Godel's Theorem

I am curious about the implications of Godel's Theorem on philosophy. Please feel free to correct me if I am misinterpreting.According to my understanding of Godel's Incompleteness Theorem, any mathematically consistent supposedly self-provable system in mathematics must jump outside of such a system to remain provable. Once the jump outside of the system is made then the system changes, so in order to remain proven, it must jump outside again, which renders it unprovable, so one must jump outside again, and so on ad-inifitum.I have often heard that the Hisenberg principle shows that the observer affects the observed, because the position of a wave-particle cannot be ascertained simultaneously with its velocity. When treated as a particle, the position may be observed, when treated as a wave, the velocity may be observed, but not visa-versa. The actual math involved is pretty easy, it just falls right out of the wave equation.I find it unusual that, when critiqing the limits of scientific knowledge, that one only hears of the Hisenberg principle, that the observer affects what is observed, and never hears of Godel's Theorem, that no self-consistent system is self provable, when both speak to the limits of using reasoning, be it inductive (in the case of Hisenberg) or deductive (when in the case of Godel).I am just curious if I am wrong in my interpretation of either. I am also curious why Godel's theorem, which has more implications concerning the ultimate unknowability of the universe, is less well known and less utilized, than Heisenberg's Principle, when it comes to defining the limits of scientific knowledge.Considering this forum, I do not want to imply that because math has found math unprovable or that science has discovered the observer affects the observed, that this automatically makes all math and science heresy to be persecuted. I just think that if my interpretation is correct, the philosophy of materialism is under severe stress and positivism is dead.I am just curious. Please feel free to comment.{Edited for spelling and other blame-the-keyboard errors}This message has been edited by anglagard, 04-22-2006 01:41 AM

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"Considering this forum, I do not want to imply that because math has found math unprovable or that science has discovered the observer affects the observed, that this automatically makes all math and science heresy to be persecuted. I just think that if my interpretation is correct, the philosophy of materialism is under severe stress and positivism is dead."

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Heh! You spend three quarters of this paragraph saying what you are not going to do, and then you go and do it!In any case, given that this work was published in 1931, and in light of continued scientific developments since I'd say there's no need for us to give up on science and return to our caves just yet!As has been thrashed out countless times on this forum, natural science doesn't seek outright proof, but instead seeks to accumulate evidence that will provide the basis for predictions. If "proof" of evolution required that we found every fossil of every being that ever lived then we'd likely never be able to "prove" it.I don't see how Godel's theorem (interesting as it is) stops humanity from deriving scientific theorems from empirical observation.This message has been edited by rjb, 04-22-2006 06:00 AM

If I get a moment I'll come back to chat about Godel, but for now I just want to point out this common but erroneous view that Heisenberg's Principle places a limit on scientific knowledge.It is not that we cannot measure say position and momentum to arbitrary accuracy; it is the fact that at quantum scales, simultaneous knowledge of position and momentum is no longer a well defined concept.This message has been edited by cavediver, 04-22-2006 06:13 AM

Thread moved here from the Coffee House forum.

Four points:

1. Neither Godel's theorem nor Heisenberg's principle have anything to do with placing limits on scientific knowledge.
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2. Heisenberg's principle places limits on some very specific types of measurements, not on scientific knowledge.
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3. Godel's theorem is mathematical, and math, unlike the real world, is not influenced by the observer.
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4. Science and math differ in that in math everything must be proven while in science nothing can ever be proven.

--Percy

Quick comment: Goedel proved many theorems. I will take it that you are referring to the Goedel incompleteness theorem (or to one of them). Not a strictly correct statement.A better statement (I hope): For any formal system sufficiently large to contain arithmetic, there are statements that can be expressed in that system, which can neither be proved nor disproved. Such statements are usually considered undecidable.The theorem itself does not say anything about "jump outside".As a mathematician, I consider Goedel's incompleteness theorems to be technical results about the limitations of formal logic. It has no consequences for reality, and few (if any) consequences for the bulk of what mathematicians study. There isn't any relation between Goedel's theorem and the Heisenberg uncertainty principle, although confusion about this is common. Goedel's theorem has no consequences at all concerning the knowability or unknowability of the universe. Your interpretation is not correct. Goedel's theorem says nothing about materialism. Positivism may well be silly (but not yet dead), however Goedel's theorem is not in any way related to the problems of positivism.

