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Sir Llama

Godel Escher Bach

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

True hilarity is his "refutation" of special relativity. He does so by reference to his linguisitic interpretation of two words in a popular book that Einstein wrote for public consumption! Truly amazing. The postmoderns strike again! He recently spammed the "Physics Forums," dredging up old posts and dropping his load. Let's hope the moderators here do not let him run rampant with his bizarre nonsense.

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

True hilarity is his "refutation" of special relativity. He does so by reference to his linguisitic interpretation of two words in a popular book that Einstein wrote for public consumption! Truly amazing. The postmoderns strike again! He recently spammed the "Physics Forums," dredging up old posts and dropping his load. Let's hope the moderators here do not let him run rampant with his bizarre nonsense.

Refutation? I doubt it, from people who have read neither the Einstein book nor Garciadiego. Who's laughing now?

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Start with the initial claim: "[the incompleteness theorem says] that logical systems cannot explain everything because there are an infinite amount of paradoxical statements where truth cannot be ascertained".

That is nothing remotely like what the incompleteness theorem says.

Next is that I don't know of any competent argument that the incompleteness theorem refutes atheism. 

Best bet is Torkel Franzen's highly recommended non-technical-friendly book 'Godel's Theorem: An Incomplete Guide To Its Use And Abuse'. 

http://www.ams.org/notices/200703/rev-raatikainen.pdf

Edited by GrandMinnow

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18 hours ago, GrandMinnow said:

Start with the initial claim: "[the incompleteness theorem says] that logical systems cannot explain everything because there are an infinite amount of paradoxical statements where truth cannot be ascertained".

That is nothing remotely like what the incompleteness theorem says.

Next is that I don't know of any competent argument that the incompleteness theorem refutes atheism. 

Best bet is Torkel Franzen's highly recommended non-technical-friendly book 'Godel's Theorem: An Incomplete Guide To Its Use And Abuse'. 

http://www.ams.org/notices/200703/rev-raatikainen.pdf

Thank you!  The review itself reveals much of the abuses I had suspected.

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While I can't speak to the book that was reviewed (in the link provided by GrandMinnnow) the reviewer is incorrect in putting forth the idea that Godel was only narrowly addressing mathematics.  Godel was a logician, and chose arithmetic because it is (or was) a, supposedly, very simple system upon which so many others rest (not only mathematics, but also Mechanics).  But by demonstrating his ideas with regards to arithmetic, he was ,in fact, indicting an entire approach to, not only mathematics, but science, philosophy, physics, language, etc., that was prevalent at the time.

Godel was very much in the thick of the Vienna Circle/Logical Positivism, and the idea of Verificationism.  But what he demonstrated was the flaws behind this approach to philosophy (i.e. that all philosophical problems can be reduced to problems of language as a system).

I was searching for an on-line copy of an essay by Jacob Bronowski (a mathematician and a contemporary) found in the book A Sense of the Future, in which he explains in detail the relationship of Logical Positivism and Godel, and the ideas surrounding the Vienna Circle.  I couldn't find a link to the essay, but a Googel search of Jacob Bronowski/Godel did come up with something that might interest Objectivists.

(As an aside, I would highly recommend reading J. Bronowski's book The Western Intellectual Tradition.)

Edited by New Buddha

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11 hours ago, New Buddha said:

Godel was only narrowly addressing mathematics

But what he succeeded to "prove" was specific to mathematical systems of logic.  No?  How can you prove his "proof" actually applies to anything else?

For example, Mechanics and physics do not encompass, and are not properly directed to anything like statements about the truth of statements, evaluation of truth of statements, self referential statements and such oddities.  An equation in physics ultimately has referents only in reality and is supposed to be a statement about physical reality which is valid if true invalid if wrong.   Physics as a science then has nothing to do with the kinds of "truths", "logic", and "self-reference" that are relevant to Gödel's proofs.

Edited by StrictlyLogical

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35 minutes ago, StrictlyLogical said:

For example, Mechanics and physics do not encompass, and are not properly directed to anything like statements about the truth of statements, evaluation of truth of statements, self referential statements and such oddities.

You are addressing the issue from a historical vacuum. The Positivists of the 1800's would very much disagree with your above statement.  And this is the historical and contemprary context in which Godel was working.

Ernst Mach (The Science of Mechanics )was hugely influential on the development of Logical Positivism (although I would argue that the LP's misinterpreted him - see The Realistic Empiricism of Mach, James and Russell).  Barnes and Noble typically has a copy of the Science of Mechanics book.  But without the proper training, most of it will be meaningless to you.  But there are some interesting issues raised in the appendix.  Mach is also the one that Einstein credits for the seminal idea of Relativity.

