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## Re: [ontolog-forum] Axiomatic ontology

 To: Pat Hayes ontolog-forum@xxxxxxxxxxxxxxxx Avril Styrman Fri, 1 Feb 2008 16:10:43 +0200 <1201875043.47a3286348754@xxxxxxxxxxxxxxxx>
 ```Quoting Pat Hayes :    (01) > Nothing there about SELF > reference.    (02) I see the self-reference here:    (03) A correspondence between statements about natural numbers and statements about the provability of theorems about natural numbers. One statement about natural numbers is, that if you add 1 and 1 together, you get one 2. Proving this requires one to understand that X is different than XX. There is a correspondence between statements about natural numbers and statements about the provability of theorems about natural numbers. Both require understanding that X is different than XX. I hope this clarifies.    (04) > >That too, but also that understanding 1+1=2 is a > >prerequisite for any reasoning in general. > > Well, first, that as stated is clearly false, as > one could reason logically without even knowing > anything about arithmetic at all. (I have written > programs which satisfy this description.) But > more to the point, what is the relevance of this > simple arithmetic fact? You introduced it into > the discussion, but I cannot see why.    (05) Can you distinguist between two things and one thing? Do you see the difference between X and XX?    (06) In order to understand arithmetics, this has to be understood. And if you prove something about arithmetic, you have to be able to distinguish between X and XX. 1+1=2 is the same as understanding the difference between X and XX. Because it is so fundamental, it cannot be proved. If one starts to prove it somehow, it is clear the the one uses the ability to distinguish between X and XX. And this alone proves that 1+1=2 cannot be proved.    (07) What did Gödel's incompleteness theorem about arithmetics teach? At least it taught that not everything can be proven. The same thing can be shown by Aristotle's example, that is only simpler.    (08) You can formalize as much as you want, but it does not make the case any more conclusive in the end.    (09) > You mean, if the conclusion of a proof is one of > its own premises, it is not a proof at all. I > agree with the spirit of this, although I'd > prefer to say it is a trivial or circular or > vacuous proof. But that is not the same topic as > self-reference.    (010) I'd say that circularity is one special case of self-reference: A->B->C->A. But if mathematical proofs are all about deriving the truth of the theorem form the axioms with the rules of inference, this cannot be a bad sort of self-reference with proofs.    (011) Avril    (012) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (013) ```
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