Turing Complete

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Re: Turing Complete

Postby Natural ChemE on January 5th, 2017, 3:20 pm 

Forest_Dump,

I think that it's just reductionism. Just like physicists are always looking for the most fundamental particles and forces to explain the universe, computationalists are looking for the most fundamental components of logic to explain math/logic/computation/thought/knowledge/whatever-you-call-it.

Wolfram spent a lot of his life researching this sorta thing, and to great effect; he ended up making Mathematica, WolframAlpha, and some other cool stuff that's really helped Science progress. So, I guess that he kept looking further, trying to find that ultimate, most fundamental understanding that everyone should base their work on.

While I don't know the particulars of how he got into it, later in his life Wolfram became obsessed with the observation that cellular automata are Turing complete. To be fair, it's a really cool observation. Since any Turing-complete system can emulate any other, this means that a very simplistic system can fully reproduce everything that our modern computers can - including 3D video games, quantum physics simulations, and even artificial intelligence.

If I had to guess, I'd think that Wolfram is trying to take his work to its ultimate conclusion. He's trying to find that great, eternal philosophy to encompass all others. And if he's successful, he'll spin it into another program that, like Mathematica, will help revolutionize Science.
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Re: Turing Complete

Postby Dave_Oblad on January 5th, 2017, 3:28 pm 

Hi Forest_Dump,

Forest_Dump wrote:it seems to me that some of this work being done, and by apparently well-respected authorities, has no clear application..

One such application might be to explain how/why we Exist.

That Math doesn't just simply describe our Universe.. that the Universe actually Exists as a sub-set of Math. IMHO.. lol.

Sorry Hyksos.. no derailment of your topic intended.

Regards,
Dave :^)
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Re: Turing Complete

Postby hyksos on January 5th, 2017, 3:39 pm 

You know what's stupid to the point of being offensive? It appears that hyksos's characterization of the historical understanding of Gödel's work is correct. Gödel was actually that limited; the man literally starved himself to death, and he had no head for things outside of his limited domain of professional expertise.

This is not my characterization and I vehemently disagree with you placing words in my mouth.

I presented an anecdote about Kurt Godel, to indicate I have read a biography about him.

At this point, you are pulling anecdotes out of my posts, taking them out of context, and harping at them as a basis to hurl insults at me.

Did you have anything to say about the more major themes I have mentioned here?
Last edited by hyksos on January 5th, 2017, 4:04 pm, edited 1 time in total.
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Re: Turing Complete

Postby hyksos on January 5th, 2017, 3:44 pm 

Sorry Hyksos.. no derailment of your topic intended.

Regards,
Dave :^)

Thanks. You are always gracious to the end. This thread has disintegrated into the mods of this forum calling me "stupid to the point of offensive". I'm expecting a ban from the website in the coming hours.

I'm handing out contact emails to some folks for future interactions. I just hope the moderators here are not so mean-spirited as to block those final private messages.

Maybe you and I will cross paths again in the future.
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Re: Turing Complete

Postby Dave_Oblad on January 5th, 2017, 4:10 pm 

Hi Hycsos,

I hope you don't seek to be banned. I think your presence on this Site adds to it. I am always thrilled by your in-sights. Controversy is good, as long as we keep our tempers in check (not easy sometimes). You have no idea how angry I have gotten sometimes over the actions of some. When I get that way.. I leave and cool off.. take it with a Grain of Salt.. and keep going. I'm certain that the Mods realize that controversy is what pays the bills by attracting an audience that is exposed to the advertising.

And mostly.. it's a place to learn (and I've learned a lot over these past 6 years). I spend a lot of time on the fringe.. and occasionally get bumped. All I can hope for is some future vindications.. which has occurred sometimes. Unless I want to talk about sports or cars.. this is my home.. the only place I have found friends where I don't get bored with mundane conversations.

Hang in there my friend..

Highest Regards,
Dave :^)

(again.. sorry for going off topic)
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Re: Turing Complete

Postby Braininvat on January 5th, 2017, 5:08 pm 

Not sure quite what NCE meant but nobody is getting banned. And this thread is fascinating. And I wish I had a free month or 2, to read Wolfram's megabook.
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Re: Turing Complete

Postby Forest_Dump on January 6th, 2017, 7:55 am 

Well I have actually learned a lot and I am glad I spent some time here on this one and I expect I will be spending much more thinking about some of this stuff as I tramp through the frosty world of snow around me checking out lynx tracks.

Hyksos did post an interesting historiography from the world of math that should simply remind people that just because there are/were great thinkers who may have been capable of inspiring great leaps forward, as measured on some scale, these individuals may have been just as flawed as the "fact checkers" like to remind us other seemingly great people (e.g. Mother Theresa or Ghandi - religious figures chosen deliberately) were. But one of the lessons for me here is that the all too common practise of quote mining, whether it be Aristotle, Carwin or Einstein, has its place but it is more important to really think about what those quotes meant in their historical context and what they mean time. We do need to acknowledge we are standing on the shoulders of giants but 1) the point to standing on the shoulders of giants is to see further than they did and 2) every past giant had a different leaning based on what they in turn were standing on and some didin't see the same things as others. So we do need to jump to different giants to get different perspectives.

