I’ll try to summarize here the main ideas proposed by Plato Demosthenes with my own words (adding my own understanding and clarifications in a way that the author can confirm or correct them). This is actually an exercise to understand the original proposal as I have to admit that I didn’t comprehend it entirely. I have added explicitly numbered questions to those points where I clearly need help to understand the argumentation.
Here we go:
So we have a theoretical universe called Q, where an undetermined number of minds, M, are contained.
QUESTION 1: Is Q just a set of M elements? Or Q is the union of M plus additional kinds of elements?
QUESTION 2: You say that any given M receives input from the universe, i.e. Q. What do you mean? Input from other minds exclusively, or from the environment.
Any given M believes that Q is based on a set of axioms {A1, A2, … , An}. That is to say that M has a truth function called T that satisfies T(axioms are the base of Q) = 1 (true).
QUESTION 3: I can’t understand the word ‘believe’ in this context. I guess it is just the application of the truth function T. However, you say T(M(a.f. is equivalent to Q))=1. Which means that M believes that {A1, A2, …, An} are the axioms that ultimately define Q. But, is there an objective (or epistemological if you want) T function?
QUESTION 4: When you say M(a.f. is equivalent to Q) = Inverse of T(1). What is the function M? Is it the belief of M, Is it a subjective T function of M?
QUESTION 5: When you say ‘a.f.’ (abstract formulation), do you refer to the set of axioms? O do you mean the application of axioms to derive some element of Q?
QUESTION 6: I am stuck here because I don’t see the correspondence of these concepts with consciousness. Where is the mathematical correlate of consciousness in your argumentation?
I Hope you can kindly answer the questions and help clarify your proposal. Thanks.Raúl Arrabales Moreno. conscious-robots.com/raul
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Plato Demosthenes
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Re:Mathematical axioms for consciousness - 2007/05/31 23:57Hmm....I'll try to answer your questions as best as I can, but chances are that I will alter some of my answers later. To start with, Q is the set of all possible minds. The universe was not given a symbol, simply mentioned, but from here on just call it U. Also, when you say an undetermined number of minds, M, are contained, do you mean that M is the number of minds in the universe, that M is the set of minds in the universe, or that M is a mind in the universe? I'll assume in the rest of this post that you mean the third choice, which is what I mean in my original post. Now to your questions: 1) I see how, by thinking that Q was the universe, this would be unclear. However, assuming that you are asking if the "universe" defined is the set of minds or the set of all things in it, the answer is the set of all things in it. 2 The way that I treat it, M (or at least the mathematical workings of M) percieves all minds as simply part of the environment. It can never be certain if anything else really has a mind, anyway. So, it uses its senses, whatever they may be, as an approximation to the universe which he uses for input. It may be that his senses can only recieve information from other minds, but it is treated here so that it can be anything (although it is assumed that his senses percieve his own thoughts). 3) This here is a part which I myself am still a little fuzzy on. I am fairly certain, though, that I use T in this case to be a truth function that is not regarded as relative. When I use the notation M(statement)= 1, I mean to say that M thinks that the statement is true. When I use the notation T(statement) = 1, I mean to say that the statement actually is true. This is, of course, based heavily on the idea that there actually is a non-relative truth. I need to go for now, but I will finish answering your questions later on. Do these help so far?
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Plato Demosthenes
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Re:Mathematical axioms for consciousness - 2007/06/01 00:46Okay, here is the rest: 4) With regards to this question, the answer is in the first post: M: A |-> s(x)s(y)s(z)… |->B(A) A: Universe |-> approximation (Universe) Universe:=anything consistent with the M – M’s input is from U(niverse)M and the output changes UM – it changes with respect to a “time variable” , and every element of it must also satisfy the s(x)s(y)s(z)… are all “symbols” B: A |-> Universe’ We could postulate that there is a string of symbols in between the two processes A and B for the same reason that we cannot really know that anyone else thinks: we see only inputs and outputsThus, T(ThereExists(BM(M(Q Equivalent R))))=1 means that there at some point exists a time when the output of M (BM, a function of the output of the senses)is a response to M thinking that Q is equivalent to R (the set of axioms he believes describes consciousness). Hopefully this long-winded explanation helps some. I'm not sure what you are asking in the second part of your question. Could you please explain? 5) By a.f., I mean the same thing as R - I should have standardized the notation to make it more accesible. And by R I mean the set of axioms that M believes encompasses consciousness. 6) Consciousness has the mathematical analogue of Q in my theory. If and only if something is conscious, then it is follows the axioms of Q. Q, however, is solved for in this, being a term which is defined in terms of different, clear cut functions and quantities. I hope that these responses answer your questions. They probably still need some explaining in of themselves, but they should be a start. So does this clarify my idea better?
