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This is a series on the book Gödel, Escher, Bach: An eternal golden braid by Douglas Hofstadter.  

Earlier diaries are here

Today, we will look at Chapter IX: Mumon and Godel, p. 246-272.

From the overview

An attempt is made to talk about the strange ideas of Zen Buddhism.  The Zen monk Mumon, who gave well known commentaries on many koans, is a central figure.  In a way, Zen ideas bear a metaphorical resemblance to some contemporary ideas in the philosophy of mathematics.  After this "Zennery", Godel's fundamental idea of Godel numbering is introduced, and a first pass through Godel's theorem is made

Some thoughts, ideas and questions:
Page 249, top:  Old Mumon said something comprehensible, he must have been greatly upset

also:  If words have no power, than what are koans? But words do have power.  

p 251, middle: Are words inherently dualistic?  Are you sure? (Answer yes or no) .... How sure are you? Answer: Very sure, somewhat sure, a little sure, not too sure, not at all sure.  Perhaps words represent categories? Maybe there are archetypes of words?  Plato thought there was an ideal world and a real one.

p 254, middle:  Cooking monk did it without words, by demonstration.  Good win for the cook!

p 254, bottom: But then, who is asking and who is answering?

p 258:  I don't really  see the point of the left and right sphere - can someone enlighten me?

p. 260: What led Hofstadter to thinking of counting I's?  Once you get this, the puzzle is orders of magnitude easier.  But how does one get this? What happens in our brains when we solve puzzles like this? Or fail to solve them?  

The key is a shift in view, from
  Is MU a theorem?
to
  Can a theorem have 0 I's?
and then to
  Can the I count be a multiple of 3?

S
p 261: A little extra hint if you don't get this:

Rule 1 doesn't change the I count
Rule 2 doubles the I count
Rule 3 subtracts 3 from the I count
Rule 4 doesn't affect the I count.

now, we next ask how each of these affects whether the I count is divisible by 3.  The key is what the remainder is on dividing by 3.

Rule 1 and 4 have no effect.  

Rule 2:
If the I-remainder (that is, the remainder after dividing the I count by 3), starts at 1, it goes to 2, then to 1, then to 2.  Because the I count itself goes something like 1  2   4   8  16....
and if you divide each by 3, you get alternating 1's and 2's.
If the remainder is 2, then the same pattern holds

so, you can't go from a "not divisible by 3" to a "divisible by 3"

Rule 3: The remainder stays the same.

--
page 263, you only need to follow the details of the math if you are interested.  The key thing is that you  CAN make such rules.  

p. 264:  You can make statements ABOUT number theory IN number theory.   Some very long number (it doesn't really matter which) will mean "17 is prime".  

p 266, middle, the bit starting "the fact is..." is central to Godel's proof.

Originally posted to plf515 on Sun Mar 22, 2009 at 03:38 AM PDT.

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Comment Preferences

  •  I think Hofstadter runs into some problems (7+ / 0-)

    trying to 'explain' the koans and Mumon's commentary.  If a koan is trying to explain something beyond words - to expose the false duality of words by allowing the mind to 'break through' - then trying to parse the meaning of the koans is a fool's errand.  If it were explicable, we wouldn't need the koan in the first place!

    There are many things I like about this book, especially in how well it overreaches to include seemingly disparate ideas: math, logic, music, visual arts, etc.  I don't think his foray into Zen is quite as convincing.

    Saint, n. A dead sinner revised and edited. - Ambrose Bierce

    by pico on Sun Mar 22, 2009 at 04:01:53 AM PDT

  •  OMG! (3+ / 0-)

    I had no idea anyone was writing about this. This was my favorite book about 25 years ago or so. Now, I have to see if I can find it again!

    Squeee!!!!

  •  Is language inherently dualistic? (1+ / 0-)
    Recommended by:
    plf515

    I'm unclear what this is supposed to mean.

    Plato thought there was an ideal world and a real one.

    He was wrong.

    All of this has happened before and all of this will happen again -- Peter Pan

    by MnplsLiberal on Sun Mar 22, 2009 at 09:06:35 AM PDT

    •  Was Plato wrong? (1+ / 0-)
      Recommended by:
      plf515

      Or was his thought wrong?

      I just upgraded internet speed. Now I can be late to the best diaries, faster.

      by mississippi boatrat on Sun Mar 22, 2009 at 09:28:48 AM PDT

      [ Parent ]

    •  Yeah, Plato was wrong about a lot of things (0+ / 0-)

      although perhaps Aristotle was wrong about more things.