I will address two other posts seperately.
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Re: Turing Complete

Postby Forest_Dump on January 6th, 2017, 8:07 am 

Natural ChemE wrote:I think that it's just reductionism. Just like physicists are always looking for the most fundamental particles and forces to explain the universe, computationalists are looking for the most fundamental components of logic to explain math/logic/computation/thought/knowledge/whatever-you-call-it.

Wolfram spent a lot of his life researching this sorta thing, and to great effect; he ended up making Mathematica, WolframAlpha, and some other cool stuff that's really helped Science progress. So, I guess that he kept looking further, trying to find that ultimate, most fundamental understanding that everyone should base their work on.

While I don't know the particulars of how he got into it, later in his life Wolfram became obsessed with the observation that cellular automata are Turing complete. To be fair, it's a really cool observation. Since any Turing-complete system can emulate any other, this means that a very simplistic system can fully reproduce everything that our modern computers can - including 3D video games, quantum physics simulations, and even artificial intelligence.

If I had to guess, I'd think that Wolfram is trying to take his work to its ultimate conclusion. He's trying to find that great, eternal philosophy to encompass all others. And if he's successful, he'll spin it into another program that, like Mathematica, will help revolutionize Science.


IMHO a great clarifying post. While I see the potential for continued "pure" research and in fact in a paradigmatic sense (a la Kuhn), I also see the clear indctive side, crossing paradigms by way of independent external testing (and falsifying) in applications in computer science and biology. I can't say I understand a lot of this (any of it?) or could judge if it is even accurate and I even find the question of whether I agree with it to be irrelevant but if I were an external funder I would probably support the research directions proposed. It smells like real science with some clear deliverables and that is something I need to think about when I mull over the question of what is science vs. philosophy etc. (because maybe Natural ChemE is just a good salesman! Ugh yeah lots to think about here.)
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Re: Turing Complete

Postby Forest_Dump on January 6th, 2017, 8:39 am 

Dave_Oblad wrote:One such application might be to explain how/why we Exist.

That Math doesn't just simply describe our Universe.. that the Universe actually Exists as a sub-set of Math. IMHO.. lol.


And here is something very different. Taken in the context of many other strings Dave Oblad has followed, it is the reductionism described by Natural ChemE but it is also very much pure positivism - it is more clearly following the path of pure deduction to the point of being so "math soaked" it is very hard-core linguistic determinism to the point of being pure literary critique with the text written purely in terms of mathematical formulas as in the poetry of form without regard to content until "we" poke our head up to look around and find ourselves unambiguously in the land of..... metaphysics and religion. I don't know how we got here (and I don't know if that is even relevant unless and until we try to trace our way back - but that is important since I think it is important to learn how to navigate the slippery slope of hyper-relativism and nihilism) but I have no doubt about where this is. And politically, it is an important place to be. How could I or anyone deny that the question of ultimate origins isn't among the most important philosophical questions of all? But, however, we got here and however potentially appealing the view and prospects for further enlightenment, to me it smells very non-science and in fact it smells very strongly of Intelligent Design with the big question being, since I do know math was invented, then who invented the ultimate and first math that set the universe into motion?! Unless (historically counter intuitive) math preexisted but was (devinely?) revealed?! Arg, again, another paradox of sorts. How can a philosophy of science that most strongly argued to be the purest even most athiestic stream, place us most solidly in the branches of philosophy least constrained by objectivity and empirical data? Lots more to think about here.
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Re: Turing Complete

Postby hyksos on January 6th, 2017, 3:52 pm 

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Re: Turing Complete

Postby Scott Mayers on January 13th, 2017, 4:36 am 

I would have responded sooner but was sick (It's time I start regularly taking those flu shots now!)

Hyksos,

I don't know what your concern is on my own take here and would prefer to not digress into a need to defend my own background. If you are trying to help others understand something you think is mistaken, I think it is more effective to use contemporary relations with those you communicate with directly in modern terms rather than debate what some past thinker thought 'correctly' or not. I hold this with all subjects in 'forums' of discussion. All that matters is if you can try your best to understand and be understood.
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Re: Turing Complete

Postby Braininvat on January 13th, 2017, 10:47 am 

Scott, understanding "prior art" in any STEM field is essential. We encourage members to be fully familiar with groundbreakers like Von Neuman, Turing, Godel, Shannon, Church, Hilbert, et al when entering a chat like this.
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Re: Turing Complete

Postby Scott Mayers on January 16th, 2017, 5:15 am 

Braininvat » January 13th, 2017, 9:47 am wrote:Scott, understanding "prior art" in any STEM field is essential. We encourage members to be fully familiar with groundbreakers like Von Neuman, Turing, Godel, Shannon, Church, Hilbert, et al when entering a chat like this.

Too much crap gets tied up into referencing others as though they have a right to 'own' ideas that are bound to be discovered independent of those being credited. If, for instance, Einstein hadn't been born, do you actually think a form of Relativity theory would not be discovered? Science is ALL OF OUR right to participate in. AND...

