On Friday, January 10, 2003, at 12:34 PM, George Levy wrote:

(Tim comment: the quoted text below is partly a mix of my comments and partly George's.)This is a reply to Eric Hawthorne and Tim May.

I've been meaning to write something up on this for a long time, but have never gotten around to it. I'll try now.

Lastly, like most "many worlds" views, the same calculations apply whether one thinks in terms of "actual" other worlds or just as possible worlds in the standard probability way (having nothing to do with quantum mechanics per se).

Good point.

Or so I believe. I would be interested in any arguments that the quantum view of possible worlds gives any different measures of probability than non-quantum views give. (If there is no movement between such worlds, the quantum possible worlds are identical to the possible worlds of Aristotle, Leibniz, Borges, C.I. Lewis, David Lewis, Stalnaker, Kripke, and others.)

Interesting. I don't know how to proceed in this area.

FIRST, let me say I am not denigrating the quantum mechanics issue of Many Worlds. I was first exposed to it maybe 30 years ago, not counting science fiction stories about parallel worlds, and even Larry Niven's seminal "All the Myriad Ways," which was quite clearly based on his MWI readings. Also, I am reading several recent books on QM and MWI, including Barrett's excellent "The Quantum Mechanics of Minds and Worlds," 1999, which surveys the leading theories of many worlds (the bare bones theory, DeWitt-Graham, Albert and Loewer's "many minds," Hartle and Gell-Mann's "consistent histories," and so on. Also, Isham's "Lectures on Quantum Theory," and I've just started in on Nielsen and Chuang's "bible of quantum computers" massive book, "Quantum Computation and Quantum Information," from whence I got the funny Hawking quote about him reaching for his gun when Schrodinger's cat gets mentioned.

So I am deeply interested in this, more so for various reasons than I was 30 or 20 or 10 years ago.

SECOND, my focus is much more on the tools than on any specific theory. I may be one of the few here who doesn't some wild theory of what the universe is! (I'm only partly kidding...we see a lot of people here starting out with "In my theory...universe is strand of beads...embedded...14-dimensional hypertorus...first person awarenesss...causality an illusion...M-branes are inverted..." sorts of theories. Some have compared our current situation to the various and many theories of the atom in the period prior to Bohr's epiphany. Except of course that various theories of the atom in the 1900-1915 period were testable within a few years, with most failing in one spectacular way or another. Today's theories may not be testable for 1000 years, for energy/length reasons. (One hopes some clever tests may be available sooner...)

When I say tools I mean mostly mathematics tools. I'm a lot more interested, for instance, in deeply understanding Gleason's Theorem and the Kochen-Specker Theorem (which I do not yet understand at a deep level!) than I am in idly speculating about the significance of QM for consciousness or whatever. (No insult intended for those who work in this area...I just don't see any meaningful connections as yet.)

And the mathematical tools of interest to me right now are these: lattices and order (posets, causal sets), the connections between logic and geometry (sheaves, locales, toposes), various forms of logic (especially modal logic and intuitionistic logic), issues of time (a la Prior, Goldblatt, causal sets again), and the deep and interesting links with quantum mechanics. I'm also reading the book on causal decision theory that Wei Dai recommended, the Joyce book. And some other tangentially related things. A lot of what I am spending time on is the basic topology and algebra I only got smatterings of when I was in school, along with some glimpses of algebraic topology and the like.

I'm using category theory and topos theory not as end-alls and be-alls, but as the lens through which I tend to view these other areas. Frankly, I learn faster and more deeply when I have some such lens. If this lens turns out to be not so useful for what I hope to do, I'll find another one. But for now, it gives me joy.

