B and D are duals of each other. If you read the B as a Box, read D as a diamond. B = [], and D = (with the previous notation) Now, the pedagogical problem, linked to the truth table of the implication, has reappeared just above, when I mention by classical logic again. I want explain you how I could have manage to hide it (which could be, or not, a good pedagogy). The idea is to take at once the D modality as primitive. In that case it is enough to define the meaning of Dp in a world \a by saying that Dp is true in \a when there exists an accessible world (from \a) where p is true. It is the equivalent of the KRIPKE IMPORTANT LINE from the preceding posts. Now take, in some multiverse, a dead-end world (terminal world, cul-de-sac world, etc.), i.e. a world from which you can access NO worlds (not even itself). In that case it is obvious that, in such dead-end world, any proposition of the form ~D~p are true, because if ~D~p was false then D~p would be true (because all worlds obey classical logic, so all propositions are either true or false). But then, D~p being true, it means there would be an accessible world where ~p is true. In particular there would be an accessible world, but that cannot be the case given that we have consider a dead-end world. So it is obvious (it should be obvious) that in any dead-end world, the proposition of the form ~Dwhatever are always true. They literally assert the non existence of an accessible world. Now we can define Bp by its dual ~D~p, and this entails, without going trough the logical difficulty (of the implication) that Bp, whatever p is, is always true is dead-end world. If you read Bp as necessary p, and Dp as possible p, you see that dead-end world are not particularly funny: in dead-end world everything is necessary and nothing is possible! Caution! Here is the Aristotelian Square in term of the D modality: 1. ~D~p 2. D~p 3. ~Dp 4. Dp You should see that ~Dp is the same as B~p, and ~Bp is the same as D~p. I give you the duality for well known modalities: B = obligation; D = permission B = everywhere; D = somewhere B = always; D = once B =
Re: The Time Deniers and the idea of time as a dimension
George Levy writes: Hal Finney wrote: http://space.mit.edu/home/tegmark/dimensions.html , specifically http://space.mit.edu/home/tegmark/dimensions.pdf . Wouldn't it be true that in the manyworld, every quantum branchings that is decoupled from other quantum branchings would in effect define its own time dimension? The number of decoupled branchings contained by the observable universe is very large. Linear time is only an illusion due to our limited perspective of the branching/merging network that our consciousness traverses. While our consciousness may spread over (experience) several OMs or nodes in that network, it can only perceive a single path through the network. Tegmark's idea of multiple time dimensions was more general than this. As with multiple space dimensions, you could travel about in the time dimensions. In relativity theory, there is a light cone that restricts which direction is forward in time. You can change your direction but are constrained to always be going forward relative to your light cone. This keeps you from turning around and going backwards in time, because you can't exceed the speed of light. However with 2 dimensional time the geometry is different and you actually go backwards in time. Your own personal clock goes forward but you can end up back before you started. I'll give you a mental visualization you might find useful and interesting. There is a conventional way to think of a light cone which is what gives it its name. Imagine a 2+1 dimensional universe, 2 spatial dimensions and 1 of time. To think of it, start with an x-y plane with the x and y axes. We'll call the y axis time, positive being upward. This is a 1+1 dimensional universe. Now imagine the lines x=y and x=-y, in other words the two lines running at 45 degrees and crossing at the origin. These can be thought of as the paths of light rays emitted by or received at the origin. Now imagine spinning the whole thing around the y axis, where the new z axis will be another spatial dimension. The crossed lines become a pair of cones that represent possible light beams being emitted from or received at the origin. These are called light cones. At each point in space we could imagine a pair of such cones existing, future and past. Objects are constrained in their movements to only be going upward, they have to stay within their light cones. Now for the variant, with a 1+2 dimensional universe: 1 spatial dimension and 2 time dimensions. Again we will start with the x-y plane, y is time, and we draw the crossed 45 degree lines. This time we spin around the x axis, to again produce two cones, but they are pointed right and left rather than up and down. In this model z is a time dimension like y, so we have 2 time dimensions. Now, objects again are constrained in their movements not to cross the cones, but the cones are pointed to the side rather than upward. This means that objects are not stuck inside the cones but are in effect outside of them and are able to move much more freely. You can see perhaps how an object could start at the origin, move in a loop in the y-z plane and return to the origin, all without ever passing through the cones. This is the nature of the 2-dimensional time explored by Tegmark. It is pretty different from the MWI. I would not say that the MWI has multidimensional time any more than it has 3 dimensional space. Technically the MWI all happens in one spacetime area, it is all superimposed and squashed together. There is merely a mathematical separation which occurs when states become decoherent, such that their future histories are causally independent. But technically they are still using the same space and time, they are just invisible to each other. Hal Finney
