Possible Worlds, Logic, and MWI

From: Tim May <tcmay.domain.name.hidden>
Date: Fri, 10 Jan 2003 16:16:46 -0800

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

> This is a reply to Eric Hawthorne and Tim May.

(Tim comment: the quoted text below is partly a mix of my comments and
partly George's.)

>
> 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.
>

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.

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
(.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
Received on Fri Jan 10 2003 - 19:21:06 PST