On 27 Nov 2012, at 11:58, Roger Clough wrote:

Hi Bruno Marchal

Then since the brain is earth, shouldn't we use intuitionist logic ?

To get money, or to build bridges, or to put a man on the moon, we can argue that this is what we do. To search aliens on other planet, or to explore the realm of elementary particles, or to search a coherent view on the mind and body, we have to bet on non constructive object, like the "others", the "unknown" etc.

Once I have the time I can give an example of a simple non intuitionist proof in math, so you can appreciate better.

Bruno





[Roger Clough], [rclo...@verizon.net]
11/27/2012
"Forever is a long time, especially near the end." -Woody Allen

----- Receiving the following content -----
From: Bruno Marchal
Receiver: everything-list
Time: 2012-11-26, 12:49:38
Subject: Re: intuitional logic and comp


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], [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 Gé°€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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