IF some are sincerely 'authoritative' in some field, they should be skilled enough to prove their own credibility by relating to the less 'authoritative' at their level.
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Re: Turing Complete

Postby Braininvat on January 16th, 2017, 1:17 pm 

Well, whoever had come up with SR/GR, we would want to understand their work before commenting on the topic, given the centrality of that work to modern physics. We don't study and cite because we are enraptured with authority or celebrity, or feel that someone "owned" an idea, but rather to be informed, and sometimes inspired, by groundbreaking work that was key in a particular field and has stood the rigorous tests of decades of experiment and observation. I see many members, Scott, who are quite patient and willing to explain aspects of their knowledge and subject them to critical scrutiny and revision. You keep striking out at some imagined orthodoxy you think is stifling you and frequently posting complaints about people's personalities. I am puzzled by this, given the abundance of very UNorthodox thinkers here (DaveOblad, Faradave, to name two) (what IS it about being a "Dave" anyway??) who seem to have found a safe haven. You may notice they put in the time to study extensively the previous work done in their fields of interest, so as to avoid "reinventing the wheel" or accidentally ignoring experimental findings that will undermine their own theorizing.
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Re: Turing Complete

Postby Scott Mayers on January 18th, 2017, 2:39 am 

I think you might be thinking I'm being more cynical than I am about 'science'. I completely honor science, philosophy, and logic. My comment IN THIS THREAD related to hyksos' assumption that I was somehow lacking sufficient background rather than to deal with content at hand. So it appeared that I was being considered incompetent as a whole rather than to be challenged for something specific he might disagree with me about.


You are confusing this also to what I had messaged you personally in regards to some suggestions because I had noticed you were transferring other's threads lately. So it was not even about me personally and I thought to speak there so as not to disrupt the threads.
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Re: Turing Complete

Postby hyksos on January 18th, 2017, 7:44 am 

IF some are sincerely 'authoritative' in some field, they should be skilled enough to prove their own credibility by relating to the less 'authoritative' at their level.


My peace offering. Scott Mayers -- Mathematics does not operate by these social rules.

Mathematics is qualitatively different from all other scientific disciplines. I could list the reasons, but the one that sticks out the most is that the discipline of Higher Math is obsessive about very precise definitions. Because this is a forum, and not a university, Scott could point at this as some sort of social ungrace on the part of hyksos --make headway with that accusation since it appears superficially true. But if anyone on this forum, (from newbie user to forum administrator,) were to take a course on higher mathematics, they would find out in flying colors, that there are very good reasons why definitions are obsessively precise. They would see this is not a hyksosian character flaw, but the nature of the discipline itself.
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Re: Turing Complete

Postby hyksos on January 19th, 2017, 2:02 pm 

Braininvat » January 16th, 2017, 9:17 pm wrote: We don't study and cite because we are enraptured with authority or celebrity, or feel that someone "owned" an idea, but rather to be informed, and sometimes inspired, by groundbreaking work that was key in a particular field and has stood the rigorous tests of decades of experiment and observation.

I totally agree with this. And there is an additional danger in this sort of thing, generally speaking.

If you want to go beyond the textbook, ya know, not just regurgitate it on a test. But you want to "hang out" with the professor after class and get into the philosophical points. Maybe even meet your math professor in his office. If you do not go back and trace the history of the discipline, and you have not even a fuzzy idea of what Godel (or Hilbert, or Leibniz, or etc) were doing, and what problems they were facing, and what obstacles they were trying to overcome by proving these landmark theorems --- lacking that historical perspective, you can easily hijack these topics and use them to proselytize.

So yes, talking to a general lay audience, wherein nobody knows the reasons Kurt Godel was writing this theorem, you could easily bewitch them enough to say stuff like "Godel's 2nd Incompleteness theorem proves that there is a God". If any of you think that extrapolation is "silly" (or what have you), believe me when I tell you it has been done before.. more than once.
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Re: Turing Complete

Postby Scott Mayers on January 19th, 2017, 9:25 pm 

You're being odd to me, hyksos. I agree with extreme logical precision. You're own lack of noticing this of me only makes me wonder where you are coming from at all.

I asked you above a linked set of questions to narrow down your own thinking at http://www.sciencechatforum.com/viewtopic.php?f=19&t=32176#p313486 to which you hadn't answered. Instead you asserted in response about some lack of understanding of the terms "complete" or "undecided". I get the impression that you are trying to assert that Godel and Turing's arguments on incompleteness and undecidability theorems do not extend to logic in general. They both actually prove the same thing but in different fields: math and computation. They both basically assure that you cannot find any UNIQUE system of formal reasoning that can completely and exhaustively prove all theorems in their domains.

NOTE FIRST THAT both pioneers used reasoning of their own to PROVE their theorems based on a logic that was dependent on the assumptions of 'consistent' laws of logic presumed. As such, if their logic proves something definitively 'true' as a theorem under such systems on the question of closure (completeness), if the theorem proves that all theorems in a subdomain of it cannot be closed when sufficient enough to cover all possible truths in it, then this has to also be true of all logic that the logic being used to show this true is trusted to cover OR that theorem is itself incomplete!

