Look around you: you see people, objects, locations. They have certain characteristics and properties, right? – if they didn’t, how could you perceive them?
What do you think is responsible for those characteristics and properties? We like to think of the world as being made up of distinct things, but they’re never really distinct.
The television, for example. It’s a box with a glass front that shows images. It has those properties because of the configuration of its components. The glass is transparent because of the arrangements and properties of the silicon and oxygen atoms that make it up. The phosphorus in the screen lights up when it’s hit with energetic electrons because of its chemical properties as well, and the filters that color the light given off are the colors they are for the same reasons. The properties of the atoms making up the television are the way they are because of the laws governing the electrons in the atoms – the same laws that govern the movement of the free electrons from the electrode, through the magnetic fields that direct them, to the screen.
Your friends, and everything else in the room, act the way they do because of the configurations of their component particles. That’s the only reason they’re different in the first place – do you think it’s possible to distinguish an electron in your friends from the electrons in the television?
Just as coherent patterns can be constructed from in the Game of Life, everything you perceived is merely being modeled by the physical world.
** You’re right – it doesn’t. What it means is that your living room, and the people in it, are part of a system that contains rules governing its evolution over time – rules that do model arithmetic. We know they do because arithmetic happens in the world!
The computation being carried out by the universe is this: “What happens next?” All events can be considered to be specific configurations of a physical system, and the rules that govern the system determine how any configuration evolves over time.
** But that’s the point – it contains all statements, not just all true statements. The system doesn’t draw any distinctions, and every statement in it is accompanied by its negation. The world described by those statements has no properties (or ALL properties – the two states are identical in every way and therefore equivalent).
** That “universe stuff” is arithmetic… and a lot of other things as well.
** Aha! You understand more than I thought you did. The universe may have Godel Statements, but it by definition doesn’t contain them – they’re outside the set of things that exist in the universe. If they were inside, the universe could emulate them, which would mean that they were the result of a series of steps governed by the universe’s laws – which is equivalent to deriving a false statement.
Whew – I’m going to have a hard time responding to you all before you respond back.
** Pay very close attention, Spiritus, 'cause I’m only going to say this once.
No.
The Life game is what models the Meta-Life game. The Meta-Life game is what’s being modeled. Note: I don’t mean modeled as in “represented in a model”; I mean the word in its active sense, which is similar to “emulated”.
Meta-Life mimics the behavior of Life.
** Um, no. Your statements aren’t even coherent – are you unable to derive meaning from standard English?
**
**
Make up your mind! Do you want to discuss the systems, or their configurations?
The Game of Life underlies a game of Meta-life no matter what configuration it’s in. Life can also support configurations that are not meta-lifes. The computer ultimately underlies them all.
It doesn’t depend or rely on human perceptions! I can’t understand why you didn’t find this obvious in the first place!
Consider the crystal goblet I mentioned earlier. Does the goblet exist? I see only a collection of atoms arranged in a particular way. If I drop the goblet, that configuration is disrupted, and the goblet is “destroyed”, but all of the atoms that were present before are still present.
Don’t the atoms model (in the active, not passive, sense) the goblet? Isn’t the goblet emulated by the behavior and relationships of the atoms?
TVAA you are showing remarkable persistance, despite some extremely lucid posts explaining where you are going wrong. The link within your cite above includes the following:
GIT applies only to derivation from axioms that can model arithmetic.
The physical universe obeys rules; the evolution of the universe over time involves these rules being applied to specific configurations of its basic elements.
This state of affairs is equivalent to making derivations from statements. “Given this, and this, we find this.” GIT therefore applies.
And I respectfully disagree with your statement about the opposing posts being lucid. The posters can write a mathematical explanation of GIT, but they can’t reason about it in normal language – this is a significant weakness.
For example, take a look at Spiritus’ explanation of what GIT implies on the preceding page (3). It’s objectively wrong – it’s simply not what the Theorem claims.
How can he reasonably claim that I don’t understand GIT if he doesn’t understand it himself?
If you were a system that didn’t understand GIT, then, by your own reasoning, you would not be able to accurately model Spiritus.
I must say, TVAA, that your thoughts seem disorganized and rather hypomanic. I’m diverging from the topic because I’m actually worried about whether you’re okay.
erislover, I know you don’t see atoms. They’re there anyway.
