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44 Conjectures of Stephen Wolfram Disproved

Posted by kdawson on from the new-kind-of-error dept.

Richard Pritches writes in to let us know that MIT errata expert Evangelos Georgiadis has disproved 44 conjectures set by Dr. Stephen Wolfram (founder of Mathematica) in A New Kind of Science. The paper was published in the latest issue of the Journal of Cellular Automata and can be read in PDF form at Prof Edwin Clark's collection of reviews of Wolfram's ANKS. "The formulas provided by Wolfram for these [44] rules are not minimal. Moreover for 8 of these cannot be minimal even by simple inspection since minimal formula sizes for 3-input Boolean functions over this basis never exceeds 5."

29 of 158 comments (clear)

  1. Slightly different boolean formula by Ed+Pegg · · Score: 2, Interesting

    That is a very inflammatory title. The page in question is: http://www.wolframscience.com/nksonline/page-884 Comparing the items in the paper to this page, there isn't much here.

    1. Re:Slightly different boolean formula by gzipped_tar · · Score: 3, Insightful

      Yes the difference are "slight"...

      but according to Georgiadis's paper, they're different in nature. Wolfram guess they're 'minimal' in size (plz see the Georgiadis paper for the exact definition) but they are discovered not to be so.

      I'm not one in the circle of CA and I don't understand all the significance about these arguments. But I don't think disproving some conjectures are "inflammatory" in mathematics. It seems some people are not satisfied with Wolfram's style (e.g. his failure in acknowledgin/interpreting other people's researches), but as for the FA it is essentially an objective argument about some mathematical facts.

      Correct me if I make a mistake.

      --
      Colorless green Cthulhu waits dreaming furiously.
    2. Re:Slightly different boolean formula by poopdeville · · Score: 5, Informative

      Zermelo-Frankel Set Theory is an axiomatization of set theory. That is to say, it is a list of axioms describing properties of any structure that is meant to be a collection of sets. There are alternative structures and alternative axiomatizations to generate those structures. (FYI, a consequence of Godel's Incompleteness Theorem is that there are infinitely distinct (in a non-trivial sense) axiomatizations of the natural numbers.)

      Since you've studied Diff Eqs, I'll give you a little example of why axioms of this kind are needed. You were studying differentiable functions. Many of their properties are due to the completeness of the real numbers. Many of their properties are due the real numbers being ordered. Some are due to the fact that the real numbers form a field. And while tools like linear algebra might be necessary to study differential equations, all the properties of differentiable functions are caused by at least one of these three (and the definition of a differentiable function).

      It turns out the real numbers can be characterized as the complete ordered field. To axiomatize the real numbers -- to write sentences from which all the others follow -- we just have to group together the completeness axiom (Every Cauchy sequence converges in the set), the field axioms, and the order axioms. If, for example, you drop the completeness axiom, you would also be writing about things like the rational numbers since they're an ordered field that happens to not be complete.

      Axioms aren't about truth. They have a specific meaning in logic, and more importantly act as a sign post to the audience saying: this is what I'm going to talk about, and how I'm going to talk about it. Of course, in practice, mathematicians don't explicitly state these axioms unless they are the subject of the paper. But they are implicitly "contained" in the jargon.

      --
      After all, I am strangely colored.
    3. Re:Slightly different boolean formula by 2.7182 · · Score: 3, Insightful

      Not to be a jerk or anything, but two years of calculus and a PDE course don't prepare you to understand all that much. In this case some course in logic and the theory of computation might be in order.

  2. English anyone?? by Racemaniac · · Score: 4, Funny

    "has disproving"
    is it that hard to write a summary without such huge errors??
    i'm not a native english speaker, and it even pokes out my eyes...

    1. Re:English anyone?? by rxmd · · Score: 5, Funny

      "has disproving"
      is it that hard to write a summary without such huge errors??
      i'm not a native english speaker, and it even pokes out my eyes...
      You have assuming that the editors can reading!
      --
      As a state gets corrupt, its laws multiply; the most corrupt states have the most numerous laws. (Tacitus, Annales 3:27)
    2. Re:English anyone?? by evanbd · · Score: 4, Funny

      If your shift key is broken, it is permissible to use your caps lock key for the duration of the required capital letter. I know that key is abhorrent to most here, but it does have uses in the event a sudden failure of both shift keys.

    3. Re:English anyone?? by rcw-home · · Score: 2, Funny

      "has disproving" is it that hard to write a summary without such huge errors??

      One word: havening.

    4. Re:English anyone?? by flyingfsck · · Score: 4, Funny

      Slashdot Engrish is to be informating and not to be laughful at.

      --
      Excuse me, but please get off my Pennisetum Clandestinum, eh!
    5. Re:English anyone?? by Sky+Cry · · Score: 4, Funny

      Surely you meaning righting, not reading!

