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A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, I realise that many ideas of NKS seem to be not so novel afterall, and contained in the Maths literature already, yet with different names.

Is NKS actually presenting novel material?

If yes, what in particular?

If not, what authors have already done this kind of work? Is NKS a "repackaging" of ideas?

Jean-Claude Arbaut
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Andrea
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    A new kind of self-aggrandisement. – vadim123 Apr 10 '15 at 14:25
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    Wolfram has a reputation as a bit of an egomaniac. So I'd take it with a grain of salt. Here are some quotes he wrote about himself.

    "Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today."

    and "Stephen Wolfram is the creator of Mathematica, and a well-known scientist. He is widely regarded as the most important innovator in technical computing today, as well as one of the world's most original research scientists."

    That's just not the kind of thing you say about yourself.

     – Gregory Grant Apr 10 '15 at 14:26
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    May be relevant: http://www.ams.org/notices/200302/fea-gray.pdf – Jean-Claude Arbaut Apr 10 '15 at 14:27
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    This may be flagged as offensive...but I've always thought of Stephen Wolfram as a bit of a douche. See Gregory's comments for more details and the litany of egotestical remarks made by Wolfram. He's why we can't have nice things...and why Wolfram|Alpha pro sucks. – Daniel W. Farlow Apr 10 '15 at 14:36
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    Mr. Stephen Wolfram certainly isn't “the most important innovator in scientific and technical computing today” or “the world's most original research scientists”... I am ! :-| – Lucian Apr 10 '15 at 18:04
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    "I suppose another feature of the book that did not endear it to some academics was the very intensity of positive reaction that accompanied its release" - http://blog.stephenwolfram.com/2012/05/living-a-paradigm-shift-looking-back-on-reactions-to-a-new-kind-of-science/ – innisfree Apr 10 '15 at 18:50
  • He also shows s a pie-chart showing the breakdown of #-star reviews NKS received. I think I understand @Magic's remark. – innisfree Apr 10 '15 at 18:52
  • He could atleast make wolframalpha free for students. – Tyler Hilton Apr 10 '15 at 20:29
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    While your distress may be justified, this question is not about S. Wolfram's personality or business model. In fact, these comments are not helpful if one is aiming at weighing the validity of certain ideas. – Andrea Apr 10 '15 at 22:30
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    @Jean-ClaudeArbaut, that review is the most helpful I have read. Thank you very much. – Andrea Apr 10 '15 at 22:32
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    There are good popular expositions of Wolfram's definite contributions to mathematics in the book. But not only is this field not as novel in Mathematics as Wolfram makes out (as you observe), there are definite problems with its physics. In particular, it would seems to be in conflict with Bell's theorem unless the cellular automata were grossly nonlocal - so not realizable in any technology for the foreseeable future and certainly quite at odds from a literal reading of how Wolfram sees this stuff computed. See http://physics.stackexchange.com/q/4200/26076 – Selena Ballerina Apr 11 '15 at 02:28

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I think the answer to this question is, unfortunately, a little difficult. As many will point out, Wolfram is beyond egotistical and that fact definitely colors the reception of the book. There is a long list of (mostly negative) reviews here. The negativity reaches its apex in the review by Cosma Shalizi. There are some positive reviews as well, though, such as the one by Rudy Rucker.

So, what is NKS? Most correctly, I would say that it is a broad and semi-popular account of Wolfram's work in cellular automata. For those in the field, it clearly builds on the work of others. There are references to this fact in the text but it could certainly be made more explicit. To more clearly see Wolfram's own contributions, you might examine his earlier collection of papers Cellular Automata and Complexity. Rucker lists several of these in his review as well. An honest assessment of his earlier contributions reveal that he is certainly an important researcher in the field and most researchers would be happy to have his body of work. Given the clear antecedents, however, I think that "New Kind of Science" is a definite overstatement. He is in no danger of winning an Abel or Nobel prize.

So, the truth of NKS is certainly somewhere in the huge gulf between a revolutionary (well) new kind of science and (as Shalizi puts it) "utter bat shit insanity".

Mark McClure
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After reading the reviews that were linked in the comments [1] and @Mark's helpful answer[2][3], I came to a a few conclusions, that answer my questions.

First, let's get something out of the way. Stephen Wolfram is definitely a polariser of opinion. Plus his way of presenting himself and his work are not in the canons of the scientific literature and that is bound to make people upset. Plus, the habit of not mentioning other people's contributions and the legal actions make it more so. But the question is not about Wolfram, the question is about the book and the ideas.

Novel Ideas

It seems to me that the most novel aspect of NKS is the proposed "paradigm shift". Namely, that to proceed in Mathematics and Physics we should abandon constructing more complicated and sophisticated theories. Instead, we should focus our attention to simple programs, such as the Cellular Automata (CA), and instead of trying to prove properties of these, we can learn about them by looking at outputs of simulations.

The validity of this paradigm remains to be proven because, according to the 3 reviews I have read, no real predictions have been made.

Another novel part of the book seems to be the "Principle of Computational Equivalence". It basically means according to [1] that there are two classes of computations: the simple/easily predictable ones, and all the "computationally irreducible", equivalent to each other. This implies that all CA complex enough, or rather, complex looking enough, are Turing equivalent. This principle, apart from not being well defined in the book, also seems to be false, according to the reviewers.

Work has already been done

Turns out a lot of work had been done on CA at the time of publishing. This has been done with the "Old Kind of Science", i.e. with proofs and all. In particular Conway's work on the Game of Life, and the fact of it being Turing equivalent. NKS does contain a lot of info and discussion about 2D CA and other programs, with links to the real world. Some of this is original, some is not.

