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Author
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Topic: Cosmogony, Holography and Causality
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Zachary Aufdemberg
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Member # 335
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posted 12. September 2003 23:53
gedanken wrote
quote:
I have little time at the moment, so this will be short.
I learned language slowly. It may have been inherent that I would learn some language communication skills even if I was alone with others who had not previously developed communication, but in my case I certainly did learn from my elders and others in my environment.
Then I remember learning about “logic” at different times, and at different levels of precision and of different aspects. I remember vaguely trying logic precepts as I learned them in terms of examples – examples which are based on real-world experience. So in essence I validated my concepts of “logic” in terms of language that I learned (by experience) by applying to my experience in the real world.
I find the notion that “logic” some how comes from some other source to be perplexing.
Don't confuse the source of your belief with its actual justification.
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PS Zachary, I certainly am not arguing simplicity at expense of sufficiency. I carefully said:
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But there are indeed views that contradict other views in some regard, but which predict the same observable (empirically verifiable) phenomena. In science there is a goal of finding “views” that are consistent across the broad scale of what is empirically verifiable, and not contradictory within that, as such a contradiction would invalidate the view as being actually “empirically verified”.
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Indeed one of the problems that occurs is that someone finds a small consistency with some observation and draws a conclusion, without checking that with the larger body of implications it would have. That is not an issue of “logic”, it is an issue of properly applying the empirical method (logic per se will not resolve the issue).
Properly applying the empirical method to perception is a matter of logic. And I agree that we should be looking for a model of broad empirical scale and scope with no internal contradictions. Logic can not resolve this issue only if there are multiple internally-consistent mutually contradictory models.
quote:
But the “scientific method” includes making the subject broadly accessible by way of observation and without regard to assumptions that cannot be validated by observational methods. The statement above seems to indicate a differentiation between using observational methods to obtain wide consistency or universality of acceptance, and “the standard scientific method”.
My point was that standard scientific methodology relies on many assumptions regarding the nature of observation and then pretends to have some sort of validation. Validation comes from logic not assumptions.
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I would say that such a case of ”empirical world … found to exist in the mind” would be of two cases: Further revelation in terms of actually observable evidence, or considerations that are philosophical in nature and not based on observable conditions that can be universally accepted. In the former we have simply turned up new observational evidence, not found the answer by “logic”. In the latter, we don’t have universally agreement – unless some alternate technique is used like enforcing universality by external means rather than self-derived observation. (There are cultures in which all members voice universal agreement to concepts that we outsiders would find foreign and contradicted by our observation.)
Your two cases aren't the only games in town since the latter example leaves no room for universal agreement based in logic, and of course truth is not determined by committee. It comes down to the problem of trying to observe the true nature of observation, observer, and observables.
[ 12. September 2003, 23:55: Message edited by: Zachary Aufdemberg ]
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Rex Kerr
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posted 13. September 2003 00:09
David wrote:
quote:
Downplaying the mental aspects of reality as mere emergent properties of matter is quite probably the worst path a person could wander down if the person is in any way serious about understanding the true nature of reality.
And included no explanation/justification whatsoever. I was rather hoping to see some justification. It's a bold claim.
Zachary states that I assume that one must have premises in order to get anywhere with rationality. Indeed I do! If Zachary wishes to give a counterexample, here is a set of premises for him to use: {}
He correctly points out that in an empirical approach, it is important to explain perception. Certainly, if you have not, your conclusions will be much more tentative than if you have.
In response to my question about alternate rules, he wrote:
quote:
No, there simply are no consistent mutually contradictory axioms of logic. In other words, the supposition that there are multiple systems of two-valued logic is impossible. Because of logic's generality, any argument to the contrary would, of course, be based in the very rules of which the alternative wishes to avoid.
Here's a potential counterexample.
Let P(x) be a proof that x is true, and let LC be the statement that logic is internally consistent. Then:
System One: NOT P(LC) is true
System Two: P(LC) is true
System One is the subject of Godel's Incompleteness theorem, which basically says that NOT P(LC) for first order predicate logic (and it's been extended to other areas). System Two is the system that must be assumed if one can start with rationality and know that rationality is justified.
So, assuming we admit the Law of the Excluded Middle (i.e. A OR NOT A is a tautology), then one of the following holds:
(1) First order predicate logic is inconsistent.
(2) The process of rationally justifying rationality is impossible.
