A new(?) mathematical approach to Consciousness

Re: A new(?) mathematical approach to Consciousness


No, an axiom is not a string of symbols -- although we may represent an axiom with a symbol.



The union of the symbols in a set must only be finite if the set is finite, but sets do not have to be finite, so no again.



No, you were talking about the set of symbols in the set of axioms -- which is completely different from the set of truths that satisfy those axioms...especially in the fact that even given a finite set of symbols, the set of truths that satisfy those axioms can easily be infinite.



No, the isomorphic meaning function you refer to should be the original mapping of axioms to symbols. It cannot be redefined later on, if there is something you are unable to represent mathematically from the original axioms then the system is not completely defined.



That does not make sense because the "derived-from" relationship has no inherent depth, as there are usually many ways to derive something using different methods, and the actual steps being used have no meaning other than the final result of proving something.



That doesn't make sense



You cannot just say that the mind is a Turing machine without proving it...which is not possible to do given the scientific knowledge of the brain to date.



The universe is everything that is real, I dunno what you are talking about



You are just making wild unsupported assertions, there is nothing mathematical to this...



No it doesn't, so far "M fully believing" is not even representable in this context.



There is no way to map a generic idea into a set of symbols. You have not even attempted to define such a language. So no.



Mathematical truth is boolean, not a real number



We do not have a set of functions and we don't know what they map to, and we can't find their inverse.



Previously you were using M as a literal, and now it's a unary function? You haven't defined it. I assume T(x) is "truth of x"...Q you have defined but I don't see how you are using it..."a.f" is not a standard mathematical abbreviation and I don't know what it is supposed to mean.

Well I am stopping here because you have made so many mistakes so far that none of the rest of it can possibly be correct



Hey, I tried...but unfortunately you have a lot of work to do cleaning this up. You need to learn how to write a proof!

I do see what you mean with regards to the first part - but most of it is simply definitions. I made some terminology - "mind" for instance - simply to make it easier (for me) to understand on some level. Even things such as "axiom", though, are (in a sense), just correspondents to the normal usages of them in mathematics. I thought that I had clarified this by defining them in the way that I did - but clearly my first post did not properly clarify. All of the definitions and things are essentially the "Given" aspect of the proof. I just gave the abstract terms names to make it easier to see the possible applications if the mind was a computable machine.

As for the deductive system, that was a complete flaw of communication on my part. What I mean is some sort of string manipulation system, say, of model theory. The n in the terminology I used was simply the minimum number of string manipulations.

Another error of communication was with the "mind". It simply maps its input to an "idea", i.e. string of symbols, to a change in the universe, via the body.

I have to go now. I appreciate the comments. Bye!

Also in response to stuh 505, (sorry for not being able to finish reply earlier), I realize that, unfortunately, it is not yet a formal proof. It is simply an idea sketch of how one would be made.

I realize that you had a lot of objections, but, with what I have said now in mind, do you have any suggestions on where to go from here?

Just to try writing it over again, and try to explain things better?

I have not the slightest clue what your idea is!

Ok, here is a shot. I see now that my original explanation was not even as clear as mud. Hopefully this version is a little bit better.

