Hi Bruno,

Since intuitional logic seems to be a form of "pure logic", inapplicable to
the outside world, can or does comp implement intuitional logic ? 

[Roger Clough], [rclo...@verizon.net]
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 <marc...@ulb.ac.be> 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?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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