(3) There are multiple inconsistent sets of axioms of logic.
This isn't quite a counterexample to Zachary's claim of not-(3), but the options other than not-(3) have equally if not more serious consequences for (what I presume to be) his argument.
Fixed in edit a P(NOT LC) that was supposed to be NOT P(LC)
Added in later edit:
gedankin and Zachary have touched upon the difference between the "order of knowing" and the "order of being", that is, that there is a difference between the way in which you come to know something and the justification for the validity of something.
If you stumble into logic--let's say it falls off the back of a truck and hits you in the head--but then use logic to prove that logic is valid, it doesn't really matter how you stumbled into it, because by definition it is true. So formally, I agree with Zachary.
However, if you can't use logic to prove that logic is valid, then it may matter how you happened across it. For example, suppose you find a box and you point it at the sentence, "NOT (A AND B) IFF (NOT A OR NOT B))". It says "true". You do this a whole bunch and it always gets the right answer. Then you write down, "This box always gives me the truth value of a sentence." You point the box at it. It says "true".
But are you justified in believing this answer?
This, it seems to me, is what we must do when applying logic to itself.
[ 13. September 2003, 00:25: Message edited by: Rex Kerr ]
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gedanken
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posted 13. September 2003 08:51
Zachary, who likes to point by point reply to me said:
quote:
Don't confuse the source of your belief with its actual justification.
Actually (if you will read the quote of mine to which this is a reply) I was providing an example to justify the very point made.
My example is that the human empirically verifies the rules of logic. That is my consistent observation, even in a logic class. Can you show a counter example of how humans acquire logic skills?
On Zachary's suggestion that there are more than two "games" in town in terms of my suggestion: I suggested that universal agreement would be obtained on a view based on determining agreement using observation, or not. I classified attempting agreement not based on observation to be "philosophy"—and suggest that having this agreement become universal will be difficult and can only be achieved by means that generally would not be considered basing the result on thinking issues through.
Reaching universal agreement is a condition of the subject being discussed (at that point). Doing so based on observation, or not, is the covering dichotomy within that universal set of classifications of the universal set in the subject. Without the classification comment, I think I have provided two cases that are mutually exclusive and cover all possibilities -- since one is the complement of the other. (You all can debate the separate point of arriving at a conclusion without it being based on any observation being considered “philosophy”.)
Perhaps we need to introduce fuzzy logic to obtain degrees that are not in either class. Oops -- are we disagreeing with the fundamentals of "logic" as normally presented? (A new concept of logic based in observation that older concepts don’t cover all cases of interest by observing our use of logic to describe real-world problems? Humm....)
On the issue of science and views based on agreement with broad empirical scale, but with different presentations that may differ in some logical issues: Multiple logical concepts that cannot be differentiated by observation all present scientifically equivalent discussions of the relationships in nature. They may not be “science” if they make predictions that are not being tested. If all the predictions are being tested (and pass broad empirical scale verification) then science cannot distinguish between the views. But these views provide no difference in terms of predictions of relationships in the physical world. In science we usually choose the simplest, because it is easier to use for the purpose of predicting and understanding such physical world relationships. (I explained this before, but I don’t see any progress.)
Can someone provide example of two scientific views that are “mutually contradictory” models which cannot be resolved by empirical methods? (In other words are such “mutually contradictory” models actually presented so as to be “scientific” without being complicated with additional, possibly philosophical, claims?)
[ 13. September 2003, 09:28: Message edited by: gedanken ]
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Zachary Aufdemberg
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posted 13. September 2003 18:45
Rex Kerr writes
quote:
Zachary states that I assume that one must have premises in order to get anywhere with rationality. Indeed I do! If Zachary wishes to give a counterexample, here is a set of premises for him to use: {}
If you mean that every rational viewpoint must be based in assumptions then you're incorrect. Let's call the view that every argument or view is based in assumptions "A". If A were correct, then to be consistent A would have to be an assumption. Let's call the preceding sentence "B". B must hold true if A does. Let's call that sentence "C". Since C is not an assumption, A can't be true.
If on the other hand you're saying that rationality sans assumptions gets one nowhere, this would have to be established rationally. The limitations and scope of logic must be defined logically before one can say when and where it does or doesn't apply (e.g. the question of whether logic constrains reality).
quote:
Here's a potential counterexample.
