Tarski Undefinability Theorem Succinctly Refuted

What is the basis for reason? And mathematics?

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Logik
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Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Sun Apr 07, 2019 9:04 am

PeteOlcott wrote:
Sun Apr 07, 2019 3:35 am
For example the current set of human knowledge could be encoded entirely
as relations between integers. When a human language is mapped to these
relations a human could talk to this omniscient machine. The machine would
still know what it knows entirely on the basis of its relations between integers.
This is very naive thinking. The "current set" of human knowlede is constantly in flux.
But lets say that we froze it. We stopped learning (clue: if your knowledge-base is frozen you are working with an 'expert' system).

You can indeed try (as best as you can) to map semantics to GUIDs. And so you will end up with a network that no human can use because we will have absolutely no indexing/lookup mechanism to traverse the network.

Weight(Atom(Oxygen)) is represented by the GUID 238748726852854254234.

You need to study some semiotics. The signifier <-> signified distinction is handy.

This also reminds me of the adage in CompSci. There are only two hard problems in this field: Cache invalidation, naming things and off-by-one errors.

wtf
Posts: 833
Joined: Tue Sep 08, 2015 11:36 pm

Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Sun Apr 07, 2019 11:49 pm

Logik wrote:
Sun Apr 07, 2019 12:23 am
Sure. In exactly the same way Evolution is a belief and not a fact.
No, not in the same way at all.

CTD is a belief. A conjecture if you will. The jury is still out regarding scientific proof.

Note that scientific proof is not the same as absolute proof. There's scientific proof for Newtonian gravity even today. The theory works really well, more than could be accounted for by luck. Likewise quantum physics and general relativity. Those are scientific facts. Which means they could be overturned tomorrow morning; but till then, we can legitimately regard them as true in the sense of being established scientific theories.

CTD is nothing of the sort. CTD is well into the speculative realm along with the multiverse and Tegmark's mathematical universe. Fun speculations and food for thought. But not established scientific theories. CTD is in fact arguably a subset of Tegmark. Tegmark speculates on the CUH, the computable universe hypothesis. People have noted that it's a great theory with one slight drawback: It's inconsistent with known physics.

I'll try to hold off on posting any more about this, since the point on which we disagree is clear enough. You are dogmatic about your beliefs. I'm agnostic on mine. It's perfectly possible that we are all execution paths in the great cosmic computation. If so then so be it. I don't think so but after all ... how would I know? I could just be programmed to think that way. And I've read plenty of Philip K. Dick.

But you're dogmatic in a tricky way. I might almost call it gaslighting. Earlier you equivalenced my view with "magic" and then freely went back and forth between computability and physics. Of course that is a perfectly sensible thing to do for someone who believes physics = algorithms. Or to be more fair, computations. [Algorithms are the abstract things you can write down as code; and computations are algorithms implemented on physical hardware, inputting energy and outputting heat].

But if you said, "As someone who believes in the Church-Turing-Deutsch thesis, I believe physics is computation." That would be clear. Instead, you freely went back and forth as if to dare me to call you on the point. That's gaslighting, not dialog. For what it's worth that's why I called you a liar. I suppose that's too strong for a friendly discussion board. I apologize for calling you a liar. I'll dial it down to micro-gaslighter and leave it at that.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by PeteOlcott » Mon Apr 08, 2019 5:35 am

Logik wrote:
Sun Apr 07, 2019 9:04 am
PeteOlcott wrote:
Sun Apr 07, 2019 3:35 am
For example the current set of human knowledge could be encoded entirely
as relations between integers. When a human language is mapped to these
relations a human could talk to this omniscient machine. The machine would
still know what it knows entirely on the basis of its relations between integers.
This is very naive thinking. The "current set" of human knowlede is constantly in flux.
But lets say that we froze it. We stopped learning (clue: if your knowledge-base is frozen you are working with an 'expert' system).

You can indeed try (as best as you can) to map semantics to GUIDs. And so you will end up with a network that no human can use because we will have absolutely no indexing/lookup mechanism to traverse the network.

