Egan's "All Topologies Model"--an excerpt

From: Tim May <tcmay.domain.name.hidden>
Date: Thu, 4 Jul 2002 09:11:45 -0700

On Thursday, July 4, 2002, at 07:21 AM, Bruno Marchal wrote:
>> (This gets into Tegmark territory, about "actual" (whatever that
>> means!) physical universes with different mathematics and then,
>> obviously, different physical laws. Egan, in "Distress," calls this
>> the "all topology model.")
>
> If we are machines I bet the physical laws emerge from all consistent
> possible Universes/Histories) and are necessarily unique! (more in my
> URL).
> I will search for "Distress". I have read and appreciate "Permutation
> City" by Egan. "All topology model"? nice! Especially because I begin
> to suspect
> a "many world interpretation" of knots and links!

I happen to have the e-text/online versions of some of Egan's novels,
including "Distress." I'll include below an excerpt describing the
fictional ATM (which I found inspirational enough to trigger my recent
interests!). This is not the only description of ATMs and MWI issues in
"Distress," but it's one of the most succinct.

By the way, the book is copyrighted "1995," so I repeat my claim that in
many ways Egan preceeded Tegmark, at least in terms of copyright.
(Likewise, I am disappointed to see Marcus Chown and other writers
giving Tegmark credit for the "stand in front of a machine gun and if
you survive then MWI is real" gedankenexperiment. Hans Moravec wrote at
length about how to win in a gambling casino by committing suicide if
one loses in any universe. I, myself, described a way to factor large
numbers by "guessing" the factors and then detonating the world's
nuclear arsenal if the guesses don't multiply together to the right
number. Moravec wrote in the late 80s on this and my own RSA spoof was
circa 1993.)

Anyway, here is a "fair use" excerpt from "Distress." I recommend people
find the book. In this excerpt, the main character, a science
journalist, is flying to an artificial island in the South Pacific where
a conference on Theories of Everything is about to occur. He is planning
to interview a mathematical physicist and cosmologist who has developed
an "all topology model" of the multiverse. The time is around 2050.

--begin excerpt from Greg Egan's "Distress," copyright 1995--

  I forced my attention back to the subject of All-Topology Models.

The concept of ATMs was simple enough to state: the universe was
considered to possess, at the deepest level, a mixture of every single
mathematically possible topology.

Even in the oldest quantum theories of gravity, the "vacuum" of empty
space-time had been viewed as a seething mass of virtual worm-holes, and
other more exotic topological distortions, popping in and out of
existence. The smooth appearance at macroscopic lengths and human
timescales was just the visible average of a hidden riot of complexity.
In a way, it was like ordinary matter: a sheet of flexible plastic
betrayed nothing to the naked eye of its microstructureómolecules,
atoms, electrons, and quarks but knowledge of those constituents allowed
the bulk substance's physical properties to be computed: its modulus of
elasticity, for example. Space-time wasn't made of atoms, but its
properties could be understood by viewing it as being "built" from a
hierarchy of ever more convoluted deviations from its apparent state of
continuity and mild curvature. Quantum gravity had explained why
observable space-time, underpinned by an infinite number of invisible
knots and detours, behaved as it did in the presence of mass (or
energy): curving in exactly the fashion required to produce the
gravitational force.

ATM theorists were striving to generalize this result: to explain the
(relatively) smooth ten-dimensional "total space" of the Standard
Unified Field Theory whose properties accounted for all four forces:
strong, weak, gravitational, and electromagnetic as the net result of an
infinite number of elaborate geometrical structures.

82

Nine spatial dimensions (six rolled up tight), and one time, was only
what total space appeared to be if it wasn't examined too closely.
Whenever two subatomic particles interacted, there was always a chance
that the total space they occupied would behave, instead, like part of a
twelve-dimensional hypersphere, or a thirteen-dimensional doughnut, or a
fourteen-dimensional figure eight, or just about anything else. In fact
just as a single photon could travel along two different paths at once
any number of these possibilities could take effect simultaneously, and
"interfere with each other" to produce the final outcome. Nine space,
one time, was nothing but an average.

There were two main questions still in dispute among ATM theorists:

What, exactly, was meant by "all" topologies? Just how bizarre could the
possibilities contributing to the average total space become? Did they
have to be, merely, those which could be formed with a twisted, knotted
sheet of higher-dimensional plastic or could they include states more
like a (possibly infinite) handful of scattered grains of sand where
notions like "number of dimensions" and "space-time curvature" ceased to
exist altogether?

And: how, exactly, should the average effect of all these different
structures be computed? How should the sum over the infinite number of
possibilities be written down and added up when the time came to test
the theory: to make a prediction, and calculate some tangible, physical
quantity which an experiment could actually measure?

