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Author
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Topic: Cosmogony, Holography and Causality
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RBH
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posted 20. September 2003 01:23
As Mark implicitly suggests, logic consists in a set of formal rules for manipulating symbols according to their shapes; it is a syntactic system for pushing symbols around without reference to any meaning the symbols might be assigned by some intelligent agent. Hence logic is algorithmic and is therefore computation in Turing's sense. (Which is not to say that every logic problem can be computed in practice: the Halting Problem lurks out there.)
At the risk of sidetracking this thread, I'll mention Inman Harvey's paper Cognition is Not Computation; Evolution is Not Search. Harvey argues that computation, as it has been understood since Turing as an algorithm operating on (the forms of) inputs to produce outputs, is an inappropriate model/metaphor for cognition. Cognition, Harvey argues, "can only be attributed to the behavior that arises from the conjunction of an organism [or a machine] and the world that it inhabits." I suspect that Harvey's view derives from his work in robotics. There one is confronted directly with the problem of acting in the world in time, and with the vacuity of abstract notions of disembodied cognition. Cognition is bound up with the organism/world system; it does not reside in one of the components of that system. To suppose the latter is a category mistake.
RBH
[ 20. September 2003, 01:57: Message edited by: RBH ]
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gedanken
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posted 20. September 2003 09:52
Mark said:
quote:
... I intuit two issues here. The first is the "mapping" issue. Correct me if I'm wrong, but I think you are proposing that, because the natural world presents what we consider gradients or gray areas, this suggests that the natural world is in some sense not strictly logical, and hence this fact requires scientists to use their judgment to discern the true state of affairs, or alternatively, just admit (at least tentatively) that there are insoluble gray areas.
I don't view this as a problem of logic. I see it as an issue of information. The propositions we have at hand may present a gray area; they may even present a contradiction or a paradox. If science confronts such a situation, I suggest that historically it seeks more or better information (or better models) to resolve the contradiction (but you can be sure that science is keenly aware that an apparent contradiction exists, and that such contradictions create discomfort).
I think that RBH’s points above are highly relevant. In fact I think that the issue raised, of cognition is not computation (taken generally, as opposed to whatever is in that paper), is exactly my point. Logic is a formal syntactic procedure that works only on something that we mentally consider as “absolute” truth. It only works approximately when truth is only approximate, and the combination rules vary from situation to situation to adjudicate that mismatch from syntactic “rule” to perception of reality.
A further example:
We don’t know if John was in the room, but everybody in the room would have been killed when the bomb went off.
How do we determine if “John” was killed? Logically we would have “John in room implies John killed”. And according to principles of logic we only have the difficulty of determining if John was actually in the room. The problem is that the real world is more complicated. The implication is not strictly true in the real world—John could have been protected by a heavy steel desk, so even the implication is not a strict mapping to the real world. Then even if we take the implication mapping as strictly relevant, if john was only partially in the room (say in the doorway) he may have been only injured due to distance, protective effects of the wall which only exposed part of John’s body to events, etc.
The real world is always more complex than strict syntactic logic allows. That is why judges and juries are supposed to use “judgment” in evaluating evidence, not strict application of rules without adjudicating with further thinking about the situation. (We would not want a crisp logic machine to act as judge! And we see judges railing against crisp logic sentencing rules that tie their hands.).
Since the mapping of syntactic logic to the real world is necessarily approximate, it is necessarily the case that apparent rules of logic are not completely accurately mapped to the real world. The issues of “missing information” do not resolve the issues of logic. They confound the issues of logic, rather than resolve them. The “missing information” only resolves issues by going beyond the logic previously stated, rendering it less than exact mapping to the real world.
I see logic (learned and validated from experience) as a feature of how our minds work, and only approximately a feature of the real world. Yes indeed it is based in features of consistency of relationships causality that we observe. I just say that “logic” per se is not a sufficiently accurate map to those relationships to be meaningful in the sense of abstracting the “syntax” of logic and calling that a property of the real world.