Then there is the work of mathematicians in complexity theory that can be said to touch the same concepts as NKS.

In the end, NKS does appear to be a repackaging of ideas, in order to call to a new paradigm of doing science. If this will work, remains to be proven.

Sources

[1] A very useful, balanced and justified review. Recommended. http://www.ams.org/notices/200302/fea-gray.pdf

[2] Negative review. The arguments about the content are convincing. http://bactra.org/reviews/wolfram/

[3] Positive review. Not as balanced in my opinion. I was not convinced by it. http://sjsu.rudyrucker.com/~rudy.rucker/wolfram_review_AMM_11_2003.pdf

Andrea
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  • A reasonable assessment! I would point out, though, that the many of the ideas that he "repackaged" are, in fact, his own. From a personal perspective, I've found his numbering schemes of CAs to be quite useful and his paper Geometry of Binomail Coefficients to be delightful. – Mark McClure Apr 11 '15 at 01:25
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    @MarkMcClure I don't think anyone would complain about Wolfram repackaging his own ideas: that is inevitable if one is to write a book popularizing those ideas. The problems come when other people's ideas are repackaged. Most of the reviews I've read agree that Wolfram did some great research on CAs in the '80s. – David Richerby Apr 11 '15 at 13:33
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The problem is that asking if something is novel in science is actually very tricky because almost every scientific idea is based on previous work. Therefore, I don't think it is very valuable to question whether something is novel or not because it most certainly is not. However, you can ask if a particular work offers something interesting or particularly noteworthy. I will enumerate a few prominent ideas explained in NKS, including previous work (which is more or less what you asked):

  1. A proposed method to analyse complex systems is to explore the computational universe to see what simple programs are able to do and document their behavior using experimental or analytic methods. Previous work: Edward Fredkin and Konrad Zuse proposed the idea that the universe might be a cellular automaton. In 1997, Schmidhuber described a Turing machine capable of computing all possible histories for the universe based on physical laws.

  2. Principle of computational irreducibility. Systems that display complex behavior can not be reduced and we need to run the experiment to see the outcome.

  3. Principle of computational equivalence. Most sufficiently complex systems are computationally equivalent.

  4. Turing completeness of Rule 110. Wolfram conjectured this in 1985. Matthew Cook (who worked as a research assistant to Wolfram) published a proof in 2000 and a legal battle ensued due to a violation of an NDA. Previous work: Conway's Game of Life, Von Neumann's idea of self-replicating machines.

  5. Conjecture that the 2-state 3-symbol Turing machine is universal. Alex Smith proved this conjecture in 2007.

There are other interesting ideas but most of them are extremely speculative and unlikely to be confirmed or falsified by evidence any time soon.

r_31415
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    Thanks for the useful reply. The Reviews I have read say that 2. and 3. in your reply are actually false. They don't present a proof in the review though. Do you know of anybody that proved that? – Andrea Apr 11 '15 at 09:39
  • Try not to put much weight on reviews (positive or negative reviews). In this case, reviews are not really appropriate because they are more journalistic in nature than scientific. As far as I know, 2 and 3 have not been proven wrong. They are incompatible with other theories or observations to the same extent that pretty much any theory is incompatible with something else. Instead, we should ask "Does the theory predict something specific?" If so, then we should be able to falsify such predictions. – r_31415 Apr 11 '15 at 15:53
  • A concrete question: NKS claims that CA rule 30 is computational irreducible, and the principle of computational equivalence would then imply that rule 30 is also Turing equivalent. (This of course assuming that being computationally irreducible means satisfying the "complex enough" criterion) .This could be falsified or confirmed by proof. Has this been done? – Andrea Apr 11 '15 at 16:06
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    NKS claims CA rule might be computational irreducible but there is no proof of this, in no small part due to the lack of a well-defined notion of what computational irreducibility actually means. There is also no proof that rule 30 is Turing complete (unlike rule 110). If it is truly a property of every "complex" system in the universe, then it would be pointless to try to define it precisely, in the same way that we can't really define inertia and instead, we generally accept it as a property of our universe. – r_31415 Apr 11 '15 at 18:44
  • @RobertSmith Actually, most of the reviews linked from this page are scientific (in particular, Lawrence Gray's review in the Notices of the AMS and the review by Cosma Shalizi, who's worked on CAs). This kind of review is such an important part of the scientific process that it's a mandatory part of getting a paper published in a reputable journal. And all the scientific reviews of ANKS that I've read (admittedly, just a handful) have found scientific flaws with the book. – David Richerby Apr 12 '15 at 12:48
  • @DavidRicherby I'm not sure. The reviews you describe are very superficial and in the case of Gray's review, it tries to compress a 1200+ page book into a 12 page review. it is certainly informative but definitely not scientific. – r_31415 Apr 12 '15 at 16:12
  • @DavidRicherby: I think that you are missing that "review" is used in two very different senses. You have "reviewing" in the sense of refereeing a book or a paper for a journal, and "reviewing" in the sense of commenting an article or book. The reviews mentioned above belong to the second category, and as such they are nowhere as "scientific" as those of the first kind. – Martin Argerami Apr 18 '15 at 05:15
  • @MartinArgerami I don't think I'm missing it at all. Although they are book-review reviews, they analyze the scientific claims in detail and find scientific errors. They would, therefore, serve just fine as peer-review reviews with a conclusion of either "reject" or "major revision needed". – David Richerby Apr 18 '15 at 09:31