Let P(x) be a proof that x is true, and let LC be the statement that logic is internally consistent. Then:
System One: NOT P(LC) is true
System Two: P(LC) is true
System One is the subject of Godel's Incompleteness theorem, which basically says that NOT P(LC) for first order predicate logic (and it's been extended to other areas). System Two is the system that must be assumed if one can start with rationality and know that rationality is justified.
So, assuming we admit the Law of the Excluded Middle (i.e. A OR NOT A is a tautology), then one of the following holds:
(1) First order predicate logic is inconsistent.
(2) The process of rationally justifying rationality is impossible.
(3) There are multiple inconsistent sets of axioms of logic.
This isn't quite a counterexample to Zachary's claim of not-(3), but the options other than not-(3) have equally if not more serious consequences for (what I presume to be) his argument.
Fixed in edit a P(NOT LC) that was supposed to be NOT P(LC)
This whole exposition implicitly affirms the self-justification of logic or as you put it "the process of rationally justifying rationality". The undecidability of formal languages of finite axiom sets was proven with logic. The fact that it's been proven shows that incompleteness affirms the consistency of the level of logic of which it was formulated.
quote:
Added in later edit:
gedankin and Zachary have touched upon the difference between the "order of knowing" and the "order of being", that is, that there is a difference between the way in which you come to know something and the justification for the validity of something.
If you stumble into logic--let's say it falls off the back of a truck and hits you in the head--but then use logic to prove that logic is valid, it doesn't really matter how you stumbled into it, because by definition it is true. So formally, I agree with Zachary.
However, if you can't use logic to prove that logic is valid, then it may matter how you happened across it. For example, suppose you find a box and you point it at the sentence, "NOT (A AND B) IFF (NOT A OR NOT B))". It says "true". You do this a whole bunch and it always gets the right answer. Then you write down, "This box always gives me the truth value of a sentence." You point the box at it. It says "true".
But are you justified in believing this answer?
This, it seems to me, is what we must do when applying logic to itself.
It hasn't been shown that logic can't be used to justify logic. Sure we can assume it to be true before it's been proven, but again let's not confuse the process of validation with validation itself.
[ 13. September 2003, 18:48: Message edited by: Zachary Aufdemberg ]
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gedanken
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posted 13. September 2003 22:19
Zachary said to Rex:
quote:
If you mean that every rational viewpoint must be based in assumptions then you're incorrect. Let's call the view that every argument or view is based in assumptions "A". If A were correct, then to be consistent A would have to be an assumption. Let's call the preceding sentence "B". B must hold true if A does. Let's call that sentence "C". Since C is not an assumption, A can't be true.
What about this statement:
X implies Y, Y implies Z. Z implies NOT X.
What does that show about the logical value of X?
Now I know that Zachary had intended us to accept his second and third statements as inherently correct, thus arguing his point as an implication that we accept. But in fact he steps dangerously close to the very type of issue considered in Godel's Incompleteness.
But in this case not too close. In fact I do not agree that “If A were correct, then to be consistent A would have to be an assumption.” (Labeled “B”). Labeled “A” is “the view that every argument or view is based in assumptions.” What I actually would agree with is the following:
Label “A’” the statement “This statement is a view AND every view is based on assumptions.”
(I made the conjunction explicit that was important but hidden in the original. Then I left out the term “argument” which allowed the whole sequence to be trivially false. We must not confuse the term “argument” with the term “view”. An argument, per se, is not necessarily a view, because an argument may not reach a conclusion for example and thus not produce what I would consider “a view”. Furthermore we must not confuse the term “view” with the term “argument”, because a view might be simply an assumption without any logic statements whatsoever, and thus is not an “argument”. Consider “I like ice cream.” That’s my view on ice cream.)
It is very possible for A’ to be of the category of statements like “this statement is false”—wherein the “truth” of the statement is different from its notion of being well formed logical statement that can have internal consistency. I would accept A’ as a “view” if we are basing A’ on the assumption that we are agreeing or using terminology that way. However if we are starting within our argument with definition that “to be a view a statement must reach a conclusion based on assumptions” then I would think that the second clause of A’ is a matter of definition, not a “view”. This would make the statement internally inconsistent like “this statement is false”. Let’s stick to the former, so that we can proceed without getting stuck on the very first statement, and we have an open question of its truth.