But while their domains are apparently limited, each of those domains of math and computation are a subset of a larger inclusive domain of all subjects collectively. If part of the 'larger' domain has some system of reasoning that could encompass those truths by the same assumptions of 'consistency', even if that metalogic is itself 'complete', those theorems in it can only be trusted on the faith of the validity of that metalogic.

Godel uses Propositional (first-order) and Predicate (dependent on Propositional logic) logic to prove a theorem on math that itself is dependent on the essential properties of Propositional and Predicate logic that cannot complete or decide all 'truths' in the domain of math. If some 'other' grandfather logic (a meta-logic) exists outside of the logic that Godel uses that STILL REQUIRES "CONSISTENCY", there would be another logic that could still possibly prove that a math-domain logic IS complete and closed. Yet, Godel already covered this or his theorem using his reasoning would be a mere sub-domain of a yet larger domain to which he couldn't speak (or decide) on whether there is or is not some consistent-based logic that could completely exhaust all math truths.

As such, no logic based on 'consistency' is itself sufficient to completely exhaust all FACTS ABOUT REALITY, including any subdomains. So Godel and Turing's theorems about completeness (and 'undecidability') are easily reduced to saying that:

You cannot find a perfectly complete system of logic that encompasses all truths in reality where it is dependent on consistency. AND, you cannot expect to BEGIN with some consistent-essential system of logic that can self-justify its own system.

What is beyond reproach of this conclusion is to some system not reliant on the assumption of 'consistency' as a meta-logic that encompasses all domains.
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Re: Turing Complete

Postby hyksos on January 21st, 2017, 9:11 am 

But while their domains are apparently limited, each of those domains of math and computation are a subset of a larger inclusive domain of all subjects collectively. If part of the 'larger' domain has some system of reasoning that could encompass those truths by the same assumptions of 'consistency',


There are two very different senses in which a domain is made larger. One of them is that you start tacking on extra axioms onto the existing ones. This has actually been done in practice. The axiom of Choice is known to be independent of the other axioms of ZFC. So you can just use the "deductive system" of ZF, which has no Choice Axiom in it. Or you can include Choice Axiom, according to taste. This domain-extending is what is usually referred to if I make a statement like "Complex Analysis is a subset of ZFC" {call this Sense I}

The second way is to define a universe of symbols and their syntax, and start talking about the system from the outside. This is what you are calling "metalogic". A deductive system appearing to be very narrow, might seem boring to us citizens of 2017, but in the 1920s this stuff was revolutionary in scope. {Call This Sense II}


even if that metalogic is itself 'complete', those theorems in it can only be trusted on the faith of the validity of that metalogic.

Luckily, incompleteness is not the same thing as invalidity.

Yet, Godel already covered this or his theorem using his reasoning would be a mere sub-domain of a yet larger domain to which he couldn't speak (or decide) on whether there is or is not some consistent-based logic that could completely exhaust all math truths.

This is the difference between the 1st Incompleteness Theorem and the 2nd Incompleteness theorem. Godel explicitly addressed the issue of larger domains in the 2nd theorem. And he did so in the {Sense I} way. The adding of extra axioms (provided they are genuinely independent of the existing ones) will always place us up into a larger system containing the previous ones. In this way we scale a hierarchy of embedded systems, each larger than the previous and containing the previous inside it.



The question for the 2nd theorem was : if we tack on enough extra axioms, will we tie up all the loose ends and happen upon a system that is complete and consistent? The theorem says No. Any system rich enough to depict arithmetic must either be incomplete or inconsistent. All of them are subject to this restraint.

You cannot find a perfectly complete system of logic that encompasses all truths in reality where it is dependent on consistency. AND, you cannot expect to BEGIN with some consistent-essential system of logic that can self-justify its own system.

In Proof Theory you are not really interested in systems of LOGIC You are more interested in consequences of axioms (usually about sets) or consequences of a language with a syntax.

I have already mentioned several issues of LOGIC which exceed the scope of Godel's theorem. The first example was an agent who acts in a world under uncertainty, and whose beliefs changes whenever new data comes in. That agent's logic is going to be very different from a static system that is assumed to contain a priori all of its consequences hanging motionless in a platonic superchest. (if you will)

The other example was metalingual reference. Ironically meta-lingual reference appears in Godel's 2nd theorem.

The third example would be the question as to whether the class of computable functions is larger than ZFC.


perfectly complete system of logic that encompasses all truths in reality

This is where the conversation goes off the rails for me. When you use phrases like "encompass all truths in reality", I believe you are no longer talking about math. Your meaning is unclear. What do you mean by "encompass"? And what do you mean by "reality" here? Do you mean the physical universe?
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Re: Turing Complete

Postby Scott Mayers on January 24th, 2017, 2:41 am 

hyksos » January 21st, 2017, 8:11 am wrote:
But while their domains are apparently limited, each of those domains of math and computation are a subset of a larger inclusive domain of all subjects collectively. If part of the 'larger' domain has some system of reasoning that could encompass those truths by the same assumptions of 'consistency',


There are two very different senses in which a domain is made larger. One of them is that you start tacking on extra axioms onto the existing ones. This has actually been done in practice. The axiom of Choice is known to be independent of the other axioms of ZFC. So you can just use the "deductive system" of ZF, which has no Choice Axiom in it. Or you can include Choice Axiom, according to taste. This domain-extending is what is usually referred to if I make a statement like "Complex Analysis is a subset of ZFC" {call this Sense I}

The second way is to define a universe of symbols and their syntax, and start talking about the system from the outside. This is what you are calling "metalogic". A deductive system appearing to be very narrow, might seem boring to us citizens of 2017, but in the 1920s this stuff was revolutionary in scope. {Call This Sense II}


Sorry for the late response, hyksos.

