On 17 Aug 2009, at 12:39, David Nyman wrote:
>
> On 17 Aug, 08:43, Bruno Marchal <marc...@ulb.ac.be> wrote:
>
>> Good intuition David. I think that at some point you are too much
>> precise, so that I can refer only to the interview of the Universal
>> Machine, and you may agree with her, perhaps by making some
>> vocabulary
>> adjustments.
>
> Thanks Bruno. How might I take part in such an interview?
I am not sure what do you mean exactly.
Today, the concrete universal machines are still very primitive, if
not already too much enslaved (full of non universal programs).
So the existing "interview" is still part of mathematical logic
exclusively. It is known today as the logic of provability, or the
logic of consistency, or the logic of self-reference.
The notion of self-reference is the natural third person notion of
self-reference as defined by Gödel in his 1931 paper. It applies to
very general notions of self-referential entity, not just machine. But
I I limit myself on correct machines.
May be you could study the second part of the sane04 paper, and help
yourself with an introductory book on the subject like the book by
Raymond Smullyan "Forever Undecided", but you may need some taste in
logic.
Textbooks exist like Boolos 1979 (which has been reedited), Boolos
1993, and Smorynski 1985. You will find the reference in my Lille PhD
thesis. Those books assumed some knowledge of mathematical logic. Good
books are Elliot Mendelson (many editions), or the book by Boolos,
Burgess and Jeffrey.
Have you follow the seventh step series, or the older but recent UDA
and MGA threads? I am actually explaining the math from scratch needed
to understand the seventh step, which is the step where the Universal
Dovetailer appears.
The understanding of the notion of Universal Dovetailer requires the
notion Universal machine, which requires the notion of computable
functions, which requires the notion of functions and related
elementary set theory.
UDA shows this: "if I am a machine" then "correct physical prediction
= a sum on self-consistent extensions".
This transforms a part of the mind body problem into a tremendously
hard mathematical body problem.
That was the goal: to show that the comp hyp reduces, if only
partially, the mind body problem into a computer science/mathematical
logico problem.
Now at first sight you get an infinite sum of infinite things, so it
could seem UDA is just a refutation of comp.
Now UDA shows *and illustrate* that machines which introspect
themselves can see that physics is the head, so to speak. So instead
of using computer science to look at the sum on self-consistent
extension, it is simpler, conceptually to directly study what a
correct universal machine can see by introspecting herself. We know
she must see the "physics" in her "head", OK? And that is what Gödel
did for correct machines known by logicians as (sufficiently rich)
axiomatic theories. This has given the logic of self-reference.
I give an exemple: Gödel shows that (arithmetical or above) axiomatic
theories can talk (prove) propositions about themselves. The
"themselves" is a third person self-reference, so that the machine is
talking in the manner of a scientific about her body, in a third
person way.
Examples
- An incompleteness theorem: it asserts that the correct machine
machine cannot prove its own consistency (own based on that third
person self-reference). This can be written ~B(~Bf) (it is not
provable that the false is not provable).
- *the* incompleteness theorem: if "I" am consistent then "I" cannot
prove "my" own consistency ("I", and "me" = 3-self).
This can be written ~Bf -> ~B(~Bf ). First person reference are more
tricky to define, and requires Theaetetus.
Now, some machine believes this, they believes in the natural numbers.
For example, they believe in Ex (x = 0), (it exists a number x equal
to the 0). and also, they believe in Ex (x = s(0)), i.e. it exists a
number x equal to the 0), and so one with s(s(0)), s(s(s(0))), etc.
But they believe also in all the induction formula:
[p(0) and Ax(p(x) -> p(s(x))] -> Ax p(x)
I translate (read the colonne vertically):
p(0) the property p is true for zero
and and
Ax for all number x we have that
(p(x) -> p(s(x)) the truth of p for x entails the truth of p for the
successor of x
-> all what precedes entails the truth of what follows
The induction formula gives an enormous power of probability. To
believe in addition and multiplication makes you already universal. To
believe in addition, multiplication and in the induction formula makes
you Löbian, and this makes you know, in a sense, that you are
universal. At the propositional level the logic has been axiomatized
soundly and completely by Solovay, and with the mathematical decor
(the so called normal modal logic) they are entirely characterized by
the formula B(Bp->p)->Bp. A formula related to a theorem by Löb.
Oops I must go. We were beginning the AUDA. It could be premature,
given that we are just at the seventh step of UDA in another thread.
Let me be short and hopefully not too much discouraging. If you want
to take part in the interview, you have to learn theoretical computer
science, math and logic. You may have an opportunity, given that I
like to teach those matter especially when people are motivated by
deep questions.
Bruno
http://iridia.ulb.ac.be/~marchal/
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