On the resolution of logical inconsistency by seeking “more information,” I agree this is the scientific process. But that is a process of finding a better approximation of our syntactic representation of scientific “knowledge” to the real world. Once again it is fitting our syntax describing the situation to the humanly generated syntactic rules of logic, based on the real world—not of the real world fitting itself to our syntactic rules! When classification fits our rules, we use it, and when it does not, we abandon them. Their being necessarily “fuzzy” is a human problem, not a problem of the real world that moves on without consideration of our categories.
Part of my point is that one can construct slightly different syntactic rules of logic, and find that those rules also apply in the real world. There is not a single syntactic set of rules that apply. Since these different rule sets all approximate aspects of the real world, but not exactly, there is no way to say that any one of them is the “real” rules of logic. The syntactic nature is from our use of language, which can be argued to be an analog process in the end, not “digital”, and thus not based strictly in a syntactic nature coming from the world itself.
[ 20. September 2003, 10:21: Message edited by: gedanken ]
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Rex Kerr
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posted 20. September 2003 15:00
I'm content to call logic a feature of the world--just not in calling it the fundamental feature without more evidence. (I'm not sure what this evidence would look like if I found it, but I do want it.)
Also, I think it is not quite fair to say that the rules of perception are isomorphically consistent with the rules of logic. Because, of course, any rules that we come up with are going to be expressed using logic, and any time there is a contradiction with a given definition of terms, we'll redefine the terms so there isn't a contradiction. (E.g. our refinement/redefinition of "inside" vs. "outside" the room.)
Saying that the rules of perception is isomorphically consistent with logic suggests that perception is isomorphically consistent with logic which in turn suggests that perception has to behave, because logic won't let it do weird contradictory things.
I think this implication has the situation fundamentally backwards. Logic is an incredibly powerful tool that we can use to systematize our perceptions. That we invent new categories to explain our perceptions (e.g. "illusion", "in the doorway", "dream", etc.) is an indication that it is our perceptions that are in control, as it were, and logic is wrapped around them to provide a framework for higher-order cognition to operate on (sort of; I don't think higher-order cognition is exactly logical either, but again, logic is so general that it is a good match).
So perhaps I actually agree with Langan's statement that
quote:
Because logic is tautological, ie, universal and reflexive, that which is logically inconsistent is not only illogical but unreal.
but only because logic apparently provides no constraints whatsoever on a system; it can be wrapped consistently around any structure. What is interesting is that the wrapping gives us predictive power in this world. But that is not a consequence of logic's universality but in spite of it.
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gedanken
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posted 20. September 2003 16:17
I decided to extend what I was saying a little.
I’m not a relativist. One might mistakenly get that impression from reading what I have said.
I would be a “relativist” in the sense of the actual way that Einstein meant his “theory of invariants” (normally known as “relativity”). In that, various “views” of reality can be translated into what one would see from any different view. Form each viewers position, one can predict largely what the other viewer would see. This is quite different from the philosophical misunderstanding of Einstein to mean that there is no single “reality” itself—when noting that there is no universal reference view. Rather that single model of reality can be viewed many ways, yet is a single reality.
And the disputes here are among ways of looking at “reality” in which all the views have a single reality. None are “relativist” views, because non claim that there is more than a single “actual” reality. The issue is of the character of that reality, e.g. is “syntax” a character of that reality.
But my points in showing that there are different kinds of logic need to be extended to show that there are also various syntactic ways to represent any one of those “logics”. Not only are there a wide variety of logic semantically that are consistent with empirical experience, but of each of those there are a very wide variety of ways to symbolize them. Even some ways may be totally image or pictorial, without a “syntactic” or “lexical” construct whatsoever.
This is my basis for arguing that there is no evidence presented here that establishes a “syntactic” basis of reality itself. (The “syntax” is in our minds and how we interpret reality, not the reality itself.) I wanted to distinguish that clearly from a relativist viewpoint that would mean that there was not necessarily a common character of that reality itself—a very different point from different viewpoints in terms of language or mental construct to represent varying aspects of that single “reality”.
