Gödel’s Theorem Needs Godel’s Law


Kurt Gödel’s incompleteness theorem came up in a debate the other night.  I usually react by hanging my head and groaning in anticipation of the chaos that eventually ensues. But on an impulse made a statement about the narrowness of its applicability in a vain attempt to avoid the conversation. It was futile. Chaos ensued.

The conversation really troubled me. Because I couldn’t defend it from memory. I couldn’t reconstruct the argument in my head.  I’ve spent time with the problem in computer science. So much so that it’s intuitive. But I could not remember how to reconstruct the salient part of the problem — the arithmetic requirement — so I couldn’t argue it. I had to go look it up again. And in doing so remembered why I can’t remember it: it’s complicated, and difficult if not impossible to reduce it to something more accessible. That’s why no one does it. 🙂 That’s why no one has done it.

Gödel’s theory is one of the most abused concepts referred to by people outside of professional mathematics. And when it is used, it’s almost guaranteed that it’s being used incorrectly. I suspect that’s because of the popularization of the idea by way of the liars paradox, which is then inappropriately applied elsewhere by analogy. But mostly it’s abused as an excuse to create arguments to defend mysticism in religion and avoidance in philosophy, and to justify any state of skepticism. Instead, it is in fact, a fairly narrow argument, related to axioms and number theory. ie: questions within axiomatic systems that are testable by the rules of arithmetic.

I do no better. I usually express it as “given any fixed axiomatic system, there are statements that are expressible that are contradictory to the claim of completeness.” Which itself is incomplete because the difficulty with Gödel’s theory is in describing its arithmetic requirements — and that description is complicated, which is why it’s never included in any definition, and by that omission leads to its spread by erroneous analogy.

This simplified definition is useful within computer science, because computers themselves are bound by Gödel’s arithmetic constraint in the first place — unlike mathematics, wherein he discussion of Gödel’s theorem must specifically address the arithmetic requirement in order for it to be narrow enough to be true.

So we have three categories of problems that help us understand Gödel’s theorem in the abstract even if the mathematical concepts are difficult to convey other than by examples that are difficult to construct: 1) the computational problem set which is by definition constrained, 2) the mathematical problem set which must be constrained, and 3) the linguistic problem which cannot be constrained. And philosophical questions are part of set 3 – impossible to constrain to arithmetic limits which are the reason incompleteness is imposed by the theorem.

The net result is that Godel’s theorem is, for all intents and purposes, never applicable to non-mathematical, non-computational propositions. Ever. But since, in casual debate, we break Godwin’s law in any conversation by mentioning Nazis about once an hour, then even if we created a new law: “The inclusion of Gödel in any philosophical discourse is sufficient proof that the argument is faulty”, we would still break it once a week. Because in the end, people of philosophical bent, are actually searching to fulfill their un-sated desire for mystical release from our inescapable requirement to reason and adapt to a constantly changing, and entirely kaleidic reality. 🙂

Here is a wonderful little criticism by From Cosma Shalizi, Assistant Professor, Carnegie Mellon University. And as such it is only an appeal to authority – again, because the proof is burdensome and inaccessible.

“There are two very common but fallacious conclusions people make from this, and an immense number of uncommon but equally fallacious errors I shan’t bother with. The first is that Gödel’s theorem imposes some some of profound limitation on knowledge, science, mathematics.

Now, as to science, this ignores in the first place that Gödel’s theorem applies to deduction from axioms, a useful and important sort of reasoning, but one so far from being our only source of knowledge it’s not even funny. It’s not even a very common mode of reasoning in the sciences, though there are axiomatic formulations of some parts of physics.

Even within this comparatively small circle, we have at most established that there are some propositions about numbers which we can’t prove formally. As Hintikka says, “Gödel’s incompleteness result does not touch directly on the most important sense of completeness and incompleteness, namely, descriptive completeness and incompleteness,” the sense in which an axiom system describes a given field.

In particular, the result “casts absolutely no shadow on the notion of truth. All that it says is that the whole set of arithmetical truths cannot be listed, one by one, by a Turing machine.” Equivalently, there is no algorithm which can decide the truth of all arithmetical propositions. And that is all.

This brings us to the other, and possibly even more common fallacy, that Gödel’s theorem says artificial intelligence is impossible, or that machines cannot think. The argument, so far as there is one, usually runs as follows. Axiomatic systems are equivalent to abstract computers, to Turing machines, of which our computers are (approximate) realizations. (True.) Since there are true propositions which cannot be deduced by interesting axiomatic systems, there are results which cannot be obtained by computers, either. (True.) But we can obtain those results, so our thinking cannot be adequately represented by a computer, or an axiomatic system. Therefore, we are not computational machines, and none of them could be as intelligent as we are; quod erat demonstrandum.

This would actually be a valid demonstration, were only the penultimate sentence true; but no one has ever presented any evidence that it is true, only vigorous hand-waving and the occasional heartfelt assertion.”

WEB

  • http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html
  • http://math.mind-crafts.com/godels_incompleteness_theorems.php
  • http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf
  • http://plato.stanford.edu/entries/goedel/#IncThe