Define Q to be the set of all items which satisfy axioms A1,A2,A3,…An, where n is a positive integer. An axiom is defined to be a string of symbols phi a, phi b, phi c, and so on. There is nothing which is an element of a symbol. The union of all the symbols of the set of axioms must be finite. Call the union the alphabet of Q. Now, earlier I stated Q to be the set of all items which satisfy the axioms. That makes no sense if the axioms are just a string of symbols, so we need to introduce a "meaning" function. This function isomorphically maps a set of symbols from the alphabet to an item y. The item y is defined as an item/concept either in this universe or derived from something which is in this universe, or is derived from something that is derived from something that is in this universe, and so on. I term this property being an element of degree n of the universe, where n indicates how many times the "derived from something" is iterated. In other words, y must be something not 100% abstract - somewhere along the way, it had to be derived from something real by an arbitrary deductive method. It could be one from model theory, proof theory, or other standard type of deduction. For our purposes, a universe is defined as any thing which responds to a "mind" ,i.e., Turing machine. The Turing machine receives its input from the universe, and its ouput alters the universe. Thus, the universe is any permutation of anything whatsoever that is consistent with the given Turing machine. Anyway - Q is thus the set of anything isomorphic in some sense to y. Furthermore, There has to be at least one Turing machine which is an element of Q. Up till this point, all of the statements have been assumptions, ones which certainly could be satisfied. Any terminology, such as universe, mind, etc., are simply terms which make clear what thoughts led to these definitions. All , however, are still simply abstract possibilities and may very well not correspond to the actual things at all. Now, what I am attempting to prove is that if we know that a "mind" M is an element of Q (say, if Q is the set of all minds) that the mere fact that M fully believes that the abstract formulation ,called a.f., (a set of axioms) is the set which Q is based on, and that we know both the equations of M and a.f., then we can derive what the actual axioms are. The proof is not meant to be within the original "axiomatic" system, but to be simply a result from standard set theory and logical postulates. Here is my idea: We know that M maps his perception of the universe ( an approximation of some sorts based on the actual state of the universe) to an "idea" (defined to be some string of symbols). The truth function maps strings to [0,1]. And equivalent maps a string to another string with the same meaning (note that it is reflexive). So, we know what all of these functions map to - thus we can find their inverse. So, because of M's belief, we have T(M(a.f. is equivalent to Q))=1
M(a.f. is equivalent to Q) = Inverse of T(1)
a.f. is equivalent to Q = Inverse of M(Inverse of T(1))
Now we are stuck - or are we? We have equivalent expressed in a different form the rest. So, change it to Equivalent(a.f.), which is the same as Equivalent(Q). Thus
Equivalent(a.f.) = Inverse of M(Inverse of T(1))
a.f. = Inverse of Equivalent(Inverse of M(Inverse of T(1)))
Now we take the set of all beta which could take place of a.f. in the equation. Call the set Y. We have established that Q must be an element of Y. However, it must also be an element of all beta which satisfy beta = Inverse of Equivalent(Inverse of M^2(Inverse of T(1))), where the two indicates iteration (Call this set Y2.) This is true because M thinks that he thinks that a.f. is equivalent to Q. In fact, Q must be in Yn for all n. Therefore, Q is an element of the intersection of Yn for all n. It also must have M as an element. So, we have greatly limited the possibilities already. It is my belief that for any a.f., there will only be one possibility for Q - but I cannot prove it, try as I might.

I think that this clarifies most of the issues you raised with regard to the original version - but there are still bound to be many more flaws in both reasoning and explantion. The point you had about the meaning function though - could you please explain? I am not sure that I fully see what you are trying to say, and I think that it might be very important. Thanks again.

I'm sorry, this reads exactly the same to me and all of my previous problems/questions with it still hold. Try doing a fresh rewrite without looking at what you had before...in fact why don't you try to describe it in plain English first since it's not very mathematical and your misuse of set notation is just making it more confusing

Plato Demosthenes,

The celebrated cognitive psychologist Piaget pointed out that "abstract modelling" like "binary logic" is one end product of the maturation of "consciousness". For such a model to embody "consciouness" itself therefore evokes the same problems of infinite regress that theists must face with the question "who made God".

Okay...here is an attempt:

Definitions:
Q:=all items that satisfy "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 Q
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 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 outputs
Earlier, we defined Q 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 , but) There Exists: Set x string |->the statement "string an n element of set"


Assumptions:

We know that there exists the string Q is equivalent to R, where R is some set of "axioms" from 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) and Equivalent(Q), 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"

Conjecture:

Q and R are the only elements of the intersection of Equivalent^-1(BM^-n(ThereExists^-1(T^-1(1))))for all n

Significance if correct:

If M is interpreted as Mind, A as the senses, its symbol-string as ideas, and B as the bodily output (be it in a following thought or physical action), and Q is the set of all minds, then one arrives at the statement that one can determine Q from a mind and its beliefs. An application might be to use an artificial intelligence which passes every Turing Test and is conclusively sentient and take its program and beliefs on the mind to program every possible mind. Using a more formalized Calculus of Self ( http://www.aleph.se/Trans/Cultural/Philosophy/identity.html ) , it becomes apparent that the creation of another mind of yours would allow you to experience what the computer you experiences. In other words, if you program Bob's computer mind to perceive being chased by a shark, then a part of Bob in real life will perceive it. The identity of Bob's mind could be determined by some method of correlating brain signals with the computational processes occurring in the computer. In fact, if you were to program the "Bob" algorithm to run on, say, 30 different computers then the majority of him would perceive that shark. The only part that wouldn't be would be a small twinge of his thoughts. Of course, it can be fairly safely assumed that any program of any mind would require vast computational resources, so parallel processing would probably be in order. All of this is assuming, however, that the human brain would not overload and go insane under the simultaneous perceptions. If no one would go insane, then one could (as long as there is some factor of "desire" in Q's axioms) program a sort of LifeLight (see Pendragon, The Reality Bug). Horrible ethical issues would be created, but would probably be solved. Some interesting experiments could also be created. What if the program it is based on is not really sentient? What if enough mind duplications are formed that they form a sort of "Meta-mind", much as the neurons of the brain collectively emerge into consciousness? Then again, the whole idea could be wrong and I have just wasted half an hour of my life.

Could you please explain Piaget's reasoning further? I actually think that what I am doing is might be just iterating his idea an infinite number of times. Also - how does it apply at all to "God"?