  3. Watch out for the roundhouse kick by Jah-Wren+Ryel · · Score: 5, Funny

    Doesn't Evangelos know that Wolfram is the Chuck Norris of Math?
    Nobody disproves Chuck Norris and lives to publish about it!

    --
    When information is power, privacy is freedom.
  4. More info by the_kanzure · · Score: 3, Informative

    Tim's cellular automata FAQ may be of some help in understanding all of this.

  5. Re:Humm... by Omnifarious · · Score: 5, Interesting

    Ahh, yes. But the great thing about math is that whether or not you have a grudge, everybody can look at the proof and see if you're right or not.

    Personally, if I were a mathemetician, I might have something of a grudge against Stephen Wolfram too. An arrogant person who hypes his own name and abilities far beyond what is justified by the available material then publishes a giant tome of half-baked reasoning that everybody fawns over because of his hyped reputation.

    --
    Need a Python, C++, Unix, Linux develop
  6. I think I speak for a lot of people here ... by mortonda · · Score: 4, Insightful

    when I say...

    Huh?

    1. Re:I think I speak for a lot of people here ... by emurphy42 · · Score: 5, Informative

      As I understand it, it's basically an abstract-logic equivalent of a Perl golf exercise.

      Given three Boolean variables (p,q,r), there are 2 possible values (T,F) per variable, thus 2^3 = 8 possible values for the combined set:

      1. T,T,T
      2. T,T,F
      3. T,F,T
      4. T,F,F
      5. F,T,T
      6. F,T,F
      7. F,F,T
      8. F,F,F

      Now consider functions f(p,q,r) whose output is a Boolean variable. Each such function can be completely described by what output it produces for each of the 8 combinations listed above, e.g.

      • f(T,T,T) = F
      • f(T,T,F) = F
      • f(T,F,T) = F
      • f(T,F,F) = F
      • f(F,T,T) = F
      • f(F,T,F) = F
      • f(F,F,T) = T
      • f(F,F,F) = F

      There are multiple ways to describe the above function, but they're all equivalent to each other because they all give the same results. Thus, there are exactly 2^8 = 256 distinct functions of this sort.

      Wolfram published a list of descriptions for all 256 of these functions, attempting to use the minimum number of symbols (p,q,r,not,and,or) in each case. Georgiadis pointed out that he could have done better in 44 cases. For instance, Wolfram labeled the function given above as Rule 2, and gave the intuitive 7-symbol representation

      f(p,q,r) = (not p) and (not q) and r

      while Georgiadis gave a 6-symbol representation

      f(p,q,r) = r and not (p or q)

    2. Re:I think I speak for a lot of people here ... by fyngyrz · · Score: 4, Informative

      First of all, mod parent up. I know you moderators suck ass and the system is broken, but come on - parent post is a precise description of TFA's issue, the first in the thread. If you're going to spend mod points at all and pretend they do something worthwhile, parent post is the place.

      Secondly, Wolfram's point - in the book - wasn't that the descriptions were minimal (even if he may have mentioned that he thought they were, which I don't actually recall), his point was that they were a complete set of correct descriptions (which I would add are functionally equivalent to the minimal ones anyway) that one can examine for certain behaviors he thought were significant. Niggling about the description not being minimal does not affect what Wolfram was trying to say to the reader. He may be completely wrong, but this kind of nit-picking can not and will not demonstrate it. It is a complete waste of everyone's time.

      --
      I've fallen off your lawn, and I can't get up.
    3. Re:I think I speak for a lot of people here ... by fyngyrz · · Score: 4, Interesting

      Aren't the original functions and new functions still equivalent?

      They are. Wolfram was saying, Look here, for each case, applying the following CA rules gets you the following output set (smaller than the input set) of behaviors. So he was observing - in another way - that the various inputs resulted in a more sparse set of results (speaking as far as they are unique with respect to each other.) Once he identified the various types of behavior possible from these sets, he showed that some were very orderly, some considerably less so, some obviously recognizable - visually - as being members of the same classes of behaviors, and some not so obvious at all except that they certainly weren't in the easy-to-recognize class.

      From there - and it gets a lot more murky - he went on to propose that these very behaviors might underlay either everything, or darned near everything. To bolster that assertion, he showed that you could find those very patterns in many interesting and disjoint places - formulas seeming completely unrelated to anything he'd talked about thus far, seashell structure and so forth. That part of the book was downright breathtaking.

      It's really a very interesting book. His conclusions far outstrip his ability to back them up, but as far as TFA goes, it's looking at the very start of his chain of reasoning and applying some nitpicking that changes nothing he said or meant to say that had anything at all to do with the proposals and justifications made in the book.