Then I agree “If A’ were correct, then to be consistent A’ would have to depend on assumptions.” (Label this “B’”)
By the way, A’ depends on the assumption that we agree on a definition of “view” as being based on some “assumption”, thus A’ does depend on some assumption. That is an assumption that comes from outside the argument, by that I mean that it’s implications are subject to question within the argument, even though it is a definitional assumption. Arguments can be generated to produce other views, based on that assumption. Even arguments that lead us to question the original assumption, because of its consequences! I’ll let Zachary attempt some statement about this if he wishes – watch out for Godel!
Now what is meant by “then to be consistent” in B’? We mean that in the larger argument presented here. B’ by itself must be taken in the context of the larger argument to be meaningful statement about A’.
Then modifying the third entry to: “B’ must hold true if A’ does.” (Label this “C’”).
We have the issue of whether C’ is true, based on what we have. Was B’ a “view”? If not, then A’ does not imply B’. Now the larger argument, e.g. statements A’ and B’ together with any possible assumptions is a reasonable assembly to consider.
Context #1: In that context we can take B’ as an indication of the whole argument including all statements referenced by B’. B’ is taking A’ as part of itself (part of the statement of B’) by its reference to A’. In that case C’ is “true”, but B’ is about A’ in the assembly of statements, thus actually referring to the entire assembly including any assumptions of the entire assembly of statements including those included by reference.
Context #2: Or we can take B’ as a lone statement. In that case is C’ true? I don’t think so, because we may have issues whether B’ is a “view” and thus implied by A’s logical “truth”. (Are we watching out for Godel? Are we asking for sets that include all subsets of that set as a member?)
Now the next equivalent:
“Since C’ does not depend on an assumption, A’ can’t be true.”
Was C’ a ‘view’? (Or was it a statement of logical tautology?)
If C’ is a ‘view’, then A’ implies C’. If not, then A’ does not imply C’. Let’s examine C’:
In either Context #1, or Context #2, C’ would be a simple statement of logical implication. Either context, it is a step in an argument. It does not itself reach a conclusion, rather it shows that a premise to its language has a logical relation to a conclusion. There is a distinction.
And it had the odd property of potentially depending on an assumption of what was the “context” meant in the wording. However that turned out not to be the case, not because each evaluation did not depend on the context assumption, rather because the result was the same in each potential context assumptions. If it had differed, however, then C’ would have depended on an assumption, rendering the issue irrelevant.
Now I must be very specific here. C’ is a statement of a conclusion of a purely logical argument. That is why I call it an “implication”. Same form as following:
X implies Y and Y implies Z. Z must hold true if X does. The last statement is a “conclusion”, but only a conclusion of a statement of a logical tautology. Go back and read the wording of C’, it was worded to use the same sentence structure as a normal “view” in normal language—let’s be precise here in our understanding because it was in fact a conclusion of a logical tautology and thus a step in a tautology. (That’s unless we are to claim that C depends on assumptions relating to the use of language and the context #1 or #2 I mentioned above.)
So unless logical implications themselves are defined as “views”, then C’ is not a view. And if they were, then I would not have made the statements I made earlier in agreement in the modified version of the statements.
In fact I think that a “view” can be simply expressing my concept of what the definitions are. And as such they depend on “assumptions”, including self-referential statements about what constitutes a view. But in essence that does not matter.
If X implies Y, and Y implies Z, is it my view that X implies Z?
Zachary has said in essence nothing more than that a mathematical (or logic) proof is per se a “rational viewpoint”. What is it a viewpoint about? I am not clear about what Zachary considers a “viewpoint”.
We must decide if a “view” includes statements of pure logic, as logic proofs or mathematical statements. If so, then I don’t make an initial statement that all “views” depend on assumptions—the issue here dissolves because I and Rex had made a mistaken assumption of definitions used by Zachary. And if pure logic statements or mathematical statements are not “views” per se, then C’ is not implied by A’ and the argument Zachary breaks down.
But I don’t think that either Rex or I would agree that logical tautologies are “viewpoints”. And logical arguments that are not based from premise to conclusion (and thus not containing “tautologies”) are even less meaningful.
And we need to further investigate this with Godel's Incompleteness theorem, as I think that has consequences in this realm.
