On 19 Jan 2012, at 18:03, John Clark wrote:
On Wed, Jan 18, 2012 at 2:30 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
"And are non computable real numbers fundamental?"
If they can not be derived from anything else, and they can not be,
then they must be fundamental.
If they exist, or need to exist. But the useful one can be derived in
the tool-kit of the self-observing digital machine.
" None occur in any theory."
Well they occurred in Turing's 1936 paper and many after it, or at
least the concept of them did, no specific non-computable number was
mentioned because none can be specified.
OK. But in that sense real number are just (total) functions from N to
{0, 1}, or N to N. Turing just shows that in some superplatonia, most
arithmetical functions are not computable.
There is a canonical bijection between real numbers, subsets of N,
functions from N to {0, 1}.
I do not assume the existence of all those objects, in the TOE, mainly
for reason of simplicity, and trying to assume the less possible. But
I have no problem if you want to assume them. I recover them in the
epistemology of natural numbers, but it does not change to ass them
(except making all proofs more complex).
If you want, I am agnostic about real numbers.
As Ludwig Wittgenstein said "what cannot be spoken about must be
passed over in silence".
Which is one sentence too much about what we cannot speak about. And
now we have two like that. No, there are four!
" In physics and math all real constant seems to be gentle and
computable (albeit often transcendent) like PI, e, gamma, etc."
I admit I'm just speculating here and might be dead wrong but maybe
the fact that physics can not exactly specify the position and
velocity of every particle and the fact that mathematics can not
specify every real number are related.
I don't think it is related. Even if space is discrete, you would
still have an uncertainty relation. Qubits are digital, but obeys to
similar uncertainty Fourier relation.
" With comp, analysis and physics belongs to the natural numbers
epistemology."
Yes but if a theory of everything is really about everything then
that is insufficient.
Well Gods and angels belong also to the numbers epistemology (that's
why I think it is better named theology).
Don't panic: by gods I mean Löbian entities which are not machine.
Some particular non computable real number, or function from N to N,
with a notion of self-reference.
Am I still missing something?
" Jacques Arsac is a french catholic who wrote a book against
mechanism. He is not solipsist, and he doubts mechanism. One example
is enough."
That is not a example that is a name. I have never doubted that
individuals, especially religious individuals, can be illogical and
simultaneously hold diametrically opposite views.
So you believe that non-comp is irrational?
But all theories are assumption.
And comp makes precisely impossible for any rational consistent
machine to ever know that comp is true. Frankly you talk a bit like
Craig here, I mean like if you knew the truth of your hypothesis.
We certainly don't know that comp is true.
"Frankly why would a non mechanist be solipsist?"
Although it can not be proven to be false no sane person can be a
solipsist, except perhaps in a philosophy classroom when they are
trying to sound provocative.
That is one reason more for saying that a non mechanist does not need
to be solipsist. You don't answer the question.
A better question would be why would anyone think it controversial
to say "things happen for a reason or they do not"?
That's a classical tautology. Personnaly I believe them about number,
but I am not sure it applies genuinely to set or functions. Now here
the word "things" and "reason" might be too vague to ascertain the use
of the excluded middel principle, but again, I would say that I tend
to agree.
But this does not explains why a non mechanist has to be solipsist.
Unless you really believe that mechanism is true and proved, so that a
non mechanist can only be a totally inconsistent. I don't think so. As
scientist we have to say that we don't know, and study the
consequences of our hypotheses. You might try to get a contradiction
from non-comp. Good luck.
Bruno
http://iridia.ulb.ac.be/~marchal/
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