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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?

wythagoras
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Motivated
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  • The matter is truth, not honesty. The idea is that with a sufficiently powerful formal system, it can either be (a) complete, and not 'know' it; or (b) 'know' its completeness and contain ≥ 1 contradiction. One way to describe this is that "truth is stronger than provability". – labreuer Nov 14 '14 at 18:22
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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 04:36
  • I'm not sure that Godel has had much of an impact at all on the even the daily lives of the vast majority of mathematicians. So far, the limitations he has put on their work have not been all that restrictive -- a small speed bump at most. Mathematics continues to grow by leaps and bounds. – Dan Christensen Nov 15 '14 at 06:19
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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](http://en.wikipedia.org/wiki/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](http://cs.stackexchange.com/questions/32845/why-really-is-the-halting-problem-important) on Computer Science Stack Exchange. – Ilmari Karonen Nov 16 '14 at 01:02
  • @David Richerby - Thanks David. What do you mean by 'axiom' systems? – Motivated Nov 16 '14 at 01:56
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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 02:03
  • "Today, tomorrow, the next day, the day after that..." Can we infer an axiom system powerful enough to express first-order arithmetic from this series of words? – Dave L. Nov 21 '14 at 03:04

3 Answers3

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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.

w128
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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 01:43
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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 02:26
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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 02:27
  • Hmmm, I'm still not convinced. I encounter claims to complete understanding of the universe in terms of some alleged formal system all the time. These formal systems virtually always include arithmetic and provability. I get the bit about how this flows to our daily lives, but when you have people arguing from determinism to concepts of moral responsibility to social policy, I ask whether somewhere along the line, a claim to completeness was made that _cannot possibly be known_. Well, unless we ourselves are composed of, e.g., non-RE axioms. Then Gödel would not criticize us. :-p – labreuer Nov 15 '14 at 02:42
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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
  • This basically says it **should** not have an effect, not that it **does** not have an effect. The intellectual effects that things like relativity, quantum dynamics and incompleteness had on people are real, whether or not this author considers them justifiable. –  Nov 16 '14 at 15:30
  • @DavidRicherby A theory that uses another theory freely can do all the things that theory does. Theoretical physics includes arithmetic. Any arithmetic fact is expected not to contradict a physical prediction. But it is far from first-order. Real analysis, a part of math physicists certainly require, immediately quantifies over relations just in defining a supremum symbol. So the theorem does not apply, but I would still not doubt it is incomplete... –  Nov 16 '14 at 18:01
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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
  • @DavidRicherby Do you really not think you can do that kind of physics experiement, at least as a gedankenexperiment? Maybe with computers that would build bigger computers whenever they ran out of power? For Goedel to apply here the **entire system would have to be first order**. Takeuti, and others still hold the hope second-order arithmetic (or at least Real Analysis) is consistent and complete. Goedel just never applies to real life. Period. Because we automatically jump over the place it matters, and right into the Reals. The effect he has on our everyday life is psychological. –  Nov 16 '14 at 18:49
  • @DavidRicherby For instance, physics might be considered consistent and complete if all undecidable statements of the physics explicitly referenced infinities. In fact, this is likely, right? Given Heisenberg, if space and time are topologically compact, we can run any approximation long enough that the error is lost in the required unknowable ranges. So if we rewrite the derivatives in wave equations as approximations, that is a model a lot of physicist would call complete. But first order logic cannot have an axiom that expresses the idea everything is finite. Second order logic can. –  Nov 17 '14 at 18:49
  • @w128 - Thanks. I accepted jobermark's answer as it was more easily understood by me. It is by no means a reflection of your answer. – Motivated Nov 18 '14 at 05:09
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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
  • What do you mean by "our trust in methodologies"? "The methodologies" relied on axioms even prior to Goedel. He only put a full stop on the search of a way to get rid of them. – Sassa NF Nov 14 '14 at 23:33
  • I don't mean by direct effect, I mean by cultural influence on decision-making. We would probably not have had postmodernism without Kuhn, nor Kuhn without Turing, nor Turing without Goedel. It is the hard line notion that tipped the balance against the faith that science does not move backward. –  Nov 15 '14 at 01:15
  • Again, the idea that "we would probably not have had postmodernism without Kuhn" is apocryphal. Kuhn was well received and promoted by postmodernist thinkers but he did not have a penultimate role in formulating postmodern thought. – selfConceivedAsEvil Nov 15 '14 at 01:57
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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 02:49
  • @jobermark - Does that mean that Goedel's theorem cannot be applied to everyday conversations or interactions? – Motivated Nov 15 '14 at 18:57
  • @goldilocks: OK, the idea that you need a critical mass of stuff to get somewhere, and if you omitted one of the sources you would fail to attain it is not mysterious. I am not attributing all of anything to a given list of people. Stop putting words in people's mouths. Those other people are not dealing in **facts** of the same order, Goedel plays a role I gave him in that it is his facts, and not his ideas that contribute to this development. He would not have been happy to see them used this way. Heisenberg's facts waited a while for validation, while math is more immediate. –  Nov 16 '14 at 15:26
  • @Motivated The context of everyday life is seldom clear enough to directly apply higher-order math to it. And Goedel only applies to a certain layer of mathematics, one with arithmetic but without named sets of sets. So you would have a hard time ever finding a real application. –  Nov 16 '14 at 15:32
  • OTOH, ideas have influence just by being there. Have you looked at Douglas Hoffstadter? He gives a good impression of how the usefulness of self-modifying self-reference is bolstered by Goedel, in an informal sense, and how it has become an important part of modern thought. –  Nov 16 '14 at 15:41
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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 01:45
  • @ChristopherE Carnap was it as a problem, and it forced him into odd assertions not really consistent with positivism: http://philosophy.stackexchange.com/questions/23922/how-did-the-logical-positivists-respond-to-g%C3%B6dels-incompleteness-theorem –  May 18 '16 at 21:16
  • @DumpsterDoofus I am not a non-mathematician. –  May 18 '16 at 21:17
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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.

BartoszKP
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  • 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 09:34
  • @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
  • @DavidRicherby Please refer to the link I've provided. "true/false" can be replaced with "provable/unprovable" - so there is an analogy. – BartoszKP Nov 16 '14 at 11:54