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Godel Escher Bach

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Sir Llama
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Hi. I have a question to any of you who's familiar with Godel's Incompleteness theorem. I've heard people claiming to disprove atheism by saying that GIT states that a logical system cannot explain everything so therefore you need an irrational faith based system to explain it all.

I thought that was a load of crap. When I looked up GIT, it said that logical systems cannot explain everything because there are an infinite amount of paradoxical statements where truth cannot be ascertained (i.e. "I am lying" or "This statement is false").

I dont' really see how this implies the existence of God. Am I missing something here, or do I have an incomplete understanding of GIT?

thanks

Sir Llama

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fallacy #1: You cannot disprove atheism, because it does not posit anything. It does not put any evidence of anything on the table to disprove. On the contrary, atheism merely acknowledges the absence of viable evidence for the existence of a supernatural being. The only way to disprove atheism's claim that there is no viable evidence is to supply some viable evidence. That is the very task GIT is trying to avoid.

fallacy #2: The definition of "explain" is to draw the connections between a statement about reality and reality itself by means of a logical system. You cannot use the word without impying that that is possible. GIT is effectively saying: a logical system cannot do what a logical system does. The effectiveness of the argument depends on its victim's superficial understanding of the word "explain" as merely "to make intelligible". If the victim would ask himself how the process of explaining works, he might stumble onto GIT's contradiction.

fallacy #3: The statement cannot be valid, because it implicitly precludes the possibility of validation. If any aspect of reality were beyond the reach of logic, there would be no way to make a distinction between valid and invalid logic. The concepts "valid" and "true" imply the accessibility of the mind to all of reality by logic in every instance.

Consequently, positing GIT is inherently self-contradictory. There is an implicit claim on objective truth in positing GIT. If not, why bother. But if truth is as conditional as GIT says it is, they will have implicitly forfeited the only viable means to validate their own statement.

Most pat answers are self contradictory. Just as there can never be absolutely no absolutes, and one can never be certain that certainty is impossible, there is no perfectly logical proof that logic is imperfect.

GuestObserver

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Firstoff, I have no idea what "Guest observer" said. Did you mean to deny that Godel's theorom was correct? Seems a bit reminicent of times when I've heard objectivists say with absolute certanty that non-local physical theories are acausal...or that probablistic motion of molecules CAN'T be fundamental.

That having been said, anyone who would use Godel to attack atheism lacks any real understanding. Godel's theorom says one thing (and it is correct in this), that any axiomatic system that includes arithmetic (so choice of religion comes from a logical system involving arithmetic?) gives rise to propositions that are undecidable without new axioms. Truth is most of these are silly self referance statements, however a few actually have some meaning. Example is the continuim hypothesis that has to do with cardinality (size) of sets and ideas of the "different sizes of infinity" (by the way, this is not some silly concept, the idea is that a set is called countably infinite if it can be put into a one-to-one correspondance with the natural "counting" numbers, and that some sets have this property and that some don't, and then another size would be if the basis of a topological space (yah that probably sounded like gibberish, its a sort of way of concisely specifying the set and certain relationships) is countable then the topological space is called second countable) Anyhow if your interested the way that you prove that the continuim hypothesis is not decidable is by constructing a model for each way and showing that it is consistent with the axioms.

Anyow I suppose the rare, rare times when some relavent proposition is undecidable bothers the mathematicians, they just kinda work around it both ways. At any rate this has NOTHING to do with life outside of extremely complex mathematical systems. I suppose when people try to invoke Godel's theorom what they are really discussing is the sort of self referance garbage like "This statement is a lie"...although really who gives a shit about such worthless statements? Hey, if you want make a rule that you don't allow yourself to issue self-refereance statements and go about your life happily. And for fun download and print a copy of Godel's theorom and hand it to the humanities majors who are trying to invoke it, my money's on that they can't get through the first page (or become extremely upset that it uses logic and isn't quite as new age as they would hope)

Tim

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My comments were not about GIT per se. They were about the logic, (or rather lack thereof) in the claims Sir Llama asked about. GIT was just the surrogate at hand. It is totally irrelevant who, if anyone, actually holds the positions. I was merely pointing out the flaws in such thinking.

