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Sir Llama

Godel Escher Bach

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10 minutes ago, New Buddha said:

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.

There's no "again" a strawman, since there is no previous strawman, let alone a strawman here.

You are welcome to define or redefine your positions. If I have misunderstood your point as being that Godel came up with incompleteness to refute logical positivism or for other philosophical purposes (or even in reaction to logical positivism), then I accept that my best attempt failed to understand whatever it is that you're saying. On the other hand, I don't see how I could be very much faulted for that, for it does seem to be a fair reading of what you actually wrote.

So, again, if that is not what you meant, then fine; and meanwhile, my points stand onto themselves (most specifically that we don't have in this thread a citation that connects Godel's proof efforts regarding incompleteness with logical positivism or any philosophy) whether or not in dispute of yours. (Of course, we know that much later in Godel's philosophical development, there may be connections with his mathematical results.) 

 

 

 

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22 minutes ago, New Buddha said:

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.


Fair enough. So, as I understand now, your view is not necessarily that Godel intended incompleteness to conflict with a certain philosophy, but rather that in fact incompleteness does conflict with that philosophy.

So, then you would have to justify that claim. Again, you would then have to show how a philosophical argument is drawn from this particular mathematical proof. 

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Asking that someone cite a source is not "moving the goal posts".

And you can't justifiably presume that I would dispute the credibility of any source not yet given. I might dispute certain sources but not others (indeed, on certain other points, you've mentioned sources that I did NOT dispute).

 

 

 

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5 minutes ago, GrandMinnow said:

Again, you would then have to show how a philosophical argument is drawn from this particular mathematical proof. 

From what I've understood from Corvini, Knapp and other ARI estimates, their approaches have been to assess how philosophy influences the math, and then how the math, in turn, transmits that philosophical influence to the culture.

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So what is the argument that the proof of the incompleteness theorem (or the theorem itself) refutes logical positivism or any particular philosophy?

As to philosophy influencing math, what philosophical influence do you have in mind regarding the mathematical proof (which - in a certain basic sense - can be formulated within computational arithmetic)? 

Edited by GrandMinnow

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21 minutes ago, GrandMinnow said:

As to philosophy influencing math, what philosophical influence do you have in mind regarding the mathematical proof


The clearest articulation of it to me was via Corvini's 2, 3, 4 and all that addressing infinities in math in set theory. The premise is that the approach to infinities Cantor used shows up as paradoxes that arise in set theory. I'll leave this as an indication, as it is a departure from Godel.

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I don't know what you mean by departure from Godel (do you mean that Godel claimed a philosophy of realism regarding infinite sets?).

In any case, neither the incompleteness proof nor Godel's proof of it rest on any notion of infinity.

As to Cantorian set theory, it is not the notion of infinity that causes paradox. Rather, Cantor himself did not have a formal theory. Later formalizations of it avoid (as far as we know) inconsistency by not including unrestricted comprehension.

And I'll give up, for now asking, the unanswered question here as to how a refutation of logical positivism is drawn from the proof of incompleteness. 

Edited by GrandMinnow

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3 minutes ago, GrandMinnow said:

I don't know what you mean by departure from Godel (do you mean that Godel claimed a philosophy of realism regarding infinite sets?).

Departure from Godel, as in not directly related to the topic. I added that to try clarifying your inquiry of philosophic influence to math.

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Thanks for that. 

But, to be clear, my point is not to ask in general about relationships between mathematics and philosophy, but rather to ask the specific question how the incompleteness proof refutes (or conflicts with, whatever) logical positivism or, conversely, what evidence is there of a certain philosophical influence on Godel's proving the incompleteness theorem. 

Edited by GrandMinnow

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I do know that Piekoff devotes attention to proof in OPAR. I also know that math is generally considered as an epitome of proof.

I don't know much about Gödel's Incompleteness Theorems (or, more extensively, their proofs), but enough to recognize that proof has to be about substantiating what is posited by his Incompleteness Theorems, much as proof in Objectivism is about identifying the logical relationship(s) between a proposition and its corresponding evidence of the senses.

Here I would want to restate your specific question: How does the proof of the Incompleteness Theorems bolster as additional evidence or serve to contradict logical positivism? (This does presume that the proof of the Incompleteness Theorems has unimpeachable veracity, to which I cannot attest.)

The deeper question, still then, lies in asking: Does philosophy derive the methodology of proof from mathematics, or does mathematics derive its methodology of proof from philosophy?

If the latter, where would this place the status of the veracity identified as being pivotal (presumed unimpeachable) earlier?

Edited by dream_weaver
clarifications

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