Zachary mentions “the question of whether logic constrains reality.” We need to be careful here (as above) to not get the cart before the horse. If we learn logic from observing reality (even the reality of learning as a Gestalt in part from others and inherently recognizing its value), then it is reality that constrains logic. I certainly think that “logic” could be an inherent part of reality, and thus reality is constrained in the way that corresponds to the language statements we humans use in “logic”. However it is not the human language statements of “logic” that constrains reality, even in that case.
quote:
The undecidability of formal languages of finite axiom sets was proven with logic. The fact that it's been proven shows that incompleteness affirms the consistency of the level of logic of which it was formulated.
Once again, we see that logic has “levels”. Logic does not by itself solve problems without assumptions. The existence of greater and greater knowledge within “levels” of logic does not alone get us closer to understanding the real world. Learning about the real world does. Using logic is required. Neither stands alone, except in the sense that logic itself can be learned from the real world using careful study.
Why so much trouble on Zachary’s implication? I thought it was important to understand how even such a simple issue is not easily resolvable without outside world experience. We must take “assumptions” from our experience to make sense of the world. Science is a refined process of using our experience to make sense of the world, and doing so in a universally sharable way.
[ 14. September 2003, 09:29: Message edited by: gedanken ]
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Rex Kerr
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posted 14. September 2003 04:23
gedankin has kindly saved me from having to write a long explanation that would have looked much like his in structure. It is gratifying to know that I am not the only one to have a certain perspective, even if that fact alone isn't indicative of its correctness.
I'd just add that I don't consider tautologies "arguments". (That was the original phrasing.)
Also, are or are not the laws of logic assumptions? In the context of the discussion of possible alternate laws of logic, I think one would have to say that they are, although I haven't thought about this deeply enough yet. In this case, tautologies would be assumption-based.
[ 14. September 2003, 04:24: Message edited by: Rex Kerr ]
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chimp
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posted 14. September 2003 17:24
Certain rules must hold for certain conditions.
In that respect tautologies are not assumptions.
Observables distinguish themselves with reference to other observables. A difference relation must hold. This is a rule that must be absolute, i.e. invariant for all reference frames. ![[Wink]](wink.gif)
I prefer to see tautologies as invariance principles.
Symmetry forms the basis of truth.
Russ
[ 14. September 2003, 17:25: Message edited by: Russell E. Rierson ]
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gedanken
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posted 14. September 2003 18:20
quote:
Certain rules must hold for certain conditions.
In that respect tautologies are not assumptions.
Yes, if you assume rules of logic as precepts of the logical analysis process, then the tautologies found within the logical analysis are not “assumptions” within that analysis. Reread this sentence. ![[Smile]](smile.gif)
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chimp
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posted 14. September 2003 21:49
The amazingly simple fact that objects can be counted, means that some mathematical structure is isomorphic to the universe.
Rex says that science has shown that math does not correspond to the world ...?
If math only approximatly agrees with the observable universe, then the mathematics of probability would agree exactly with the real universe.
http://www.hep.upenn.edu/~max/toe.html
It seems that people have a bone to pick with Max Tegmark, but at least he is asking the right questions.
The question becomes: what is a first principle that agrees with the universe, yet, it is not a useless circular definition...
[ 14. September 2003, 21:50: Message edited by: Russell E. Rierson ]
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Rex Kerr
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posted 14. September 2003 21:59
Counting is very cool. It's nice that it works. However, as I mentioned before, that doesn't necessarily mean that the universe is isomorphic to mathematics; it means that mathematics is potentially as general as our ability to perceive.
I certainly hope that all of the interesting questions that we ask about the universe can be answered in a mathematical framework. I operate on that assumption on a daily basis as a scientist, and I see extensive evidence that it often is the case.
However, metaphysically, I cannot justify assuming that it is the case except in the trivial sense of every possible perception being finitely listable, and hence isomorphic to some mathematical structure. (This is not an interesting case, as the mathematical structure gives no insight into reality.)
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chimp
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posted 15. September 2003 01:15
Those finitely listable perceptions are all relational, and the relations ARE the mathematical structure.
So this universe could be a finite subset of all possible mathematical structures.
A needle in a haystack?
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Aliet Jacob
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posted 15. September 2003 03:48
David Garrett states, correctly, that I have failed to understand the CTMU - isn't that why I was asking him questions - to gain some understanding with the help of a more knowledgeable friend like Garrett?
