On 26 Nov 2012, at 02:00, Platonist Guitar Cowboy wrote:
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.
Intuitionists just avoid the use of (P v ~P). In fact when we program
a computer to compute a total computable function in a way we are
assured that this is the case, we do intuitionist mathematics, and
"conventional programming" is a form of intuitionistic math. The same
when an engineer build machines.
But like modal logics, there are a lot of intuitionistic system, and a
lot of different philosophies associated to it.
For an intuitionist a statement like ExP(x) presuppose a mean of
constructing the n such that P(n), and this is guarantied if you don't
use P v ~P. Classical mathematics, which use freely P v ~P, do
unconventional programming, like in artificial intelligence, where we
allow to a machine to explore some infinite realm, if she wants too
(but this is a 3p crashing, and you have to unplug the machine if you
want to use it again).
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 though.
The "might" is provable for sound machine, even provable by
themselves. Somehow Gödel's theorem justifies that our intuition of
arithmetical truth is deeper than our explicit belief we can have.
Someone ideally immortal and interested in proving as much
arithmetical truth as possible is condemn to add infinitely often new
axioms and/or new inference rules, and realized that no finite theory/
machine can ever embrace the complete arithmetical truth, the
intuition will precede the formalization, and always been a bit
dissatisfied, with respect to that totality.
But "here" to be sure "intuition" is not directly related to the
intuitionist one (but still indirectly).
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".
Yes. Well I hope the teacher gave some argument/proof. But the proof
has to be understood intuitively, always. In science, everything is
based on common sense, even if it happens to get counter-intuitive
consequences.
So it's more like a proof than intuition?
All proof, when given to a human, will require some intuition from the
part of the human trying to understand the proof. With the first order
logics, that intuition can be based on no more than the intuition of
numbers or digital machines.
You need not much than the understanding that if dominoes are placed
in a way such that if dominoes n° n fall entails that dominoes n° n+1
falls, then if dominoes n° 0 falls, they will all fall.
Like you have to know what prime is,
You need to know the definition.
natural numbers are infinite etc.,
Not really. It helps and short the proof, but you don't need to know
about the infinite, nor to know that there is an infinity of naturals
numbers, in the ontology. In the epistemology, you need to know
everything, it is infinitely rich and has an unbounded complexity.
if natural numbers are infinite than there will always be one more?
You can just agree that if x is a number then s(x) is a number, that
if x is different from y, then s(x) is different from s(y), etc.
Things like that. "Infinite" in the comp TOE is already a Löbian
"hallucination", helping the machine to speed herself in the struggle
of the arithmetical complexity. But it is not real, as what is real is
just 0, s(0), s(s(0)), etc. and the laws to which they obey.
So this you can fomalize and state loosely in language,
No you can't. It is not obvious, but that is the failure of logicism.
Our belief in numbers is intuitive, and cannot be justified in any
formal way, without using the intuition to get the meaning of it. It
is because the "air" is very simple, probably innate that we can miss
the difficulty. We cannot define the intuitive natural number in any
finite theory, yet we use that intuition all the time, and without any
worry (and rightly so I would add).
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.
I guess the use of intuition here is general. It relates to the fact
that we cannot define "finite" in first order logic. We can do this in
second order logic, but the definition of second order logic is based
on the intuition of finite. Logicism has failed, and most
mathematician understand that the base of math relies on intuition/
common sense, which cannot be formalized (although it can be meta-
formalize, assuming comp and Theaetetus).
Cowboy regards :)
;)
Bruno
http://iridia.ulb.ac.be/~marchal/
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