From: Lee Corbin (lcorbin@tsoft.com)
Date: Thu Jun 20 2002 - 03:39:02 MDT
Hal writes
> I mean the argument in the message you were replying to, which was
> specifically about how computer programs might exist in a Platonic sense,
> how the conscious beings they create might be just as "alive" as we
> are, and then how such a worldview still allows for human action to have
> meaning. None of this relied on believing that static data would contain
> conscious beings, just abstract computer programs, which are not static.
Sorry if I've gotten confused over what you are claiming. Correct this:
1. We both agree that it's best to speak of large integers and
other mathematical entities as "existing".
2. We agree that computer programs (as codings or patterns
of symbols) also exist in a Platonic sense.
3. We agree that computer programs in that sense are (each one)
equivalent to a set of integers, because each could be coded
up as an integer, and so the platonic reality of integers
and programs imply each other.
4. We agree that when such programs (or integers, when properly
interpreted) when executed in some kind of hardware could be
conscious.
5. You conceive that it is possible that some programs or integers
are alive whether or not they "execute" in time and in real
hardware, and I think it extremely unlikely.
6. You go on (where I do not) in supposing that *if* 5 is true, then
human actions could still have meaning. (I may have to concede
that---more about it another time.)
> > What I want to know is "How explicit is the cause and effect
> > that connects the gels?". Notions of causality aren't so
> > easy to deal with themselves, so far as I know.
>
> I agree that this is a difficult issue. I have a couple of thoughts
> on it.
>
> One idea is to think about a mathematical proof as an abstract object.
> Proofs have a kind of causality to them as well. Each step has to
> rely on specific earlier steps, on axioms, and on rules of inference.
> In principle each step can be fully annotated so that the proof is
> mechanically checkable, as is done sometimes in computer-generated proofs.
>
> Can you imagine an abstract proof as existing in a timeless sense,
> and yet somehow the abstraction captures the notion of causality which
> must exist for the proof to be valid?
Well, I think so. That is, proofs in symbolic logic appear to have
this quality, if I understand you right. For example, taking the
system where there are 3 axioms in propositional calculus
L1 (A -> (B -> A))
L2 ((A -> (B -> C)) -> ((A -> B) -> (A -> C)))
L3 (((~A) -> (~B)) -> (B -> A))
and one rule Modus Ponens the following proof *exists* that (A -> A)
((A -> ((A->A)->A)) -> ((A->(A->A))->(A->A))) L2, where B is (A->A)
and C is A
(A -> ((A->A) -> A)) L1
((A-(A->A)) -> (A->A)) (1),(2) MP
(A -> (A->A)) L1
(A -> A) (3),(4) MP
(taken from "Logic for Mathematicians", by A.G.Hamilton).
Proofs in formal propositional logic have a certain inevitability
about them, mostly I think because of the enumerability of all
possible statements (propositions). So if you have a machine that
just cranks out all the possible statements, then your proof for
anything provable appears sooner or later. (First, it appears
implicitly where it would take a human to link together all the
statements constituting a proof, but later the whole proof itself
appears together when you set the machine to generating all
possible proofs.)
All these proofs can be transparently coded into large integers,
of course, and so also exist platonically just as do all integers.
But your key point is about causality, and your pointed question
was (sorry for the digression) "Can you imagine an abstract proof...
capturing the notion of causality which must exist [in order that]
the proof be valid?"
I can't come up with a good demonstration, but I'm convinced that
some representations would make the causality so manifest that
the answer to your question must be yes. Perhaps some scheme
whereby the next step of the proof is a definite substring of
the first step, and all subsequent steps are nested like that.
> If so, perhaps your stack of gels could be thought of in the
> same sense, not as simply a set of disconnected layers, but
> conceptually joined with the rules that lead from one layer
> to the next.
But the very, very simple rules of Life *do* instantly and
undeniably point to the next generation from the current
one! The births and deaths could scarcely be more plain.
Since continuing in the direction that I'm going here
leads inevitably via (a) separate the gels and distribute
them all around the universe (b) bisect each gel (c)
encode each gel as an integer etc etc... to the Theory
of Dust (and so thereby to my current disapproval),
Let me instead take you at your suggestion to see where
it leads. Suppose on each gel of a Life calculation
we attach some threads between consecutive generations.
Since a death occurs if 4 or more adjacent squares are
occupied, we could have 4 or more black threads leading
from the squares surrounding a currently alive pixel
to the same square (though empty) in the next generation.
Likewise, we could have three white threads leading up
from surrounding empty squares to a currently vacant square
indicating physically how it becomes a birth in the next generation.
Then maybe your idea takes this form: cut all the threads,
and the gels are just as lifeless as I've always claimed;
keep the threads, and the causality is now so manifest
that they're conscious.
> These kinds of models suggest that causality can exist in
> a timeless sense. So perhaps it is not so impossible to
> imagine consciousness existing in the same way.
The threads do make it seems physically compelling in a
way that the mere Rules of Life perhaps don't (make the
connections between the gels quite so manifest). I don't
know; on the one hand it seems that I could still stretch
the threads around galaxies, and still bisect and trisect
the boards (gels) without disturbing this so-called
causal connection, and continue with the reductio ad
absurdum. But on the other hand making the connections
actually physical instead of logical is appealing.
Right now, of course, each state of your brain is connected
by the august Laws of Physics with the next...
Lee
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