On 26 Nov 2012, at 12:34, Roger Clough wrote:
Hi Bruno,
Since intuitional logic seems to be a form of "pure logic",
inapplicable to
the outside world,
Why do you say that? On the contrary, most people believe that
intuitionist logic is the logic most suited for the application in the
real world. I tend to think that woman and engineers are intuitionist
by nature. The believe in what they can construct, where a classical
logician extends its beliefs into what is impossible to not exist,
even if we can't construct it. Intuitionism is Earth logic, classical
logic is Heaven logic.
can or does comp implement intuitional logic ?
Yes, in many ways, but what is nice is that we get freely an
intuitionistic subject associated to the machine, by applying the
theaetus' definition of the knower (true justified belief) with
"justified belief" interpreted by the machine (sound) provability
ability. It is the []p : Bp & p definition that I mentioned earlier.
But this can'be made clear without doing a bit more of logic. This
works because Bp -> p, although true, is unprovable by the machine, so
that from both her 3p and 1p self views, they behave differently. Bp
and Bp & p leads to two veru different view on the Arithmetical truth
that he machine can be aware of.
Bruno
[Roger Clough], [[email protected]]
11/26/2012
"Forever is a long time, especially near the end." -Woody Allen
----- Receiving the following content -----
From: Platonist Guitar Cowboy
Receiver: everything-list
Time: 2012-11-25, 20:00:12
Subject: Re: arithmetic truth
On Sun, Nov 25, 2012 at 5:10 PM, Bruno Marchal <[email protected]>
wrote:
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.
I still have difficulty with intuition as "ability to understand
something no reasoning" in this loose linguistic sense and how
mathematicians frame that. When Kleene makes this precise in "The
Foundations of Intuitionistic Mathematics"... this is a bit too much
for cowboys with guitars, but for some reason I am intrigued.
�
But as a non-logician, I have some trouble wrapping my brain around G
鰀el 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.
That is so strange and amazing. Especially that weird parallel to
music. And "might" is a very large word there to me because don't
composers or mathematicians of, I'll say vaguely, "similar
approaches to their craft" already agree on certain facts about
objects and their properties already? I know, I do with certain
musicians/composers, without total certainty though.
�
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.
But isn't this like informally stating that Euclid proof "there will
always be larger prime". So it's more like a proof than intuition?
Like you have to know what prime is, natural numbers are infinite
etc., if natural numbers are infinite than there will always be one
more? So this you can fomalize and state loosely in language, but
what the student dreamed at the last concert he enjoyed is not. It
is not clear to me why the prime statement is intuition.
Cowboy regards :)
�
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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