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Raúl
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Re:Mathematical axioms for consciousness - 2007/06/07 17:37Thanks for your answers Plato Demosthenes! I think that I got a better understanding of your proposal…
Ok, we have a universe called U, where a set of minds (called Q) live. A single mind contained in Q is called M. The universe U contains both Q (the set of all minds M’s) and everything else. Any given M is actually more than a mind because it is (as I understand from your former posts) a situated agent. M has senses and can use them to perceive both the environment (which could be determined as a subset of U) and its own body. I assume M has a body because otherwise it couldn’t have any senses. Therefore, M is situated in U. I’d say that M has an internal model (depiction) of his environment.
You said that M can ‘sense’ his own thoughts. I assume that means M is endowed with a mechanism for introspection. But, to what degree? What do you mean by thought? High-Order thoughts? It could be just that M has an internal model (depiction) of his own self…
As I understand it, the universe U has a truth function that can decide on the truth of some facts within U. This looks like a tool for observation of U from outside. Is that your aim?
So there is a fact which is that M believes that R is the set of axioms that encompasses consciousness. But, every M in Q believes in the same R?? When I say M, I refer to a specific M within Q. The problem is that the set of axioms R1 that M1 believes as the root of consciousness is different from R2, which is the belief of M2, and so forth.Raúl Arrabales Moreno. conscious-robots.com/raul
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Plato Demosthenes
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Re:Mathematical axioms for consciousness - 2007/06/07 23:01Closer - Q is the set of axioms that all of the minds must satisfy; a sort of definition of consciousness. M is one of the minds that satisfy Q.
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Plato Demosthenes
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Re:Mathematical axioms for consciousness - 2007/06/27 18:48Here is your prior post with the appropriate concepts replaced (by the way, I see now where you got the idea that Q was the set of all the minds - it was a horrible typo that I had in the start of my responses to you, and continued using that notation. With all the typos that I've made and mixed terminology, combined with the obscurity of the idea itself, it is a wonder that anyone could make heads or tails of it at all!): Ok, we have a universe called U, where a set of minds (satisfying Q) live. A single mind contained in U satisfying Q is called M. The universe U contains both the set of all minds M which satisfy Q and everything else. Any given M is actually more than a mind because it is (as I understand from your former posts) a situated agent. M has senses and can use them to perceive both the environment (which could be determined as a subset of U) and can use its own body to interact with both the other minds and the other things in the universe, to varying extents. This includes itself. M has a body because otherwise it couldn’t have any senses. Therefore, M is situated in U. I’d say that M has an internal model (depiction) of his environment.
You said that M can ‘sense’ his own thoughts. I assume that means M is endowed with a mechanism for introspection. The degree of introspection does not matter. By thought, I simply mean some string of symbols (unspecified - the sort of "letters of Mentalese", or it could even be the neuronal (or whatever other system any alien or artificially intelligent system might have) properties of itself). It could be just that M has an internal model (depiction) of his own self, depending on how general Q is.
As I understand it, the universe U has a truth function that can decide on the truth of some facts within U. This looks like a tool for observation of U from outside, however it could be equated in some instances to the relative truth on the inside.
So there is a fact which is that M believes that R is the set of axioms Q that encompasses consciousness. When I say M, I refer to a specific M satisfying Q.