Recommended by Shalizi

  • Michael Arbib, Brains, Machines and Mathematics [A good sketch of the proof of the theorem, without vaporizing]
  • George S. Boolos and Richard C. Jeffrey, Computability and Logic [Textbook, with a good discussion of incompleteness results, along with many other things. Intended more for those interested in the logical than the computational aspects of the subject — they do more with model theory than with different notions of computation, for instance — but very strong all around.]
  • Torkel Franzen, Gödel’s on the net [Gentle debunking of many of the more common fallacies and misunderstandings]
  • Jaakko Hintikka, The Principles of Mathematics Revisited [Does a nice job of defusing Gödel’s theorem, independently of some interesting ideas about logical truth and the like, about which I remain agnostic. My quotations above are from p. 95]
  • Dale Myers, Gödel’s Incompleteness Theorem [A very nice web page that builds slowly to the proof]
  • Roger Penrose, The Emperor’s New Mind [Does a marvelous job of explaining what goes into the proof — his presentation could be understood by a bright high school student, or even an MBA — but then degenerates into an unusually awful specimen of the standard argument against artificial intelligence]
  • Willard Van Orman Quine, Mathematical Logic [Proves a result which is actually somewhat stronger than the usual version of Gödel’s theorem in the last chapter, which however adds no philosophical profundity; review]
  • Raymond Smullyan, Gödel’s Incompleteness Theorems [A mathematical textbook, not for the faint at heart, though the first chapter isn’t so bad; one of the few to notice the strength of Quine’s result]
  • To read:
  • John C. Collins, “On the Compatibility Between Physics and Intelligent Organisms,” physics/0102024 [Claims to have a truly elegant refutation of Penrose]
  • Rebecca Goldstein, Incompleteness [Biography of Gödel, which seems to actually understand the math]
  • Ernest Nagel and James R. Newman, Gödel’s Proof [Thanks to S. T. Smith for the recommendation]
  • Mario Rabinowitz, “Do the Laws of Nature and Physics Agree About What is Allowed and Forbidden?” physics/0104001

12 responses to “Gödel’s Theorem Needs Godel’s Law”

  1. I’m very sorry to hear that the only references you seem to hear about Gödel’s work turn out to be misinterpretations and abuses, but I must make clear that I disagree, and explain why.

    The applicability of Gödel’s work is by its nature limited only to fields that are subject to logic, and thus, they aren’t really limited at all. His Incompleteness Theorems would be included.

    Now, the impact of his Theorems were subtle, but drastic; they didn’t really tell us anything we didn’t intuitively know anyway. So using them to justify skepticism, mysticism, and sloppy philosophy is indeed a gross abuse. But what they did do is force a paradigm shift upon the mathematical positivists of his time.

    But the most useful contribution Gödel made wasn’t his Incompleteness Theorems; that’s just what gets the most attention. The Gödel numbering system he devised in order to come up with those Theorems has grand potential for checking consistency and axioms of theorems in all fields. The only limitations are practical rather than technical (Translating some linguistic theorems into an arithmetic form can be nigh impossible).

    But the constraints you put upon computational, mathematical, and linguistic problems are quite simply false. Any problem, formula, or theorem can *potentially* be reduced into arithmetic terms; it’s just a question of how long it would take.

    I had serious trouble with this phrase: “Given any fixed axiomatic system…” EVERY system is axiomatic. It’s the axioms that make the system. And EVERY axiom is fixed, or it’s not an axiom; it’s a theorem. Once you change axioms, you’ve moved to a different system.

    And this quote: “This would actually be a valid demonstration, were only the penultimate sentence true; but no one has ever presented any evidence that it is true, only vigorous hand-waving and the occasional heartfelt assertion.”

    Sort of misses the point; it’s not possible to prove that the sentence is true. It’s only possible to prove that there is no proof for that sentence. The idea was never meant as proof of the difference between computers and minds; merely a thought-provoking find on the matter: If human minds are capable of perceiving truths that cannot be proven, which Gödel’s Second Incompleteness Theorem *seems* to indicate, then that would give them a cognitive ability very distinct from a machine. That is, however, a very big IF.

    Don’t be too hard on those who attempt to use Gödel’s Theorems philosophically. Gödel himself was a philosopher; epistemology was his passion. He intended it to be put to such use. Correct their mistakes (I get the impression your work is cut out for you), but don’t try to say that Gödel’s Theorems have no viability outside of mathematics. It’s a discredit to the man’s genius.

    -E.S.T.

    • I’ll respond on two levels. First, that I’m not diminishing his contribution. I’m simply expressing the fact that THE INCOMPLETENESS THEOREM is inapplicable outside of arithmetic limits, because it is CAUSED by arithmetic limits. So I don’t know how you jumped to the broader conclusion. But that’s something you did not I. 🙂

      You start out with “Godel’s Work” The set of all G(…). Where I start out with G(T1,T2) Godel’s theorem in the vernacular, the Incompleteness Theorems in the formal. So you’re making a criticism that is inapplicable to my statement. So that’s the first point.

      Second, you state: “But the constraints you put upon computational, mathematical, and linguistic problems are quite simply false. Any problem, formula, or theorem can *potentially* be reduced into arithmetic terms; it’s just a question of how long it would take.”

      Actually no. You’re welcome to try to prove it. But this is precisely the problem with misinterpreting the theorem. Arithmetic operations are extremely narrow. THe fact that linguistic problems cannot be subject to this constraint is part of the proof itself. And this is where people abuse it. Godel’s theorem is a limitation on computational systems. Abuses of it usually come from tech nerds. Which, since I’m kinda one of them, and originally fell into the trap myself, and have had thousands of them working for me, I just run into it a lot.

      Of course if you can come up with a proof that you’re right, or even a solid example, I’m sure the world would be fascinating to to hear it. 🙂

      • “So I don’t know how you jumped to the broader conclusion. But that’s something you did not I. :)”

        Fair to say. Allow me to narrow it down, then.

        Gödel’s Incompleteness Theorems were built around the arithmetization of syntax (Gödel numbering). Specifically, starting with the liar’s paradox, but noticing that it led him in circles, Gödel instead came up with the self-referential sentence G that states that G cannot be proven, which produces quite different results from Sentence L which asserts that L is false.

        Now, what is a linguistic problem but a problem with a different syntax than mathematics? The whole basis of the Gödel numbering system that provided the results of Gödel’s Incompleteness Theorems was the translation of syntax (linguistic expression) into arithmetized form. Every linguistic statement is syntactically based, and can be reduced into mathematical terms. Even if the expressed terms are vague (hello, variables) they are still subject to mathematical deduction if they were syntactically non-contradictory.

