Quoting Pat Hayes <phayes@xxxxxxx>: (01)
> Nothing there about SELF
> reference. (02)
I see the selfreference 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
> selfreference. (010)
I'd say that circularity is one special case
of selfreference: 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
selfreference with proofs. (011)
Avril (012)
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