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I was curious if there was any site or book that dealt exclusively with wolfram's physics research. In particular I want to study his theory of quantum gravity and how it can emerge from a simple rule and the relating subject of time as a mobile automaton. I was very intrigued by the research he listed on the subject in NKS, but want to delve deeper into the research. Anyone who has read NKS should know what I am writing of.

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I don’t think Stephen Wolfram has developed a working theory of quantum gravity, particularly one based on CAs. It appears that the vast majority of people who believe that fundamental physics can be explained using CAs are people outside of the physics community who have little formal understanding of superstring theory or loop quantum gravity. Before you can have a theory of quantum gravity you need to have a theory of gravity or general relativity, but before you can even have a theory of general relativity you need to have a theory of special relativity or Lorenz transforms. Wolfram needs to explain how the continuous nature of Lorenz boosts and rotations can be modeled by discrete CAs in an elegant manner. Certain CAs like rule 110 display universality and so in principle can model anything, but this doesn’t lead to any new insights into physics. There is also an issue with the quantum mechanical side of quantum gravity when Bell’s theorem is considered. How can CAs based on local rules model the non-local nature of quantum entanglement?

Personally I think that CAs are beautiful and that it would be wonderful if Wolfram did demonstrate a CA rule that could model fundamental physics. It has been pointed out that quantum cellular automata (QCA) don’t seem to have these limitations, but Wolfram only mentions them once in NKS and that is in the context of quantum computers. I suspect that QCAs have been around even longer that Ulam’s CAs; Heisenberg investigated a type of QCA he referred to as Gitterwelt (grid world) for modeling systems of electrons.

Wolfram’s discussion of the combinatorial structure posets as a basis of space-time leads me to believe than he is trying to create a bridge from CAs to Lee Smolin’s work in loop quantum gravity; Smolin is one of the scientists acknowledged at the beginning of NKS. If Wolfram has actually discovered a CA rule that generates posets, then that is a significant mathematical discovery in of itself. There is currently no known generating function for generating the posets combinatoric structure. So for further background on quantum gravity I would highly recommend Smolin’s book Three Roads to Quantum Gravity.

Last edited by Daniel Geisler on 12-02-2004 at 07:17 PM

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http://en.wikipedia.org/wiki/Poset

I also looked up Smolin in the index of NKS, and found nothing. What page is this on?

It really seems as though Smolin and Wolfram are talking about the same thing.

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Mark Suppes

Last edited by Mark Suppes on 11-27-2004 at 03:10 AM

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Smolin is mentioned in the NKS’s Preface on page xiii
A posets are also known as partially ordered sets. Wolfram provides a nice discussion about posets on page 1040 of NKS with page 1041 showing diagrams of posets with n unlabeled elements A000112. Any relationship between set elements that is reflexive, antisymmetric and transitive is a partial order relationships. Posets are produced by applying partial order relationships to sets. Consider the set {a,b,c,d} where a ≤ b ≤ d, and a ≤ c ≤ d. This defines a partially ordered set because ≤ (the less or equal to relationship) satisfies the criteria for being a partial order relationship.
Reflexive: a ≤ a is true.
Antisymmetric: a ≤ b and a ≠ b implies that b ≤ a is false, alternately
a ≤ b and b ≤ a implies a = b
Transitive: a ≤ b and b ≤ c implies a ≤ c.

Posets are closely related to partitions; defining a ≤ b ≤ d and a ≤ c ≤ d leaves the question of where b ≤ c or c ≤ b open and partitions the set into three equivalence classes {{a},{b,c},{d}}. A chain is a totally ordered set, for example the relationships a ≤ b ≤ c ≤ d imply that the set {a,b,c,d} is a chain. Posets are based on set theory and combinatorics; many different combinatorial structures can be constructed from them by using category theory and umbral calculus.

Posets are important in special relativity because causality is a partial order relationship. The relationship a ≤ b ≤ d expresses that event a preceded event b which in turn preceded event c. Event b is in the future light cone of event a and thus event b may causally be dependent on event a. The relativistic interpretation of a ≤ b ≤ d and a ≤ c ≤ d is that b ≤ c or c ≤ b is based on the frame of reference from which event b and event c are observed.

A good way to get a sense of what posets are is to look at where they fit into the Mathematics Subject Classification 2000 (MSC2000). David Rusin’s Mathematical Atlas website covers the MSC2000 and graphically depicts the interconnections between the different areas of mathematics. While the primary category for posets is 06A06, the categories 03E02, 03E04, 05A18, 05A40, 05E25, 06A07, 06A11, 18B35, 54F05 are also relevant.