I wrote a fair amount here last summer about topos theory, intuitionistic logic, notions of time evolution, and the work of Baez, Smolin, Markopoulou, Crane, Rovelli, and about a half dozen others. This remains a core interest, with some interesting (but not worked out, IMO) connections with QM (cf. the papers of Isham and Butterfield, and I. Raptis, and even some Russians). Bruno is more advanced than I am on the logic, as I have only gotten really interested in it recently. (I studied some logic out of Stoll, Quine, etc., and one of my best profs was Ray Wilder, a leading metamathematician of the 1950s. But I always thought logic was "obvious, but grungy in the details." More akin to bookkeeping, in other words. It took my realization that all is not what it appears to be in Quine and Tarski and for me to realize that nonstandard logics may have deep significance for our views of the universe or metaverse. Reading Lee Smolin's "Three Roads to Quantum Gravity" and Greg Egan's "Distress" were what did it for me, triggering me to start buying and reading books on category theory, topos theory, modal and nonstandard logic, and so on.

(Again, I currently have no pet theory of what Reality is. But I'm happy to be building a base of tools to be able to more intelligently comment later. Having a pet theory is not so important.)

THIRD, here's an explanation of some of the names I mentioned in the paragraph you quoted:

"Aristotle, Leibniz, Borges, C.I. Lewis, David Lewis, Stalnaker, Kripke"

* Aristotle. Famous of course for "A or not-A" classical logic (although modern logicians typically refer to "classical logic" as that of Frege, Russell, Tarski, et. al., that is, the logic which took shape around 1900). But even Aristotle knew the limitations of "A or not-A." Consider the proposition "There will be a sea battle next month." Either true or not true, right? Aristotle understood that not only do we "not yet know" the answer, but that in a deep sense there can BE NO ANSWER at this time, the time we are making the statement. If in fact the answer is "Yes, there will be a sea battle next month," or the inverse, then this implies nothing we do today can change this truth.

Aside: Part of the "the answer must be yes or no" intuition we initially have comes from our observation that in a month we will have either answered the question in the affirmative or the negative, that this is our Bayesian experience with recording history. "In a month we'll have our answer one way or the other" is what we have always experienced. This is the "honest observers will always agree" point that Smolin makes at one point, which can be interepreted in topos-theoretic terms as a time-varying set or a set with a subobject classifier. "The moving hand of time writes, and having writ, moves on." But Aristotle was anticipating modal logic, the logic of "it must be the case" (square) and "it may be the case" (diamond).

* Leibniz. He also dealt with possibilities, with "possible worlds." A possible world is a world which, for example, does not violate any laws of logic. (There are some possible worlds which violate our known laws of physics, some which obey laws of physics, etc. Differing gradations of "possible" are often used.)

Voltaire later wrote about "this is the best of all possible worlds," but Leibniz deserves the credit for thinking deeply about the idea.

Aside: Possible worlds are all around us. Fiction, "what if?," even planning. More on this below.

* Borges. I mentioned him because of his seminal "Garden of Forking Paths" story. He was not the first to write about alternate histories...I'm not sure who wrote the first recognizable story in this genre. Probably as old a concept as any.

* C.I. Lewis (I hope I got his initials right...can't check right now) was an American logician who formalized modal logic around 1920. One of his systems, or that of his collaborators perhaps, was the "S3" (and "S4") systems which Bruno often mentions. I won't define these here, as the Web has better and more detailed explanations than I can give here.

It was shown in the 1930s that some of Lewis' modal logic systems correspond exactly to some of the logic systems which arise in Intuitionistic logic (Brouwer, Heyting). Godel proved this, and then Marshall Stone showed some representation theorems linking such geometric ideas as the topologies of open sets and their algebras to the logics described by Lewis and others. (Some of these results I discussed last summer, such as simple examples of why "not (not A) is not the same as A." Saunders Mac Lane gives a good example of this in "Mathematics Form and Function," and I gave an example last summer of how "not (not Past) is not the same as the Past, in terms of light cones and causal reasoning. Prior, Goldblatt, and others have interpreted Stone's work in terms of the nature of time. I find this fairly compelling, though it is only a _facet_ of reality, not a theory of reality.

* David Lewis is a much more recent logician, who died just a couple of years ago. He is sometimes caricaturized as the guy who believes that there "really are" worlds in which unicorns exist, that there really are worlds in which Germany won the Second World War, etc. (By the way, I'm rereading Phillip K. Dick's important novel, "The Man in the High Castle," about just such a possible world.)

