Stephen Wolfram's A New Kind of Science is now available online. (You have to register to read more than a few pages, but it's free.) I decided to take the opportunity to browse it a bit. Let me quote the opening for you:
Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world. My purpose in this book is to initiate another such transformation, and to introduce a new kind of science that is based on the much more general types of rules that can be embodied in simple computer programs.
In case you missed it: Wolfram is comparing himself to Isaac Newton. He did this more explicitly when he spoke at Chicago (on part of his grand tour of universities, where he must have been saddened to see how few people shared his enthusiasm that a "paradigm shift" was underway.) Wolfram is smart -- he had received his Ph.D. in physics from Caltech by age 20, and had published papers in theoretical particle physics at 15. What he seems not to realize is that being smart does not make one right.
If you've read much about the book before, I'm probably not saying much new below, although I haven't seen much criticism of Wolfram's ideas about particle physics before. (This is probably because he continually fails to provide enough details to suggest that he has any theory at all.)
Wolfram's central point is that simple computer programs can generate complex behavior. While his arrogance and disdain for the capabilities of other people is obvious, it must extend pretty far if he thinks that no one has thought of this before. See Leo Kadanoff's review for a discussion of the history of cellular automata. Kadanoff is himself very smart; he was one of the inventors of our modern understanding of quantum field theory. (I'm referring to his work in relation to the renormalization group.) Kadanoff's review is fair, and even forgiving in ways that I would not be. For instance, he mentions Wolfram's statement that symmetries are usually not important in phase transition problems as a mistake that will probably be fixed in later editions. I'm more inclined to see it as a sign of a bigger underlying problem (I'll get to this in a moment.)
Not having read the book myself, I decided to take advantage of its posting online to peruse the section on fundamental physics. I'm surprised to find Wolfram saying that his idea that the universe is a computation is "bold." It's certainly been claimed before, notably by Edward Fredkin. In fact, Konrad Zuse, who built the first working programmable computer, speculated on the universe as a discrete computation. Beyond its lack of originality, I think the idea is facile and uninteresting until it has been fleshed out thoroughly. The early development of modern physics had people thinking of the universe as being like clockwork. In the 19th century people looked for mechanical explanations of physics, and thermodynamics developed by thinking about engines. I'm no expert on the history of physics, but it certainly seems that current technology has often served as a useful metaphor or example. It's no surprise then that people start thinking about the universe as a computer, and it could even be useful. But it should be acknowledged that this device is a tool, possibly a useful one, but not a profound truth. Otherwise this insight is no more interesting than a Matrix-style "Dude, what if we're all living in, like, a computer?" So, we have to look more closely at Wolfram's ideas to decide if there is any utility in them.
Unfortunately Wolfram has published little detail on his ideas about fundamental physics. Kadanoff notes that specific elements of Wolfram's ideas "emulate previous two dimensional quantum gravity theories and integrable systems work," and calls the chapter "exciting... but not yet science." Wolfram's footnotes say that he has developed more formalism for these ideas, but did not publish it as ANKOS was meant for the general public. Still, no such publications seem to be forthcoming. Wolfram appears to think physicists will be so excited by his skeletal ideas that they will rush to fill in the details themselves. His case is not nearly compelling enough for this to happen.
Wolfram notes that cellular automata can have clusters that resemble particles. This idea is not new, as anyone who has played with John Conway's Game of Life will realize. These are, in some sense, the discrete equivalent of solitons in differential equations. (That is, waves that propagate without changing shape, historically first observed in a canal.) Wolfram's general idea is that the universe consists of a network of lines, with "tangles" in this network representing particles. This is an intriguing idea, though it's worth noting that discrete approaches to space, like spin networks, have generated more interesting physics before Wolfram's book was published. Unfortunately, as Wolfram has stated this idea, it's not apparent to me that it has any chance of working. It is telling that he has so much difficulty with simple things like explaining what moving particles look like in this model (pg. 529). He speculates that faster-moving particles should have more nodes. How low-energy Lorentz symmetry -- in which, in different reference frames, a moving particle can appear to have any speed below the speed of light -- could emerge from such a model is puzzling. His attempt at an answer seems to be that different frames view different "slices" of the "coat" of nodes and links surrounding a particle. This sounds like a kluge, barring further details. In fact this lack of observable symmetry seems to be the biggest problem with his proposal: it discards the most important lesson of 20th century physics, which is the vital importance of symmetries in defining physical theories. Aside from Lorentz symmetry, there are various gauge symmetries corresponding to electromagnetism, the weak force, and the strong force, and the role of these in Wolfram's model is totally unexplained.