More important, they are ubiquitous flaws. Being able to spot them in the countless other arguments you will encounter if you persist in your interest in Objectivism will save you a lot of time and intellectual energy.

While I'm at it, those self-reference statements smell a lot like they could be fallacy #4.

The words "all knowledge is contextual" spring to mind, and if I am applying them correctly, the statements are not paradoxical. To have a specific meaning, a statement has to have a context:

This statement is false (that I just read). I am lying (in the previous sentence). What is so paradoxical? In order to make the statements paradoxical, you have to supply a context that makes the statements actually refer to themselves, in which case, they would be self-contradictory, not paradoxical. Or, you would have to consider the statements sans context. Of what use could it be to ponder contextless statements?

Actually it could be useful to comedians. Abbott and Costello elevated context-dropping to a high art in "Who's on first?". Oh yeah, I almost forgot ... the ability to pass off contextless statements in the guise of profundity can get you a philosophy chair in the university of your choosing.

GuestObserver

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My comments were not about GIT per se. They were about the logic, (or rather lack thereof) in the claims Sir Llama asked about. GIT was just the surrogate at hand. It is totally irrelevant who, if anyone, actually holds the positions. I was merely pointing out the flaws in such thinking.

More important, they are ubiquitous flaws. Being able to spot them in the countless other arguments you will encounter if you persist in your interest in Objectivism will save you a lot of time and intellectual energy.

While I'm at it, those self-reference statements smell a lot like they could be fallacy #4.

The words "all knowledge is contextual" spring to mind, and if I am applying them correctly, the statements are not paradoxical. To have a specific meaning, a statement has to have a context:

This statement is false (that I just read). I am lying (in the previous sentence). What is so paradoxical? In order to make the statements paradoxical, you have to supply a context that makes the statements actually refer to themselves, in which case, they would be self-contradictory, not paradoxical. Or, you would have to consider the statements sans context. Of what use could it be to ponder contextless statements?

Actually it could be useful to comedians. Abbott and Costello elevated context-dropping to a high art in "Who's on first?". Oh yeah, I almost forgot ... the ability to pass off contextless statements in the guise of profundity can get you a philosophy chair in the university of your choosing.

GuestObserver

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  • 1 month later...
Guest Eric Lanser

I noticed this phenomenon in my symbolic logic class first term this year.

When self-referencing statements, even in mathematics, (i.e. "the set of which contains all sets which don't contain themselves") turn out to be self-contradictory, this does nothing to refute logic, or language or mathematics. It merely shows that the statement is invalid. (Or some rule of the system doesn't correspond to reality, as seems to be the case with the example above.) It is a contradiction, dressed up not to be too obvious.

If I said "this is a square circle" that sentence would merely reflect that it is unconnected to reality. With most self-referencing statements, the context contradicts the statement. In principle, this is the same thing. Such statements do nothing to refute language or logic.

Since human beings possess the ability of abstraction and free will, they are able to make mistakes. They are able to provisionally imagine (and speak and write about) what is actually a contradiction. For instance, they can say "imagine a set of numbers which contains all sets which don't contain themselves." Once they see that such a set is impossible (if said set did contain itself, then it wouldn't, and if it didn't, it would) then you simply can't imagine it. Until you recognize the contradiction, you can prepare you mind to imagine a square circle.

You can imagine exactly the parts that don't contradict each other. Your ignorance of the identity of the object allows you to imagine what about it you do know. Once you discover the aspects of the identity that are self-contradictory, you can't imagine it anymore. You could never imagine the contradiction, but could imagine something like the object in some respect without the contradiction. This ability to "sort-of" imagine the object is what lends credibility to the idea that the object (or set) actually exists. It is only on this basis that one could think that a contradiction exists (and therefore the field where the contradiction exists is invalid). The important thing to keep in mind is that it is exactly the contradictory elements (once recognized as such) that can't exist in one's imagination.

