On Sunday, November 24, 2002, at 04:26 AM, Colin Hales wrote:
Hi Folks,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.")
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).
***Isnít 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?
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!
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.)
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.
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 observerís scope is, I concluded, simply meaningless as the
math is defined by the observerís 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.
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.
"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