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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."
"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...
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.
Tim's cellular automata FAQ may be of some help in understanding all of this.
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
when I say...
Huh?
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...
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.
Wait a minute, that's what I do!
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.
"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!
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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.
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.
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.
s/Disproved/Improved/
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.