 

Edited by New Buddha

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You have not persuaded me that Gödel showed anything applicable outside of mathematics, can you provide a concrete example?

We are all allowed to make bald assertions here, but for now I will trust the reviewer.  That said I'd be interested to hear what you concretely have to say about this.

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(1) Maybe I missed it, but I don't see in that review a claim that Godel was addressing only mathematics. Whether or not Godel had in mind subject matter other than mathematics when he devised the proof of the incompleteness theorem, the article (as far as I an tell from skimming it after having read it only long ago) reports only Franzen's own views of the import of incompleteness and even there, in that review, it is not claimed that Franzen entirely ruled out that incompleteness may have import other than in mathematics - rather only that certain claims of a certain kind of non-mathematical import do not hold up (I would have to reread Franzen's book to see whether this extends beyond what is said about him in the review to also what he wrote in the book).

(2) Just to be clear, pretty soon, Godel moved decidedly away from the Vienna Circle. (I would have to refresh my memory by looking up what his notions in that regard where at the time of the incompleteness proofs.)

 

 

Edited by GrandMinnow

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I would like to see specific quotes from Godel's writings, or at least specific quotes from credible writers about Godel, in which logical positivism is credited as a source for the incompleteness theorem.

And I am very suspect of Saint-Andre's claims (1) that Wang said the incompleteness theorem pertains to ANY [all caps added] consistent formal theory of mathematics or that NO [all caps added] formal mathematical theory can be both consistent and complete (though maybe Wang did allow himself to make these gross oversimplifications) (2) that Wang said that Godel claimed his remarks about society are a generalization of the incompleteness theorem

Meanwhile the essay includes this doozy of a piece of flat out misinformation:

"for most of his life Gödel did not continue to work in logic and the foundations of mathematics":

That is hilariously wrong. You might as well say that after 'The 39 Steps' Hitchcock pretty much stopped making films. 

 

Edited by GrandMinnow

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1 hour ago, StrictlyLogical said:

But what he succeeded to "prove" was specific to mathematical systems of logic.

Why do you put 'prove' in quotes? The proof of the incompleteness theorem is an unassailable mathematical proof; it can carried out with assumptions and inference means no greater than those of computational arithmetic itself. 

Edited by GrandMinnow

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This may help (I'm using 'finitistic' conveniently, not necessarily Godel's or Hilbert's own terminology): 

Hilbert hoped (indeed, expected) that there would be a finitistic proof of the consistency of analysis. 

Godel started work to devise such a proof. But in doing that, he realized that instead he could come up with a finitistic proof that if arithmetic is consistent then it is incomplete (perforce that if analysis is consistent then analysis is incomplete) and moreover this provides a finitistic proof that if arithmetic is consistent then there is no finitistic proof of the consistency of arithmetic (perforce that if arithmetic is consistent then there is no finitary proof of the consistency of analysis or set theory), thus that Hilbert's hope (indeed, expectation) was destroyed.  

Edited by GrandMinnow

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39 minutes ago, GrandMinnow said:

Why do you put 'prove' in quotes? The proof of the incompleteness theorem is an unassailable mathematical proof; it can carried out with assumptions and inference means no greater than those of computational arithmetic itself. 

I am unsure of the "kind" of logic he used to prove the conclusion.

 

If his logic is divorced from reality in the same way which creates this "paradox": https://en.wikipedia.org/wiki/Paradoxes_of_material_implication#Paradox_of_entailment

I cannot know what is meant by the term "proof", and I stand by my use of quotes, and mean no disrespect by them.

That said, IF I HOLD that all knowledge at its base is reality, and all abstraction for the purpose of knowing that which is (rather than the arbitrary or that which we wish), and there are no paradoxes in reality, then what constitutes "logic" and "proof" to me (it sounds like I'm being subjective... but I'm being diplomatic) is different from what certain types of mathematical "logicians" (and yes I use quotes again) hold as "logic" and "proof".

 

A certain type of logician holds inconsistent premises imply any conclusion.

I (and perhaps certain other types of logic) hold that inconsistent premises imply .. nothing.

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I can't go into the technical details, but in basic terms, the assumptions and logic in the proof can be reduced to nothing more than those of computational arithmetic. Not only is the proof within ordinary mathematics, it's within an even more restrictive criteria: constructive and finitistic. 

To question the methods of the proof would be tantamount to questioning the methods that make your computer do what it does or even merely those of algorithms for computations on plain counting numbers.

As to the explosion principle (a contradiction implies any statement), I don't necessarily want to go into another discussion on it (I've discussed it so many times on forums that it is tedious now), but it too is basic Boolean logic. One can propose non-explosive logics (logicians do study that also), but impugning ordinary mathematics - via certain philosophical or even everyday concerns - really misses the point.

