From: Lee Corbin (lcorbin@rawbw.com)
Date: Thu Mar 13 2008 - 22:40:12 MDT
Stuart wrote
----- Original Message -----
From: "Stuart Armstrong" <dragondreaming@googlemail.com>
To: <sl4@sl4.org>
Sent: Wednesday, March 12, 2008 5:49 AM
Subject: Re: The GLUT and functionalism
> Since we're bringing in mathematical isomorphisms between systems (via
> the hash function and the game of life),
Please see the second message in this thread, posted by me, on
Tuesday March 11, 2008 at 9:17 PM.
> let's consider a simpler
> example: differentiating functions. Let S be some finite (though
> possibly very large) set of polynomial functions, closed under
> differentiation. The derivative is a map from S to itself. Via a hash
> function f, this is then seen as an abstract map from f(S) to itself.
What Stuart is getting at, for those not too much into math,
is that the operation of "taking a derivative" can be seen as
analogous to "calculating the next state" in a TM (Turing Machine)
(e.g. general computer), or "calculating the next generation" on
a Life Board, where I should say that by a TM I am not talking
about the abstract set of quadruples or quintuples, but the
idealized version of a little machine that runs up and down a
tape. (No more references in this post to TMs.)
> Then what? Well, differentiation is connected with multiplication and
> addition of functions. We can choose S to be sufficiently rich to have
> many examples of this. Via f, we get some complicated relations on
> f(S), and differentiation interacts with these relations (similarly,
> localness can be phrased in terms of the image of the hash function;
> it's just a lot more complicated).
>
> So far, I see no reason to consider that S and f(S) should be treated
> differently. Sure it's more complicated in f(S), but consciousness is
> complicated anyway.
I think, by the way, that below you do identify a difference, and
you may just be saying the opposite here for effect. (?)
Before going further, I think that Stuart's mathematical example is
nothing short of glorious.
> I see only two ways of distinguishing S and f(S), and the equivalents
> in the game of life: 1) The question asked and 2) The implicit
> infinity.
But S, you said, was a finite subset of the infinite set of all polynomials.
I think that this is important later (below).
> 1) The question asked.
> When you ask "what is the derivative of this function?", you are not
> asking the equivalent (but hideously complicated) [analogous] version
> in f(S). In fact that hideously complicated question has no meaning
> unless we know its simple equivalent.
Has no meaning? It's still sort of a one-to-one function, (except for
the top polynomals, the top dogs, as it were, where there is no
inverse function, and except that 0 is the final "state", the final end
of taking derivatives of elements of S, and except that g(x) + c
has the same derivative as g(x)). But I deliberately suggested
the concept of "one-to-one" to help us understand the analogy in
that each state gives rise to a unique next state, just as in Conway's
Life, and just as in deterministic computer programs.
> Similarly, when we ask "what is consciousness", we expect the answer
> to have meaning, while the hash function equivalent is meaningless.
I really do totally---and admiringly---agree with where you are going
with this. I just am concerned about inserting the notion of "meaning"
here without further ado; also, my hunch is that it's an unnecessary
encumbrance.
> We can even formalise what we mean by meaning. "What is
> consciousness" is a complicated question, but we know the
> outlines of the answer (and it is not: the average wavelength
> of the light hitting Io during a solar flare).
Very good and amusing example :-) I do so wish everyone
would embroider their compositions with examples that enable
the poor reader to check his understanding and gain conviction
from redundancy. But that's another rant.
> The answer, whatever it is, will have simpler, quasi answers -
> incomplete but informative. The hash function equivalents
> will not be simpler than the hash function equivalent to the full
> question.
Darn. I'm afraid that you're losing me here. What question? "What
is consciousness?" Oh, dear. I hope that your development here
is not contingent on arriving at an answer to that! And what does
it mean for an answer to have simpler quasi-answers?
> 2) The implicit infinity.
> Implicit in the definition of differentiation is the fact that we
> could differentiate any polynomial (with, say, rational coefficients).
> The definition of differentiation to f(S) does not extend to infinity
> in this way; in fact, there is no evident extention of f(S).
What? Even if you have infinitely many polynomials, each has a
derivative, the only difference being that there is no "top dog".
Strategically, would it have been better to stick with a crippled
definition of derivative, oh, something along the lines of either
"we are not going to consider S to be infinite", or "a qDerivative
is a derivative of a polynomial function, but the domain of
qDerivative is by definition finite"? Or something like that?
> So our definition of "differentiation" somehow covers an infinite
> amount of cases, though it is defined finitely. A GLUT could not do
> this.
Yes, well, GLUTs were never intended to, at least not in the applications
to which I ever put them (nor, if memory serves, anyone on the FoR
list, the SL4 list, or the Extropy list).
I think that the whole infinity digression is a red-herring to your otherwise
brilliant analysis (and I don't often call what other posters do "brilliant"),
reserving that appellation strictly to apply to my own missives only.
Gadzooks! I'm eager to go away now and start thinking about how
your hash function of the polynomials does destroy "locality" (as I
originally used it here in this thread), and does destroy the causality
and information flow (in a certain sense that you've made somewhat
clear, at least to me) from one polynomial to the next.
> Stuart, proud sprouter of random thoughts since 2008.
Your modesty is endearing, but so far as I can tell, entirely undeserved :-)
Lee
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