I want to give a solid example of a time-varying set and how it relates to possible worlds (and even to MWI).
Consider that cat being chased by the dog. Is the cat in the world in which he will escape the dog or is in the world in which the dog catches him? One answer is "That lies in the future. We don't know yet." Another is: "All knowledge is Bayesian. Based on his running in the past, I'll lay odds of 5 to 2 that he'll escape the dog." A MWI-flavored version is: "There are many worlds in which he escapes, many where he doesn't. We'll only see one of the possible worlds." And then there's the strictly Boolean, determinist point of view: "The cat is in one of the two possible worlds you describe. A sufficiently powerful being or computer able to calculate all of the factors, including the wind speeds, the slipperiness of the stairs, and so on, knows which world we are in." I think the last point is actually the most naive of all of the views, as it simply "punts" the question and asserts that some omiscient, omnicomputing entity knows the future...or, regardless of any such being, that the future is determined. (This gets into free will issues, obviously.) In the other approaches, the Boolean "cat is in one world or in other world" is replaced by a time-varying set: -- "We don't yet know which world the cat is in, or which world we are in along with the cat, but in a few minutes we'll know for sure." (And everyone will agree on this...there will be no disagreement amongst honest observers as to whether got away or got caught by the dog.) I give this example to show that we don't need quantum weirdness to show how useful/important time-varying sets are, and how the logic of reality can be "non-Boolean becoming Boolean," how the time morphism (passage of time) results in assignment of an event to one of N possible worlds. In other words, the naturalness of Heyting logic instead of Boolean logic. In other words, topos logic. (Aside: I don't claim that more and more powerful computes and analysis tools don't help us to either determine which universe we are in--by making predictions of stock movements, or weather, or wars, or the escape of that cat--or even by helping us to make our own changes which change the future. My ideas are not firm on this, but I think computational and cognitive power relates to how far forward in time this "knowability" extends. In the case of a billiard table, knowability may extend a few seconds into the future, for accuracy within some range. In the case of planetary motions, many millennia, for some accuracy. In the case of the cat....? My point about the "omniscient" model being a "punt" is that it simply defines omniscience as being enough to have complete knowledge. There is no evidence that such omniscience is possible, not even with all the computer power in the universe.) --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