I was thinking of your second understanding in part at least. The broader "Propositional Calculus", for instance, treats propositions as variables that would include a more general (broader) domain.

Another might be to extensively broaden a binary valued (true/false) logic to multivariables.

In Boolean Algrebra, for instance, rather than define

A + B = (A OR B)
A * B = (A AND B)
-A = NOT A

we can define these for a multivalued system as:

A + B = min(A, B),
A * B = max(A, B),
-A = P minus A, P = N minus 1

even if that metalogic is itself 'complete', those theorems in it can only be trusted on the faith of the validity of that metalogic.

Luckily, incompleteness is not the same thing as invalidity.

The limit of being able to test validity exhaustively on the domain given, that is.

Yet, Godel already covered this or his theorem using his reasoning would be a mere sub-domain of a yet larger domain to which he couldn't speak (or decide) on whether there is or is not some consistent-based logic that could completely exhaust all math truths.

This is the difference between the 1st Incompleteness Theorem and the 2nd Incompleteness theorem. Godel explicitly addressed the issue of larger domains in the 2nd theorem. And he did so in the {Sense I} way. The adding of extra axioms (provided they are genuinely independent of the existing ones) will always place us up into a larger system containing the previous ones. In this way we scale a hierarchy of embedded systems, each larger than the previous and containing the previous inside it.

The question for the 2nd theorem was : if we tack on enough extra axioms, will we tie up all the loose ends and happen upon a system that is complete and consistent? The theorem says No. Any system rich enough to depict arithmetic must either be incomplete or inconsistent. All of them are subject to this restraint.

Moot points. Even if you extend a logic, it is either to make it MORE BROAD or LESS BROAD (limiting domain). But whether math is limited or extended of some metalogic, as long as there is ANY consistent system that can include all math, the theorems hold. In other words, even if we could find ANY system that can cover all math problems even if it means including a larger domain, such a system still could not cover all math truths (a subset of all truths).

You cannot find a perfectly complete system of logic that encompasses all truths in reality where it is dependent on consistency. AND, you cannot expect to BEGIN with some consistent-essential system of logic that can self-justify its own system.

In Proof Theory you are not really interested in systems of LOGIC You are more interested in consequences of axioms (usually about sets) or consequences of a language with a syntax.

I have already mentioned several issues of LOGIC which exceed the scope of Godel's theorem. The first example was an agent who acts in a world under uncertainty, and whose beliefs changes whenever new data comes in. That agent's logic is going to be very different from a static system that is assumed to contain a priori all of its consequences hanging motionless in a platonic superchest. (if you will) [/quote]
The underlined "uncertainty" implies incompleteness and potential inconsistency.

The other example was metalingual reference. Ironically meta-lingual reference appears in Godel's 2nd theorem.

The third example would be the question as to whether the class of computable functions is larger than ZFC.

Regardless of your interpretation, as long as Godel's 2nd theorem holds for general cases even IF the first one was limited, that 2nd theorem holds for his OWN use of logic to prove these theorems. It is the same thing trying to hold the certain claim that "nothing is absolutely certain" because it would have to hold for this statement too. If you can't find closure of a system using a 'consistent' logic by proof using a consistent logic system, then the problem lies in assuming 'consistent' systems. Godel's theorems are "consistent-limited" systems. You need a Trivial or Paraconsistent or ?X? logic to completely cover all truths.

perfectly complete system of logic that encompasses all truths in reality

This is where the conversation goes off the rails for me. When you use phrases like "encompass all truths in reality", I believe you are no longer talking about math. Your meaning is unclear. What do you mean by "encompass"? And what do you mean by "reality" here? Do you mean the physical universe?

I thought you understood I was referencing ALL reality, not simply math. ?? Yes, the theorems hold for the physical laws of science too. As long as they are consistent logics, Godel's system holds true for all such logic.
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Re: Turing Complete

Postby hyksos on January 24th, 2017, 4:35 am 

Scott,

I honestly and swear on my mother's grave that I am 1.) Not trying to flame you. 2.) Not trying to insult you. I do not know any other way to say this to you in a way that is "nice". So I will just say it, and you can accept my words or not.

Your reply is a mish-mash of confusion between logic and and these things called deductive systems. You keep saying things like "When Godel extended logic" and you refer to "Godel's system". These phrases make no sense. I have little indication that you have understood anything I have said to you in the previous post. I could go line-by-line but I don't see the point, because it was just be me repeating this same message over and over again.