Rex quotes Langan:
quote:
Because logic is tautological, ie, universal and reflexive, that which is logically inconsistent is not only illogical but unreal.
I don’t personally think I agree with this. My example of the person who is part way in the room is a good example. It is inconsistent in the sense that “logical” statements that require the person to be outside the room are partly true, and statements that require the person to be inside the rule are partly true. Logic cannot resolve these issues. More information does not resolve those issues either, because the more information has to be in terms of a more general structure of the world itself, not more information to clarify the given logical statement “he is in the room” with greater certainty one way or the other. So in at least that sense, that the person is part way in the room is inconsistent with “logic”. Does that make the situation not “real”?
Take the following simplistic argument I have seen presented against common descent:
The descendents of a member of a species is also a member of that species. Common descent means that descendents of one species are members of a different species and not the current species of the descendent. Therefore common descent is false. QED
So clearly common descent is not logical! Common descent is “not only illogical but unreal”.
[ 20. September 2003, 17:01: Message edited by: gedanken ]
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Rex Kerr
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posted 20. September 2003 22:40
If a person is partly in a room, the statement
"The person is in the room"
is false, and the statement
"Not (the person is in the room)"
is true. In a simplistic model, "not (X in Y)" and "X not in Y" and "X outside Y" all mean the same thing. So if
"The person is in the room" is false
then
"The person is outside the room" is true.
However, we then notice that these premises are giving us contradictions when people stand in the doorway. So we then retract the premise
"X is outside Y" iff "not (X is in Y)"
and create a new category called "partly inside" and add the premise
"not (X is in Y)" IMPLIES ("X is outside Y" xor "X is partly inside Y")
It's kind of clunky, but it fixes the first set of contradictions. So it's not really logic that's at fault here (aside from not allowing as simple of descriptions as one would hope), it's our axioms. But this is no surprise, given the extraordinary flexibility of logic, and its utter dependence on axioms/premises to say anything about anything.
[ 20. September 2003, 22:42: Message edited by: Rex Kerr ]
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gedanken
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posted 20. September 2003 23:48
Rex,
You have implemented a step in working with fuzzy logic in recognizing discreet areas of importance. For if one was to use some simple piecewise linear degree of “truth” approximate rule, one could have a fuzzy “truth” function that is fully 1 for your new “inside and not in doorway” class, and fully 0 for new “outside and not in doorway class”. In the transitional “in doorway” one could approximate truth value with some linear measure of how far through the door—and then calculate fuzzy “truth” values as an approximation with partial truth values for implications showing up only when in the doorway. We can resolve the “contradictions” you mentioned with this method as well. X_Outside = (1 – X_Inside), so in the doorway poses no contradiction since both Inside and Outside can be partially true.
But of course even that model would be too crude an approximation for many aspects or relationships we may want to explore. There is no reason, for example, for the “linear” approximation to be a good fit to consequences we may encounter, nor have we provided a good means to measure the truth value for an actual human in the doorway, any more than the original crisp or stepwise logical truth function.
In fact that is the problem. Creating the “in the doorway” class does not really solve the problem. Because it was not the logical problem of the conflict of outside and inside descriptions that was at heart in the first place. Rather it was that the combinatorial explosion of logical conditions that arise as one considers more refined details.
This combinatorial explosion is actually the downfall of “fuzzy logic”, just the same as “crisp logic”. Both attempt a simplification of an even more complex underlying structure. Both attempt approximation models of the real world, attempting to describe as accurately as possible what happens under varying circumstances. They both provide very useful “fictions” which can be adjusted to have substantial predictive value. (And my point from other posts that there is not a single “logic” with a single “syntax” that is inherent in some degree of mapping to our real-world experience.)