Last edited by Daniel Geisler on 11-29-2004 at 05:39 PM

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Thank you greatly but I've already read 3 Roads to Quantum Gravity as well as the introduction to Loop Quantum Gravity, doubly special relativity, and How Far are we from a theory of quatum gravity? Lee's backround indepdent approach is one of the most practical I've seen and Penrose's spin network as well as Wolfram's nodes networks can serve as a generalized form of Loop quantum gravity, including direct correlation to the Wilson loops involved, though I do not believe Loop quantum gravity is nearly as fundamental as Wolframs' theory of the universe. The non-local nature of quantum entanglement(Bell's experiment) is explained by Wolfram as information traveling through the node network over many nodes, which cannot be observed directly by us as observers part of the network. As far as the specifics such as Lorentz transformations are concerned, the universe is itself compuationally irreducible, therefore one would have to idealize the universe to fit only the lorentz transformations in order to have any chance of finding a rule, then hope that the rule achieves other observed characteristics of our universe . I'd like to thank you for taking your time to explain posets and other forms of more abstract branches of mathematics which I have very little knowledge of. I've read a little on group theory, which from what I know about the axioms involved is somewhat like set and ring theory, at least I think. Wolfram dosn't use CA's for his theory because the y are obviously far too rigid. I think you're confused with Edward Fredkin of MIT on his interpretation of the final theory of the universe using CA's . That's a very interesting fact about Heisenberg by the way.

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Quotes are from Michael Kreutzjans.

… I do not believe Loop quantum gravity is nearly as fundamental as Wolframs' theory of the universe.

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I understand your viewpoint now. I agree that wolfram's approach is very raw and hasn't probably made any achievements yet, though is almost identical to the spin network envisioned by Penrose in that it follows the basic node rules. Daniel, has Stephen published anything else on his theory of everything using simple programs? I want to see if Stephen has more than what he wrote in NKS. Also, what's a good place to find information on combinatorics? You seem to have great knowledge of this topic.


Thank you

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To be honest, I think that everyone’s approach is very raw. I dispute the statement that Wolfram hasn’t made any real achievements. The problem is that he tends to polarize people to such a great degree that it is almost impossible to find an objective assessment of his accomplishments. Stephen Smale and Ralph Abraham advocated the fundamental unity of different dynamical systems back in the sixties; particularly those of maps (discrete iterated functions like the Mandelbrot set) and flows (the solutions to partial differential equations that form the basis of classical physics). But it was Wolfram’s work that fired peoples imagination and got a large number of scientist to take the application of iterated functions in physics seriously. He librated physicists from the tyranny of continuous mathematics; I say this as an advocate of continuous mathematics. Fredkin’s advocacy of a computational view of physics definitely preceded Wolfram’s, but once again it was Wolfram who mobilized large numbers of scientist to begin taking the computational view of physics seriously. These facts in no way diminish the importance of Wolfram’s predecessors. Newton talked about standing on the shoulders of giants. People hear this and think Newton was being modest, but an examination of the history of physics and mathematics shows that Newton simply spoke the truth. I think Wolfram is one of the great educators of our time, his involvement with Mathematica has accelerated the development of powerful symbolic mathematical programs by many years. I consider his support of Eric Weisstein’s MathWorld to be a notable accomplishment in of itself.

As to post-NKS physics articles from Wolfram, my understanding is that there are none. I contacted folks at Wolfram Science around two months ago to see if there was any hope that Wolfram would be publishing new material soon. I was told that he is very focused on Mathematica 6 right now, but the people at Wolfram Science do a great job of keeping folks up to date on Wolfram’s current activities. I have found that the best way to keep up with Wolfram’s activities in to subscribe to MathWire and periodically check NKS Forum’s News & Announcements area.

I don’t have great knowledge of combinatorics, but certainly have knowledge of great combinatorist and combinatorial resources. I initially learned about combinatorics and a number of important combinatorist by reading the biographies of Paul Erdos. These led me to the Neil Sloane’s On-Line Encyclopedia of Integer Sequences (OEIS) which literally changed my life. The only reason I sound knowledgeable about combinatoric is that I make heavy use of OEIS. Stanley’s two volume set Enumerative Combinatorics is considered the definitive work on combinatorics. Wilf’s book generatingfunctionology is the most important work on generating functions which are the heart of combinatorics.

If you are interested the connection between combinatorics and fundamental physics then you want to understand combinatorial species, combinatorics from the perspective of category theory. While you might assume that this approach is less accessible that the others mentioned, I personally find the opposite to be true. While Wolfram was my dominant scientific influence in the Eighties, I have found John Baez’s work with category theory and physics more useful in the Nineties. His series of articles This Week's Finds in Mathematical Physics is a real treat and amazingly accessible. While Baez discusses combinatorial species from time to time, I am particularly impressed with Philippe Flajolet’s work. Most of what I understand about combinatorics comes from Flajolet. I am particularly fond of Random Generation of Combinatorial Structures, written by Philippe Flajolet, Paul Zimmermann, and Bernard Van Cutsem. This paper was the foundation for combstruct, an amazingly powerful combinatorial program written for Maple by Zimmermann. Did you ever hear of PGP (pretty good privacy) encryption written by Paul Zimmermann? Yep, same guy. Zimmerman also is one of the main people researching the posets combinatorial structure seen in NKS.

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