I have already mentioned several issues of LOGIC which exceed the scope of Godel's theorem. The first example was an agent who acts in a world under uncertainty, and whose beliefs changes whenever new data comes in. That agent's logic is going to be very different from a static system that is assumed to contain a priori all of its consequences hanging motionless in a platonic superchest. (if you will)

The underlined "uncertainty" implies incompleteness and potential inconsistency.


It absolutely implies no such thing. But again, you are not even using these phrases correctly. You totally overlooked that I had also underlined (and capitalized) issues of LOGIC. Because you are constantly posting erronious things about the application of Godel's theorems to logic. I understand clearly why these two things are different, but your posts indicate a confusion.

I thought you understood I was referencing ALL reality, not simply math. ?? Yes, the theorems hold for the physical laws of science too. As long as they are consistent logics, Godel's system holds true for all such logic.

I am not the confused person here.

Listen. Bertrand Russell and Alfred North Whitehead discovered that all branches of mathematics, which were once considered separate disciplines, were all united under the same umbrella. That umbrella is something called Zermelo-Fraenkel-Choice Set Theory. By the time of Hilbert and Godel, proof theory had developed far enough that mathematicians could "step outside" of ZFC and refer to is as a "system". A "system" contains symbols, sentences, well-formed-formulas, and deductive translations between them. (To be more formal and academic with you, because that's important right now) we should laboriously refer to ZFC as a deductive system.

Completeness
If a deductive system is "complete", then all the sentences within the system which are true can be derived from its axioms by deduction. (If s is a logical consequence of the system, then there exists a proof of that fact).

Consistency
If a deductive system is "consistent", then for all sentences s, which contain a proof showing that s is true, there will not exist a proof showing that s is false.

Theorem
If a sentence s is in a deductive system U has a proof, then s is a theorem in U.

Incompleteness
If U is incomplete, there will exist an s in U that is a true statement which is not a theorem.

Undecidable
A sentence s is "undedicable" if and only if "s is true" is not a theorem in U, and "s if false" is not a theorem in U. In most working cases, both of these must be proven. In some cases in history, both proofs are proven decades apart by different people.

Independent
A sentence s is "independent of U" if and only if s is undecidable in U and s is consistent with the axioms of U.

The above glossary should go a long way. I have warned you earlier in this thread, and I will warn you a second time. Wikipedia is absolutely terrible for this topic -- they write about all this stuff in an "ultra-modern" form, which assumes the reader has had exposure to Model Theory. (I do not doubt that Godel's work as a mathematicians has , in the ensuring decades, been formalized as 'obvious things' in Model Theory. ) But generally, the wikipedians are being obtuse!



In your reading of the material surrounding this topic, you will undoubtedly come upon something like :
"Godel showed that the Continuum Hypothesis is independent of ZFC."

Armed with the above glossary, you should be able to parse this sentence now. Let CH=Continuum Hypothesis. You should see that CH exists as a well-formed sentence of ZFC. Yet ZFC cannot be used to prove CH is true. Further, CH does not refute ZFC.
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Re: Turing Complete

Postby hyksos on January 24th, 2017, 4:52 am 

So I was banging around the wikipedia article on Foundations of Mathematics.

https://en.wikipedia.org/wiki/Foundations_of_mathematics

Then I arrived at the section titled "Philosophical consequences of the Gödel's completeness theorem". The very first sentence reads:
Gödel's completeness theorem establishes an equivalence in first-order logic between the formal provability of a formula and its truth in all possible models.

Now. Let this be a warning to both Scott, and everyone else reading this. I don't doubt that sentence there is authoritative. But holy smokes! There is no way you will understand what that sentence says unless you have taken graduate level mathematics courses.

I'm not even joking.

This stuff has something to do with a branch of math called Model Theory, where you do things like consider varying "interpretations" of a collection of infinite sentences, and declare them all equal if they sustain a certain mapping to truth values. People who talk like this wear tweed jackets and turtle neck sweaters, and refer to "...my distinguished colleague at Cornell" in idle conversation.

So again, (for the third time) do not try learn Proof Theory from wikipedia.

To Scott, I suggest very strongly that you take at least some university courses on Proof Theory. It might be listed at your uni as "Introduction to Mathematical Logic", but this will depend on your institution. If money and time are an issue, contact the instructor and find out what textbook they use. Get a copy. If the first three chapters mention "Well-formed formula" and/or "Sentential logic" then you know you have the right book.

(edit: Also, Scott, you may need a little bit of exposure to Abstract Algebra, depending on your circumstance)
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Re: Turing Complete

Postby someguy1 on January 24th, 2017, 11:45 am 

hyksos » January 24th, 2017, 2:52 am wrote:So I was banging around the wikipedia article on Foundations of Mathematics.

https://en.wikipedia.org/wiki/Foundations_of_mathematics


hyksos » January 24th, 2017, 2:52 am wrote:So again, (for the third time) do not try learn Proof Theory from wikipedia.


You quoted Wikipedia to come to that conclusion? Now that's funny.

I wouldn't want to get in the middle of your entertaining pissing match with Scott, but this is a non sequitur to exceed all the others in this thread. Which I promised myself I'd stay out of. But this latest ... who are you to say who can post what in a discussion forum? There's a model if and only if the axioms are consistent. This can be explained very simply. But this thread has derailed into an insult fest. Perhaps you could go back to your OP and try to explain what you have in mind, rather than getting derailed by personal annoyances and laying down rules for what ideas can be uttered in your presence.