Let’s go back to the doorway and bomb problem again. Now when is “John” actually “IN” the doorway? With respect to bomb fragment trajectories, we have a very different meaning of “IN” the doorway, from say a simplistic model where we check for any portion of John’s body crossing a line. (Which line, anyway? Was it a region defined by the thickness of the wall and straight lines drawn across the doorway? If John was virtually all inside the room, but his little finger was just slightly poking into the “doorway” region, is he still “in the doorway”? How about the tip of John’s shoe? Or was his shoe actually part of “John”? How about his umbrella? His hair? If he left a piece of his hair in the doorway, is he still “in the doorway”? A dog might be described as “in a territory” if his sent markings still lingered.)
The problem of course is that the logical rules describing consequences were themselves approximations to a more complex world. Wherein being in the “shadow” of the doorway was just as good as being “IN” the doorway or “IN” the room, up to a significant distance away from the door when the bomb went off. (That “shadow” defined in part by the bomb’s position.) Each little detail as one refines the problem shows problems with the logical rule set that was initially posed. The refinement never ends—we never really get the “correct” rule.
And the situation is the same with membership in species. Just it is perhaps even more complicated.
Fuzzy logic does not solve these problems, because it is simply another approximation theory. Approximate a control system with a proportional feedback rather than a “bang bang” (e.g. “logical”) feedback, and one may get a better control system. Or one may simply get a more complex system, with fractal attractor related instability within the certain region, and a “bang bang” control outside that region after all.
In fact, Fuzzy Logic can produce brittle resolutions to contradictions, like suggesting X_Inside = NOT X_Inside, which has a brittle solution of 0.5 Inside, 0.5 Outside. These are posed in fuzzy logic books to suggest that an improvement of some essential characteristic has been achieved. But one need simply construct a three way overspecified logical contradiction, and it has no solution once again even in single dimensional continuous “truth” values. One can even construct logical constraints that have “imaginary” truth values that solve, but not real. With combinatorial explosion of details, there is little comfort in having a brittle resolution to a 2-way overspecification when N-way overspecifications will abound.
My point was never to suggest that “fuzzy logic” was a better model, nor that “logic” is not a good model for many approximations to real-world relationships. Rather it is to indicate that there is not a single such “logic” model that is useful, and that all such models have failures when trying to apply them to the real world. Since a variety of such “logic” models are useful and map well but not perfectly to any situation, we have no “isomorphism” demonstrated from a particular logic model to the real world. (Certainly not a “syntactic” identification.)
Also one must note that “predictive power” comes from generalization. Such generalization can use one of the ordinary crisp “logics”, or any number of fuzzy logic descriptions or combination rules. The usefulness of such a “logic” combined with the particular set of premises of the “theory” or concept gives that concept usefulness and “predictive power” that we can use in decision making with incomplete information. That is useful because we can never gather all the exact details about any situation we may be interested, we must inherently examine limited data. Generalization is a powerful technique—wherein the “logic” of choice helps in processing models of the real world.
[Some edits above, this added]
You will notice that I am not recognizing “logic” as a single pattern of rules, by the way, even for “crisp” logic. As philosophers and mathematicians discover various cases that provide inconsistencies with some mental construction (modeled on a way of thinking about the real world), they propose new logic rules, such as recognizing class level set distinction, etc. I am simply lumping in various “fuzzy logics” into the mix. But there may be other resolutions to some of the problems. Already in standard “logic” text books one finds dealing with fuzzy conditions (not so recognized) with other means of argument that are not deductive and are considered “persuasive”. Problems of “computability” and also Godel-like issues are relevant. Rex already recognized various logics like “first order predicate logic”, and that there are various axioms of the various logics. The possibility of certain fuzzy logics is that they don’t recognize the Godelian problems in the same way, and thus may correspond more closely to how humans resolve such seeming contradictions to draw some sort of generalization or inference anyway in the presence of incomplete or overspecified information.