        Arithmetic operations are FAR from narrow. They’re simply the most basic reduction of more complex operations. Just as you can break down 9×4 into 9+9+9+9, complex mathematical operations can be broken down into simple arithmetic. It’s simply tedious and impractical.

        Take any syllogism. Replace the terms with variables. Voila. You’ve just taken a linguistic expression and made it mathematical.
        All lizards are reptiles. A?B.
        All geckos are lizards. C?A.
        All geckos are reptiles. C?B.

        Mathematics is the foundation of computation, language, and knowledge itself. Trying to say that basic arithmetic operations(the foundation of mathematics) are *narrow* is so backwards that I can’t even conceive of how you drew that conclusion. I mean, they’re “narrow” in that you reduce more complex operations down to them, but that fundamental nature is exactly what makes them so broadly applicable.

        Now I’ll grant you that reducing some linguistic problems to mathematical terms is very, very difficult. Practically impossible, even.

        …But if you can show me what linguistic problem is not subject to the axioms of mathematics (including first-order logic), and I might start to buy what you’re saying.

        Gödel’s Incompleteness Theorems were logically deducted; they thus apply to any system that is subject to logic. Same axioms (as derived from Gödel’s Completeness Theorems), same system. Different aspects of it.
        You can’t say they don’t apply to a field without saying that field is illogical (which in some cases may be perfectly valid). And you can’t say someone’s misusing them without saying their argument is illogical (which it may be). But to say they don’t apply to anything but mathematics is disingenuous. They may be *irrelevant* in most other fields, but it still technically applies.

        Gödel himself came up with a strange loop result (inconsistent completeness) in Einstein’s Relativity equation, which ties in pretty closely with his Incompleteness Theorems (He could have reached that result without them, but they made it clearer how he reached it). He had clear *philosophical* motivations for reaching the Incompleteness Theorems in the first place. He knew they were applicable outside of mathematics quite well.

        And just because I suspect a miscommunication, please clarify your definitions of “computational”, “mathematical”, and “linguistic”. I want to make sure I’m not misinterpreting tech jargon (Hi, tech nerd, I’m physics nerd, nice to meetcha).

  2. “But if you can show me what linguistic problem is not subject to the axioms of mathematics (including first-order logic), and I might start to buy what you’re saying”

    It’s not what I’m saying.
    And that’s not what Incompleteness says.
    Incompleteness specifically refers to addition.
    Torkel Franzen’s book is probably the best full treatment of the subject. I think chapter 8? refutes your assertions.
    I’ve attached a list of authors to the article.
    No, language is not subject to mathematical constraints except in the most reductio of circumstances.
    We can express mathematical relations in language, thereby limiting language. However, language is not limited to mathematical constraints.
    I am on the way to the airport so I must run but I’ll catch up with this later.

    • I look forward to your return. In the meantime, let me leave something for you while I’ve the time to spare.

      First, How is this statement: “But if you can show me what linguistic problem is not subject to the axioms of mathematics…”
      Not comparable to this statement: “…language is not limited to mathematical constraints.”
      You said that’s not what you’re saying, but… it sure looks like it is. If you could clarify, I’d appreciate it.
      Second: “Incompleteness specifically refers to addition.”
      Incompleteness has to do with the fact that no formal system (it specifies strong formal systems, but weak ones are just as subject) can be both consistent and complete. It can only be one or the other. If it is complete, it is inconsistent (you create a strange loop paradox), if it is consistent, it is incomplete (you cannot prove all of your theorems) and specifically, you cannot prove a theorem asserting the system’s consistency.

      Gödel’s Incompleteness Theorems merely provided mathematical proof to the necessity of undecidable theorems in a consistent system, and that one of those undecidable theorems will be the consistency of the system. In what I’ve read, it has far less to do with addition than it does with self-reference. You’ll have to explain that one to me. I don’t get it. Gödel reached his result through deduction (subtraction) not induction. How is Incompleteness all about addition?

      And while I am indeed very interested in Franzen’s book, I can’t help but notice the poor reviews it gets.
      You might try looking at Hofstadter’s “Gödel, Escher, Bach” book. It takes its time to explain things, but offers greater clarity as a result, I think.

      Gödel’s Incompleteness Theorems were far more epistemological than mathematical. And as a result, they applied to all sciences. According to Einstein, “Science without epistemology is – insofar as it is thinkable at all – primitive and muddled” (Goldstein 29). Without the philosophical basis a scientist bases his results upon, he cannot *know* that he knows anything.

      “No, language is not subject to mathematical constraints except in the most reductio of circumstances.”

      Which is exactly what I’m saying. I’m merely adding the assertion that it is possible to reduce lingual statements to such forms without losing any information. And if both the lingual and arithmetic expression of a problem contain the same information, then both are subject to the same rules, among which would be Gödel’s Incompleteness Theorem.

      Even the most vague, complex lingual statements have mathematical bases as at their core. Nothing can even be *known* without math.

      To know something, one must quantify it. To quantify it, one must measure it. To measure it, one must compare it. To compare it is to examine a difference. To examine difference is basic arithmetic subtraction.

      A lingual statement would have to contain something the stater did not know in order to not contain math. And to say “I don’t know” is still the assertion that you *know* you don’t know. Even if the measurement upon which you draw your knowledge is vague, emotionally clouded, intuitive rather than analytic, or merely unintelligent, all measurements are still taken in the same mathematical way.

      So I posit that language *is* limited by mathematical constraints. There are rules of grammar, rules of logic, and rules of *knowledge* that all have mathematical basis.
      To say that there are some problems too complex for mathematics is to *seriously* underestimate mathematics.

      -E.S.T.

  3. Ok, so at this point you’ll need to give me an example that proves that everyone else who writes on this is wrong and you’re right.