For the record I haven't read most of these posts, but I did closely read your OP (actually your second post) and I would only say that nobody here is in a position to throw stones.

ps -- One should not take Wikipedia's confusing exposition as evidence that you need to wear any particular type of clothing. I am baffled as to your weird locutions in that regard. Do you have some kind of axe to grind with academia? Please, remember why you started this thread. It was to explain, not fight, make fun of people's clothing choices, or to say that a simple idea like Gödel's completeness theorem requires grad level math. It does not. Nor, frankly, does grad level math help much, mathematical logic sadly being a backwater in most mainstream math departments.

hyksos » January 21st, 2017, 7:11 am wrote:"Complex Analysis is a subset of ZFC" {call this Sense I}


This is exactly the kind of thing I'm talking about. What you wrote makes no sense. In no way is complex analysis a "subset" of ZFC. Or if it is, you will have to explain it to me. You are slinging words in very careless and imprecise ways then getting on your high horse when others do the same.

hyksos » January 24th, 2017, 2:35 am wrote:Listen. Bertrand Russell and Alfred North Whitehead discovered that all branches of mathematics, which were once considered separate disciplines, were all united under the same umbrella. That umbrella is something called Zermelo-Fraenkel-Choice Set Theory.


My God. No. No on the history, no on the math. No. In any event, how could something be both a foundation and an umbrella? It's not even a good metaphor.
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Re: Turing Complete

Postby hyksos on January 24th, 2017, 2:23 pm 

Which I promised myself I'd stay out of. But this latest ... who are you to say who can post what in a discussion forum?

I am not saying who can post what on a forum. I am picking up loud and clear that Scott has no formal training in the subjects in which he is trying to expound upon. AND THAT DOES NOT WORK IN MATHEMATICS. YOu know this. It might work for painting, for physics, for cosmology, for biology, for programming, for motorcyle maintenance you name it. Most of those things in the world can be picked up by osmosis and self-teaching. I won't deny. Mathematics is not one of those things. Higher math is even worse in this regard.

Please, remember why you started this thread. It was to explain, not fight, make fun of people's clothing choices, or to say that a simple idea like Gödel's completeness theorem requires grad level math.

Luckily I never made this claim even once in anywhere in this thread.

What I will say, more clearly now, is that I am picking up (using my spidey-sense) that Scott Mayers got his training in higher mathematics from reading the internet. I link wikipedia only to illustrate that whenever mention a whole entire sub-discipline in mathematics, or a branch of mathematics, that the reader can see from wikipedia that this branch of math does indeed exist and holds in some cases very long articles. But I am not pointing to such as a place where the person can go and learn about it.

Learning mathematics is not a passive activity. It requires an active and engaged learner (who in most cases must work the problems in the back of the chapters to really "get it"). You know this. I know this.

I have never said to single person on this entire forum that "you must hold a degree in order to talk to me". Never. Not once-- ever have I said that or implied that.

What I did say is that for some reason or another, the authors of wikipedia articles presume the reader is already comfortable with Model Theory. I have no indication in any slight indication at all that any person on this entire forum is trained in Model Theory. I certainly am not. I don't blame anyone for not knowing Model Theory. But I am educated to enough to know how advanced and how highly abstract Model Theory is. You can't even understand the beginning chapters of a textbook on Model Theory unless you have already taken many many courses in very abstract forms of mathematics. These facts have nothing to do with who is smart and who is dumb -- it is simply the nature of the discipline.


I don't who the f&%k you are, "someguy1" but if you want to sit in this thread, and be Scott Mayer's personal mathematics tutor -- have at it. He's all yours. Don't let me stop you. Tutor him to your own delight.
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Re: Turing Complete

Postby hyksos on January 24th, 2017, 2:27 pm 

For the record :

I am being extremely patient.
I am being extremely helpful.
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Re: Turing Complete

Postby someguy1 on January 24th, 2017, 2:31 pm 

hyksos » January 24th, 2017, 12:23 pm wrote:What I will say, more clearly now, is that I am picking up (using my spidey-sense) that Scott Mayers got his training in higher mathematics from reading the internet.


I have the same sense about you. Your OP and your most recent post that I responded to were highly inaccurate.

hyksos » January 24th, 2017, 12:23 pm wrote:I don't who the f&%k you are, "someguy1" but if you want to sit in this thread, and be Scott Mayer's personal mathematics tutor -- have at it. He's all yours. Don't let me stop you. Tutor him to your own delight.


Nicely avoiding the direct challenges I gave to you. Your remarks about complex analysis and Russell and Whitehead were so far off the mark I had to call them out. Normally I wouldn't bother, just as I didn't bother to give a line-by-line refutation of virtually everything you said in your OP. It would be tedious. But when you attack another poster (whom I am not necessarily defending) with statements that are profoundly misinformed, that's not right. You should have some humility and self-awareness. Frankly your remarks about Russell and Whitehead show that you didn't even bother to read the Internet, since the Wikipedia entry on PM explicitly refutes your claim that they said anything about ZFC.
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Re: Turing Complete

Postby hyksos on January 24th, 2017, 2:37 pm 

I have the same sense about you. Your OP and your most recent post that I responded to were highly inaccurate.