Does that clarify your previous questions some? From now on, let's use the following definitions to keep our terminology unified: Definitions: L:=all items that satisfy the set Q of "axioms" A1,A2,A3,...An, where: n is a positive integer each "axiom" is a string of "symbols" s(a),s(b),...s(q), where there does not exist anything which is an element of a “symbol” the union of all of the symbols must be finite logical symbols (first order predicate, modal, etc.) are all symbols logical axioms of a deductive system are included M is an element of L M is a Turing machine, or some similar function M: A |-> s(x)s(y)s(z)… |->B(A) A: Universe |-> approximation (Universe) Universe:=anything consistent with M – M’s input is from U(niverse of)M and the output changes UM – it changes with respect to a “time variable” , and every element of it must also satisfy the seperate "Laws of Physics" A'1,A'2,A'3.... s(x)s(y)s(z)… are all “symbols” B: A |-> Universe’ We could postulate that there is a string of symbols in between the two processes A and B for the same reason that we cannot really know that anyone else thinks: we see only inputs and outputs Earlier, we defined L as all items that satisfy the “axioms” – yet nothing can satisfy a set of symbols. Thus, we define: “Meaning": symbols |-> p p is an element of UM, or derived/abstracted from UM, or derived/abstracted from something that was derived/abstracted from UM…etc. derived – logically deduced using logical axioms abstracted- if P(v)=c for all v an element of G, then the abstracted version applies it to a different set than G For further use, T(truth): symbols |-> [0,1] Equivalent: f |->h Meaning(f)=Meaning(h) Here is one that I am not quite sure how to define as a function (it really shouldn’t be, but) There Exists: Set x string |->the statement ”string an element of set”
Assumptions:
We know that there exists the string Q is equivalent to R, where R is some set of “axioms” on some set of “symbols”, within M’s processing We know M We know R
Method:
By assumption, T(ThereExists(BM(Q Equivalent R)))=1 ThereExists(BM(Q Equivalent R)) is an element of T^-1(1) BM(Q Equivalent R) is an element of ThereExists^-1(T^-1(1)) (Q Equivalent R) is an element of BM^-1(ThereExists^-1(T^-1(1))) Now, we know that Q Equivalent R means Equivalent(R)= Q and Equivalent(Q) = R, so Both Q and R are elements of Equivalent^-1(BM^-1(ThereExists^-1(T^-1(1)))) We know that M “thinks” that it “thinks” that Q is equivalent to R, So Both Q and R are elements of Equivalent^-1(BM^-2(ThereExists^-1(T^-1(1)))) We can continue the line of reasoning, ending with “Both Q and R are elements of the intersection of Equivalent^-1(BM^-n(ThereExists^-1(T^-1(1))))for all n”
Hopefully we can use this to standardize the haphazard notation and symbols that I have been using in my replies. Sorry for the confusion that terminology has caused you!
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Raúl
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Re:Mathematical axioms for consciousness - 2007/06/29 11:34Thanks for the clarification! It is always difficult for me to understand mathematical notation. That is the reason why I tried to rewrite the argumentation myself. So thanks for rewriting it again with the fixed notation. I think I understand better your proposal now. However, I think I am still missing some of the meaning. For instance,
- What is the exact nature of the symbols s(a), s(b), etc? - I don’t get this: M: A |-> s(x)s(y)s(z)… |->B(A). What is the exact mathematical meaning of “|->” operator? - I can’t see how do you integrate physics in the model. Is there a relation between axioms A1, A2, A3 and laws A’1, A’2, A’3?
Sorry if I ask a lot of questions… I am just trying to understand the whole idea.Raúl Arrabales Moreno. conscious-robots.com/raul
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Plato Demosthenes
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Re:Mathematical axioms for consciousness - 2007/06/29 17:04The symbols are really just squiggles or diagrams or notations of some sort which are used to represent something. Formally, I have defined them as being something for which: there does not exist anything which is an element of a “symbol”; the union of all of the symbols must be finite; and that logical symbols (of some sort - first order predicate, modal, etc.) are all an element of the symbols for the given axiomatic system. Do you have any other defining properties of a "symbol" that I could use to formally clarify what I am doing?
The -> "operator" is really just a functional notation. In modern mathematics, a function f is represented by f: X->Y, i.e, that it maps the set X to the set Y. If f(x)=x^2 for all real numbers, then both the X and the Y would be the set of reals. Depending on the usage, the sets X and Y can have a certain order (a portion of X might be [1,2,3] and the corresponding portion of Y would be [1,4,9]) or they can just indicate what sort of sets f is acting on.
No, the axioms A'1, A'2, ... are simply stating that for the Universe UM, there are certain laws or rules which it follows/satisfies. It really is not overly crucial to the argument, but it does constrain UM from being one out of an infinite (aleph 1) array of options. Even if those "laws of physics" contained probabalistic equations like quantum mechanics, you would still know that only the one universe was being observed by you. These axioms, for our universe, would end up being the Theory of Everything that physics is currently searching for (or, at least, would profoundly limit the axioms, if the resulting TOE is too general).
Sorry if I ask a lot of questions… I am just trying to understand the whole idea. No need to be sorry. Besides, your questions are clarifying my thoughts, as well as yours.
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Raúl
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Re:Mathematical axioms for consciousness - 2007/07/02 19:40Thanks for the clarification. So these symbols are the ones presented as input to M (considering M as a Turing Machine)?
That leads me to wonder about the kind of computation that M can perform. Being a Turing Machine like function, M cannot perform any computation. I mean, there are certain computation problems that cannot be solved by a Turing Machine. Did you think about that?Raúl Arrabales Moreno. conscious-robots.com/raul
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