Weight(Atom(Oxygen)) is represented by the GUID 238748726852854254234.

You need to study some semiotics. The signifier <-> signified distinction is handy.

This also reminds me of the adage in CompSci. There are only two hard problems in this field: Cache invalidation, naming things and off-by-one errors.
So I guess that you are unaware that a significant amount of this is already done?
https://en.wikipedia.org/wiki/Cyc

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Tarski Undefinability Theorem Reexamined

Post by Logik » Mon Apr 08, 2019 7:35 am

wtf wrote:
Sun Apr 07, 2019 11:49 pm
CTD is a belief. A conjecture if you will. The jury is still out regarding scientific proof.
Note that scientific proof is not the same as absolute proof. There's scientific proof for Newtonian gravity even today. The theory works really well,
That's because you are looking in the wrong place ;)
The scientific proof is that science produces knowledge that can be re-used.
The scientific proof is that we are able to build models of the universe.

If the universe can be described with logic, then it can be (partially) modeled in Mathematics.
If you can model it in Mathematics - you can execute the model on a computer.
wtf wrote:
Sun Apr 07, 2019 11:49 pm
more than could be accounted for by luck. Likewise quantum physics and general relativity. Those are scientific facts. Which means they could be overturned tomorrow morning; but till then, we can legitimately regard them as true in the sense of being established scientific theories.
Precisely.

And if the scientific method stopped working tomorrow we could discard Shannon's information theory - but for now, we keep it.
Because we use it for statistical inference.

wtf wrote:
Sun Apr 07, 2019 11:49 pm
CTD is nothing of the sort. CTD is well into the speculative realm along with the multiverse and Tegmark's mathematical universe. Fun speculations and food for thought. But not established scientific theories. CTD is in fact arguably a subset of Tegmark. Tegmark speculates on the CUH, the computable universe hypothesis. People have noted that it's a great theory with one slight drawback: It's inconsistent with known physics.
Two things we are most certainly missing each other on. In fact, that is the hugest gap.

You fail to distinguish a bunch of things:
1. What the universe is from what we SAY about it.
2. What the universe is from what we THINK about it.
3. What the universe is from the SYMBOLS we use to describe it.

All this time we are discussing "the universe", you do understand that we are talking about the map, not the territory, right?
Those are the implications of model-dependent realism, constructivist epistemology.

You and I know nothing of the universe. In fact - my datapoint for "arrogance", is people who lack the self-awareness to recognise that they are talking about their PERCEPTION of the universe not the universe itself.

CTD is not food for thought. CTD is the theory of thought. Hence my remarks on metacognition.

If you don't understand how you primary measurement apparatus for doing science works, then your results may as well be nothing but luck.
wtf wrote:
Sun Apr 07, 2019 11:49 pm
People have noted that it's a great theory with one slight drawback: It's inconsistent with known physics.
See. I don't understand how mathematics can be "inconsistent" with physics.
The theory says that the universe CAN be described with mathematics, not that it IS described by known mathematics.

If the mathematics is inconsistent with reality. FIX THE MATHEMATICS.

Physicists have always looked to Mathematicians to give them the language necessary to describe the phenomena they are observing, and by and large it seems Mathematicians were always ahead of the curve. It seems that what you are saying by "mathematics is inconsistent with physics" is that Mathematicians have run out of language.

What computer scientists are offering instead is the skills to INVENT the language you need to describe your experiences. From 1st principles.
wtf wrote:
Sun Apr 07, 2019 11:49 pm
I'll try to hold off on posting any more about this, since the point on which we disagree is clear enough. You are dogmatic about your beliefs. I'm agnostic on mine. It's perfectly possible that we are all execution paths in the great cosmic computation. If so then so be it. I don't think so but after all ... how would I know? I could just be programmed to think that way. And I've read plenty of Philip K. Dick.
:lol: :lol: :lol:

You can't be agnostic about the fact that you have beliefs. How do "beliefs" work?
wtf wrote:
Sun Apr 07, 2019 11:49 pm
But if you said, "As someone who believes in the Church-Turing-Deutsch thesis, I believe physics is computation." That would be clear. Instead, you freely went back and forth as if to dare me to call you on the point. That's gaslighting, not dialog. For what it's worth that's why I called you a liar. I suppose that's too strong for a friendly discussion board. I apologize for calling you a liar. I'll dial it down to micro-gaslighter and leave it at that.
I can't read your mind - I don't know which argument will best connect the dots for you.