I am university-trained in higher mathematics. Some of the books from those years are still on my shelf. I could photograph them in front of the screen.

Nicely avoiding the direct challenges I gave to you. Your remarks about complex analysis and Russell and Whitehead were so far off the mark I had to call them out.

They were not far off the mark at all. Your only complaint was the the metaphor of an umbrella was bad. Oh!!! -- well pardon me for using metaphors that pet your fur in the wrong direction.
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Re: Turing Complete

Postby hyksos on January 24th, 2017, 2:42 pm 

Frankly your remarks about Russell and Whitehead show that you didn't even bother to read the Internet, since the Wikipedia entry on PM explicitly refutes your claim that they said anything about ZFC.

For the record, ladies and gentlemen. What this someguy1 is complaining about is historical hair-splitting. Principia Mathematica (the magum opus of Russell and Whitehead) did not yet ground the set theory in what we know today and refer to as ZFC. That's harping on an historical point, and "technically" its wrong -- but it is mere anecdotal detail.

I can convince anyone that this is trollish hairsplitting. It has nothign to do with the existing discipline of math, and its like we are complaining about the name of Napolean's wife's hairdresser. PM showed that set theory was going to be the foundation, and this was formalized later on by Ernst Zermelo. Yes yes yes -- I don't deny any of this. someguy1 is splitting hairs between Set Theory and axiomatizable theories of set theory. That's really hairsplitting and only historians would get upset by this.

This troll is not going to be satisfied until I give a year-by-year blow-by-blow biography of Russell. someguy1 has picked his favorite subject, which is undoubtedly the development of Set Theory in history of modern math, and now he is going to use his pet subject as a weapon to troll and attack me. I wasn't born yesterday. I already see where this is going. But someguy1, if you must go through this little psychological process with me, do what you must.

Scratch your little itch. Go ahead.
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Re: Turing Complete

Postby someguy1 on January 24th, 2017, 2:43 pm 

hyksos » January 24th, 2017, 12:37 pm wrote:
I have the same sense about you. Your OP and your most recent post that I responded to were highly inaccurate.

I am university-trained in higher mathematics. Some of the books from those years are still on my shelf. I could photograph them in front of the screen.



Try reading them instead. Your remarks about Russell and Whitehead are laughable. As is your claim that complex analysis is a "subset" of ZFC. That's not even wrong. I see that your ego does not allow for the possibility that your exposition, if not your understanding, is wildly off the mark in many respects. I objected to your superior attitude with respect to Scott, since one could easily take the same tone with you. Your education means nothing. What you write here is all I know of you, and most of what you wrote in your OP was historically and mathematically wrong. I didn't bother to jump in till you started attacking others.
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Re: Turing Complete

Postby hyksos on January 24th, 2017, 3:18 pm 

Oh yes well pardon me for "subset of ZFC". Let me be more precise so as to your cure the troll's little tummy ache on this fine point. I did not mean to imply that Complex Analysis is a sub-discipline of set theory. That would be silly.

ZFC has a universe of deductive consequences. Complex Analysis is a particular branch of the universe of those consequences. That fact is not useful to the work-a-day mathematician writing in functional analysis. But it might be a fact that is very useful for someone working in foundations.

This is how you would do that. Functions that map the complex plain into the complex plain are particular instantiation of a type of function called holomorphic functions. Instead of looking at such function geometrically (as we do in elementary graphing of real functions) lets instead say that a "function" is in fact just a set of pairs , such that

F = {
(a0+b0i,c0+d0i) ,
(a1+b1i,c0+d1i) ,
(a2+b2i,c0+d2i) ,
.
.
.
(an+bni,cn+dni) }

Where a+bi is in the domain and c+di is in the range. So a holomorphic function is a set with a particular pairing that will be true of all the elements of that set. These procedures might be used for certain types of proofs within complex analysis, but this is very rare. The above set will be uncountably infinite, and so will its (compact) subsets, the n is used only to indicate continuation of a list.

The properties of differentiability and continuity in holomorphic functions are such strong restraints, that even elementary topology can represent a far vaster and richer variety of sets. Further, Set Theory itself would be able to define even more exotic sets with looser and looser restraints.

We humans write big fat books about holomorphic functions, because for cultural and social reasons, those kinds of functions (read: sets) are useful to us. Set Theory also allows for sets which are alien and totally useless in any practical context. So in this sense, Set Theory is a larger 'thing' /// Is the word "thing" okay with you someguy1? I don't want to give anyone a tummy ache here. I would lke to say it is a bigger Oh I dunoo..."Class" of objects than the sub-class of holomorphic functions? No? Do these metaphors bother you in some way?

YOu tell me how you would precisely express the fact that set theory contains holomorphic function/sets as a smaller portion of its larger class. I think the notion is being expressed very well, so I will leave it up to you to pick the perfect word for it so as to avoid upset stomach.
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