Plato-D,

Sometime has passed since I studied Piaget. The inheritors of his analysis of the development mathematical thinking in the child would seem to be Lakoff and Nunez (Where Mathematics Comes From).

http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From

The implication is that mathematics is a product of "consciousness" not some independent (transcendent) realm of knowledge which can be used to model its own source process. This objection to your endeavours is therefore analogous to the atheists challenge to theists who want "a deity" as a source of "creation" involving the regress is "who created the creator?".

Even if you reject Piagetian inspired explanations of mathematical thinking there is a plethora of material on "consciousness" attempting with limited success to describe its nature.

http://consc.net/online.html

There are even deflationists like Maturana who who argue that "human consciousness" is merely a complex aspect of "the general life process" which in turn has been modelled in terms of non-linear "Systems Theory"

http://en.wikipedia.org/wiki/Systems_theory

In short, whereas I applaud your efforts in modelling "something" what that "something" is may have little to do with the total range involved in consciousness studies. I believe in fact your model is predicated philosophically on "naive realism" which assumes there is a world "out there" of which we are "conscious". I think you will find that many writers agree that "consciousness" is a non-dualistic process in which "isomorphisms" have no meaning.

Sorry, fresco, for the long delayed response. I lost track of this site for a while there. Anyway, though you are right in that I believe firmly in Platonic Ideals of sorts, I do not see how that has bearing on my idea. Even if mathematics WERE solely to be a product of the mind, my idea would still give rise to a concrete thought of what a mind is anyway. I also agree that the human mind, as part of human physiology and behavior, could theoretically be described by systems theory - but here again, though it may give a slightly different interpretation of the results, would keep the results unchanged. First and foremost, a human mind can be approached in two ways: from the level of neurons (which is dependent on the levels of DNA, molecules, atoms, sub-atomic particles, theories of everything, etc.) or from the level of thought. They both must, neccesarily, converge to one thing in the end - but they still can give different interpretations. From the neuron level, consciousness is explained by collisions of atoms and electromagnetic fields - yet on the thought level, it is described as the interaction of thoughts, ideas, and concepts. To be clear, however, this is not dualism - the "thoughts" are, in essence, simply generalized physical notions and events. The same principle is shown in all social sciences, and even most physical ones: a star, cell, pig, or acid is really just a collection of superstrings or {insert favorite Grand Unified Theory's fundamental item here}, interacting on its own level with its own rules. But, it still makes sense to study the orbits of stars, the structure of cells, the population of pigs, and the characteristics of acids. Doing so is not separating the world into two distinct substances: cells and else, one is just generalizing certain large clumps of the else. Secondly, though systems theory is making great progress, the general notion of human is still unknown. With regards to your comment that I am assuming that there is an "out there" world, I think that you have again misinterpreted me. I simply state that there is something where we receive input from, and someplace where we receive output. That place might be ourselves, but it still must come from somewhere. Isomorphisms are, again, a generalized concept. Given a property P (or a set of properties N), anything isomorphic to one item with those properties is something with those same properties. I'm not sure, though, what you meant in your previous post by "isomorphisms". Could you explain?

By "isomorphisms" I mean mappings from between "internal states" and "external states". Non-duality imples there is no dichotomy between "internal" and "external". This has a direct bearing on concepts such as "input" and "output"

In short we differ on the acceptance of fundamental axioms. For example from a devils advocate position I could argue that "neurons" tell us as little about "consciousness" as "wires" do about computer programming. (Indeed the choice of analysis models for "neural networks" can range anywhere from simple switches though finite state machines to quantum events of a "non-local" nature). Furthermore, that "devil" could even argue that we don't know what "an event" is, because the event window is determined by the ad hoc aspirations of "the observer". In other words (non dualistically) all "things" require "a thinger" such that your "something" and "some place" become deconstructed. (I have in mind here some ideas from "second order cybernetics" which deals with the observation of observation and relates to rhe genetic epistemology of Piaget).

Ultimately what matters is agreement btween ourselves about a paradigm or "semantic field" for "consciousness". This is notoriously difficult and having dipped into the range of the literature cited previously I feel as though I am looking from some small vantage point at some of your potential blind alleys. I therefore suggest you also look at some of these papers and perhaps report back if you find a reference in direct support of your views.

LATER EDIT

I note that I did not spell out my central problem with your original post which implied a "proof". Static set theory and the binary logic of persistent "properties" themselves seem to be predicated on "naive realism" thereby rendering a concept of "proof" vacuous. On the other hand I do agree with some of your ideas which imply "nested" levels of description since this is taken up by the Santiago theory of cognition in which "thought" is an epiphenomenon of "general life processes". The nested structures of "cell", "organism", "society" etc are vertically ineractive such that no descriptive level is ever "sufficient" without reference to the next level.

