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44 Conjectures of Stephen Wolfram Disproved
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."
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.
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
As a state gets corrupt, its laws multiply; the most corrupt states have the most numerous laws. (Tacitus, Annales 3:27)
For particularly small values of "everyone" of course.
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.
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."
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.
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:
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.
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)
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.