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I want to write a philosophical essay centered about Goedel's incompleteness theorem. However I cannot find any real philosophical consequences that I can write more than half a page about. I read the books of Franzen (Incomplete guide of its use and abuse) and Peter Smith (Introduction to Goedel's Theorems). I really cannot find any philosophical discussion topic which which is really a consequence of the incompleteness theorems. I tried the mind vs. machines debate (e.g. http://users.ox.ac.uk/~jrlucas/mmg.html) a little, but one can find to many arguments against the proposition that Goedel's incompleteness theorems make statements in this debate (as in Franzen's book).

So I would be grateful if someone could direct me into interesting philosophical (or mathematical) implications or further directions I could write about.

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    I do not think the theorem has any philosophical *consequences*. However, it can motivate some philosophical *questions*, and enrich the discussion of others.2012-04-13
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    In fact, Torkel Franzen's book is all about the fact that people keep trying to draw philosophical conclusions from Goedel's Theorem, and that this is an "abuse" of the theorem. That is, he argues that it has very few "philosophical consequences."2012-04-13
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    Yes I know. But I decided to write an essay (for a course philosophy of science) about the incompleteness theorems and their philosophical consequences. so i need something where there is no counterargument that can be stated in two sentences2012-04-13
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    @Dooro: I could choose to write an essay about the philosophical consequences of the Axioms of Group Theory. My decision to write an essay on the topic does not imply that there are any such consequences (even less that there is one "where there is no counterargument that can be stated in two sentences".) In short, your decision to do something does not, by itself, change the nature of reality.2012-04-13
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    I think the entire idea that you can write an essay in Philosophy of Science about Goedel's theorem and its implications is what Arturo/Andre are criticizing.2012-04-13
  • 1
    You might look into the campaign to systematize mathematics behind Principia Mathematica and how that hope was crushed by the incompleteness theorems. Also, it lead to theory surrounding the Halting Problem and eventually the fact that the Continuum Hypothesis was independent of ZFC was revealed, which has subsequently precipitated varieties of metaphysical skepticism about the nature of infinity, the foundation of mathematics, our axiom systems, the relation of the human mind to mathematical truth, etc.2012-04-13
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    [Also see here](http://math.stackexchange.com/questions/54982/why-bother-with-mathematics-if-godels-incompleteness-theorem-is-true/).2012-04-13
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    @Arturo Magidin: of course you cannot generalize it to each topic in mathematics. In my opinion (and so it was in my course) goedel's incompleteness theorems are amongst the top 10 theorems in mathematics you can write a philosophical essay about.2012-04-13
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    If you intend to write about mathematical philosophy it's fine. If, on the other hand, this is a "real world philosophy" implications I beg you to abandon the idea. Using mathematical theorems in non-mathematical environment (where the objects are not "ideal") is more than wrong and misleading. It's plain demagogy, using tools that the layman (and often the user) does not comprehend or understand. It also enforces the idea that mathematics is somewhat related to the real world (especially abstract parts of it, like logic) which is not very true in modern context. Food for thought.2012-04-13
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    @Dooro: You missed my point; it was not about "generalizing to each topic in mathematics." My point was that just because you think it is a subject worth writing about does not mean that there is something there to be written about. Your beliefs and opinions about the subject do not shape the subject. You may very well think it is *the* top theorem to write about in terms of philosophical implications. That does not imply, in and of itself, that the theorem *has* philosophical implications. And Torkel Franzen makes a pretty strong case that it *doesn't have any.*2012-04-13
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    These comments are amazingly short-sighted. There is actually a *wealth* of legitimate philosophical consequences of Godel's theorems. See [my answer from below](http://math.stackexchange.com/a/131553/26927).2012-04-14
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    @EMS: And you yourself quote statements indicating that Penrose's use of Goedel's theorem is not particularly legitimate. That it shows up (usually spuriously) in philosophical discussions does not mean it has philosphical consequences. I would say your lengthy answer *supports* the claims that you dismiss as "amazingly short-sighted".2012-04-14
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    That sounds like confirmation bias to me. Because I think that very complicated arguments can be made to counter against one the most talented physicists of the last century, you think it supports the idea that Godel's theorem *obviously* doesn't have Penroses' alleged consequences. Hrmm.. Though I do believe Penrose can be argued with, I think one can only do so if willing to really think about the philosophical extent of Godel's theorem. It's not simple to say humans can't reason about the formal system they are. And, to boot, none of this says anything about the Kritchman and Raz paper.2012-04-14
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    @EMS: Do kindly point out where I said "obviously". I can't find that word anywhere. I mentioned an entire book by someone who was quite knowledgeable (Franzen), not some kind of "obviousness". He takes pains to explain the issues. Simply because someone invokes Goedels' Theorem and it takes hard work to debunk the invocation does not mean the invocation was an instance of valid "philosophical applications." Personally, I don't think the expectation paradox is a particularly *philosophical* issue, so I didn't see why I should address it. We disagree; fine. No need to put words in my mouth.2012-04-14
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    @Arturo That's true; you did not say the word 'obvious', but here you are extrapolating from the fact that Penrose can be disagreed with to claim that it means Godel's theorem has no philosophical implications. For that matter, Darwin's theory of evolution can be argued with (like any scientific theory), so should we believe it has no philosophical applications? Any given alleged philosophical application of evolution can be disagreed with, often successfully, so we are to conclude that there exist *no* applications to philosophy? That seems to be the reasoning you're giving.2012-04-14