I could have said "As somebody who believes in CTD".....
Or I could have said "As somebody who is a Model dependent realist" (and I DID say that). I think all knowledge is models. And computation is model building.
Or I could have said "As somebody who is subscribed to Constructivist epistemology (and I HAVE said that - and you HAVE recognized that) then it follows: the world is independent of human minds, but knowledge of the world is always a human and social construction.

I see all three of the above arguments as equivalent in weight. What I don't know is which argument is closer to your background knowledge.

Because I can argue my position from 10 different perspectives (hint: the mathematical ideal of consistency is flawed). Like any good physicist has more than one theory which explains a phenomenon, so should any good philosopher have more than one framework (perspective!) to look at things.

But here is the TL;DR. Physics produces models of the world in the language of Mathematics. As physics discovers new phenomena (ones that we have never experienced before), phenomena for which we have no language for - physicists go looking to Mathematicians for the language to express this experience. The output of Physics is Mathematics.

So you can trivially falsify my CTD "belief" (as it pertains to being the theory of the mind, NOT the universe) then.

Point me to any physics knowledge which is not-computable e.g cannot be described in language (logic).

P.S Here is an alternative hypothesis to your "gaslighting or micro-gaslighting". You got triggered. Which is perfectly expected - since I am doing science on your mind. Metacognition...

Mathematics is just language. A man-made language.
Last edited by Logik on Mon Apr 08, 2019 10:07 am, edited 3 times in total.

Logik
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Joined: Tue Dec 04, 2018 12:48 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Mon Apr 08, 2019 7:44 am

PeteOlcott wrote:
Mon Apr 08, 2019 5:35 am
So I guess that you are unaware that a significant amount of this is already done?
https://en.wikipedia.org/wiki/Cyc
Great! So where/how do I query it?

I expect it to work MUCH better than Google when I type in an arbitrary question in English, German, French, Portuguese, Spanish, Chinese or Russian.

The above is rhetorical.

I have seen the list of success stories and it strikes me mundane and rather unimpressive. It's domain-specific, not general-purpose. It's orders of magnitude smaller and less complex than the engineering behind Google.

So it took Cyc 35 years to build a small-scale, domain-specific tool.
Whereas it took Google 20 years to build a global-scale, general-purpose tool.

I think you are failing in terms of adoption, scale and complexity. Mostly because Google is free and Cyc is not.

If you want Cyc to be representative of the current set of HUMAN knowledge - you have to get all of HUMANITY to use it.

You could've just told us that you are selling something all along ;)

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by PeteOlcott » Mon Apr 08, 2019 3:38 pm

Logik wrote:
Mon Apr 08, 2019 7:44 am
PeteOlcott wrote:
Mon Apr 08, 2019 5:35 am
So I guess that you are unaware that a significant amount of this is already done?
https://en.wikipedia.org/wiki/Cyc
Great! So where/how do I query it?

I expect it to work MUCH better than Google when I type in an arbitrary question in English, German, French, Portuguese, Spanish, Chinese or Russian.

The above is rhetorical.

I have seen the list of success stories and it strikes me mundane and rather unimpressive. It's domain-specific, not general-purpose. It's orders of magnitude smaller and less complex than the engineering behind Google.

So it took Cyc 35 years to build a small-scale, domain-specific tool.
Whereas it took Google 20 years to build a global-scale, general-purpose tool.

I think you are failing in terms of adoption, scale and complexity. Mostly because Google is free and Cyc is not.

If you want Cyc to be representative of the current set of HUMAN knowledge - you have to get all of HUMANITY to use it.

You could've just told us that you are selling something all along ;)
Can we get back to the point of the title of this thread?