As to GIT more specifically

Wouldn't a system be entirely solvable given that some self-evident axioms exist? Existence is such a system. See Aristotle for at least one axiom's validation. B)

This has been touched upon but, notice that in order for GIT (or any theory) to be logically valid, logic must be valid. Any theory claiming the impossibility of certainty, objectivity, etc. are self-refuting as they make certainty of, objectivity about, or knowlege of themselves impossible.

Eric Lanser

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  • 1 year later...

Has anyone read this book? It is suggested reading in the curriculum I will be entering next year, so I want to get a start on it.

Each semester I like to read several books and connect ideas I learn from the books with course material. I am about to order my personal reading material from amazon, and I would like to see what books you all recommend (they don't need to be Objectivism).

So far I am getting:

OPAR

Zen and the Art of Motorcycle Maintenance (My father has been telling me to read this book for years, I might as well check it out)

Godel, Escher, Bach: An Eternal Golden Braid

I am taking a double major next year in a program called Minds and Machines. The program entails one study area for the mind (I am doing psychology with an emphasis on neuroscience), and one area of study for the Machines part (I am doing electrical engineering). I will be very gracious for any books you can suggest that would corroborate well with my areas of interest.

thanks!

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Douglas Hofstadter is very intelligent and is a wonderful writer but his ideas are very, very wrong. You should get familiar with his work considering the field that you are entering but, as I have recommended to you elsewhere, be wary...

Zen and the Art of Motorcycle Maintenance is trash if you want my opinion. Unlike Hofstadter, Pirsig is not a good writer and the philosophy that he puts in his novels is piecemeal and schizophrenic.

I wouldn't really recommend anything other than a good neuroscience book. Cognitive science has no good analysis since everyone basically agrees about the same fundamentally wrong view of consciousness. For instance, "Minds and Machines" may as well be "Apples and Rocks" but, as you will see, this comparison has basically sat unchallenged since it was first formulated. Indeed, comparing the mind to a computer has been around as long as computers have. If you really pressed me, I would recommend John Searle's Rediscovery of the Mind with reservations.

More than anything else, you should read as much Objectivism as you can. Without a philosophical shield, your mind will be penetrated with all kinds of wrong premises especially in your case with the major that you are taking.

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Has anyone read this book? It is suggested reading in the curriculum I will be entering next year, so I want to get a start on it.

Each semester I like to read several books and connect ideas I learn from the books with course material. I am about to order my personal reading material from amazon, and ...

Be sure to order from Amazon through the Amazon box on the forum page of this website, to benefit ObjectivismOnline!

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Douglas Hofstadter is very intelligent and is a wonderful writer but his ideas are very, very wrong. You should get familiar with his work considering the field that you are entering but, as I have recommended to you elsewhere, be wary...

Zen and the Art of Motorcycle Maintenance is trash if you want my opinion. Unlike Hofstadter, Pirsig is not a good writer and the philosophy that he puts in his novels is piecemeal and schizophrenic.

I wouldn't really recommend anything other than a good neuroscience book. Cognitive science has no good analysis since everyone basically agrees about the same fundamentally wrong view of consciousness. For instance, "Minds and Machines" may as well be "Apples and Rocks" but, as you will see, this comparison has basically sat unchallenged since it was first formulated. Indeed, comparing the mind to a computer has been around as long as computers have. If you really pressed me, I would recommend John Searle's Rediscovery of the Mind with reservations.

More than anything else, you should read as much Objectivism as you can. Without a philosophical shield, your mind will be penetrated with all kinds of wrong premises especially in your case with the major that you are taking.

thanks for the reply, I will definetely check that book out.

I also plan on focusing most of my energy on Objectivism. I believe the best way to tackle this is to read OPAR while reading the other material. That way I can connect ideas and see how they relate to Objectivism.

I will be reading Pirsig's book regardless, as it has certain sentimental values to me. It makes me happy to see me father happy, because for a long-time he was depressed that his children all "failed." My mother and stepfather brainwashed me and my brothers to believe my father was evil. Later I found out he lived out of his van so he could afford the child support my stepfather was using to support his drug habbit. When I ended up in jail, and my brothers were following the same path, I realized how much pain we put him in, and I want him to die a happy man. Now I do whatever I can to make sure he will live the rest of his life with a strong, relationship with his children. You see, even though I don't share his enthusiasm with that book, he found it helpful when he was living in a world of shit, so I would at least like to see how it helped him.