A second point here is that these varying “logics” don’t really in themselves resolve the issues of combinatorial explosion of detail, which is a constant feature of the real world. Rather we tend to ad hoc create new logics when we discover that the situation merits a new way of looking. (Thus “logics” are a feature of our thinking process and language use, as opposed to natural extensions of the real world.) Set/Class distinction arose when we wanted to do a certain type of generalization, realized that a previous logic had a problem. “Evidence theory”, “fuzzy logic”, various axioms of crisp logics—these are all attempts to deal with different situations that are encountered in a variety of attempts to model the real-world. They often occur when we extend the results of our rules to more distant “predictions”. And of course our ability to predict is always becomes less certain as the distance or time of the event becomes further from our consistently analyzed area of expertise. Neural networks attempt a different way of drawing the very rules in a fuzzy manner form the experience directly. They also exhibit the same failures in kind as our logic based predictions—though they become quite immediately obvious because of the number of strange “logics” that they produce.
(Nice post, by the way. It helped me to clarify some issues where I was unsure myself—I may actually have understood something that I can apply in some on/off again research.)
[ 21. September 2003, 11:05: Message edited by: gedanken ]
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Erik
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posted 21. September 2003 16:53
While I have no interest in defending the so-called CTMU, I am a bit puzzled by the manner in which the CTMU (or the CTMU supporters' argument) is criticized via criticism of logic and mathematics. Rex Kerr gave an example of how naive logical reasoning about statements involving the notion of being "inside a room" may have to be modified to accomodate the possibility of being "partially inside a room". The point of formalized logical and mathematical reasoning, however, is not to directly provide us with the correct properties of any predicate (e.g. the predicate P(x,y) = "x is inside y") we are interested in. The real utility of logic and mathematics is threefold:
1. Logical reasoning enforces consistency in our knowledge, postulates, and conclusions. Many of our thoughts are probably incoherent or inconsistent. Formalizations helps weed out the inconsistencies.
2. By insisting on rigour, we can discover and remove non sequiturs.
3. The human mind is often poor at detailed reasoning that requires many steps. By automating reasoning, logic and mathematics can transform difficult problems into routine problems. Many problems that could only be solved by geniuses in the 17:th century can be solved by mediocre college students in the 21:st century.
An example of 1 & 3 in effect is the analysis of many-body mechanical systems. Anyone can understand how an individual pendulum or an individual spring behaves. However, deducing the unique behaviour of a many-body systems that is consistent with behaviour of the individuals parts is a task that requires mathematical models.
Much earlier (top of page 4) Rex Kerr also wrote: "But Russel, the universe is notorious for not quite matching our abstract rules for it. "Sort of matching" doesn't count as an isomorphism. If f=ma has abstract "real" existence, it has a funny way of showing it at v=0.99999c.
That we can find more complex mathematical descriptions of the behavior of the universe is a fine scientific assumption. I simply don't think we are justified in assuming that this must be the case in our metaphysics."
I disagree. Mathematics has an extreme expressive power. It can model things we can never imagine. The universe may not be isomorphic to any simple mathematical structure, but it seems obvious that it is isomorphic to some (possibly complicated) mathematical structure, simply because mathematics can express everything that is both consistent conceivable and more (i.e. we can drop the "conceivable" requirement).
Furthermore, the history of physics shows a progression towards ever more general and accurate mathematical models. General relativity and quantum physics are both extremely accurate. Indeed, some of the predictions of the latter have been verified to an accuracy of 1 part in 10^19. (A subscription may be required to access the linked site. It is an online article of Physics Today, July 2001, about BCS theory and Josephson junctions.) This level of accuracy and generality even raises considerable hopes that our universe is isomorphic to a simple mathematical structure.
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Rex Kerr
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posted 21. September 2003 18:33
I have been intending to criticize absolute certainty, not logic and mathematics. However, if people want to be absolutely certain that logic and mathematics occupy some fundamental role in the nature of the universe, then if I disagree my arguments will necessarily be pointing out potential discrepancies between math&logic and our observations of the universe.
I am not sure whether Erik actually disagrees with my statement, even though he says he does, as I agree with everything that he said.
On the surface of the earth, a 10kg cannonball will behave as a particle (as opposed to a wave) with much better accuracy than one part in 10^19, and it will accelerate according to Newton's laws to within a few parts per million. Making ever more accurate observations of a well-understood phenomenon is not necessarily the way to notice that one's theories are imperfect.