    I’ll take an example and work it through. But even Kripke had to play a game of redefining the problem in order to find a solution.

    So lets get away from generalizations and get to a specific.

    • “Ok, so at this point you’ll need to give me an example that proves that everyone else who writes on this is wrong and you’re right.”

      I’m not the only one writing from this opinion. Gödel himself was seeking proof of the a priori nature of natural law when he came up with the GITs. Carnap supported the translation of physical observations: “…the logical analysis of physics, as part of the logic of science, is the syntax of the physical language” (Rodriguez-Consuegra 89). Hofstadter also expresses the opinion that the nature of the necessary strange loops exposed by GITs is at the core of intelligence, and understanding them is necessary to the development of any creative analogizing algorithm.
      I’m hardly alone in this opinion. Countless highly-approved scholars share it.

      As for examples, I can come up with rudimentary examples, like the syllogism earlier, on the fly, but anything more detailed will be, naturally, more difficult. I don’t see how those simple examples don’t cover the argument as needed, so tell me what criteria exactly you want me to fit an example into, and I’ll do my best.

      That said, you don’t need to drop an apple to say that it’s subject to the law of gravity. It’s not so much generalizations that I’ve been trying to make as establishing fundamental principles which are universally applicable.

      Getting down to basics isn’t necessarily non-specific.

      For how GITs apply to physics, check out this lecture from Stephen Hawking, especially the conclusion. http://www.damtp.cam.ac.uk/events/strings02/dirac/hawking/ I’m not a huge fan of Hawking, but he’s no idiot.

      For how it could apply to law (oh yes, law) look up Gödel’s lost loophole. An inconsistency he (reportedly) found in the Constitution apparently had no mathematical solution. Just as GIT says it wouldn’t (Though I would prefer incompleteness to inconsistency in law, of all places…).

      I’ll try to come up with more later. But if I’m to come up with valid translation of lingual syntax into arithmetic (which was Gödel’s whole point behind Gödel numbering…), I need you to come up with a lingual sentence which is not subject to the first-order logic that Gödel based his GITs on, and explain how, and why. I think the burden of proof is at least somewhat shared, here.

      Let me just clarify our theses, however:
      Mine: All statements in any language are potentially translatable into Gödel sentences, (though it might take eons in many cases) because they, too, are subject to first-order logic. This makes them likewise subject to GITs, however irrelevantly.
      Yours: Language is not limited to mathematical constraints, (because they can contain non-mathematical information that would be lost in reduction to mathematical terms). They are thus not subject to GITs (Parenthetical statement is my inference, correct me if I misunderstood).

      So, I’ll work on examples for my thesis, and you try to come up with one for yours. Just a warning: I’m not terribly practiced at arithmetization of syntax at all, so it might take me a while. I’ll get it to you as quick as I can, though! (I might enlist the aid of one of my professors…)

      -E.S.T.

  4. I might be wrong but I am pretty sure Hawking’s quote is given as an example of how the findings can be misinterpreted, and that he’s wrong. I’ll have to look that up in Franzen.

    This seems like a poor use of my time. But I”m game. I might learn something. So Ill play.

    But I’m pretty sure that once you create a ‘sentence’, you’ll have to also produce a set of defined axioms, and apply them to that sentence, and when you do the problem will become obvious. 🙂

  5. 1) The theorem relies upon natural numbers and addition. That is all it proves. It is effectively an essay on the limits of computational systems.
    2) re: Hawking in ch4.4 of Franzen. “…a Metaphor at best…”
    3) The strange loop is also just a metaphor by which an intellectual can talk to the common populace metaphorically about the concept of cognition.
    4) The Lost Loop likewise is just a metaphor.
    5) The metaphors are just that — Metaphors. No evidence exists to support them. The reason we can prove GIT, is because arithmetic operations are narrow.
    6) My Sentences to choose from, with the last the most interesting:
    “Her beauty inspires me with both profound passion and humble awe”
    “I prefer happiness and beauty to roses and wine, and John the opposite”
    “There is a difference between risk and uncertainty”
    “Mary wants to be wholesome girl, a loving wife and a devoted mother”
    “I have a ‘penchant’ for writing gregorian hymns.”

    The futility of pursuing the argument should be obvious from the content of the sentences themselves. But you are welcome to try. While godel’s numbering, thanks to prime numbers, makes it possible to represent such things in ‘code’, the second problem is representing the content, the third is subjecting it to axioms to which the GN can be manipulated, and the fourth is the loss of content.

    I kind of suspect that your professors would prefer not to comment on this problem. But if they did, they would argue that no, godel’s theorm does not apply outside of arithmetic scope, but that it is a useful metaphor for educating people on the limits of our axiomatic systems as a means of expressing the totality of our knowledge.

    But I have enjoyed this conversation so thank you for playing. 🙂

    • “It is effectively an essay on the limits of computational systems.”

      …Like mathematics.

      “…a Metaphor at best…”

      I unfortunately don’t have access to Franzen’s book at this time (unless you know of a free online version), so I can’t really get the context of how he substantiates the claim. Still, it’s worth noting that he disagreed with Hawking’s assessment.

      “The strange loop is also just a metaphor by which an intellectual can talk to the common populace metaphorically about the concept of cognition.”

      … Dude. I’m not saying them to sound smart or ‘intellectual to the common populace’, here. The terms are clearly defined in Hofstadter’s book and Pujel’s essay. They’re not metaphors or even vague, and have nothing to do with cognition at all anyway.
      Strange Loops are simply operations that when followed in natural logical procession, return to the initial quandary. That they are a necessary element of any system *is* proved by GITs.
      The Lost Loophole (not ‘loop’) was simply the term given to the supposed logical error Gödel found in the Constitution. It might not have been a very clear application of GITs, but that doesn’t make it a ‘metaphor’. And given that it was Gödel himself who found it, your assertion seems pretty absurd.