Hmmm, I would definitely like to know the truth or falsity of the Continuum Hypothesis...And as a weak(ish ) Platonist, that certainly concerns my Universe!

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I would definitely like to know the truth or falsity of the Continuum Hypothesis...

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Heh. Design an experiment.I know people who have enough trouble accepting the Axiom of Choice.

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"Religion is the best business to be in. It's the only one where the customers blame themselves for product failure."
-- Ellis Weiner (quoted on the NAiG message board)

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According to my understanding of Godel's Incompleteness Theorem....

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As one member has already pointed out, Godel proved that in any logical system that can produce ordinary arithmetic, there are statements that can neither be proved nor disproved.He also proved that no logical system that can produce ordinary arithmetic can prove its own consistency.The first means that one cannot come up with a finite set of axioms by which all other mathematical statements can be proven or disproven.The second means that there is always the possibility that mathematics as we know it might be inconsistent. Probably not, but we cannot completely be sure of it.Unless you are a Platonist, though, these theorems have only limited implications for real life.-

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I have often heard that the Hisenberg principle....

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What the Heisenberg Uncertainty Principle actually does is tell us that there are certain questions that make no sense. Of course, this has always been true; "What color is angular momentum", of course, is a nonsensical question. The Heisenberg Uncertainty Principle simply adds to the number of nonsense questions; "What is the exact postion and the exact momentum of this particular electron at this particular time?" is another question that makes no sense (even though it seems reasonable to people who have not been trained in quantum mechanics -- and even to many people who do understand quantum mechanics).

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"Religion is the best business to be in. It's the only one where the customers blame themselves for product failure."
-- Ellis Weiner (quoted on the NAiG message board)

Isn't there something that makes the truth or falsehood of the continuum hypothesis relative to your mathematical standpoint?I'm dreadful at formal logic, set theory and anything remotely associated with Cantor, so I could be completely wrong.

It has been shown that the Continuum Hypothesis can neither be proven nor disproven in Zermelo-Fraenkel set theory. So you are allowed to either accept the Continuum Hypothesis as an additional axiom, or you may accept the negation. Potentially one could end up with very different systems of mathematics, although I am only familiar with mathematics accepting the Continuum Hypothesis.Is there any practicing mathematician who uses the negation of the Continuum Hypothesis in her work?

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"Religion is the best business to be in. It's the only one where the customers blame themselves for product failure."
-- Ellis Weiner (quoted on the NAiG message board)

Thanks Chiroptera.
Funnily enough I asked because I was reading something by Penrose which mentioned (in an aside) that some that the standpoint that certain people take the standpoint that it isn't true.
Which suprised me, I didn't know you could just reject it (or anything) because of your standpoint.All it mentions apart from this, is that those who do reject it tend to also reject the axiom of choice.This message has been edited by Son Goku, 04-22-2006 12:56 PM

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All it mentions apart from this, is that those who do reject it tend to also reject the axiom of choice.

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Heh. That's interesting. I do know of one serious mathematician (very well respected in differential geometry, too) who does not accept the Axiom of Choice. I never thought to ask his opinion on the Continuum Hypothesis.

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"Religion is the best business to be in. It's the only one where the customers blame themselves for product failure."
-- Ellis Weiner (quoted on the NAiG message board)

The Axiom of Choice is certainly not as sacrosanct as it once was. A pure math aquaintance of mine spent his PhD on topoi that excluded it. And some have conjectured that we need to consider the space of all topoi in the hunt for our Theory of Everything! But I have enough trouble just with ZF...

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The Axiom of Choice is certainly not as sacrosanct as it once was.

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God. I would hate to live in a universe without the Hahn-Banach Theorem.

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"Religion is the best business to be in. It's the only one where the customers blame themselves for product failure."
-- Ellis Weiner (quoted on the NAiG message board)