BTW This looks interesting as a third party contribution to the discussion.
http://www.cs.bham.ac.uk/~axs/misc/consciousness.rsa.text

Ah...I'm think that we have two very different kinds of isomorphisms in mind. The isomorphism to which I am referring to has no bearing on internal or external, merely properties of some sort between two things (real, abstract, in the mind...anything).
As to your devil, I would agree, though half-heartedly, on both accounts. I do not see the connection to the proof, though - could you please spell it out a bit clearer next post, I evidently am not seeing what you are seeing.
As I said in my first (?) post, I'm not sure on the strength of the relevance to the actual mind that my theory has. However, as I am not yet finding a way to prove that Q and a.f. will be the only remaining members of the set, the issue on degrees of sentience that I believe you are referring to can not be settled yet. For all that we know, it may turn out that the axioms of Q are in a way such that there are different degrees of truth to them? By the way, leaving aside the possible implications about the mind the proof might have, do you see any way to prove that Q and a.f. are the only members of the union? I think that once that is settled, the possible relevance to consciousness will be far more clear.

Over several years of thinking about "consciousness" I have a major problem with communing with any classical set theoretical terms like "union". I therefore am unable to comment on the coherence of your model if you see it resting on such a term. Again, I urge you to read some of the literature like "Where mathematics comes from" in which both "mathematics" and"consciousness" seem to have a common origin in organismic action. This implies that for any model to be descriptively adequate it must reflect the dynamics of of an organism "far from equilibrium" (see references to "autopoiesis"). Such dynamics imply that binary truth functions (or indeed "truth" itself) are insufficient to describe the interactional flux.(Fuzzy logic comes a little closer perhaps).

I believe that what is actually needed in modelling consciousness is a closer look at "identity" with respect to its dynamics. From a transcendental point of view there is no "constant self"....rather a committee of argumentative aspects of "ego". Indeed "self" could be simply a "social device" held loosely together by one's "name". Further thoughts on "identity" include the point that any two items are necessarily both "similar" and "different". (Trivially..."similar" because they are both objects of comparison and "different" because there are two.) What matters therefore is for a model to encompass functionality of "similarity" and "difference" with respect to action.

I am however encouraged by the particular reference you cited on "self identity" which at least pays lip service to some of these ideas.

Well, that being said, do you have any suggestions to make it more applicable to the actuality of consciousness? Any extensions, perhaps, of my basic ideas which would, in your opinion, correlate my theory to reality a bit more? Would you, possibly, have any ideas which would take into account the "fact" that mathematics stems from the mind, and not the converse?

I think you should have another look the reference on "self identity" and attempt to extend it with respect to a concept of "functionality" of a "self consciousness" concept in cognate organisms. For this you might bear in mind the Santiago theory of Cognitition (Maturana et al). I don't think any reference to "mind" (human or otherwise) is helpful. ("Mind" is not mentioned in current thinking on the origin of mathematiocs..."action" is) You might also find the work of Fritjof Capra (e.g The Web of Life) to be useful in pulling together some central ideas.

Once again, this is a difficult area to agree about...a point which is underscored by the "third party"reference at the end of one of my previous posts. It may be the case that mathematical modelling per se is doomed to fail to be descriptively adequate for central aspects of "consciousness" (I am thinking of lessons from Godels incompleteness theorem here), but it seems that dynamic non-linear models (e.g catastrophe theory) are looking more useful than static ones.

Well...it does not seem to me that the Santiago theory of Cognition is really relating much to the study of consciousness. Yes, it most certainly is a process of life - few would dispute that. The theory also claims that mind is a process - few modern philosophers would dispute that, either. But it does nothing to explain what sort of process the mind is; what role it has in life; what it even is - let alone issues on duality, realism, or AI. I may be missing some key parts to the theory, but it currently seems to me as though I have little to gain from it. And though the Web of Life has some interesting views on the immune system, my prior comments still apply.

Yes, it is a very difficult area to agree - so little hard fact is known that countless, mostly incompatible, theories abound.

As for Godel's Incompleteness Theorem - it does not limit mathematical representations of human thought in any ways that I know of (disregarding the [in my opinion utterly false] so-called proof that Penrose gave stating that the mind was above mathematical description. Anyway, even if it were in some, as of yet unknown, way a limit to mathematical theories of consciousness, then all that would mean is that humans are at least as complicated as numbers - hardly a crippling downfall of those theories.

The nice thing about mathematics is that both dynamic and static systems are, essentially, the same. So, really, it won't matter in the end which formalism of it is used - they are equivalent.

Anyway, my apologies if, in 200 years or so, it turns out that you were right all along! Do you know of any other theories that I might be interested in?