Garrett had earlier failed to answer my question regarding what global level means and I sought an explanation as to why he believes reality is atemporal at that level. Incidentally, Garrett had earlier asked Langan what atemporality means and he claims Langan answered his question. But that claim is inconsistent with Garrett's inability to even spell the word correctly and inability to offer an answer to the related question that I asked him. Its also a bit puzzling that Langan could choose to offer a private answer to a question that was asked publicly - a question that was also on topic. Perharps Garrett could be kind enough as to share Langans explanation with us? It will go a long way in helping me understand CTMU.
Garrett instead, claims that Rex and me "failed to understand" the working of SCSPL and complains that he feels compelled to explain the entire course of communication theory and its models to us. Something he is not willing to do. I never asked David Garrett to explain SCSPL's mechanism to me. However, I would like to remind Garrett that Langan explains the working of SCSPL without "explaining an entire course of Communication theory and its models" precisely because SCSPL is NOT "an entire course of Communication theory and its models". I have great difficulty understanding how Garrett has managed to find such an exact isomorphism between "an entire course of Communication theory and its models" and SCSPL. Perhaps he can be kind enough to explain.
I respectfully ask Garrett to explain why he thinks its a dangerous to state that mental is simply a higher order emergent property of the physical. And it would also be very kind of Garrett to explain what he understands "the true nature of reality" to be. I will be forever grateful if he shares his deep, well-thought insights concerning reality's nature. I remain humble.
I would like to point out that the fact that we can count things means two things:
1. That there are discrete entities in our world.
2. That reality is stable.
Mathematics is a tool we have designed to help us understand and deal with discrete phenomena. If there is a great correspondence between mathematical structures and said phenomena, cool. That means we designed a good tool - we shouldnt be transfixed and awed about it. Maths did not design itself. It has, over the ages, been refined and ex[panded and has found wide application in science.
We can congratulate ourselves that something we developed is isomorphic to the real universe to some degree. Or we might as well be enraptured that language is actually capable of describing reality - and express wide-mouthed marvel that language can actually do that.
Then later state that reality is a language. You do know that, per Langan, language is a mathematical structure don't you? And that Language is self-generative (per computational theory) hence infocognitive reality is chacterised as a SCSPL.
I think its a horribly flawed approach to take while trying to understand reality. It confuses whats being described with its descriptors, then the described is supplanted with the descriptor then the descriptor is examined and treated like an actual thing.
Which, with due respect, is misleading, unjustified and obfuscatory.
[ 15. September 2003, 04:40: Message edited by: Aliet Jacob ]
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Rex Kerr
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posted 15. September 2003 04:58
There's a lot more elegance in f = GmM/r^2 than in
"at t=10827518023815.1927519851855162s there is particle 1923856198632891654913750982708651 at x=912651982639898151235.1982561962397845619571962
y=92578237587827010617.1928349812638956183261559
z=2345278791761028475119.192837498716378615123544;
at t=10827518023815.1927519851855163s the center of its distribution has moved to
x=912651982639898151235.1982561962397845619571915
y=92578237587827010617.1928349812638956183261715
z=2345278791761028475119.192837498716378615121151" over and over again for the entire universe.
However, it is a structure of the latter type that we know can be isomorphic to (our perceptions of) the universe, not the former. The latter need have no symmetry, little invariance, etc..
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chimp
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posted 15. September 2003 14:35
Why are the symbols confused with the abstract "mental" perceptions, that the symbols represent?
The abstract contains the concrete. The general contains the specific. Rex and Aliet seem to be saying that the mathematical correspondence to our objective universe is just a "cosmic lucky break" to quote CML
The subjective abstract rules correspond to the objective reality. Don't confuse the written symbols with their abstract "real" existence.
Does a nonlinear universe, give us the "strange loop" of Godelian self-reference? The clear cut distinction between the active transformer and the passive transformed is no more? The state vector is not the passive victim. It fights back. The fusion between the operator and the state vector what completes the self-referential feedback control circuit, which becomes the mechanism of free will ...?
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gedanken
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posted 15. September 2003 15:19
The problem is that when people suggest a mapping, a symbolic to reality congruence, then someone else suggests that the symbolic is some way controlling the reality.
And then if the suggestion is made that the symbolic does not control reality, that is taken (or suggested) as a statement that there really is not a congruence.
But if one is careful in argument to keep definitions clear, one realizes that there is no such implication (in either case). Could I suggest understanding the skepticism of those with different positions? (And producing carefully reasoned evidence if a claim is made?)
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