Still, mathematics has such expressive power as to make some isomorphism between (our perceptions of) the universe and a mathematical structure a certainty. (One possible proof would be constructive and involve building a giant look-up table.) One has reason to hope that it would be more elegant than this.
However, hoping and knowing for certain have very different metaphysical consequences. Although I have particular complaints about CMTU as opposed to other assumed-certainty theories, it is really the unreasonable certainty that I find most troubling.
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gedanken
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posted 22. September 2003 00:01
I think that Eric, Rex, and myself would find ourselves in very close agreement on all relevant points.
As per Rex’s discussion, I also was not intending a criticism of “logic” per se, but rather dealing with issues of hyper-certainty. For example in my example somewhat above, Rex came up with a quite reasonable solution to many perceived difficulties, using conventional “logic” principles, which when applicable to premises makes the standard logical argument completely usable and valuable.
The issue has revolved around a rather subtle issue of the degree to which specifics of logic are constructs of the mind or consciousness. That they are extremely important for us to use, and are found to be very consistently relevant to real world problems does not make “logic” less a human construction.
The nature of the mapping of the “logic” expression to aspects of the physical universe is the subtle question.
quote:
I disagree. Mathematics has an extreme expressive power. It can model things we can never imagine. The universe may not be isomorphic to any simple mathematical structure, but it seems obvious that it is isomorphic to some (possibly complicated) mathematical structure, simply because mathematics can express everything that is both consistent conceivable and more (i.e. we can drop the "conceivable" requirement).
Here, to echo Rex’s point, the mapping may be very very accurate—yet is it strictly “isomorphic”?
I find science to be an art of abstraction. All abstraction removes characteristics of the specific and generalizes. And furthermore specific situations seem subject to combinatorial explosion in their details, as we try to refine those small “errors” that make any mapping seem accurate but not absolutely and completely accurate.
The fact of the accuracy of mappings, found under the art of scientific application, shows that the concepts in many mathematical models have a very important descriptive power in relationships in the physical universe.
But the combinatorial explosion in the inaccuracies in any measurement of any such mapping dispute the absoluteness or separate and Platonic existence of the strict isomorphism. (At least they render the issue to be philosophical in nature, as opposed to directly scientifically verifiable.)
After Langan’s OP, he also made reference to “syntax” of logic, and claims of the physical universe having a “syntactic” nature.
I think that Rex, RBH, and my presentations were intended to question the “syntax” component. Most of the disputes or apparent “criticisms” of logic were really intended to demonstrate that there are more than one syntax, e.g. a huge multiplicity of syntactic representations that could map to properties of the physical universe. These do not dispute the importance or usefulness of any one of those “syntactic” methods, and Eric’s points are completely consistent with our intended meanings in my opinion.
But with multiple, even contradictory in part axiom sets for various “logics”, that a single logic is strictly isomorphic in some absolute sense to the universe is being questioned. And if the syntax of “logic” per se cannot be completely, entirely, and universally described, one can hardly then argue that such a syntax exists in a strict “isomorphism” to the physical.
PS I agree with Rex almost universally in his last several posts—even when seeming to present some issues.
ALSO with regard to Eric’s quote above, there is one more issue I might bring up. That there is a “simple” strict isomorphism from a mathematical relation to the universe is indeed an interesting question that does not have a clear answer. That has been the physics “Holy Grail”—not clearly going to exist. That many approximations have great expressive power has not given an inductive “proof” that the final fully accurate relationship will be found. (But of course that is not to say that it would not be found, either.) In fact if we use an “inductive” argument from experience of Physics, we find that we keep finding sub-structure within structure. So will the next refinement still have sub-refinements? Perhaps ad infinitum?
[ 22. September 2003, 00:17: Message edited by: gedanken ]
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chimp
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posted 22. September 2003 02:24
Perhaps there is no escaping the fact that language[mathematics] corresponds to the perceptual universe, in that language describes "things" and things themselves are representable by identity operators. Even if the theory is not completely constructed due to Godellian incompleteness, it must have an identity, such, that it may be represented as ...a variable.