      Personally, I could give the south end of a northbound rat if he has a high opinion of himself, or not. What I appreciate is that he provided me with a great read, thought provoking on the one hand, and as far as I could tell, without running into anything I knew that would make me question his approach or conclusions. I've got a good general science background, strong engineering and technical design skills, passable math, and am very creative; I do fairly well at finding little - occasionally large, usually not - holes in new science books and have a huge collection of such volumes full of snippy little annotations of my own (and just for my own benefit, on re-reading.) Aside from a really wonderful book on fuzzy logic which to this day I know of no faults with, this book stood out as almost uniquely solid in the specific area he was clearly trying to explain his thoughts on. I consider it money and time well spent.

      --
      I've fallen off your lawn, and I can't get up.
  7. Re:!Clearly the lack of posts by hung_himself · · Score: 4, Insightful

    is directly proportional to the perceived knowledge required to post.

    You must be new around here. When it comes to biology, everyone seems to think they are experts. Because there are so many computer people here, at least when it comes to math, more of them know that they know nothing...

  8. Re:Humm... by ceoyoyo · · Score: 5, Funny

    For particularly small values of "everyone" of course.

  9. Re:Humm... by IgnoramusMaximus · · Score: 5, Insightful

    Whose name is he supposed to hype? Yours?

    Nobody's.

    And no hype either.

    That is because the supposed subject of all this is Science. And hype and personality cults are to science as money is to politics: corrupting, destructive, counter-prodctive forces.

    Reason, peer review, rigourous analysis, unassailable demonstration of proof, etc are the ways of science, not ascension to prominence via grooming oneself for mass-media "stardom" by boggling the "minds" of the rather feebly-minded general public.

  10. MIT Errata Expert, eh? by mathcam · · Score: 4, Funny
    "...so basically I'm really good at being right about things that other people are wrong about."

    Wait a minute, that's what I do!

  11. That's egg on his face. by yotto · · Score: 3, Funny

    "The formulas provided by Wolfram for these [44] rules are not minimal. Moreover for 8 of these cannot be minimal even by simple inspection since minimal formula sizes for 3-input Boolean functions over this basis never exceeds 5."

    Oh, SNAP!

    --
    Pulp Audio Weekly - Geek News and Reviews
    1. Re:That's egg on his face. by mcg1969 · · Score: 5, Informative

      That's not as much of a SNAP! As you think. The reason that "simple inspection" reveals that the formulas are not minimal is because, in an earlier paper, the same author demonstrates that 5 boolean operators are sufficient. So it actually took a bit more than "simple inspection" to get there.

  12. The paper is not as hostile as the citation by mcg1969 · · Score: 4, Insightful

    The author of the article, Evangelos Georgiadis, has participated in two of the "New Kind of Science" summer schools (2003, 2005; the link above is from 2003). I must suspect, then, that he is somewhat sympathetic to Wolfram's work, and his papers are not intended to be hostile attacks. Indeed, his paper really doesn't read that way, from my perspective as an academic; it is simply a correction of errors. Indeed, if anything, this work tends to buttress Stephen Wolfram's basic point (whether it is true or not) because it further reduces the complexity of CA implementations.

    1. Re:The paper is not as hostile as the citation by mcg1969 · · Score: 5, Informative

      If you go to the NKS Forum, you can find quite a few contributions by the author of this paper, and many of them are error corrections or other disputes with the content. To try it yourself, go to the search page and type in "Evangelos Georgiadis" into the "Search by Author" field, select "Show results as posts", and click "Perform Search."

      I think if you read through the posts yourself you'll see his overall interest seems to be in improving the text, not tearing it down. In fact, one of the threads he created is called "Further Improvements and Errata."

  13. What's good for the goose by mcg1969 · · Score: 2, Insightful

    Likewise, the fact that Stephen Wolfram is an arrogant blowhard should not prevent people from making a reasoned assessment of his work. And that is, in my view, what seems to be happening. Sure, Wolfram is hogging some undue spotlight right now. But his work is absolutely useless unless it can be reproduced, verified, built upon, and applied by others. Give it 20-50 years and we'll see what happens. My prediction is that Wolfram's claims about the work, in particular its wide applicability, will be proven to be wildly overstated. But my prediction is as valuable as the bandwidth it is transmitted upon.

  14. Re:Humm... by IgnoramusMaximus · · Score: 3, Insightful

    The fact that various people continuously try to remodel Science into a contest of egos and popularities does not change the fundamental fact that Science itself is in the long term immune to such tactics.

    And those who attempt it end up, sooner or later, with the only scientific title they deserve: "Crackpot", their "theories" having been ground into dust by the slowly, unglamorously, mundanely, steadily turning wheels of the scientific method.

  15. Simple title correction. by ksw2 · · Score: 4, Insightful

    s/Disproved/Improved/

  16. Full text of "A New Kind of Science" is online by slacka · · Score: 2, Informative

    You can read the full text of "A New Kind of Science" online for free at http://www.wolframscience.com/nksonline/toc.html