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 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.
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

> 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/ <http://iridia.ulb.ac.be/~marchal/>
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