Re: [ontolog-forum] Axiomatic ontology

[Pat Hayes <phayes@xxxxxxx>]

At 11:12 PM +0200 1/31/08, Avril Styrman wrote:

> > >This is copied from Wikipedia:
> > >
> > >  Gödel specifically used
> this scheme at two levels: first,
> > >  to encode sequences of symbols representing formulas,
> and
> > >  second, to encode sequences of formulas representing
> proofs.
> > >  This allowed him to show a correspondence between
> > >  statements about natural numbers and statements about
> the
> > >  provability of theorems about natural numbers, the
> key
> > >  observation of the proof.
> >
> > And this does not mention self-reference anywhere, right?
> >
> Yes, it strongly seems that it
> does.

As it does not, in fact, I am at a loss to explain how you get this impression. The passage is referring to goedel numbering, which is a fairly elaborate algorithm for associating each _expression_ and each formal proof with a unique number, in such a way that statements about proofs and sentences (such as that this sentence is the conclusion of that proof) can be made exactly equivalent to statements about numbers, statements which can be stated and proved in formal arithmetic. Nothing there about SELF reference.

> > >
> > >If the key idea is not self-reference, then what is it?
> > >
> > >Suppose that I'm totally wrong. This does not change the
> fact
> > >that proving 1+1=2 requires understanding the difference
> of
> > >I and II. It does not change anything if Gödel took a
> longer
> > >road and included conventions used by modern
> mathematicians.
> > >It is still the same old story: self-reference.
> >
> > Honestly, you have no idea what you are talking
> > about.
>
> Being arrogant does not help solving issues. It won't
> kill you to try to be polite, even if you disagree.

I'm not being arrogant, but there comes a point when the level of mutual comprehension is so utterly beyond what is necessary for a rational conversation to take place, that no normal response is possible. It's not that I disagree with you: I believe, from the evidence so far, that your grasp of the topic is so weak that there is nothing in what you say to agree or disagree with. And there are no 'issues' connected with the Goedel proof: it is a thoroughly investigated piece of modern mathematics, now over half a century old.

> > I see where you are coming from
> (that in
> > order to understand a the formal proof one needs
> > the same cognitive or intuitive abilities as one
> > needs to simply understand that the conclusion is
> > true.So the formal proof does not itself add
> > anything to our pre-mathematical intuition. Do I
> > have that more or less right?)
>
> 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.

> but this is NOT
> > what the Goedel theorem is about: its conclusion
> > is that certain things that are, in a sense,
> > clearly true cannot be proved *at all*, even from
> > the obvious premises; and it establishes this by
> > showing that one can always construct a sentence
> > which has the logical structure similar to the
> > liar paradox: this sentence is unprovable. If
> > this were provable, the whole system would be
> > inconsistent: so (assuming that the system is
> > consistent) it cannot be: but then what it says
> > is correct, so it is in fact true. True, but
> > unprovable. The really clever part of the proof
> > was to show that one can do this is any axiomatic
> > system for arithmetic, so arithmetic is itself
> > undecideable. This last part was done using
> > Goedel numbering, which is what the above quote
> > from Wikipedia is all about.
>
> And what you described above was only a round-route
> of saying that to prove that 1+1=2 requires
> self-reference.

No, it isn't. Read it again, or read any of the popular books about Goedel's theorem. It has nothing whatever to do with self-reference (or with proofs being trivial or circular, see below.) And it does not refer to the provability of simple facts: it establishes the NON-provability of some rather complex arithmetic facts.

> It is the _same_ as that it
> cannot
> be proved at all.

Not in the sense of 'prove' that is used in the Goedel proof (and throughout mathematics.)

> If a proof of X requires self-
> reference, then it is not a proof at all.

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.

Pat

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