Re: White Rabbits and QM/Flying rabbits and dragons

From: Marchal <marchal.domain.name.hidden>
Date: Thu Nov 25 08:07:47 1999

Russell Standish wrote;

>If you mean by "1st person WR problem" why we shouldn't expect the
>world to get increasingly bizarre as we get older, then I would answer
>that this is exactly what I would expect. The reason I don't see white
>rabbits is that I'm still relatively young, so expect that the third
>person WR explanation should still apply to me. Of course, I would not
>expect that other people would report 1st person white rabbits to me,
>which neatly explains why they don't exist in the literature.

I guess you are talking about usual senility. That argument doesn't
convince me. More about that later, when I will explain more clearly
the 1-white rabbit through the UDA.

>Maybe this explanation is still rather too woolly for the more
>rigourous of us, but surely it must be the kernel of the
>explanation. A more satisfactory theory would be able to make some
>predictions about the rate of departure from a a normal lawlike environment.

Indeed.

>I'd prefer to avoid the use of the word _set_, as set has a specific
>mathematical meaning which may or may not be relevant in this case,
>however I believe this is what I mean. Now, currently I have a picture
>that such a collection of consistent mathematical theories (including
>the completion obtained by allowing the input program to grow to
>infinity (\aleph_0 of course) - and allowing the machines to execute
>for infinite time) pretty much exhausts the range of mathematical
>theories that might be interesting. (It would certainly include N, for
>example). So what exactly do you mean by an orthogonal collection of
>mathematical theories? One that cannot be constructed by means of
>(in)finite combinations of the original collection?

Such combinations will alter the measure. Worst, how will you climb
the ladder of more powerfull theories, how will you be sure those
are consistent ? Will you choose set theories with the axiom of choice
or with the axiom of indeterminacy: the mathematics are not the same,
the word "measure" has different meaning in each. Etc.

>I really feel I need to understand your thesis chapter 5. However, I
>get thrown off the horse within the first couple of pages.
>
>i) You say that assuming a minimum of "pop psychology"
> COMP_n \Rightarrow \neg\Box COMP_n
>
>In English, I interpret this as saying that COMP_n implies the
>possibility that COMP_n is false. (Using the definition \Diamond p
>\equiv \neg\Box\neg p.) What is the source of this remarkable
>statement?

There are two roots for that 'remarkable statement':
1) An arithmetical roots is just Godel's second incompleteness theorem:

Read "comp_n" as "I am consistent", then Godel's th. implies:

  (I am consistent) \Rightarrow (not-provable (I am consistent))

Which is indeed equivalent to

  (I am consistent) \Rightarrow (consistent (I am not consistent))

Consistency is a sort of arithmetical possibility. And it is indeed
a remarkable fact that the "inconsistency" of a machine is relatively
consistent with that machine. You can add to the axioms of Peano
arithmetic that "arithmetic is inconsistent" and you get a consistent
although very unreasonnable arithmetical theory. (BTW this is yet another
example of a mean to build orthogonal theories).

This can also be nicely illustrated with some Kripke model, but I will
not do it here (because I would need drawings). More on that latter
when everybody has read Chellas book ;-)

2) The second roots is the duplication part of the UDA. Read the box
as 'I can communicate scientifically that'
In that case if I duplicate at some level n a person which does not
believe in comp_n, then the doppelganger will understand that he will
never be able to communicate (scientifically) to the original person
that `he' has survived the duplication. That is comp_n is not
scientifically communicable.
>From the original person point of view the very fact that the
doppelganger will appear to him *as* a doppleganger will confirm
his belief that comp_n is could be false: he would have been killed
in case the original ("me" he says) would have been destroyed.

There is a relation between 1) and 2) which is that Godel's theorem
implies that "if I am a machine then I cannot know which machine I am".
Penrose did not understand that point in his book "the emperor's new
mind" but he did eventually understand it in his second book
"the shadow of the mind". This is an almost unanimously accepted
statement concerning Godel and machine philosophy.
Intuitively you can see the link by interpreting a "reconstituted"
person in some environment as a consistent extension of that person.
This explain the importance of the concept of "consistent" extension
for the search of the measure on the possible computational
continuation. I think that this could be used to clarified some
points in your discussion with Alastair.

>ii) You introduce the interpretation of \Box p as Bew(`p'), where the
>latter is the Goedel sentence corresponding to p. How does this
>interpretation work?

No. The Goedel's sentence is "p iff \neg Box p". (Box = Provable)
The sentence "p iff Box p" is the Henkin-Lob sentence. The crazy
happening is that the Henkin-Lob sentence is true and provable.
This is not yet important. The important thing here is to realise
that \neg Provable \neg p = Consistent p, and vice versa:
     \neg Consistent \neg p = Provable p.

>I'm not trying to criticise these - merely understand them.

I appreciate very much.

Bruno
Received on Thu Nov 25 1999 - 08:07:47 PST