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I came across a simplified description of Gödel's theorem and the discussion touches on a concept of honesty (truth?) and completeness. How does Gödel's theorem apply to everyday interactions?

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    It does not. – Mauro ALLEGRANZA Nov 14 '14 at 19:59
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    @Mauro ALLEGRANZA - Could you explain why it does not? – Motivated Nov 15 '14 at 4:36
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    @Motivated First off, because Goedel's incompleteness theorems only apply to axiom systems that are powerful enough to express first-order arithmetic. This is a significant restriction and is invariably omitted from pop-sci invocations of the theorems. – David Richerby Nov 15 '14 at 19:03
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    Turing's proof of the unsolvability of the halting problem is essentially a computational variant of Gödel's first incompleteness theorem. There's a related question about the practical importance of the halting problem on Computer Science Stack Exchange. – Ilmari Karonen Nov 16 '14 at 1:02
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    @Motivated Google is your friend, or ask a separate question. A 500-character comment is way, way too short to answer that. – David Richerby Nov 16 '14 at 2:03
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It may never affect your everyday life, but it has weakened our trust in rigid logical methods, as a culture. If even mathematics cannot attain to this kind of complete coverage of a domain, there is a good reason to think we habitually overvalue the role of rules in science.

I think that the shift toward seeing more of the human side of scientific inquiry, and admitting that it is deeply affected by personal faith, was unchained by the brake this kind of result put on logical positivism.

It is in effect the first post-modern fact. Even if you don't go down the whole trail of postmodernism, it keeps the bug in your ear that says absolute modernism strives for more than can be realistically attained. Sociology, faith, human nature, etc. really do matter in the end, and will not just be steamrolled by the sheer power of any system.

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    I like this, but I think you are attributing way too much to Goedel and the sinking of logical positivism by saying these have in a significant way "weakened our trust in methodologies as a culture", or that they have all that much to do with the rise of postmodernism, although it is an interesting parallel (which is why I like this). – selfConceivedAsEvil Nov 14 '14 at 22:37
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    Do you have a citation for the claim that Goedel's theorems have "weakened our trust in methodologies"? – David Richerby Nov 14 '14 at 23:29
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    Put another way, one could draw up a list including e.g., Freud, Heisenberg, Husserl, Kafka, etc. and claim a similar historical significance for what you've attributed purely to Goedel and Turing. – selfConceivedAsEvil Nov 15 '14 at 2:49
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    Unfortunately, an understanding of Godel's incompleteness requires a fairly deep understanding of axiomatic systems, which renders the vast majority of philosophers functionally incompetent as far as being able to say anything substantive on the matter. Hell, even I don't understand it, nor do I feel safe making claims about what it does or doesn't say, and I'm an MIT alum. There is a sad history of cranks and loons wrongly attributing a variety of outlandish claims to Godel's theorems, and I suspect that the work of most non-mathematicians on the topic will fall into this category. – DumpsterDoofus Nov 16 '14 at 17:27
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    The Gödel result did not put a brake on logical positivism. That's not at all reflective of the history of positivism. See, e.g. Ronald N. Giere, “From WissenschaftlichePhilosophie to Philosophy of Science.” – ChristopherE Nov 17 '14 at 1:45
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Here's what Jordan Ellenberg, a professor of mathematics at the University of Wisconsin, has to say about this topic in his Does Gödel Matter? article:

What is it about Gödel's theorem that so captures the imagination? Probably that its oversimplified plain-English form—"There are true things which cannot be proved"—is naturally appealing to anyone with a remotely romantic sensibility. Call it "the curse of the slogan": Any scientific result that can be approximated by an aphorism is ripe for misappropriation. The precise mathematical formulation that is Gödel's theorem doesn't really say "there are true things which cannot be proved" any more than Einstein's theory means "everything is relative, dude, it just depends on your point of view." And it certainly doesn't say anything directly about the world outside mathematics, though the physicist Roger Penrose does use the incompleteness theorem in making his controversial case for the role of quantum mechanics in human consciousness.

So the short answer to your question seems to be that it doesn't, and that extreme care should be taken not to misuse or misrepresent the theorems.


Edit: given the high number of upvotes this answer has received, I should point out that I'm by no means an expert on the subject, and that an alternative, more in-depth explanation by someone who knows more would be highly appreciated.