      But let’s not get off track. If you don’t want to look up those terms, that’s fine, but don’t tell me I’m talking out of my rear without some evidence. I was all set to enjoy this thing; don’t go ruining it. You’ve got plenty of material to work with without resorting to that sort of thing; heck, my thesis is a heck of a lot harder to buy than yours.

      “While godel’s numbering, thanks to prime numbers, makes it possible to represent such things in ‘code’, the second problem is representing the content, the third is subjecting it to axioms to which the GN can be manipulated, and the fourth is the loss of content.”

      That’s all pretty much the same problem: The second and fourth pose the same problem exactly; the third poses no problem at all (axioms can be as arbitrary as you like, the only difficulty is convincing people that the axioms you apply in your demonstration are the same ones that apply in the real world, which is almost entirely a matter of opinion).

      “But if they did…”

      He did.

      “But if they did, they would argue that no, godel’s theorm does not apply outside of arithmetic scope…”
      Pretty much; but nice going for putting words in his mouth before you’d received them. Yeesh.

      Here’s what he said: You can come up with almost any ambiguously-phrased sentence that cannot immediately be converted into Gödel numbers. But if you can find out *exactly* what each term meant, at *exactly* the time the utterer said it, then it can *potentially* be translated – but you can’t prove it until you do it. So while he stated that he was of the *opinion* that couldn’t be translated, there was no way of knowing for certain until someone did it.

      He gave me this example: “I love that girl”. You’d have to quantify each term and axiom. What is love (baby don’t hurt me…sorry, couldn’t resist), what is a girl, who is speaking, and who exactly is he/she referring to as “that girl”. Essentially, it doesn’t have *enough* information to be translated into a Gödel number. It’s not in danger of *losing* information.

      Now, given the plastic nature of language, the sentence could mean different things to different people. Languages are robust; arithmetic is austere. There’s less standardization of meaning in common language. So first you have to overcome that hurdle by finding out exactly how the utterer, and no one else, defines their terms and axioms.

      Next, you have to define those terms *in context*. The utterer could say “I love that girl” at one moment, and say “I love my dog” another, and ‘love’ would have very different meanings, even though it was the same person that said it. So that’s the second major hurdle.

      Then, once you have all the terms polished and refined to the strictest degree, you can potentially translate it.

      For example, say I define “I” as the speaker. This is a specific person. No ambiguity there.
      “Love” as “valued by the speaker more than the speaker’s self”. Not a definition the speaker would necessarily use in every context, but if it can be confirmed that it is exactly what the speaker meant in this context, it is valid and specific.
      “That” is simply referential; it specifies the next term. It can be essentially thrown at as the translation of the next term would cause it to specify itself.
      “Girl” as the specific human female the utterer was referring to.

      The axioms, as I said, can be applied arbitrarily, but one has to make a convincing case that they are the same ones that apply in the context of the true model that the sentence operates in and that they’re relevant to the operation being performed with the translated expression, whatever it may be. You could insert the axiom “God exists”, but unless you’re having a theological debate, it has no bearing on the equation. So I can’t really apply axioms without knowing how I’m using the expression, I don’t think. At least, it would be pretty pointless to do so.

      As you can probably tell, my prof didn’t give me any instruction on *actually* Gödel numbering anything (though I’d hoped he would, I caught him at the end of his office hours and I’m sure he wanted to get home); the only instruction he gave me is that to convert an ambiguous sentence into a Gödel number, all ambiguity must be removed – but he gave an exception. Using the antithesis of a definition, as an exclusionary term, one can use ambiguous terms to cover a broad spectrum (i.e. using a specific antithesis, e.g. hate, giving it a specific definition, and defining love as ‘not hate’), or similarly, using all possible definitions (that would operate under the axiom that there are no other possible definitions, though). Then the ambiguity can be kept, and still translated. An interesting idea, really.

      So in conclusion, it is a perfectly rational assumption to operate under that a given sentence cannot be translated into a Gödel number, until it is done.
      It is, in my opinion, also a rationally valid assumption to operate under that it can be translated; and there is similarly no way I can prove it until it is done.

      So the issues are this:
      1. Is information lost when ambiguous terms are refined into specific ones?
      2. And are the axioms applied in the arithmetic operation of the numerical expression valid in the situation the lingual expression refers to?

      I would make the argument that the answer to #1 is no. It simply has to be specified that the definitions are applied to that *specific* instance from that *specific* speaker. Which means that the information has to be found before it is applied to the translation, of course… but if it wasn’t specified beforehand anyway the information was lost *before* the translation.
      However, #2 is case-to-case, and it can never be “proven” in an arithmetic way. Axioms are kind of like that. You don’t “prove” an axiom; you merely apply it, and others can agree with it or not.

      So if you can refine your terms, and agree upon the axioms, then it can be translated without losing information.

      And if they can be translated while containing the same information, then that information is subject to GITs. It doesn’t really *mean* much that it is, but it is.

      My prof may have agreed with you on the application of GITs, but he at least considered my thesis a valid thought experiment, and worth pursuing. He didn’t try to say he could prove it wrong, only that he thought it was wrong. And I don’t really think you can prove it, either, any more than I can prove I’m right in all cases (though I still think I’ll attempt to prove myself right on some of your sentences, at least until I get tired).

      And one thing to consider: Even if I could hypothetically prove that GITs are universally applicable…
      … would it really change anything anyway? Do they really tell us anything we didn’t already know on a non-mathematical level?

      -E.S.T.

      • EST: …Like mathematics.”
        CURT: That’s again, an analogy. It is not a proof. And until you put it to practice you’re operating metaphorically, not scientifically.

        EST: I unfortunately don’t have access to Franzen’s book at this time (unless you know of a free online version), so I can’t really get the context of how he substantiates the claim. Still, it’s worth noting that he disagreed with Hawking’s assessment.