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    Or, maybe to be more mathematical, what about, say information theory? [Here is a draft of a paper](http://people.seas.harvard.edu/~ely/ThingsThatStartWithB.pdf) that I wrote on applications of information theory and the machine learning idea of boosting to philosophy. Should we also say that Leslie Valiant's work on evolvability has no applications to philosophy? I think you should read the Aaronson paper that I linked below. Just because one person invokes something and it gets debunked through hard work doesn't mean there are *no* applications.2012-04-14
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    That's totally disingenuous reasoning, and I didn't think it was unreasonable to extrapolate the "obvious" that I claimed was part of your argument above.2012-04-14
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    You might want to read Incompleteness by Rebecca Goldstein. It's excellent, and if I recall it has some good discussion of the philosophical ideas motivating Godel's work.2012-04-14
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    @EMS: Please take this to chat.2012-04-14
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    @EMS: But Goedel's theorem only says that a first-order theory which is strong enough to develop the natural numbers and is effectively enumerable cannot be both consistent and true. Can you show me something in this "real world" which you can prove to me to be such theory? What about intuitionistic, $<\omega$-order theory which is not strong enough to hold Goedel's requirement? What if *that* is the true logic of the universe? What if there is no "true logic"? How can you apply Goedel's theorem correctly then?2012-04-14
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    (I have not read all the comments) You could write an essay about how people erroneously draw philosophical consequences from the theorem. The irony would be sweet and there would be a sense of originality in that you'd be relaying your own attempts and the attempts of others.2012-04-14
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    Look, no one is disputing that people can draw erroneous conclusions from Godel's theorem. But the questions you ask about alternate logics and so on are exactly why it is valid to ask philosophical questions about Godel's theorem. Do you believe that human mathematicians can "see" the consistency of whatever axiomatic theory they are equivalent to? A lot of people think the answer is *yes*, even some mathematicians. I think a person can argue the answer is no, especially if you believe in computationalism or reductionism, but it's *by no means* a settled philosophical question *at all*.2012-04-14
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    Another link [is here](http://home.mindspring.com/~presting/). This is more speculative, which is why I didn't put it in the answer below, but it is a great attempt to show how incompleteness arises naturally in decision theory, and in particular w.r.t. [Newcomb's paradox](http://en.wikipedia.org/wiki/Newcomb's_paradox). This is a *highly philosophical* question, and it appears that incompleteness relates to it non-trivially. Overall, I'm advocating that more credence ought to be given to the proposition that the incompleteness theorems have legitimate philosophical side effects. That's all.2012-04-14
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    @EMS: To claim that the kind of analysis of Penrose's argument can be compared with "arguments" about Darwin's theory that are bandied about (are you talking about creationists? or about *social darwinism*?) is rather beyond the pale. If you are talking about the arguments put forth by creationists/et al, that's disingenuous. If about social darwinism, that is not Darwin's. If that is the kind of strawmen positions that you need to take to make your points (when you aren't putting words into other people's mouths and thoughts into other people's heads), then we have nothing to discuss.2012-04-14
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    @EMS: And no, I am not extrapolating from "Penrose can be disagreed with" to "this means Goedel's theorem has no applications." Did I not mention at least one major source for my conclusion (Torkel Franzen's book)? Funny you keep ignoring it and instead just assert I am being "short-sighted", that I make claims I did not make, or that I am basing my conclusions on things you brought up a posteriori. Ironically, that's the kind of shoddy argument I see all the time when people try to find "philosophical consequences" to Goedel's theorem.2012-04-14
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    I'm not referring to creationists. I was more referring to the arguments that a pure Bayesian reasoner could be viewed (just philosophically, not biologically, due to computability arguments) as a fixed point of Darwinian evolution. I do not consider creationist claims valid enough to even merit a response. And you still haven't responded on points about Kritchman and Raz except to say that you don't think it's philosophical. Well, that's great, but doesn't jive with the math community at large.2012-04-14
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    And I willfully admit that I haven't read Franzen's book. I should do so given what you are saying, but all my peers who study decision theory laugh about that book. They essentially consider it a useless straw-man type argument against really dumb uses of the incompleteness theorems. I'm not advocating dumb uses; I'm advocating legit uses (legit in the sense that they have been deemed legit by the same peer review process that deems any other math research legit).2012-04-14
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    @EMS: Franzen addresses Penrose in that book, so I guess your peers consider Penrose's use of Goedel's theorem "really dumb". As to your views of what "the math community at large" opines or not, they don't "jive" with my impression of what they consider to be philosophical or not. As to your second sentence, frankly, it sounds much like the nonsense of Derrida invoking topology. In my experience, I've seen a *lot* of philosphers wax and wane about Goedel's Theorem, with lots of other philosphers commenting approvingly; and with mathematicians who know better snickering at their nonsense.2012-04-14
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    Be closed-minded about it if you want to, but the bulk of the uses of Godel's theorems in philosophy come *from* mathematicians, especially decision theorists. It's a non-trivial part of the philosophy of mathematics. You are certainly free to disagree with it if you want, but it's nothing but disingenuous to claim that there are virtually no philosophical ramifications of Godel's theorem.2012-04-14

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