Re: Torkel Franzen on truth

On Dec 11, 6:18 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
Newberry says...

Are you saying that mathematics is an empirical science?

There are two aspects of mathematics: (1) Deciding what
mathematical principles are useful, interesting, consistent,
etc. (2) Deriving what follows from those principles.

The first aspect is certainly empirical. The second
aspect is definitely not empirical, but it is something
that we can easily program a machine to do.

No, I'm not saying that. I'm saying that there is no reason
to believe that a machine could not understand the consistency
of PA in the same way humans do. Certainly Godel's theorem
doesn't suggest otherwise.

You sound like a broken record.

Well, you keep saying the same false things over and over,
so I feel inclined to correct them by saying the truth over
and over. If it's boring, then stop. Stop saying false things,
or at least give more of an argument as to why you believe them.

Put it in the form of a syllogism: Give your starting assumptions,
and show how your conclusion is supposed to follow. Unfortunately,
you always leave out the crucial step:

Given:

(1) Humans believe that PA is consistent.
(2) The only proofs of the consistency of PA
rely on mathematical principles that go beyond PA.
(3) The only proofs of the consistency of those
principles rely on yet other principles.
(4) etc.

I grant all of that. But the conclusion
"Therefore, human abilities surpass those
of any machine" does *not* follow from those
facts. Do you know what it means for one statement
to follow from other statements? At least in one
characterization of it, a statement C follows
from hypotheses H1, H2, etc. if there is a
sequence of statements S_1, S_2, ... such that
each S_j is either a hypothesis, or it is a
theorem of pure predicate logic (no nonlogical
axioms), or it follows from earlier statements
by logical deduction.

Do you actually think that the conclusion
"Human abilities surpass those of any machine"
follows in this sense from Godel's incompleteness
theorem, together with the observation that
humans believe that PA is consistent? If so,
demonstrate it.

No, I believe that this follows; either
1) We do not know if PA is consistent (I mean we know zilch, we do not
even have a good reason to presume that it is consistent.)
2) The human mind can perform non-computable functions
3) There exists a formalization of arithmetic that can prove its own
consistency. (That would explain our intuition that PA is consistent.)

I did put my argument in syllogism.
a) Humans are certain that PA is consistent
b) No formal proof of PA has any cogency (TF explicitly admitted this)
c) Machines are capable only of formal proofs
d) Hence humans can perform feats no computer can
But you chose to ignore it.



.

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