        CURT: well, you will need to understand the theorem better. I recommend simply trying it. But Franzen’s fine as well. He uses a case by case example.

        EST: “The strange loop is also just a metaphor by which an intellectual can talk to the common populace metaphorically about the concept of cognition.”
        … Dude. I’m not saying them to sound smart or ‘intellectual to the common populace’, here. The terms are clearly defined in Hofstadter’s book and Pujel’s essay. They’re not metaphors or even vague, and have nothing to do with cognition at all anyway.
        Strange Loops are simply operations that when followed in natural logical procession, return to the initial quandary. That they are a necessary element of any system *is* proved by GITs.

        CURT: well I know, but they ARE metapors. That’s the purpose of Hofstadter’s book: both GEB and “I’m a strange loop” (or whateve rit’s called) they are both books that attempt to create analogies so that his concept of consciousness can be understood. But it is a metaphor. Nothing but a metaphor.

        EST: The Lost Loophole (not ‘loop’) was simply the term given to the supposed logical error Gödel found in the Constitution. It might not have been a very clear application of GITs, but that doesn’t make it a ‘metaphor’. And given that it was Gödel himself who found it, your assertion seems pretty absurd.

        CURT: Yes, but it’s simply silly. That wasn’t a loophole it was well understood at the time. It’s just a simple observation of hte fact that any legal document can be iteratively modified by small statements which render the whole meaningless. It’s nothing to do with godel numbers or the proof. I just don’t understand what you’re getting at. This is not meaningful.

        EST: But let’s not get off track. If you don’t want to look up those terms, that’s fine, but don’t tell me I’m talking out of my rear without some evidence. I was all set to enjoy this thing; don’t go ruining it. You’ve got plenty of material to work with without resorting to that sort of thing; heck, my thesis is a heck of a lot harder to buy than yours.

        CURT: I dont need to look them up, becuase they’re well known popular works and concepts. But they are simply analogies. You can’t put a proof together for them they way that you can put a proof together for arithmetic.

        “While godel’s numbering, thanks to prime numbers, makes it possible to represent such things in ‘code’, the second problem is representing the content, the third is subjecting it to axioms to which the GN can be manipulated, and the fourth is the loss of content.”

        That’s all pretty much the same problem: The second and fourth pose the same problem exactly; the third poses no problem at all (axioms can be as arbitrary as you like, the only difficulty is convincing people that the axioms you apply in your demonstration are the same ones that apply in the real world, which is almost entirely a matter of opinion).

        “But if they did…”

        He did.

        “But if they did, they would argue that no, godel’s theorm does not apply outside of arithmetic scope…”
        Pretty much; but nice going for putting words in his mouth before you’d received them. Yeesh.

        Here’s what he said: You can come up with almost any ambiguously-phrased sentence that cannot immediately be converted into Gödel numbers. But if you can find out *exactly* what each term meant, at *exactly* the time the utterer said it, then it can *potentially* be translated – but you can’t prove it until you do it. So while he stated that he was of the *opinion* that couldn’t be translated, there was no way of knowing for certain until someone did it.

        He gave me this example: “I love that girl”. You’d have to quantify each term and axiom. What is love (baby don’t hurt me…sorry, couldn’t resist), what is a girl, who is speaking, and who exactly is he/she referring to as “that girl”. Essentially, it doesn’t have *enough* information to be translated into a Gödel number. It’s not in danger of *losing* information.

        CURT: the act of translating it to such a number would require losing information that was present when spoken or written. Since taste is immeasureable, then it cannot be cast quantitatively.

        EST: Now, given the plastic nature of language, the sentence could mean different things to different people. Languages are robust; arithmetic is austere. There’s less standardization of meaning in common language. So first you have to overcome that hurdle by finding out exactly how the utterer, and no one else, defines their terms and axioms.

        CURT: OK. you are saying that hypothetically speaking, despite lacking any evidence, we could somehow capture all meaning (despte the fact that humans cannot define concepts such as justice and freedom that they use daily), and we could capture all the axioms necessary to constrain that system sufficiently to prove incompleteness.

        EST: Next, you have to define those terms *in context*. The utterer could say “I love that girl” at one moment, and say “I love my dog” another, and ‘love’ would have very different meanings, even though it was the same person that said it. So that’s the second major hurdle.

        Then, once you have all the terms polished and refined to the strictest degree, you can potentially translate it.

        For example, say I define “I” as the speaker. This is a specific person. No ambiguity there.
        “Love” as “valued by the speaker more than the speaker’s self”. Not a definition the speaker would necessarily use in every context, but if it can be confirmed that it is exactly what the speaker meant in this context, it is valid and specific.
        “That” is simply referential; it specifies the next term. It can be essentially thrown at as the translation of the next term would cause it to specify itself.
        “Girl” as the specific human female the utterer was referring to.

        The axioms, as I said, can be applied arbitrarily, but one has to make a convincing case that they are the same ones that apply in the context of the true model that the sentence operates in and that they’re relevant to the operation being performed with the translated expression, whatever it may be. You could insert the axiom “God exists”, but unless you’re having a theological debate, it has no bearing on the equation. So I can’t really apply axioms without knowing how I’m using the expression, I don’t think. At least, it would be pretty pointless to do so.

        As you can probably tell, my prof didn’t give me any instruction on *actually* Gödel numbering anything (though I’d hoped he would, I caught him at the end of his office hours and I’m sure he wanted to get home); the only instruction he gave me is that to convert an ambiguous sentence into a Gödel number, all ambiguity must be removed – but he gave an exception. Using the antithesis of a definition, as an exclusionary term, one can use ambiguous terms to cover a broad spectrum (i.e. using a specific antithesis, e.g. hate, giving it a specific definition, and defining love as ‘not hate’), or similarly, using all possible definitions (that would operate under the axiom that there are no other possible definitions, though). Then the ambiguity can be kept, and still translated. An interesting idea, really.