A = identity
Truth = A V ~A
Whatever the mathematical structure that corresponds to material reality is, it must be governed by an invariance principle. The general contains the specific. Now the question becomes: is the law of excluded middle from Aristotelian logic, powerful enough and general enough to be the invariance principle that we seek?
Sure, fuzzy logic concepts also correspond to what we percieve, but those fuzzy logics can be explained as symmetry invariances, just as Aristotle's law can be:
A V ~A
Is an invariance principle:
_A_|_~A_|_A_V_~A
_T_|_F___|__T
_F_|_T___|__T
which is a symmetry:
(T|F) = (F|T) = (T)
Logic forms symmetry groups with generalized n-valued logic:
Generalizing:
T = A
F = B
? = C
? = D
MV-logic = {A = T, B = F, C = ?, ...Z = ?, ...n = ? }
1valued logic
A = A
2valued logic
(A|B) = (B|A) = A
3valued logic
(A|B|C) = (B|C|A) = (C|A|B)
= (C|B|A) = (B|A|C) = (A|B|C) = A
4valued logic
[A|B|C|D]=[B|C|D|A]=[C|D|A|B]=[D|A|B|C]
=[D|C|B|A]=[C|B|A|D]=[B|A|D|C]=[A|D|C|D]
=[A|C|B|D]=[C|B|D|A]=[B|D|A|C]=[D|A|C|B]
=[B|A|C|D]=[A|C|D|B]=[C|D|B|A]=[D|B|A|C]
=[A|B|D|C]=[B|D|C|A]=[D|C|A|B]=[C|A|B|D]
=[D|B|C|A]=[B|C|A|D]=[C|A|D|B]=[A|D|B|C]
= A
The most general identiy distributes over all other differentiated forms.
Tautologies of *generalized* logic are "invariant" under choice of truth value since they are always true.
Certain invariances must hold for certain conditions. For example, conservation of energy, conservation of charge. @@ = 0
Space is relational.
Mass-energy relates in space-time. Of course, cognition utilizes a different type of processing than a "Turing machine".
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gedanken
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posted 22. September 2003 12:49
Russel, I think you are implying that continuous value fuzzy logic would also "reduce" to a crisp logic for some symbols for absolute "true" and absolute "false".
The problem of course is that we have been identifying problems that don't fit with the "absolute" true category in the first place. Therefor the logical evaluations of the different logics in those cases would be different.
Can you show me a case in the physical universe in which an absolute "truth" applies? Describe a case of observation of the universe and a combination of logical rules applied, wherein there is no possible "fuzzyness" or dispute about the premises and logical inference.
--
By the way, the "turing machine" problem gets to the point, as RBH has posted some interesting links that you might want to investigate. How can humans use "logic" and yet get beyond computability limitations? Either they must be limited by the computability limitations--or humans must be able to deal with observation by going beyond computable "logic". Which is it?
[ 22. September 2003, 12:52: Message edited by: gedanken ]
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Rex Kerr
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posted 22. September 2003 20:30
Personally, I see no evidence that humans are not bound by computability limitations, but do see evidence that we compute. Thus, I tentatively assume that we are limited.
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Mark
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posted 22. September 2003 20:44
Very good posts. Of course, if Langan construes the Real as a language (as he does), he requires a syntax. He chose deductive logic as his syntax, claiming (I think) that this accounts for the continuity and coherence that we perceive in the universe. He also claims (I think) that this syntax provides the common medium making perception and cognition possible.
Some cognition models are computational, some are holographic. Logic would seem to provide a syntax for computational models of cognition, but I am not so sure about the holographic. Is cognition syntactic, or "whole"?
Ged, I agree that various logical systems deliver some degree of "mapping," and strict isomorphism is difficult to confirm, regardless of the system that is employed. However, I don't think the CTMU is addressing morphological isomorphism; I am under the impression that the isomorphism of the CTMU is intended to be at the syntactic level only, and this would implicitly allow for discontinuities at the morphological level (such as species, etc.).