Oh brother. I think this is a new low for GD: someone is actually suggesting that their opponent is mentally ill.
That’s not hypomania, that’s frustration, you dolt.
I’ll try to make this simple enough for you all to understand:
(Are you paying attention, Spiritus?)
A system that includes ‘events’ can always be shown to be equivalent to a set of operations. The operations describe how one state of the system leads to another.
If we posit a finite universe, GIT doesn’t apply, because finite systems can only represent a limited number of configurations and require a finite number of statements to describe them. They’re not complex enough to include all of arithmetic.
However, this means that there are fundamental limits on what goes on in these systems. The universe might not include the statement that a given subsystem is vulnerable to; a Godel statement for this subsystem might be too long to be represented by the finite universe, for example.
If we don’t assume that the possible interactions of this system with other things are limited (in other words, if we accept the possibility of an infinite universe), we’re forced to consider everything.
Anything that can be modeled – anything that can be represented as a system of operations – will then be vulnerable to GIT.
The representation of this system will have statements that it cannot derive but are true anyway.
All this time, all this effort, and the best you can come up with is “It sucks?”
TVAA, that’s not a summary of the theorem, or a discussion. That is GIT. If you can’t translate that into plain English, then why should we believe that you’re capable of reasoning about it in any language?
Well, this merely proves beyond a doubt that you have been using the word inconsistently, since that was not an element of the definition that you offered. In case you are too busy thinking great thoughts to remember, the complete definition that you offered was: A model is a system – a set of interactions – that manifests properties similar to those of another system. This is inherent in the dictionary definition of the word.
You can become as snide and condescending and insulting as you wish, it does not change the fact that you are being sloppy with language. You asked me to provide an example of your inconsistent or vague use of language. I have done so.
If that angers you, well, frankly I don’t give a rat’s ass hair.
Not coherent? Shall I parse the sentence for you? I guess so, since you seem sadly unfamiliar with the language you demand that we all speak.
[ul][li]So – Interjection indicating comprehension.[/li]
[li]**either ** – conjunction introducing two or more structures to be linkd with the conjunction “or”.[/li]
[li]**you misread the question ** – clause indicating a lack of reading comprehension by TVAA[/li]
[li]and – conjunction used to indicate an additional element in a structure[/li]
[li]anwered whether some future, post-disruption, Meta-life game would be a model of the life game – the second part of the structure tied to “you misread the question”. Verb phrase in the past tense indicating that TVAA answered a question with information about a Meta-life game that had been disrupted. The implication of this phrase, paired with the clause “you misread the question” is that I did not ask a question about a Meta-life game that had been disrupted.[/li]
[li]**or ** – cunjunction required to complete the structure introduced by “either”[/li]
[li]else idiomatic completion of “or”. Blame my mother for being a midwestern girl.[/li]
[li]you have added another requirement – second listed element in the “either . . . or” structure. Clause indicating that TVAA has added another requirement. Contextual information is necessary to understand that the additional requirement is a further restriction on the definition of “model”.[/li][/ul]
If you do not find my statements coherent, then I suggest that you put some more study into your chosen language of expresion.
Why are you asking me? You are the one throwing “model” around with vague and inconsistent references.
I asked whether Meta-life**(A)** can model life**(B)**. Your answer is no, though none of the definitional elements for “model” that you have proposed would eliminate this use. This is a question about specific instantions of a life game and a meta-life game. You answer that "The Life game is what models the Meta-Life game. ", but the only additional definitional test that you offer is "Meta-Life mimics the behavior of Life."
Why?
In what sense does Meta-life**(A)** “mimick” life**(B)** that is not perfectly symmetrical with life**(B)** “mimicking” Meta-life**(A)**?
In the generic, sure, but I asked you about Meta-life**(A)** and life**(B). Life(B)** does not underlie Meta-life**(A)**.
Obvious? I do not find it obvious that measurement is a meaningful word in the absence of an interpretive agent. Please tell me exactly how you mean this “plain English word” to be understood in tha absence of a conscious agent doint the measuring.
Feel free to place lots of exclamation points in the answer. I am sure that they will make your case look ever so much more convincing.