But Lewis was no dummy. In his books "Counterfactuals" and "On Possible Worlds" (book not handy to me where I am typing), he makes the case for "modal realism," arguing that nothing is gained by adopting the conceit that the world we "are" in is the "actual" world and that other possible worlds are of lesser status or are wholly fictive. I think he has a point.

He argues that some possible worlds which we actually know to be unachievable in any actual reality are of more importance to us than more achieveable worlds. For example, the possible world of ideal geometric shapes. We know that we can make a rope or a pencil line more and more ideal by straightening out its kinks, narrowing the line, etc. At the end of this refinement process, this nested series of improvements (shades of Stone's topology, by the way), lies the possible world of the straight line or the perfect plane. And so on, for all of our ideals. Some call this "Platonia."

Where this stops being obvious is where our next guy, Saul Kripke, comes in.

* Kripke. Saul Kripke looked at certain linguistic problems and found that the semantics of possible worlds provided important answers. I confess that I have not found his one book, "Naming and Necessity," to make this point as clearly as I would like, so I am relying mostly on what others have said about his work.

The semantics of possible worlds is having a lot of significance for AI, especially in scene and language understanding, planning, etc. (An example would be a robot explorer on Mars attempting to understand what it is "seeing" by constructing various possible worlds and comparing sensory data against those "mental models" and trying to determine "which world are we actually in?"

Aside: Much of science and everything we do is connected to trying to determine which of many possible worlds we are in. Assuming there is "the" world is not so useful. Agents of bounded rationality, or agents in a world of limited and finite information, must test their notions of which world they are actually in. Connections to philosophy of science, falsfifiability, decision theory, etc. are pretty obvious, and this is why the work Kripke did in the 1960s has had an enormous effect.

OK, these were some of the names I mentioned. Bruno probably knows of some I missed. (Oh, I left out Stalnaker here. He's a contemporary of David Lewis and has similar, though slightly less "radical," ideas.).

FOURTH, and last for now, what does all this stuff have to do with MWI and QM?

One of the reasons people take to the Many Worlds Interpretation (or are repelled...) is that it suggests that the other realities, the possible worlds, are somehow more "real" than the mere figments of some novelist's imagination are, or more real in some tangible sense than the possible worlds of Platonia are. And perhaps we can even _communicate_ with these worlds (a la the novel by James Hogan in the mid-90s) or even _travel_ to these worlds (a la Barnes' "Finity" and many other novels and stories).

The formalism of the theory makes "splitting on quantum events or measurements" look more "physically plausible" than just hand-waving about realities where Germany won the Second World War might seem.

However, I argue that the views are not different at all. Any _possible_ world, of the sort where the laws of physics are obeyed but specific details are diferent, is formally indistinguishable from a possible world where things branched a different way than in the world we find ourselves in. (I think this can be formalized more precisely, probably in terms of sheaf theory and related topological/logical ideas, but I'm a long way from attempting this formalization myself.)

And of course in an Eganesque or Tegmarkian way ("Distress" with "all topologies model" and Tegmark with "everything" theory, respectively), one can extend the possible worlds to those with different laws of physics (talked about even earlier than Egan and Tegmark, of course) and even with different laws of mathematics (naturally reflected in toposes, which are like universes of mathematics, pocket realities).

In other words, the quantum branchings are not needed to give the same piquancy to the idea of parallel realities.

Unless, of course, one thinks the quantum branchings are "real" in some ways that merely conceivable branchings are not. Which may turn out to be the case...and Deutsch (or is it Deutch?) would argue that Young's double slit experiment already tells us that the other branches already _do_ (must, modal square) exist and that coherence between many realities is maintained for at least a while. Entanglement, in other words.

I'll wrap up here.

I hope this better explains where my interests lie and what I think is a fertile area for mathematical and logical work. Of course, the lack of experimental verification (aside from the above point, which is an interpretation, subject to alternative interpretations) makes the situation we are now in quite interesting and quite frustrating.

Best wishes,

--Tim May

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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