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Thread: Our beloved ******

  1. #41
    Dioklecian's Avatar
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    In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

    Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths ... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.


    Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.

    He proved it impossible to establish the internal logical consistency of a very large class of deductive systems - elementary arithmetic, for example - unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves ... Second main conclusion is ... Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set... Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set.

    The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:

    Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
    Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
    Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
    Now Gödel laughs his high laugh and asks UTM whether G is true or not.
    If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
    We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
    "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."
    Think about it - it grows on you ...

    With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics ...

    Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.

    All consistent axiomatic formulations of number theory include undecidable propositions ...

    Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ...

    How can you figure out if you are sane? ... Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is "peculiar' or not, given that you have only your own logic to judge itself? I don't see any answer. I am reminded of Gödel's second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent.

    The other metaphorical analogue to Gödel's Theorem which I find provocative suggests that ultimately, we cannot understand our own mind/brains ... Just as we cannot see our faces with our own eyes, is it not inconceivable to expect that we cannot mirror our complete mental structures in the symbols which carry them out? All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally.
    Well I am back. How's everyone? Don't have as much time now, but glad to see some of the old gang are still here.

  2. #42
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  3. #43
    Dioklecian's Avatar
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    Quote Originally Posted by implied
    Quote Originally Posted by Transigent
    Quote Originally Posted by Dioklecian
    Exactly, the austrian INTJ
    I was just gonna pick some random type and try to state it, but it seems he had some problems that are "Alpha" to the extreme negative side (poisoning, paranoia, sickness)
    haha. some of these things sound like my dad (or stories i've heard about my grandfather, INTp) in a negative state, actually.

    a (very) sick gamma NT acting like their quasi/contrary, maybe?
    No he was INTJ I think.
    Well I am back. How's everyone? Don't have as much time now, but glad to see some of the old gang are still here.

  4. #44
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    Godel I mean. At his citizenship swearing ceremony (EINSTEIN was his witness) he tried to prove that a fascist group could take over the uSA from some loophole in the constitution.
    Well I am back. How's everyone? Don't have as much time now, but glad to see some of the old gang are still here.

  5. #45
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    asdlfkj;lkj
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  6. #46
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    Quote Originally Posted by implied
    Quote Originally Posted by Dioklecian
    Quote Originally Posted by implied
    Quote Originally Posted by Transigent
    Quote Originally Posted by Dioklecian
    Exactly, the austrian INTJ
    I was just gonna pick some random type and try to state it, but it seems he had some problems that are "Alpha" to the extreme negative side (poisoning, paranoia, sickness)
    haha. some of these things sound like my dad (or stories i've heard about my grandfather, INTp) in a negative state, actually.

    a (very) sick gamma NT acting like their quasi/contrary, maybe?
    No he was INTJ I think.
    if that was his regular state, i suppose. i never bothered learning much about godel at all! he did seem like a real wet blanket most of the time, from what i've read!
    Have you decided on INTP then for yourself?
    Well I am back. How's everyone? Don't have as much time now, but glad to see some of the old gang are still here.

  7. #47
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    asdlfkj;lkj
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  8. #48
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    Quote Originally Posted by implied
    i've stuck with that type for a while, really!
    Thats' good. What music do you like?

    What books?

    What movie?
    Well I am back. How's everyone? Don't have as much time now, but glad to see some of the old gang are still here.

  9. #49
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  10. #50
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    (1) there is logic
    (2) and causality
    (3) but the world is very complex and it's fucking difficult to
    • (a) know all the relevant facts
      (b) know their meaning, priority over each other etc.
      (c) and to know which caused which,

    (4) which is why it's important to not only lean on your logic and on your ability to understand causality, but to also focus on perceiving reality.

    Seems that FDG is arguing for points (1) and (2), Transigent for point (4). Point (3) seems to be the missing link between your arguments.

  11. #51
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    Quote Originally Posted by implied
    Quote Originally Posted by Dioklecian
    Quote Originally Posted by implied
    i've stuck with that type for a while, really!
    Thats' good. What music do you like?

    What books?

    What movie?
    oh dear. haha.

    i like stuff like da ali g show/david cross/chris rock/similar comedians. like some dark stuff too (some french films - c'est arrive pres de chez vous, irreversible, alphaville, those were all cool.) stuff like being john malkovich and eternal sunshine of the spotless mind (i dig most charlie kaufman screenplays, it seems.) sofia coppola stuff is great, too. oh and happiness.

    books i listed on myspace:
    the unbearable lightness of being, breakfast of champions, lolita, a heartbreaking work of staggering genius, antes que anochezca, fragments: a lover's discourse, american psycho, vladimir mayakovsky's stuff, the story of the eye

    musically i'm all over the map, but i guess i tend towards electronic stuff and the whole electronic scene, some hiphop/rap, punk, indie, jamaican stuff, country, emo, whatever you can name, i think i've probably been into it at some point in time.

    i have been all over the place as far as music/books/movies are concerned.
    It sounds great, what do you like about Milan Kundera?
    Well I am back. How's everyone? Don't have as much time now, but glad to see some of the old gang are still here.

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