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gedanken
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posted 22. September 2003 21:56
Mark,
Then with many syntaxes that could represent the same semantics, we don't even have a mapping to syntax. (Point being that the syntaxes are not isometric to each other, even if the semantics are, therefore can't even define a single "syntax", much less provide an isomorphism to the real world. Multiple non-isomorphic structures cannot be “isomorphic” to another, only at most one. And if one, why not the others? Furthermore it would have to be a sub-graph or subset isomorphism, as it does not describe all of reality, more of a homomorphism. But not a "homomorphism" either, since a single principle is found in many physical instantiations at a semantic mapping level. Homomorphism of the real world to the logic syntax does not apply either, as the entire real world is not mapping to the syntax.)
So overall I don't see a "syntactic" match, even if I see approximate relationships.
But of course my point on "species" was that there would really be a nearly topological (but not really) continuum from one species to another. Syntax has a hard time with transitions that must be made but don't exist in the real world. (It is of course our rules and perceptions that make the distinction later when there was no noticeable distinction at an earlier point.)
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Rex,
I suspect once again we are in complete agreement--but we must be careful on definitions in our communication. It depends entirely on what we mean by "computability".
One meaning can be the strictly limited Turing machine recognizer, which has very definite limitations. However I don't consider the analog recognition and presentation skills to differ from a different type of "computation", just that the definition is not of the same "computability". That analog computation capability of the human is something I am confident can be replicated (in kind if not isomorphically) in machine mechanism. (I know we will get a lot of argument here.) So in that different sense we don't have that distinction.
I think that Russel is discussing a brittle low order "logic", one that is "computable" by the former and more limited Turing machine.
(Now of course one may ask about Turing machine simulating an analog system--which I believe it could fully do. The problem is not once again in the means of computation, but in the definition of a successful computational result. An analog result does not look like a symbolic logic output from a theorem proof. The symbolic representation may be limited, as conventional Turing machine interpretation, yet the analog interpretation not so limited. I would have to expand on this to make it more clear--probably not here.)
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Looks like an interesting paper:
On the Nature of Explanation: A PDP Approach, Paul M. Churchland, Physica D 42 (1990) 281-292, North Holland. Also in Emergent Computation, edited Stephanie Forrest, MIT 1991, North Holland, and apparently also in A Neurocomputational Perspective. MIT Press.
The paper discusses what explanatory understanding consists in, but using a neural network (e.g. analog) rather than symbolic (and thus syntactic approach. I'll tell more when I've read it, if it turns out to be of more interest.
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One last issue: How would those who were supportive of this “isomorphism” judge the discussion of Dennett and computability and human vs. machine reason? (Thread Ray Kurzweil’s Impossible Vision, which also discussed “frame problem” and Daniel Dennett’s “Robot’s dilemma”.) Because it would be very interesting if the same people thought that human reason had to go beyond that which was “computable” by a logic machine in that thread, and then here thought that the properties of the universe were isomorphic to the syntax processed by the “logic machine”. If the universe and the rules of logic are isomorphic, why is it so difficult to understand how the robot could use the rules of logic to figure out the problems, then?
Also that thread, though a little tangentially, discusses issues of “logic” in a different way that is of some relevance here. I also appears to discuss the issue of “computability” there that I mentioned in previous division of post—so I don’t need to repeat here.
[ 23. September 2003, 01:00: Message edited by: gedanken ]
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Rex Kerr
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posted 23. September 2003 00:58
I actually mean computability in the Turing sense. Any analog system can be simulated to arbitrary precision by a digital one. A Turing machine can perform any computation that a digital system can.
People seem to have mixed up their innate sense of the superiority of human cognition and their frustration with not being clever enough to implement artificial intelligence with the idea that computability is the barrier.
Cognition of the gaps isn't any more scientific than God of the gaps. It may be that computational limitations can never fill some gaps and that those must be bridged by non-computational cognition. However, I have yet to see any argument that doesn't essentially boil down to, "Human cognition is so cool, and I can't figure out how it works!"
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