As to the paradox of material implication, we can dispense it easily: Let 'P->Q' be merely an abbreviation for '~(P & ~Q)'. I.e. 'if P then Q" (in the specific context of sentential logic, which itself is merely a variation of basic Boolean logic) is merely a way of saying "It is not the case that P is true while Q is false". The use of this in sentential logic is no more than plain Boolean logic. It's what is used for the programs that make your computer do what it does. One can have whatever philosophical objections to such logic, but then it is odd that one doesn't object to it when it makes your computer do what you tell it to do.

I mean, when you do a search for results on, say, "NOT(Washington and NOT-Jefferson)" as saying "Any hit you give me on Washington must also be a hit on Jefferson", you don't have philosophical objections to that Boolean logic, right?*

*To be fair, ordinarily that search would be an odd one to conduct, since it would bring up every hit that does not even include Washington. But my point is that, while not necessarily of much use, it isn't logically prohibited to want to see only results in which either Washington does not occur or both Washington and Jefferson do occur. 

A better example: For my business, I want a combined list of all my customers who don't order shovels with all my customers who order both shovels and gloves. That is merely "If S -> G", and has the paradox of material implication with it. 

 

 

Edited by GrandMinnow

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1 hour ago, GrandMinnow said:

Meanwhile the essay includes this doozy of a piece of flat out misinformation:

"for most of his life Gödel did not continue to work in logic and the foundations of mathematics":

That is hilariously wrong. You might as well say that after 'The 39 Steps' Hitchcock pretty much stopped making films. 

 

No.  You are wrong.  I can sort multiple sources (all of which I'm sure you will say are not credible), but from Wiki:

Gödel became a permanent member of the Institute for Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.[21]

During his many years at the Institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed time-like curves, to Albert Einstein's field equations in general relativity.[22] He is said to have given this elaboration to Einstein as a present for his 70th birthday.[23] His "rotating universes" would allow time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric (an exact solution of the Einstein field equation).

2 hours ago, GrandMinnow said:

(2) Just to be clear, pretty soon, Godel moved decidedly away from the Vienna Circle. (I would have to refresh my memory by looking up what his notions in that regard where at the time of the incompleteness proofs.)

Godel was never really a member in "good standing" with the Vienna Circle.  He was pretty much on the preiphery.  Meaning that he knew his ideas conflicted with Verificationism.  Thus I said that he was "in the thick of the Vienna Circle and Logical Positivism", and not that he was a Logical Positivist.  And, again, I can quote multiple sources, but I'm sure that you will find them not credible.

Edited by New Buddha

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2 hours ago, StrictlyLogical said:

You have not persuaded me that Gödel showed anything applicable outside of mathematics, can you provide a concrete example?

You are missing the point.  Positivism/Logical Positivism, which was the dominate school of thought in Godel's time (roughly speaking, since Comte) and was shown by Godel to be methodologically wrong.  And P/LP was not just limited to mathematics.  It's methodology was supposed to explain everything: mathematics, history, psychology, economics, aesthetics, mechanics, politics, sociology, etc.

Edited by New Buddha

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So what's the big deal about the incompleteness theorem anyway?  I am under the impression that the particular set of "truths" that are left out are self referential... essentially empty statements with no referent.

Such statements that state nothing about anything (other than its own empty self) are probably useless, if reality is the goal of knowledge, the standard of all truth, and the referent and purpose of abstraction.

Is this an unfair evaluation?

 

http://rationalwiki.org/wiki/Essay:Gödel's_incompleteness_theorem_simply_explained

 

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(1) The incompleteness theorem was in 1930 (1931?). That's well before 1946. Godel made major contributions in logic (cf. the Stanford Encyclopedia article) well after 1930. Also, foundations of mathematics includes philosophy regarding mathematics. That later Godel turned a lot of his attention to physics and philosophy does not contradict that he did important work in logic, mathematics, and in foundations of mathematics well after the incompleteness theorem. Moreover, I would need to check on the first publication dates, but Stanford cites papers in logic as late as 1970.

(2) I didn't dispute that Godel was not an avid proponent of the ideas of the Vienna Circle. Indeed, it was my point that whatever affinity he might have had for the Vienna Circle, he soon enough moved on from it.

 

 

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1 minute ago, New Buddha said:

You are missing the point.  Positivism/Logical Positivism, which was the dominate school of thought in Godel's time (roughly speakjing, since Comte) and was shown by Godel to be methodologically wrong.  And P/LP was not just limited to mathematics.  It's methodology was supposed to explain everything: mathematics, history, psychology, economics, aesthetics, mechanics, politics, sociology, etc.


