Hi Cowboy,
On 24 Nov 2012, at 19:52, Platonist Guitar Cowboy wrote:
Hi Everybody,
At several points the discussions of the list led us to hypothesis
of arithmetic truth. Bruno mentioned once that the basis for this
hypothesis was quite strong, requiring studies in logic to grasp.
You might quote the passage. Comp (roughly "I am machine", with the 3-
I, the body) is quite strong, compared to "strong AI" (a machine can
be conscious).
Although the comp I use is the weaker of all comp; as it does not fix
the substitution level. But logically it is still stronger than strong
AI.
But arithmetical truth itself is easy to grasp. Even tribes having no
names for the natural numbers get it very easily, and basically anyone
capable of given sense (true or false or indeterminate, it does not
matter) to sentences like
"I will have only a finite number of anniversary birthdays", already
betrays his belief in arithmetical truth (the intuitive concept). So I
would say it is assumed and know by almost everybody, more or less
explicitly depending on education.
But as a non-logician, I have some trouble wrapping my brain around
Gödel and Tarski's Papers concerning this.
Well, this is quite different. It concerns what machine and theories
can said about truth. This is far more involved and requires some
amount of study of mathematical logic. I will come back on this,
probably in the FOAR list (and not soon enough, as we have to dig a
bit on the math needed for this before).
What I do see is that Tarski generalizes the notion and its
difficulties to all formal languages: truth isn't arithmetically
definable without higher order language. Post attacking the problem
with Turing degrees also resonates with this in that no formula can
define truth for arbitrarily large n.
My question as non-logician therefore is: don't these results weaken
the basis for such a hypothesis or at least make it completely
inaccessible for us?
No, it is totally accessible to us, but by intuition only. You can be
sure that music is very similar. We are all sensible to it, but to
explain this is beyond the formal method; neither a brain nor a
computer might ever been able to do that.
Now, comp needs only the sigma_1 truth, which is machine definable, to
proceed. I use the non-definability of truth only to see the relation
with God, and for the arithmetical interpretation of Plotinus, as the
numbers themselves will have to infer more than Sigma_1 truth
(actually much more).
But it is clear that consciousness is also not definable, yet we have
all access to it, very easily. It is the same for arithmetical truth.
The notion is easy, the precise content is infinitely complex, non
computable, unsolvable, not expressible in arithmetic, etc.
Only "philosophers" can doubt about the notion of arithmetical truth.
In math, both classical and intuitionist, arithmetical truth is
considered as the easy sharable part (even if interpreted
differently). COMP is strong because "yes doctor" involves a risky
bet, and the Church thesis requires a less risky bet but is still
logically strong, but the Arithmetical realism is very weak: it is
assumed by every scientists and lay men, and disputed only by
philosophers (and usually very badly).
I have never heard about something like a student abandoning school
and thinking his teacher is mad when he heard him saying that there is
no bigger prime number. It *is* a bit extraordinary, when you think
twice, but we are used to this.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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