My degree program will not be focusing on computer and brain analogy's. I am focusing on designing very efficient computer/human interfaces. It is now common knowledge that neurons and silicon chips can interact back and forth with eachother (See Fromherz's research from the Max Planck Institute). My primary goal in this degree is to see the relationship to which we can connect biological systems with electronic systems.

Here is the program if you are interested http://www.cogsci.rpi.edu/minds_machines/

BurgessLau: I will make sure I order through this website :pimp:

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Your course of study is exactly what I think I want to do as a career.

I am currently studying electrical engineering at Drexel University, a coure I choose because of my interest in Robotics. I have experienced an increasing desire to see robotic systems integrated with the human nervous system.

I am currently undecided about whether or not I will choose to dedicate my life to this. If I do, perhaps our paths will cross somewhere down the line.

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Your course of study is exactly what I think I want to do as a career. 

I am currently studying electrical engineering at Drexel University, a coure I choose because of my interest in Robotics.  I have experienced an increasing desire to see robotic systems integrated with the human nervous system. 

I am currently undecided about whether or not I will choose to dedicate my life to this.  If I do, perhaps our paths will cross somewhere down the line.

Imagine that, you are the first person I have met that shares this interest. This profession is very small, and I am quite sure that if you choose to enter this field we will meet some day, either through lectures or research. I am quite the proactive person, and where ever this technologies epicenter seems to be - I will flock to that area to indulge myself in the available information.

I am fully convince the potential exists to take the advantages of man and machine and create a stronger organism. The first biomachine is just over the horizon IMO. To bad we will probably be dead by the time all the really interesting events in this field start taking place, such as the consequences of developing the technologies.

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Douglas Hofstadter is very intelligent and is a wonderful writer but his ideas are very, very wrong. You should get familiar with his work considering the field that you are entering but, as I have recommended to you elsewhere, be wary...

I think you give Hofstadter too much credit. The book of his that lc is asking about here is one of the worst books I've ever read--not just in terms of bad ideas, but also the ridiculous writing style. The fact that his book garnered any critical praise is worthy of the "Horror File" in my opinion.

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I wasn't thinking of Escher in particular but some of his more recent writings. I don't give him a lot of credit but I would much rather read him than many many other philosophers.

I haven't read any of his more recent stuff. Escher was bad enough that I wouldn't care to read any of his other work.

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  • 2 months later...

Godel's Incompleteness Theorem states that, given a collection of axioms powerful enough to describe arithmetic, there are statements describable in the language of the system which are true, but not provable in terms of the axioms.

A lot of my math buddies are really upset by this, since they're used to "truth" and "provability" being the same thing-- that is, if something isn't true, you can find an instance of it being false (a counterexample). Some even go so far as to say that it disproves the Law of Excluded Middle.

If Godel's Result applied to reality, I might agree with them. However, it does not. Reality isn't based upon a set number of axioms-- you can't "derive" reality from just "A is A" (which is a common misconception that I struggled with for a long time). Sense perception is also axiomatic, and there's no limit to the kinds of things that one can observe. So trying to apply Godel's Theorem to reality is just being Rationalistic.

So, I actually take GIT as a nice refutation of Rationalism-- there is no guarantee that you can prove every single statement to be true or false with a limited number of assumptions (axioms), and truth and falsehood has meaning outside of the mathematical system in which one happens to observe them, i.e., it has meaning in reality. You may have to include outside information to determine whether something is true or false, even if you can describe it in the language of the system.

Needless to say, GIT does not prove the existence of God.

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Nice points, Nate.

My favorite quote from Goedel is in the German edition of his writings (Werke, book 1, pg. 369), where, referring to his own work, Godel says that he "had not established any boundaries for the powers of human reason, but rather for the possibilities of pure formalism in mathematics." Very revealing.

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Related thread: Liar Paradox

My favorite quote from Goedel is in the German edition of his writings (Werke, book 1, pg. 369), where, referring to his own work, Godel says that he "had not established any boundaries for the powers of human reason, but rather for the possibilities of pure formalism in mathematics." Very revealing.

