Re: Algorithmic Revolution?

From: Tim May <tcmay.domain.name.hidden>
Date: Sun, 24 Nov 2002 09:39:50 -0800

My caveat before commenting: I'm an opinionated person, but I really
don't have any particular theory of everything to share with you. No
dreams theory, no soap bubble theory, no 18-dimensional cellular
automaton theory. I'm currently doing a lot of reading in logic, topos
theory, quantum mechanics, and math in general. I'm more interested
right now in really grasping why the Kochen-Specher says what it says,
for example. Some of you are way ahead of me. Anyway, these are some of
my caveats before plunging in.

On Sunday, November 24, 2002, at 04:26 AM, Colin Hales wrote:

> Hi Folks,
>
> I have chewed this thread with great interest.
>
> Our main gripe is the issue of emergent behaviour and the mathematical
> treatment thereof? Yes? This is the area in which Wolfram claims to
> have
> made progress. (I am still wading my way through his tome).
>
> ***Isn’t the 'algorithmic revolution' really a final acceptance that
> there
> are behaviours in numbers that are simply inaccessible to "closed form"
> mathematical formulae? - That closed-form mathematics cannot traverse
> the
> complete landscape of the solution space in all contexts?

And I believe Charles Bennett said all this more clearly and
convincingly with his "logical depth" arguments. (See, for example, his
long essay in an excellent book called "Fifty Years of the Turing
Machine.")

The argument goes like this: suppose one is walking on a beach and
finds a gold watch. The watch has many moving parts, many non-natural
things like the crystal, the case, and much internal structure. (I'm
deliberately mixing in parts of Dawkins' "Blind Watchmaker," as the
arguments are closely related.)

The watch shows evidence that a lot of "processes" have run for a very
long time. Not computer processes, but processes of manufacturing the
components, of fitting them together, of learning what doesn't work and
what does work, and of a serious industrial infrastructure.

Or consider an e. coli organism. Something like 4 GB of genetic
material, measured in bits and bytes (if I recall this correctly...it
doesn't affect the argument if I am off by some factor).

The "complexity" of the e. coli genome is, by some measures, just this
4 GB. But nearly all 4 GB strings (which is a very, very large number!)
produce dead organisms. In the space of 4 GB strings, some relatively
small "patch" of them produce functioning, reproducing organisms like
e. coli.

Both the watch and the e. coli appear to have been the result of a lot
of shuffling and processing of the "apparent" number of bits.

Charles Bennett calls this "logical depth." This is closely related to
algorithmic information theory, where the shortest description of a
string (or other object) is essentially the program or process which
produces the object. Bennett has placed more emphasis on the "depth" of
a series of iterated processes, but the idea is basically the same.
(And there may have been good syntheses of the ideas in recent years...)

Another way to look at this, metaphorically, is in terms of compression
of a spring. The evolutionary pressures and differential reproduction
rates with e. coli, or with watches!, takes a spring and puts more
energy into it...the energy to do things later. Even a fixed-length
string, like 4 GB in e. coli, can be seen as being "compressed" in this
sense. More and more logical depth is compressed into a string of fixed
length. (Imagine a program in a competing robot, perhaps in one of
those "Battlebots" arena shows, where the program is perhaps, by
coincidence, limited to about 4 GB of Pentium 4 main memory. The
program gets shuffled and changed, via either genetic algorithms (GA)
or genetic programming (GP) or whatever. The same 4 GB program space
("string," seen abstractly) gets more and more capable.

A cellular automaton can also have high logical depth. In fact, back
when I read (some of ) Wolfram's earlier book on CAs, "Cellular
Automata and Complexity," I believe it is called, this is the viewpoint
I was reading it from.

But the fact that cellular automata exhibit this kind of logical depth
does not mean that gold watches and e. coli are proof that the universe
is a cellular automaton!

Q.E.D.

Now, pace Zuse, Fredkin, Lloyd, and all the others, it may in fact be
the case that a Theory of Everything somehow involves CA-like
computations or interactions at perhaps the Planck scale.

But nothing in Wolfram's recent book is at all convincing to me that he
has shown this in any meaningful sense. The phenomena he has been
experimenting with are at least 25-30 orders of magnitude away from the
Planck scale. Believing snowflakes, crystals, and sea shells accrete
material in CA-like ways, which I think physicists and biologists have
been convinced of for a long time, does not mean the universe is in any
meaningful sense itself a cellular automaton.

(And, pace Gleason's Theorem and the aforementioned Kochen-Specker
Theorem, and the work of Bell, I am suspicious for other reasons that a
purely local theory of the universe, one based on CA-like iterations,
can be consistent with quantum mechanics. "No local hidden variables"
and all that.)
>
> My own approach has been to regard emergence as the repositioning of
> the
> observer of a system such that the mathematical descriptions you have
> been
> using fall over/cease to be relevant. The idea that the math can
> seamlessly
> transcend an observer’s scope is, I concluded, simply meaningless as
> the
> math is defined by the observer’s scope. The prejudices of our
> position as
> observers are therefore automatically destined to be embedded in our
> descriptors of things.
>
> If this is the case then one cannot overlook the use of computers or
> the AIT
> approach if you need to study, understand and replicate real-world
> phenomena
> (in particular, MIND) that transcend the boundaries of emergence.
>
> Will the historians look back on our obsession with closed form math
> and see
> it as the machinations of mathematical youth? Para *** above is the
> clincher
> and I have been unable to distil a definitive stance from all the
> writings.
> Clues anyone?

There are many phenomena which have no closed-form, simple description.
That watch on the beach clearly is not going to have some master
differential equation describing it. And e. coli is not likely to have
some simple theory behind it: it emerged/evolved as the result of many,
many interactions with other e. coli, with a complex environment of
chemicals and proteins, and the resulting code is "packed" with a lot
of stuff.

All of these things are interesting from an information theory point of
view, an AIT point of view, and other mathematical and philosophical
points of view.

But the evidence is slim that these things have anything at all to do
with what's going on 30 orders of magnitude away in space, time, and
energy, down where perhaps spin foams are bubbling with instantons,
where perhaps wormholes are opening and closing, where perhaps
Kalabi-Yau topological structures are vibrating or whatever it is they
do.

Fascinating stuff, to be sure.


--Tim May
"He who fights with monsters might take care lest he thereby become a
monster. And if you gaze for long into an abyss, the abyss gazes also
into you." -- Nietzsche
Received on Sun Nov 24 2002 - 12:45:00 PST