        CURT: all you have is a vague theory that the limits imposed by very narrow arithmetic rules is extensible to more complex systems. But you don’t have a proof of it. It’s just an analogy. It’s not scientific. THat’s the whole point.

        EST: So in conclusion, it is a perfectly rational assumption to operate under that a given sentence cannot be translated into a Gödel number, until it is done.
        It is, in my opinion, also a rationally valid assumption to operate under that it can be translated; and there is similarly no way I can prove it until it is done.

        CURT; So you mean that you are willing to act on faith that an unproven metaphor may someday be found applicable despite the narrowness of the original theorem. Thats an aesthetic statement not a rational or scientific one. 🙂

        So the issues are this:
        1. Is information lost when ambiguous terms are refined into specific ones?
        2. And are the axioms applied in the arithmetic operation of the numerical expression valid in the situation the lingual expression refers to?

        I would make the argument that the answer to #1 is no. It simply has to be specified that the definitions are applied to that *specific* instance from that *specific* speaker. Which means that the information has to be found before it is applied to the translation, of course… but if it wasn’t specified beforehand anyway the information was lost *before* the translation.
        However, #2 is case-to-case, and it can never be “proven” in an arithmetic way. Axioms are kind of like that. You don’t “prove” an axiom; you merely apply it, and others can agree with it or not.

        So if you can refine your terms, and agree upon the axioms, then it can be translated without losing information.

        And if they can be translated while containing the same information, then that information is subject to GITs. It doesn’t really *mean* much that it is, but it is.

        My prof may have agreed with you on the application of GITs, but he at least considered my thesis a valid thought experiment, and worth pursuing. He didn’t try to say he could prove it wrong, only that he thought it was wrong. And I don’t really think you can prove it, either, any more than I can prove I’m right in all cases (though I still think I’ll attempt to prove myself right on some of your sentences, at least until I get tired).

        And one thing to consider: Even if I could hypothetically prove that GITs are universally applicable…
        … would it really change anything anyway? Do they really tell us anything we didn’t already know on a non-mathematical level?

        -E.S.T.

        CURT: I hope this was fun for you in some way. Thanks for playing.

  6. Darn it, man, I was trying to chill out after the last post, but you’re missing my points by such a wide margin again that I’m getting so frustrated that I’m just giving up after this.

    “That’s again, an analogy. It is not a proof. And until you put it to practice you’re operating metaphorically, not scientifically.”

    I was not saying computational systems were *similar* to math. I was saying mathematics *is* a computational system. That’s no metaphor. That’s an assertion. If you’re going to rebut it, rebut it for what it is. It’s not allegory, simile, or metaphor. The purpose of the assertion was simple: Do you or do you not agree that mathematics is a computational system?

    “I recommend simply trying it.”

    Hawking “tried” it. He tried applying GITs to physical results. They matched up. He drew the conclusion that physics was therefore subject to GITs, or something like them. As an induced prospect, it’s not something you can really falsify without showing results that *don’t* match. He drew a conclusion from physical evidence; you have to shoot it down with physical evidence.

    “well I know, but they ARE metapors.”

    Metaphors for what? You’re not explaining your statements. They’re growing less and less coherent. Hofstadter’s “strange loops” are just an observable mathematical occurrence. I don’t see how that’s a metaphor for anything. It’s only called “strange” because of the impression it leaves of going in one direction to end up at the beginning. He tries to argue that this impression says something about cognition, not that it’s a *metaphor* for cognition. He simply asserts that understanding them is vital for the production of creative analogical algorithms – which is what his research deals in. Meaning, he applies GITs outside of ‘regular’ mathematics on a regular basis.

    “This is not meaningful.”

    It’s an example of self-reference in a system (a legal system) which causes it to collapse. It’s simple, yes. Pointless, probably. But the fact that you can analyze such a thing in a legal system displays that there are non-mathematical systems subject to GITs. It’s only intended as corroborative evidence. That’s all.

    “I dont need to look them up…”

    …Really? ‘Cause I get the distinct impression you know next to nothing about them. The concept of strange loops may be bandied about a bit in your circles, but the lost loophole is an obscure bit of *legal trivia*. And frankly, neither concept is particularly well-known. The writings involved have a very niche market in people like you and me; math nerds aren’t exactly mainstream. And your repetition that strange loops cannot deal with anything except cognition tells me you never read the book.
    And frankly, it’s just academically lazy to say “I don’t need to look them up,” when someone says your knowledge is incomplete. Even when you’re sure, you double-check to see if you missed something. I’ve never had a single professor who would not look up a reference when I challenged their knowledge, and I’ve had some pretty arrogant ones. That’s basic academic rigor, there.

    “You can’t put a proof together for them they way that you can put a proof together for arithmetic.”

    As I already said, yes you can, and they *did*. Gödel’s first IT was a strange loop, *by definition*, and it has proven vital to Hofstadter’s work. Gödel himself derived the self-reference in the loophole. These *have* had proofs written.

    “the act of translating it to such a number would require losing information that was present when spoken or written.”

    Yes, but not when heard or read, which is the same standards applied to most recorded knowledge. The information was lost *prior* to the translation from language to numbers, because it lost most of its information in the translation from thought to language. If you want the information that was present when spoken or written, hunt it down from the speakers or writers themselves, and get them to strictly define their terms. It’s far from unfeasible.

    “…despte the fact that humans cannot define concepts such as justice and freedom that they use daily…”

    One can define them contextually. In a given case, one could define “justice” as “an eye for an eye”, while in another given case, one could define freedom as “Not having to pay sales tax”. The meanings of our words are context-dependent, and I already addressed that issue. And beyond that, people can be made to clarify their statements, as I said before. People have ideas the can’t put into words, I get that. But that only means the information was lost *because the words could not contain them*, not because numbers couldn’t.