That is indeed a great quote.

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That is indeed a great quote.

Most people do not read Goedel beyond his most notable papers, and what he reveals about himself is sometimes astounding. Many do not realize that, aside from his work in logic and mathematics, Goedel also wrote five papers on relativity; three of which were published in his lifetime, and two previously unpublished essays now appearing in Goedel's "Collected Works."

Goedel's interest in relativity was sparked not only by his growing friendship with Einstein in the mid and late 1940s, but mainly by Paul Schlipp's request in 1946 to have Goedel contribute an essay for the volume Schlipp had planned in honor of Einstein's seventieth birthday. The original title Goedel had selected for the essay was "The Theory of Relativity and Kant." (!)

Goedel became completely absorbed with relativity, and his continually broadening investigation led to his first paper on the rotating universe, An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation, Reviews of Modern Physics, 21, pp. 447-450, 1949. In addition to the essay Goedel finally prepared for Schlipp's volume on Einstein, Goedel wrote an expanded paper on rotating universes, published in 1952, Rotating Universes in General Relativity Theory, Proceedings of the Internatonal Congress of Mathematicians, Cambridge, Mass., pp. 175-181. Whereas Goedel's 1949 paper was geared towards phsyicists, the 1952 paper was addressed to mathematicians.

To give some idea of just how obsessed Goedel became about relativity, cosmology, and his rotating universe, in his 1952 paper he states that "a directly observable necessary and sufficient criterion for the rotation of an expanding spatially homogeneous and finite universe" was that "for sufficiently great distances there must be more galaxies in one half of the sky than in the other half."

Indeed, after Goedel's death, two bound notebooks were found in which Goedel had recorded the angular orientation of galaxies. Clearly Goedel had made an initial effort towards providing what he thought was the evidence for rotation which he wrote about in his 1952 paper. And, according to Freeman Dyson, for many years, almost up to his death, Goedel would call Dyson to ask about any new observational evidence for rotation. Dyson would always tell Goedel there was no such new evidence, but Goedel still did not give up the possibility for his rotating universe. (It's not bad enough that the universe is supposed to expand, but it is supposed to rotate too? :dough: )

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That is indeed a great quote.

Most people do not read Goedel beyond his most notable papers, and what he reveals about himself is sometimes astounding. Many do not realize that, aside from his work in logic and mathematics, Goedel also wrote five papers on relativity; three of which were published in his lifetime, and two previously unpublished essays now appearing in Goedel's "Collected Works."

Thanks for this information. I was having a discussion with an aquaintance about this subject and this info will come in handy. I'd like to say for the record that I read every one of your sceince posts and look forward to each day's new ones with enthusiasm. Your knowledge is near encyclopedic. I really wish that you would write a book on physics and offer an Objectivist foundation for the discipline. In time you could be ranked up there with the very men you study and admire.

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Thanks for this information. I was having a discussion with an aquaintance about this subject and this info will come in handy. I'd like to say for the record that I read every one of your sceince posts and look forward to each day's new ones with enthusiasm. Your knowledge is near encyclopedic. I really wish that you would write a book on physics and offer an Objectivist foundation for the discipline. In time you could be ranked up there with the very men you study and admire.

I very much appreciate your thoughts. Thank you.

As I mentioned in this post, I am starting out with a book on Einstein and relativity.

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  • 1 month later...
Godel's theorem has been disproved.  Here is the link:

Ryskamp, John Henry, "Godel's Theorem Disproved" (January 19, 2005). http://ssrn.com/abstract=651382

Wow. I'm impressed. In only four sentences you disproved "Goedel's Theorem." And this Goedel disproof is even more succinct than your recent disproval of special relativity. :) What's next on your disproof agenda? How about disproving Euclidean or Riemannian geometry?

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Wow. I'm impressed. In only four sentences you disproved "Goedel's Theorem." And this Goedel disproof is even more succinct than your recent disproval of special relativity.  :)  What's next on your disproof agenda? How about disproving Euclidean or Riemannian geometry?

Wow, that was really hilarious. I started laughing when reading the "In only four..." sentence, and it just got better after that...

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