Wolfram's notes supply brief summaries of gauge invariance, the Standard Model, and conserved quantities in physics. They even mention dualities in field and string theories in which solitonic objects in one theory are particles in a dual theory. There is only one apparent mistake I see in his summaries of physics, namely a statement that supersymmetry unifies quarks and leptons with gauge bosons (the superpartners of quarks and leptons are scalars, of spin 0, not gauge bosons, which have spin 1), and this is not a big deal. Despite these notes, Wolfram does not seem to address how these ideas could fit into his theory. Somehow particles have associated with them a means of transforming under gauge groups. One could think about placing information about these properties on the nodes or links of the network, but Wolfram's formulation in which the universe is just some network evolving according to certain rules is probably unable to capture this information. His idea may have merit for geometry. General relativity tells us that geometry and gravity are intimately related, and it is natural to wonder if geometry is fundamentally discrete, so this is not surprising. It is much trickier to see how the other, non-gravitational, forces fit in. And the easiest route to addressing this is essentially through lattice gauge theory, a well-established theory that Wolfram can't lay claim to. Perhaps this is why he doesn't bother to address the issue at all.
I see one way out of this criticism, but I don't think it's a good one. This is the observation that gauge symmetries describe redundancies in our physical description of a system. One can fix a gauge and work happily there. In other words, it may be argued that the formalism of gauge theory is a human construct, not an aspect of the natural world. There is some reason to go along with this. One might have dual theories that are precisely equivalent but that appear to have different symmetry groups. So, Wolfram might claim that the symmetry groups are emergent properties of behavior described by one of his networks with tangles. This is not entirely implausible. But, if true, it is difficult to see how his approach would render anything calculable, or offer much insight into the world. If we must do long computer simulations to extract information, we haven't really gained any understanding. This is a general problem with Wolfram's philosophy; he argues that standard techniques will hit a wall and we must use computation anyway. He points at indecidable problems. But, fundamentally, science is about gaining useful understanding of the world, not about crunching numbers. I don't see his approach helping us here. Even if the underlying physics of everything (whatever that would mean) is cellular automata, our equations are much more comprehensible, and simulating these higher-level equations is easier (and more interpretable) than simulating whatever the underlying CA would be. Suppose for a moment we find an underlying CA, or other discrete system, describing the universe. Then what do we do with it? Simulate it? The simulation would be slower to evolve than the actual universe. What could we ever do with such a thing? We're unlikely to be able to determine whether it truly does describe the universe, because the time needed to run it to get anything resembling what we know about the universe would be prohibitive. We're better to stay in the realm of less grandiose, but more useful, theories. They're testable. They're science.
I think I'll close before getting too carried away in philosophical nonsense. As for the "Principle of Computational Equivalence" and Wolfram's claims about mathematics, I'll point you to Lawrence Gray's review in the Notices of the AMS and to Scott Aaronson's review. The latter also critiques Wolfram's idea that physics is fundamentally not quantum, and argues that his model of fundamental physics probably violates Bell's inequality. To my mind this is not devastating; quantum mechanics can likely be tacked onto the model after the fact. The idea of space-as-network has been used in other quantum theories like loop quantum gravity. Wolfram's own insistence that the universe is not fundamentally quantum is probably wrong, but this is not reason enough to reject entirely his thoughts. He does seem to think of his attempt at explaining quantum behavior classically as one of the things that distinguishes his model from loop quantum graivty, so this criticism does make his ideas seem even less original. However, I think the puzzle of how particle physics works in his model is a more troubling issue, and he will have to address it if he wants to have a hope of convincing physicists to take him seriously. Let me close with this quote from Kadanoff's review:
From my reading, I cannot support the view that any "new kind of science" is displayed in NKS. I see neither new kinds of calculations, nor new analytic theory, nor comparison with experiment.
P.S. Also of interest is this page of humor related to the book. See especially "A New Kind of Review" from the amazon.com site.Posted by Matt at February 6, 2004 02:53 PM