Hi Bruno,
----- Original Message -----
From: Bruno Marchal
To: everything-list.domain.name.hidden
Sent: Thursday, May 14, 2009 10:35 AM
Subject: Re: 3-PoV from 1 PoV?
Hi Stephen,
On 13 May 2009, at 22:20, Stephen Paul King wrote:
snip
By relagating the notion of implementation, to Robinson Arithmatic, etc., one only moves the problem further away from the focus of how even the appearence of change, dynamics, etc. obtain. The basic idea that you propose, while wonderfully sophisticated and nuanced, is in essense no different from that of Bishop Berkeley or Plato; it simply does not answer the basic question:
Where does the appearence of change obtain from primitives that by definition do not allow for its existence?
[BM]
Because you can define in arithmetic, using only addition and multiplication symbols, and logic, the notion of computation, or of pieces of computation, like you can define provability (by PA, by ZF, or by any effective theory) already in the very weak (yet Turing universal) Robinson Arithmetic.
[SPK]
Ok, Robinson Arithmatic is " ... or Q, is a finitely axiomatized fragment of Peano arithmetic (PA). ...Q is essentially PA without the axiom schema of induction. Even though Q is much weaker than PA, it is still incomplete in the sense of Gödel."
http://en.wikipedia.org/wiki/Robinson_arithmetic_Q
It does not tell me where the assumption of implementation of the addition and multiplication obtain. Just because one can define X does not mean that one has produced X; unless we are assuming that the act of defining a representational system is co-creative of its objects. Are we to consider that an object, physical or platonic, is one and the same as its representations?
Oh, I forgot, it has been proposed that a book containing a symbolic representation of Einstein's Brain is equal/equivalent to Einstein's Mind. OK! ... Moving on.
[BM]
You can entirely define in arithmetic statements of the kind "The machine x on input y has not yet stop after z steps". The notion of "time" used here through the notion of computational steps can be deined entirely from the notion of natural numbers successor (which can be taken as primitive or defined through addition and multiplication).
[SPK]
Ok, time (pun intended!) for a thought experiment. I go the Library of Babel and pull out the "Einstein's Brain" and bring it home with me.
I sit down and ask it: "what are your latest thoughts on the nature of Unified Fields?". How long am I going to wait before I realize that I will never get an answer?
You might say:"Stephen, you are going about it all wrong! You have to first create a well-formed question in the language of "Eintein's Brain" and then look up the appropriate responce inthe book."
I answer, "Ah, So "Einstein's Brain" can answer my question after all; it can only sit there on the table until I opening and use my own computational implementation to get my answer."
So where is "Einstein's Mind"? Nowhere...
What is it that distinguishes a random sequence of scratches on a plane of sand from the sequence of symbols of the equation representing the Grand Unified Theory of Everything? Well, one person might say: " I can read the one that is an equation..." Meaningfullness necessitates a subject to whom meaning obtains. Computational states, symbolic scratches or patterns of concurrent neuron firings or distributions of voltage potentials, mean something because their existance is such that situations would be different otherwise for some system other than that of the states, scratches, patterns, etc..
Remember the notion of Causation?
X is the Cause of Y if and only if X would not occur without the occurence of Y. David Deutsch defines it more pointedly: "...an event X causes an event Y in our universe if both X and Y occur in our universe, but in most variants of our universe in which X does not happen, Y does not happen either." The trouble is that unless there exists a unique measure on the "space" where in events are coded in the Universe of possible statements or sentences of Robinson Arithmatic, it is undecidable if X happens or Y happens because one can not distinguish between actual computational steps that would generate a means to distinguish X from Y or strings that code some other computational string. Remember how Goedel numbering works... Only if the number of possible statements that can be coded with the same string are computationally isomorphic (generate the same output per input) can one obtain a means to distinguish X from Y, but if we require this it will be no longer possible to code any variants of our universe. Variants would not be allowed. Without the possibility of variants, how does one obtain a notion of contrafactuality?
To claim that the ordering of natural numbers from the notion of succession allow for us to obtain a coherent notion of the ordering of events time is fine, but only if there is one and only one possible successor per any given number. The notion of Causation, inherent in the notion of "time" requires there to be more than one possible successor or else there would be no notion of "variants", ala Deutsch's statement above or any contrafactuals. In a Mathematical Set, one can is not free to have uniqueness or excluded middle rules apply only when convinient for the theorists. Axioms and Rules are Universal or Nothing.
The identification of Time as a "dimension", usually represented as the Real number line does not allow for anything like the variants that we obtain in QM theory. As a matter of fact, events prior to the specification of 1) a basis and 2) a specific experimental condition exists as quantities that have a complex number value. Complex numbers DO NOT have a natural ordering. Thus the notion of Time as a Real number Lines is Not an a priori defined notion, it is strictly a posteriori and thus not available as a means to assing successional overing to events.
[BM]
If you prefer, I could tell you that in arithmetic we have a very notion of time: the natural number sequence. Then we can define in arithmetic the notion of computation, and the notion of next step for a computation made by such or such machine. And from that, we can explain how the subjective appearance of physical times and spaces occur.
[SPK
That would make sense only if we could show how numeric succession, which is a form of "x is less than y, thus y succeds x", is equivalent in all situations for sequential stepping in a computational string. If we have to allow for some form of process for our notion to be coherent then we have not eliminated some primitive form of change from our theory; we have merely pushed it into a corner and hope that no one notices. Again, we can not derive any form of change from a system that does not allow for its existence.
[BM]
UDA explains why we have to proceed that way, and AUDA explains how we can do, and actually, it has been done concretely. Of course the extraction of physics is technically demanding. I should test on new machine the quantum tautologies (and some people are trying recently to do so, we will see). Up to now quantum mechanics confirms the comp self-referential statistics.
You should keep in mind that, due to incompleteness, from the point of view of the machine, although Bp, Bp & p, Bp & Dp, Bp &Dp & p, all define the same extensional provability notion (G* knows that), they differ intensionally for the machine, and, for the machine they obeys quite different logic. The incompleteness nuances forces the arithmetical reality to *appear* very differently "from inside". The Theatetical knower Bp & p, for example, gives a knowledge operator, and can be used to explain why machine can know many things, but also why they can not define knowledge, why the first person knower has really no name, etc. The logic of Bp & Dp & p gives a logic of qualia, or perceptive fields, etc.
Don't hesitate to ask question. Normally UDA is much simpler to understand than AUDA. I will reexplain the step seven to Kim, soon or later.
[SPK]
I would like to better understand tyhe notion of a Theatetical Knower. Could you link to a discussion of it?
Bruno
Time is an illusion, but the illusion of time is not an illusion.
It is a theorem that all self-referentially correct machines are confronted with such an illusion, and they make precise discourses about them. UDA forbids to take such arithmetical machine as mere zombie, or you have to abandon the comp hypothesis.
http://iridia.ulb.ac.be/~marchal/
[SPK]
I do not wish for any one to abandon Comp, I wish only to show that it is a key fragment in a larger system. We can easily show how "static" systems emerge from dynamics ones. The converse is difficult at best. When we assume that "Becoming" is fundamental, "Being" is show to be identical to the automorphisms. A change in a system that leaves it invariant is identical to a non-change, but to neglect the fact that a change is possible is to nullify the entire notion of meaningfullness.
Onward!
Stephen
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Received on Fri May 15 2009 - 16:33:19 PDT