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    The more scientists claim the world is mathematical, the more Gödel's theorem has to say about the world. Typically, when scientists tell us "what the world is really like", they depend on mathematics. This holds most stringently for those who predicate statements on the extreme precision of quantum field theory, which is almost entirely mathematics. – labreuer Nov 15 '14 at 1:43
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    As you're no doubt aware, scientists depend on math to quantify phenomena; one can understand "what the world is really like" - e.g. have a notion of atoms, genetics, galaxies, light polarization, photoelectric effect etc. - without using any math at all, or still not understand much despite complex, useful math. At any rate, I agree that math can obviously play a large part in daily lives, but Gödel's theorems refer to certain formal systems only, so I believe your first statement is too general - one needs to be very specific how and why his theorems may apply (cont.) – w128 Nov 15 '14 at 2:26
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    (cont.) I.e. just because math is involved it doesn't mean that Gödel's theorems are inherently relevant, as explained in the article I cited. Besides, phenomena such as quantum field theory don't seem to be what the OP's post is referring to, i.e. truth in daily lives. But indeed it would be interesting to see in what kind of mathematical models Gödel could be used legitimately ... – w128 Nov 15 '14 at 2:27
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    @labreuer Theoretical physics is a system that uses arithmetic; Goedel's incompleteness theorems apply to systems that can express first-order arithmetic. – David Richerby Nov 15 '14 at 19:10
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    @jobermark If you can express second-order arithmetic, you can certainly express first-order arithmetic. But the use/express distinction is real and crucial. Sure, e.g., special relativity is expressed using arithmetic but, for Goedel to apply to special relativity, SR would have to be able to express proofs about arithmetical facts. So, for example, you'd have to be able to translate formulae of first-order arithmetic into physics experiments that would determine the truth/falsity of the formulae. – David Richerby Nov 16 '14 at 18:27
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For those who are interested in theoritical automaton and the philosophy of computation, the converse of church turing thesis plays a nice interesting effect on godel's limitations.

Check this out:http://blogs.msdn.com/b/gpalem/archive/2007/02/02/establishing-the-existence-of-uncountable-number-of-accelerated-turing-machines.aspx

  • For those who gave negative rating, look around you, all the computers you are seeing in "real life" are based on the theory of automaton, which godel's theorem applies to in a straight forward manner. The question, loosely stated, is: "can everything be automated". And the best definition of what is meant by "automated", is given by "Turing's notion of computation". Now, go back and refer to that blog article and you would see why it has direct relevance to human life. And if someone still believes the link is irrelevant, good luck :) – Gopalakrishna Palem Nov 16 '14 at 14:24
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An almost real-life example of the simplified explanation you've referred to could be procedures in a huge corporation, if they are complex enough. Imagine a procedure:

A procedure that doesn't comply with The Company's mission must not be followed

Now, imagine a coffee-drunk, inexperienced employee at 5AM accidentally modifies company's mission statement by adding this sentence:

The Company doesn't allow procedures with description starting with the capital 'A'

Should now all the procedures that don't comply with company's mission (for example being obsolete, after earlier modifications of policy's mission) be followed or not?

This is of course an instance of the liar paradox. While this doesn't express the whole of Gödel's theorem it is closely related.

The described situation is not strictly real-life as it probably haven't occurred in reality :) However, systems of procedures may be viewed as formal systems, and when they become complex they often have problems with consistency and completeness.

  • Should there be controls to minimize such occurrences? I am curious what it has to do with the conversations and interactions humans have with one another. – Motivated Nov 15 '14 at 18:12
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    First, how is this situation "real life"? Second, what does it have to do with Goedel's incompleteness theorems? The first question is rhetorical. To answer the second one, you need to explain, among other things, how your example relates to axiomatic systems that are powerful enough to express first-order arithmetic. – David Richerby Nov 15 '14 at 19:02
  • @DavidRicherby: I expect the wording allowed in the imagined procedures is capable of expressing first-order arithmetic (given that it embeds enough English to describe any mathemetic procedure), and that it is capable of stating axioms which then lead to unprovable truths (which can also be stated). However, this answer is demonstrating a logical paradox instead. – Neil Slater Nov 16 '14 at 9:34
  • @DavidRicherby Thanks for your comment. I've addressed the problems you indicate in my edit. Hopefully it's at least a bit better now :) – BartoszKP Nov 16 '14 at 11:46
  • The liar paradox tells us that there are statements that are neither true nor false. Goedel in a nutshell is that there are true statements that have no proof. They don't really have a lot to do with each other. – David Richerby Nov 16 '14 at 11:48

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