No. The atoms make up the goblet. In plain old English, we don’t usually say that the bricks “model” the house or that the players “model” the team.
You defined as model as: a system . . .that manifests properties similar to those of another system. The system of atoms makes up the goblet. It does not satisfy the relationship “another system” to the goblet.
No. The goblet is the behavior and relationships of the atoms, so long as those relationships and behaviors conform to what a human being would recognize as a goblet.
I assume that you refer to the following exchange: it was one of the things that I had set aside until your promised “drawing th elines” proof, but since you seem to have your heart set on misrepresenting my posts I suppose I shall have to correct your mistakes once more.
If you look carefully at my statement, you will see that I follow the noun phrase “true statements” with the restrictive prepositional phrase “in a Peano Axiomatization”. For a statement to be demonstrated true “in a Peano Axiomatization” it must be proved “in a Peano Axiomatization”. Proving it in system Y does not demonstrate that it is true “in a Peano Axiomatization”.
While you are certainly correct that in some other system I can prove any statement that is true (but unproveable) “in a Peano Axiomatization” this is irrelevant to the question of whether a proof “in a Peano Axiomatization” exists for some statements that are true “in a Peano Axiomatization”. Thus, my statement that no possible proof exists for some statements that are true “in a Peano Axiomatization” is strictly correct, since such a proof can only take place “in a Peano Axiomatization”.
And , as long as you have made me turn my attention to this exchange
Since you made that statement in a context of “extended systems” rather than Peano Axiomatizations, I must observe that it betrays a pretty stunning lack of imagination. Any statement can be proven in a system that you allow to be arbitrarily extended with new axioms.
What was that phrase you uses again? Oh, yes, "I can’t understand why you didn’t find this obvious in the first place!"
:rolleyes:
Because I do.
And because from the evidence presented thus far you do not.
Yes. Was my attention what you were craving all along?
I assume that the “argument” that you finish up with does not represent your “draw the lines” proof, so I will refrain from further comment upon it.
You’re not familiar with the active verb “to model”? As in striking a pose, or entering into a specific configuration?
Perhaps you’d feel more comfortable with the verb “to emulate” – the meanings are similar.
** If the grid of life(A) is sufficiently large, Meta-life(A) could model life(b), but that modeling would be Meta-meta-life(A).
Argh. ‘A’ model is a noun. ‘To’ model is a verb. Can’t you tell the difference?
My statement is not an additional definitional test. It’s merely a statement about Meta-life’s properties.
** There don’t need to be any measurements by any interpretive agents for there to be properties. Thus, the presence or absence of interpretative agents is irrelevant.
**
Actually, no. The second statement is correct if the phrase “in a Peano Axiomatization” modifies ‘true’ and not ‘demonstrated’, but the third sentence is wrong. It may be the case that the proof of a theorem’s truth cannot be demonstrated “in a Peano Axiomatization”, but it might be true but unprovable. Adding additional axioms might permit such a proof to be shown – a proof that demonstrates that the theorem is indeed “true in a P.A.”.
** Who said anything about arbitrarily? If the axioms are consistent with the previous axioms, it’s not the case that anything can be proven.
To you, perhaps. Others realize that the interactions between the system’s elements necessarily follow some principles, and those principles can be represented by a set of operations.
I dislike your presentation of GIT, ultrafilter. The version I linked to is much better, both in terms of overall clarity and reduced jargon.
GIT shows that no finite group of axioms (sufficiently powerful to represent arithmetic, which requires several basic operations) is capable of generating proofs for all statements true relative to those axioms.
There is no disagreement about this, yes?
Then the disagreement is about the statement that changes in the world over time are equivalent to the evolution of a computational system, right?
I am. The definition you originally offered was for a noun. As I said, this is one example of your inconsistent use of terms.
I care not at all, so long as you use your chosen term unambiguously and concistently. Thus far, you have not done so.
Why? The behavior of Meta-life(A) is already similar to the behavior of life(B). You need to choose a single consistent definition for model and apply it appropriately in all cases.
I can. You are mistaken if you think that my question represents a confusion on that score. Why you might think that, I have no idea. As usual, you offer little in the way of substantive support for your declarations.