    “…and we could capture all the axioms necessary to constrain that system sufficiently to prove incompleteness.”

    You only need *one* axiom to prove incompleteness in a system, and usually it’s easier the simpler the axiom is. Why do you think Gödel started with something as absurdly simple as the Liar’s Paradox? Indeed, it would be difficult with something more complicated to find axioms that would make it easy to find Incompleteness, but it’s really not necessary; the question was not whether or not you could *prove* incompleteness in an extra-mathematical system, but whether or not the concept *applied* to it. You don’t need to find the incompleteness to prove that it *must* be incomplete. You only need find the factors that make it incomplete, i.e. “In any formal system strong enough to contain Principia Mathematica…” You get the idea.

    I’ll break it down Barney style: GITs apply in mathematics, a system.
    Systems are defined by their axioms; that’s what everything in them comes from.
    If other systems are subject to the same axioms, then GITs apply to them, too.
    My only assertion is that other systems are subject to the same axioms as mathematics. That everything is subject to the rules of mathematics.
    This cannot be proven or disproven. But, it does have much in the way of circumstantial evidence. One’s judgement according to this evidence is the crux of what brings about one’s opinion on the matter.

    Incompleteness was obvious in the weaker formal systems of other fields *long* before it was proven in the strong formal system of mathematics. Logical positivists were of the opinion that math was the exception to the rule, that it could be ironed out so as to be both complete and consistent. GITs just showed them they weren’t as special as they thought.

    Take philosophy: Everyone knows that if you ask “why” enough times you hit a question to which the answer cannot be proven or disproven.
    Law: No case can ever be *absolutely certain* that aliens didn’t abduct all parties and replace their memories, can they?
    Physics: Prove the Big Bang Theory. Just try.
    History: Prove that William Shakespeare was actually Christopher Marlowe.
    Art: Prove that your art is beautiful.
    I could go on.

    *Every* field knew that there were limits to what they could prove, but they kept on anyway. Math just found out it had to do the same.

    Coming up with axioms to make an incomplete system is *easy*. Coming up with enough axioms to make the incompleteness as difficult to find as it was in math? Not so much.

    “all you have is a vague theory that the limits imposed by very narrow arithmetic rules is extensible to more complex systems. But you don’t have a proof of it. It’s just an analogy. It’s not scientific. THat’s the whole point.”

    It’s actually a pretty specific theory, but that’s irrelevant. Explain to me why I need proof of *that*, of all things. That’s purely a matter of opinion. You can’t prove or disprove that the laws of mathematics govern everything. It’s the axiom that my whole thesis rides on; you don’t prove axioms – you prove that theorems are valid under them. Which is exactly what I’m trying to do and is a perfectly valid logical method.

    Think about it anyway: In order to prove it, I’d have to find it impossible to disprove, right? In order to disprove that math applies to all systems, I’d have to disprove it – using math. Which, obviously, would prove no such thing. So in order to “prove” it, I have only to state that there is no system which is not subject to math, and then invite anyone to find a way to disprove that statement – without using math.
    So I can’t “prove” it, technically, because I can’t prove it impossible to disprove… but the opposing argument has it tougher, since each time they attempt to disprove it, they only end up demonstrating it, unless somehow they find a way to prove something without using math. You’ll forgive me if I say, “That’ll be the day”.

    “So you mean that you are willing to act on faith that an unproven metaphor…”

    … You keep using that word. I do not think it means what you think it means.

    “…may someday be found applicable despite the narrowness of the original theorem. Thats an aesthetic statement not a rational or scientific one.”

    Yes, yes it is. Because it cannot be proven that math governs all, as I just demonstrated. There is, however, in my opinion, a boatload of evidence to support that idea, some of which I’ve already given. So I take it on faith – but I don’t need very much.
    My point, however, is that your stance is *also* one of faith – that math does NOT govern all. Why quibble over our axioms when we’re debating the validity of our theorems?

    You seem to have missed my entire point. I wasn’t debating the validity of my thesis under *your* axioms; I was debating its validity under mine. I know enough about model theory to know I wasn’t even working with your model. Nor did I intend to.

    I arrived at the conclusion that neither of our theses could be proven because they rested upon undecidable axioms (which, by definition, most axioms are anyway). Your thesis (inapplicability of GITs outside mathematics) was provable under your axiom (non-universal nature of mathematics) and mine (universal applicability of GITs) was provable under mine(everything subject to math).

    Because, quite frankly, I can’t see you coming up with your thesis under any other axiom. If math governs everything, including knowledge and communication, then everything is reducible to mathematical terms without losing information, because of of the information contained is mathematical. The only difficulty would be in finding it all; not in preventing *loss* of information.

    So I have to ask: If not everything is subject to the rules of mathematics, then what is? Only math? Or is it a more broad, but still incomplete scope?

    Can you even see beyond that axiom? Every attempt I’ve made to explain my model to you fails. It’s getting me nowhere. What I thought might be a productive, intelligent, rational debate quickly devolved. You didn’t answer my questions when asked, or really even substantiate any of your statements. I frankly suspect you’re not interested in reading what I have to say or what I reference at all anyway. If so, that’s your business, but don’t string me along, then.

    I’m pretty disappointed, really. Given how much better-informed you seemed to be on the topic I expected to learn something. The only thing I’m really coming away with is the title of Franzen’s book that I now want to read. I was expecting a LOT more. I’m a student; I generally assume the person I debate with knows more about an advanced topic like this than I. And you probably do. But as such, you *should* be able to explain it better than I. And…you haven’t.

    So I’m done. Maybe we got something out of this, maybe not. But I can’t help but feel like I’ve spent way too much energy on something that turned out to be a waste of time.

    Sorry. I thought it would be fun, but it hasn’t turned out so great. I hope you had a better experience.

    Thanks for “playing”.

    -E.S.T.

Leave a Reply