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Tarski Undefinability Theorem Reexamined

Post by PeteOlcott » Mon Apr 08, 2019 3:40 pm

wtf wrote:
Sun Apr 07, 2019 11:49 pm
Logik wrote:
Sun Apr 07, 2019 12:23 am
Sure. In exactly the same way Evolution is a belief and not a fact.
No, not in the same way at all.

CTD is a belief. A conjecture if you will. The jury is still out regarding scientific proof.

Note that scientific proof is not the same as absolute proof. There's scientific proof for Newtonian gravity even today. The theory works really well, more than could be accounted for by luck. Likewise quantum physics and general relativity. Those are scientific facts. Which means they could be overturned tomorrow morning; but till then, we can legitimately regard them as true in the sense of being established scientific theories.

CTD is nothing of the sort. CTD is well into the speculative realm along with the multiverse and Tegmark's mathematical universe. Fun speculations and food for thought. But not established scientific theories. CTD is in fact arguably a subset of Tegmark. Tegmark speculates on the CUH, the computable universe hypothesis. People have noted that it's a great theory with one slight drawback: It's inconsistent with known physics.

I'll try to hold off on posting any more about this, since the point on which we disagree is clear enough. You are dogmatic about your beliefs. I'm agnostic on mine. It's perfectly possible that we are all execution paths in the great cosmic computation. If so then so be it. I don't think so but after all ... how would I know? I could just be programmed to think that way. And I've read plenty of Philip K. Dick.

But you're dogmatic in a tricky way. I might almost call it gaslighting. Earlier you equivalenced my view with "magic" and then freely went back and forth between computability and physics. Of course that is a perfectly sensible thing to do for someone who believes physics = algorithms. Or to be more fair, computations. [Algorithms are the abstract things you can write down as code; and computations are algorithms implemented on physical hardware, inputting energy and outputting heat].

But if you said, "As someone who believes in the Church-Turing-Deutsch thesis, I believe physics is computation." That would be clear. Instead, you freely went back and forth as if to dare me to call you on the point. That's gaslighting, not dialog. For what it's worth that's why I called you a liar. I suppose that's too strong for a friendly discussion board. I apologize for calling you a liar. I'll dial it down to micro-gaslighter and leave it at that.
Can we get back to the point of the title of this thread?

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Mon Apr 08, 2019 4:17 pm

PeteOlcott wrote:
Tue Apr 02, 2019 9:48 pm
Tarski Undefinability Theorem Succinctly Refuted
https://www.researchgate.net/publicatio ... ly_Refuted
I have stated it over and over again (and I keep demonstrating it).

Provide a logical statement which you think should evaluate to True or False.

IF I am allowed to control the value of any of your inputs (e.g the context) I will invent you a system (input to your framework) in which your theorem evaluates as False when you expect it to be True. AND a system in which your theorem evaluates as True when you expect it to be False.

I will turn any refutation into a proof, and any proof into a refutation.

Such is the logical consequence of Type 1 Chomsky Grammars Context sensitive grammars

Such is the logical consequence of Model-dependent realism

You need to interpret the meaning of symbols in EXACTLY the same framework as the one which assigned them meaning.

This is input/data validation 101 stuff. https://en.wikipedia.org/wiki/Data_validation
Failing to validate your inputs results in Memory corruption: https://en.wikipedia.org/wiki/Memory_corruption

This allows me to control your interpretation system.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by PeteOlcott » Mon Apr 08, 2019 5:00 pm

Logik wrote:
Mon Apr 08, 2019 4:17 pm
PeteOlcott wrote:
Tue Apr 02, 2019 9:48 pm
Tarski Undefinability Theorem Succinctly Refuted
https://www.researchgate.net/publicatio ... ly_Refuted
I have stated it over and over again (and I keep demonstrating it).

Provide a logical statement which you think should evaluate to True or False.
Within the precise context of the provided proof
3) x ∉ Pr if and only if x ∈ Tr // page 275

http://liarparadox.org/Tarski_Proof_275_276.pdf

Logik
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Joined: Tue Dec 04, 2018 12:48 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Mon Apr 08, 2019 5:01 pm

PeteOlcott wrote:
Mon Apr 08, 2019 5:00 pm
Within the precise context of the provided proof
No - you don't get to prescribe that.