Whatever the merits of those remarks, they don't at all evidence that Godel devised the incompleteness proof as a way to refute logical positivism or any other philosophy or to advance any particular philosophy. 

As I mentioned, the incompleteness proof came from Godel's effort to address a certain mathematical problem.

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1 minute ago, GrandMinnow said:

(1) The incompleteness theorem was in 1930 (1931?). That's well before 1946. Godel made major contributions in logic (cf. the Stanford Encyclopedia article) well after 1930. Also, foundations of mathematics includes philosophy regarding mathematics. That later Godel turned a lot of his attention to physics and philosophy does not contradict that he did important work in logic, mathematics, and in foundations of mathematics well after the incompleteness theorem. Moreover, I would need to check on the first publication dates, but Stanford cites papers in logic as late as 1970.

(2) I didn't dispute that Godel was not an avid proponent of the ideas of the Vienna Circle. Indeed, it was my point that whatever affinity he might have had for the Vienna Circle, he soon enough moved on from it.

Every time I make a point, you just ignore the source or move the goal post.  It's tiring.

Godel was disenchanted with the way that his incompleteness theorem was being bastardized and misinterpreted.  This prompted his focusing on other fields.   Something which you say is a "doozy" and minsinformation.

How many times can you be wrong and not admit it?

 

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10 minutes ago, GrandMinnow said:


Whatever the merits of those remarks, they don't at all evidence that Godel devised the incompleteness proof as a way to refute logical positivism or any other philosophy or to advance any particular philosophy. 

As I mentioned, the incompleteness proof came from Godel's effort to address a certain mathematical problem.

Again, a straw man argument.  I never claimed that he did.  He was working with the dominate idea(s) of the time and reached a conclusion that conflicted with it.

Edit:  And many people misinterpreted it as a belief some how that all knowledge is flawed or incomplete.  This is what disenchanted him.

Edited by New Buddha

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"I am under the impression that the particular set of "truths" that are left out are self referential... essentially empty statements with no referent. " [Strictly Logical]

That is incorrect. The undecided statements may concern all kinds of matters in arithmetic. The undecided statement (the "G" statement) used to prove the theorem itself may be INTERPRETED as denying its own provability, but the statement itself is a plain statement about natural numbers. It's only by Godel's ingenious work that he hooks up the plain mathematical formula with the matter of unprovability. Again, the statement itself is a basic (though complicated) statement about natural numbers. Moreover, the second incompleteness theorem shows that such statements also evidence that a finitistic consistency proof is not possible. Moreover, by later developments, we find that there is no general solution to Diophantine equations, which is a very basic matter of interest to mathematics. (It's basically to say that there is no algorithm to determine whether an arbitrary equation of the nature as in a high school algebra class has a solution.) And further questions about arithmetic also shown undecidable.

It's a lot better to actually investigate the mathematics and context of the incompleteness theorem than to rely on woozy oversimplifications and actual mischaracterizations of it found in many Internet entries. 

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16 minutes ago, New Buddha said:

Every time I make a point, you just ignore the source or move the goal post.  It's tiring.

Godel was disenchanted with the way that his incompleteness theorem was being bastardized and misinterpreted. This prompted his focusing on other fields.   Something which you say is a "doozy" and minsinformation.

 

 

I have not moved any terms of the question. My replies are exact to each point. And I have not ignored your sources; on the contrary I've addressed them, such as with this one. 

Next, please cite your source that Godel went on to other questions in logic and mathematics because of his supposed fatigue with misunderstandings of incompleteness (it's possible that is true, but I am curious as to your source). 

Godel went on to the relative consistency of the axiom of choice and the continuum hypothesis. That was well after he proved incompleteness. And other research he did in the field. It is quite correct to say that the writer of he essay is plainly incorrect on subject. And there is no moving of goal posts by me in that.

Edited by GrandMinnow

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7 minutes ago, GrandMinnow said:

And there is no moving of goal posts by me in that.

I was replying to the below quote:

2 hours ago, GrandMinnow said:

Meanwhile the essay includes this doozy of a piece of flat out misinformation:

"for most of his life Gödel did not continue to work in logic and the foundations of mathematics":

That is hilariously wrong. You might as well say that after 'The 39 Steps' Hitchcock pretty much stopped making films. 

Now you want me to:

9 minutes ago, GrandMinnow said:

Next, please cite your source that Godel went on to other questions in logic and mathematics because of his supposed fatigue with misunderstandings of incompleteness (it's possible that is true, but I am curious as to your source). 

This is the very definition of "moving the goal post".  And, of course, once I site the source, you will (again) discredit it.

I'm getting tired of this, so I'm backing out of the post.

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