Re: [ontolog-forum] Terminology and Knowledge Engineering

[Christopher Menzel <cmenzel@xxxxxxxx>]

On Wed, Jan 25, 2012 at 6:15 PM, Rich Cooper <rich@xxxxxxxxxxxxxxxxxxxxxx> wrote:

Dear Chris,

 

Thanks for the erudite (if somewhat cockeyed) explanation of how you interpret Gödel’s work. 

I do realize that it is a useful rhetorical strategy to lob a pejorative adjective into a discussion without providing any actual arguments to warrant its use (vide the current GOP primary debates), but I would be more than happy to have anyone with reasonable expertise in logic weigh in on just how "cockeyed" my (utterly bland and standard) characterization of the theorem was. It was, moreover, not an "interpretation" of Gödel's theorem, any more than "There are infinitely many prime numbers" is an "interpretation" of Euclid's theorem.

 

Personally, I never use Robinson Arithmetic,

Neither does anyone else. That is because its purpose is not to be "used" for workaday calculation, its purpose is to show the bare minimum of arithmetic that is needed to generate incompleteness. It is a theoretical lower-bound.

 

but the diagonalization Gödel invented for his proof is one which shows a more creative view of math than previously stated by earlier mathematicians.  It is that diagonalization which he pioneered.

Actually, Cantor was the first person to introduce diagonalization, about 55 years before the appearance of Gödel's proof. As to how diagonalization "shows a more creative view of math" than anything in the previous history of the discipline, the claim is rather preposterous. You don't think, e.g., the creation of the calculus or the invention of group theory might measure up? Look, this is just the sort of thing that I object to in some of your posts. While you seem to have some good things to say in your own area of expertise, you are prone to wander beyond the boundaries of your expertise and start throwing out grandiose claims that have no real connection to actual historical or scientific facts.

 

The word “theorem” is well known to all high school geometry students.  Gödel shows that there are theorems which cannot be reached by diagonalization.  You can use you own words here, I don’t mind if you insist that a theorem is formally defined.  But in Geometry classes, they don’t formally define theorems; they just state them and then prove them.  I use the word in that sense.  That is, a theorem is an FOL _expression_ plus embedded arithmetic.

Not in anyone's universe but your own. A theorem of a system is a statement that is provable in the system; that is a universally accepted definition. If you want something less "ultraformalized": a theorem is a statement that can be proved by a chain of valid inferences from a given set of assumptions. Provability (in some system) is what makes a statement a theorem (of that system). A statement cannot rightfully be called a theorem until it has been proved.
 

But if you wish to ultraformalize Gödel, you are welcome to do so.  I don’t find it worth the effort since nearly everyone on this list already understands Gödel and bringing him up is simply a reminder to the list members.

I have no idea how many people on this list have taken the time to study Gödel's theorem in sufficient detail to be able to state it properly; I'm guessing not a huge majority. But that is perfectly OK; it is perfectly OK that YOU don't understand it; understanding it is not necessary for doing a lot of good work in many aspects of ontological engineering. But saying that Gödel's theorem is something that it decidedly is not as if one understands it is not OK; it has the potential to mislead others into false or careless beliefs and needs to be called out for what it is. Do you think it would be OK to characterize Newton's second law as, say, "Force is inversely proportional to mass"? Why not? After all, that statement uses the words "force" and "mass" just like the actual law. Your statement of Gödel's theorem is no less wrong. Gödel's theorem is one of the greatest achievements in the history of science. State it accurately or don't state it at all.

 

It isn’t necessary, IMHO, to be so formal and careful when the point being made is so simple:

You can't make a point by saying something that is egregiously wrong. At least, not the point you want to make.
 

Thanks for your views on this.

I did not state any "views", i.e., opinions, in my previous post, at least in regard to Gödel's theorem. I pointed out the objective errors in your statement of the theorem and I provided a correct statement of the theorem. I did express my view of the value of Franzen's book. It would be strongly in your interest to read it.
 

Do you see any relationship between the terminology issues discussed here and Gödel’s work?

There is no more relationship to be found between those issues and Gödel's theorem than exist between them and Euclid's theorem. Seeking such a relationship is precisely the sort of abuse of Gödel's theorem that Franzen discusses in his book.

-chris

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