Your language is context-sensitive - it interacts with the environment.
It's an open system.

There is the framework in which it is synthesised and the framework in which it is interpreted.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by PeteOlcott » Mon Apr 08, 2019 5:27 pm

Logik wrote:
Mon Apr 08, 2019 5:01 pm
PeteOlcott wrote:
Mon Apr 08, 2019 5:00 pm
Within the precise context of the provided proof
No - you don't get to prescribe that.

Your language is context-sensitive - it interacts with the environment.
It's an open system.

There is the framework in which it is synthesised and the framework in which it is interpreted.
I am saying that I am requiring that the context is the original proof.

If you want to dodge the question by changing the context then your answer
will be construed as useless.

Logik
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Joined: Tue Dec 04, 2018 12:48 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Mon Apr 08, 2019 5:30 pm

PeteOlcott wrote:
Mon Apr 08, 2019 5:27 pm
I am saying that I am requiring that the context is the original proof.
What does that even mean? Can you map out a concrete consequence or an expectation given your requirement?

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by PeteOlcott » Mon Apr 08, 2019 7:42 pm

Logik wrote:
Mon Apr 08, 2019 4:17 pm
PeteOlcott wrote:
Tue Apr 02, 2019 9:48 pm
Tarski Undefinability Theorem Succinctly Refuted
https://www.researchgate.net/publicatio ... ly_Refuted
I have stated it over and over again (and I keep demonstrating it).

Provide a logical statement which you think should evaluate to True or False.

IF I am allowed to control the value of any of your inputs (e.g the context) I will invent you a system (input to your framework) in which your theorem evaluates as False when you expect it to be True. AND a system in which your theorem evaluates as True when you expect it to be False.
Why the Hell would you think that you get to specify the context?
The context is already provided as the Tarski proof.
http://liarparadox.org/Tarski_Proof_275_276.pdf

Tarski is unable to evaluate this as true or false within his theory:
3) x ∉ Pr if and only if x ∈ Tr // page 275

By providing three truth predicate axioms I show a simple way to eliminate
undecidability in all formal systems including his own theory.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Mon Apr 08, 2019 7:55 pm

PeteOlcott wrote:
Mon Apr 08, 2019 7:42 pm
Why the Hell would you think that you get to specify the context?
Where the hell did I say that? I said that YOU don't get to prescribe the context.

Either the context is reality and your truth is universal.
Or your truth is contextual e.g domain-specific.

Domain-specific truth leaves you with a decision-problem.

Am I using the correct theory in the correct context?
PeteOlcott wrote:
Mon Apr 08, 2019 7:42 pm
Tarski is unable to evaluate this as true or false within his theory:
3) x ∉ Pr if and only if x ∈ Tr // page 275
Great! So we are talking about domain-specific truth.

Being unable to evaluate something as true or false within a theory is the same as undecidability.
PeteOlcott wrote:
Mon Apr 08, 2019 7:42 pm
By providing three truth predicate axioms I show a simple way to eliminate
undecidability in all formal systems including his own theory.
I need a yes/no answer from you on this one. Are you claiming that your axioms (if satisfied) guarantee universal truth?

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Mon Apr 08, 2019 8:20 pm

PeteOlcott wrote:
Tue Apr 02, 2019 9:48 pm
Truth Predicate Axioms
(1) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
(2) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (False(F, x) ↔ (F ⊢ ~x))
(3) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (~True(F, x) ↔ ~(F ⊢ x))
So, as best as I can understand your grammar/syntax I have defined the above in Ruby as follows:

https://repl.it/repls/FlawlessDeafeningControlflowgraph

Code: Select all

def implies(a,b)
  return ( (not a) or b ) 
end

def true_func(f,x)
  implies(f,x)
end

def false_func(f, x)
  implies(f, (not x))
end

def not_true_func(f, x)
 not (implies(f,x))
end
What happens next?

There are just definitions - no consequences. The values for F and x are not specified.

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