On one level, the obvious response to both questions was: "Use whatever
gives the right answers" but choices which did that were hard to
find . . . and some of them smacked of contrivance. Infinite sums were
notorious for being either intractable, or too pliable by far. I jotted
down an example remote from the actual tensor equations of ATMs, but
good enough to illustrate the point:

Let S=l-l+l-l+l-l+l-...
Then S= (1-1)+(1-1)+(1-1)+...
=0+0+0. ..
=0
But S=l +(-1+1)+(-1+1)+(-1+1)...
=1+0+0+0...
= 1

83

It was a mathematically naive "paradox"; the correct answer was, simply,
that this particular infinite sequence didn't add up to any definite sum
at all. Mathematicians would always be perfectly happy with such a
verdict, and would know all the rules for avoiding the pitfalls and
software could assess even the most difficult cases. When a physicist's
hard-won theory starred generating similarly ambiguous equations,
though, and the choice came down to strict mathematical rigor and a
theory with no predictive power at all ... or, a bit of pragmatic
side-stepping of the rules, and a theory which churned out beautiful
results in perfect agreement with every experiment ... it was no
surprise that people were tempted. After all, most of what Newton had
done to calculate planetary orbits had left contemporary mathematicians
apoplectic with rage.

Violet Mosala's approach was controversial for a very different reason.
She'd been awarded the Nobel prize for rigorously proving a dozen key
theorems in general topology theorems which had rapidly come to comprise
a standard mathematical toolbox for ATM physicists, obliterating
stumbling blocks and resolving ambiguities. She'd done more than anyone
else to provide the field with solid foundations, and the means of
making careful, measured progress. Even her fiercest critics agreed that
her mathematics was meticulous, beyond reproach.

The trouble was, she told her equations too much about the world.

The ultimate test of a TOE was to answer questions like: "What is the
probability of a ten-gigaelectronvolt neutrino fired at a stationary
proton scattering off a down quark and emerging at a certain angle?" ...
or even just: "What is the mass of an electron?" Essentially, Mosala
prefixed all such questions with the condition: "Given that we know that
space-time is roughly four-dimensional, and total space is roughly
ten-dimensional, and the apparatus used to perform the experiment
consists, approximately, of the following..."

Her supporters said she was merely setting everything in context. No
experiment happened in isolation; quantum mechanics had been hammering
that point home for the last hundred and twenty years. Asking a Theory
of Everything to predict the chance of observing some microscopic event
without adding the proviso that "there is a universe, and it contains,
among other things, equipment for detecting the event in question" would
be as nonsensical as asking: "If you pick a marble out of a bag, what
are the odds that it will be green?"

Her critics said she used circular reasoning, assuming from the very

84

beginning all the results she was trying to prove. The details she fed
into her computations included 50 much about the known physics of the
experimental apparatus that indirectly, but inevitably they gave the
whole game away.

I was hardly qualified to come down on either side . . . but it seemed
to me that Mosala's opponents were being hypocritical, because they were
pulling the same trick under a different guise: the alternatives they
offered all invoked a cosmological fix. They declared that "before" the
Big Bang and the creation of time (or "adjoining" the event, to avoid
the oxymoron), there had been nothing but a perfectly symmetrical
"pre-space," in which all topologies carried equal weight. . . and the
"average result" of most familiar physical quantities would have been
infinite. Pre-space was sometimes called "infinitely hot"; it could be
thought of as the kind of perfectly balanced chaos which space-time
would become if so much energy was poured into it that literally
everything became equally possible. Everything and its opposite; the net
result was that nothing happened at all.

But some local fluctuation had disturbed the balance in such a way as to
give rise to the Big Bang. From that tiny accident, our universe had
burst into existence. Once that had happened, the original "infinitely
hot," infinitely even-handed mixture of topologies had been forced to
become ever more biased, because "temperature" and "energy" now had a
meaning and in an expanding, cooling universe, most of the "hot" old
symmetries would have been as unstable as molten metal thrown into a
lake. And when they'd cooled, the shapes into which they'd frozen had
just happened to favor topologies close to a certain ten-dimensional
total space one which gave rise to particles like quarks and electrons,
and forces like gravity and electromagnetism.
By this logic, the only correct way to sum over all the topologies was
to incorporate the fact that our universe had by chance emerged from
pre-space in a certain way. Details of the broken symmetry had to be fed
into the equations "by hand" because there was no reason why they
couldn't have been utterly different. And if the physics resulting from
this accident seemed improbably conducive to the formation of stars,
planets, and life . . . then this universe was just one of a vast number
which had frozen out of pre-space, each with a different set of
particles and forces. If every possible set had been tried, it was
hardly surprising that at least one of them had turned out to be
favorable to life.

85

It was the old anthropic principle, the fudge which had saved a thousand
cosmologies. And I had no real argument with it even if all the other
universes were destined to be forever hypothetical.

--end excerpt--


--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 Thu Jul 04 2002 - 09:25:24 PDT