Tell me, does TVAA English preserve the relationship between the verb and the noun? Is that which models a system a model of that system? If so, Meta-life(A) can model (verb) life(A) by the definition you have offered for model (noun). Meta-life(A) manifests properties similar to those of life(B).
You obviously feel that the above statement is incorrect, which means that you obviously are not applying the definition of model that you have claimed.
Your statement contradicts the definition that you have offered for model (noun).
This is maddening. Do you even bother to follow the context of these conversations. Here is how this particular discussion has progressed:
:rolleyes:
You want to know how you use language in vague and inconsistent ways? Look above. All I have been trying to do is get an unambigous statement from you about measurement as it relates to “similarity”. Now, I also have problems with the treatment of simlarity as a first-order property, but we can’t even get to that part of the discussion because you keep dancing back and forth about measurement. I asked you how is measurement meaningful absent an interpretive agent. You replied “there don’t have to be any measurements.”
I don’t care. That wasn’t the question I asked.
I also don’t care because you had previously told me that there can be[sup]1[/sup] measurements and that those measurements do not depend upon an interpretice agent.
Why don’t you simply answer the questions that I ask? It really would make communication much easier.
The statement is correct. As it happens, the modifier does apply to the word “true”. The statement would also be true if the modifier applied to the word “demonstrated”, of course, since it would then be a tautology. “Demonstrated true” is synonymous with “proved”.
No, it is not.
May be? It quite explicitly is the truth, since we are talking about the statements whose existence is proven by GIT.
That is not correct. the only way to prove that X is “true in a P. A.” is to use only the tools available to a P. A. to demonstrate X. If you have to use additional axioms, then all you can prove is that X is true in a P. A. +Y.
Perhaps here we are hitting a difference of usage, though. You may be thinking of things like the ability to prove that a Peano Axiomatization is consistent by alternate axiomatic sets (such as Gentzen’s transnfinite induction). I can see how that might lead you to say that “The system defined by a P. A. is consistent” would be a statement true in a P. A. that has been proved in an extended (alternate, in this case) axiomatic set. That isn’t really correct, though.
“The system defined by a P. A. is consistent” has not been proven to be a true statement within a Peano Axiomatization. It has been proven true in a different logic altogether. The truth of the statement within a P. A. remains unknown.
To illustrate, consider the axiom of choice. It is very easy to extend ZF in order to prove that the axiom of choice holds. But it is also easy to prove the converse. Obviously, we cannot take either proof as a demonstration that the axiom of choice is or is not true in ZF.
You said the systems could be extended to prove statements true. You did not require that the system be extended consistently. For that matter, G2IT proves that we cannot know from within our new extended system whether said system actually is consistent.
No. Your formulation is imprecise. GIT allows for inconsistent axiomatic sets to prove all true statements. I also would like to know exactly what you mean by “represent”, since in the past you have declared the ability to perform calculations sufficient to invoke GIT.
No.
Before I go further, please clarify for me whether you actually have already posted your “draw the lines” proof.
[sup]1[/sup][sub]Or at least objected to the statement that similarity cannot be measured.[/sub]
It occurred to me this morning that we are really getting off on a tangent with the whole “in a Peano Axiomatization” discussion. Now, I would be happy to continue it as I am confident in my interpretation, but it has nothing to do with GIT and cannot possibly affect your “draw the lines”.
[ul][li]We both agree that the implication of GIT is that no proof within a P. A. can exist for all true statements in a P. A. That is where GIT ends.[/li][li]We both agee that it is possible to prove the statements that are unproveable in a P. A. by extending the axiom set. That has nothing to do with GIT, but we both agree on it anyway.[/li][li]We disagree on whether those proofs in extended languages can be rigorously said to prove results “in a Peano Axiomatization”, but that is irrelevant to your proof since you are arguing that the Universe itself is a system that is bound by GIT. Since the whole Universe is your system, and you argue that any property the system emulates is a property of the system as a whole, there is no “axiom outside the box” that could be used to extend the language and prove an “unprovable in the Universe” statement.[/ul][/li]So, this looks like a big “non-issue” to me. That doesn’t mean we can’t keep fighting about it, of course, but I thought I should point out that this particular element of contention has no bearing on the OP or on any proof of the OP that you might offer.