      •  I still don't understand (1+ / 0-)
        Recommended by:
        plf515

        What it is supposed to mean that language is dualistic. I suspect it is not. I doubt that language can be reduced to a binary representation. If that's what's intended.

        All of this has happened before and all of this will happen again -- Peter Pan

        by MnplsLiberal on Sun Mar 22, 2009 at 10:31:46 AM PDT

        [ Parent ]

        •  The 'language is dualistic" line comes from GEB (0+ / 0-)

          and it's part of Hofstadter's attempt at discussing Zen.

          I think his discussion of Zen is largely a failure; unlike his discussion of the three main topics of his book, which is brilliant.

        •  Meaning of dualistic ... (1+ / 0-)
          Recommended by:
          plf515

          ... is not the common definition. It doesn't mean that words provide two or multiple meanings. It means (and I hope my paraphrase is meaningful and helpful) that words are used to define things in relation to other things, that they work by dividing things into categories. Thus, in words, the world isn't one thing, no thing is one thing. For Hofstadter's discussion of this, see page 251.

          I agree that words are dualistic, but there's really no way around them. For example, the chef's way of saying what a vase is without using it's name, by kicking it over, is maybe a bit better than the Monk's saying it wasn't a wooden shoe. But only a little. After all, the demonstration was only a way of miming the words, "This is a vase." The method still uses words, still imparts the dualism of categorizing a vase from everything around it.

          I guess we think in both words and symbols and translate back and forth, but isn't the thinking itself dualistic? Makes me think of reports of people with split brains(from injury, virus, etc). It's as if the individual is two people -- one verbal, one non-verbal -- who seem unaware of each other. (There are stories of wrestling matches between the left and right hands of such individuals.) Would the non-verbal one be enlightened?  

  •  Just unearthed GEB (3+ / 0-)
    Recommended by:
    N in Seattle, Ahianne, plf515

    so I'll have to start playing catchup reading your earlier diaries - thanks for the handy link!

    ...it is unfortunate that the opposition to the Democrats in this country now consists entirely of crazy people. - NNadir

    by RunawayRose on Sun Mar 22, 2009 at 10:57:55 AM PDT

  •  Of words, sounds, patterns, sets and ....ZEN.... (1+ / 0-)
    Recommended by:
    plf515

    Of words, sounds, patterns, sets and ....ZEN....

    (first, my common disclaimer ...)

    (again), not having read the book, but I am, none the less, inclined to share my thoughts on a few items of particular note and relevance to me in my own life path and training, namely, this topic of words and sounds.

    Okay, first ... I would like to offer up an axiom/teaching, a piece of lore, that we use/employ in our training in certain ancient martial lineages ... this being that ... "words/sounds have power."

    And, certain sounds have more power than others, or rather, certain sounds have more power than others with respect to certain goals, or intentions, or purposes. The most common one that many/most people have heard of today is AUM, though, as I was taught, the correct pronunciation of this word is IEAOUM ... in other words, all five vowels (from the ancient language of MU (Sumeria) must be enunciated for this word/sound to have its true power, for its true/full purpose/power to be awakened/channeled. It is said that this word/sound, if enunciated correctly, is the same vibrating frequency as the earth. It is also said that this is the same vibrating frequency of cat's purring ... ahhh, correction, of Pooties purring, (and some people actually (naively) think pooty diaries are not important, hah! ... lol) ... anyway, it is also said/taught (in Shinto Mikkyo (tantric) lineages) that at the gates of heaven, stand two guardian demons, in the shape of dogs ... which, btw, you will see depicted at the gates of Shinto temples, and one of these guardian demons has his/her mouth wide open, thus enunciating an "AHHHH", sound, and the other has his/her mouth closed, thus enunciating a "hmmmm", sound ...

    and so, together, they are jointly making the eternal sound of ...
    IEA-OUHMM-IEA-OUHMM-IEA-OUHMM-IEA-OUHMM-IEA-OUHMM-IEA-OUHMM

    the "IEAA" is the masculine expanding,
    the "OUHMM" is the feminine contracting,

    expansion~contraction, which is the cosmic breath of the universe,

    a shadow, or a doorway, if you will, of the, to the, multi-verse,

    in Japanese, the term for these practices of chanting sounds is KOTO-DAMA,

    ... sacred vibrations ...

    And, by doing them, we not only awaken something within us, and something surrounding us, but at an even more base level, we are training ourselves to be in harmony with the uni-verse, with the earth, with the cosmos.

    Now, when we practice these chanting (energy channeling) tantric disciplines, it is taught that one must summon these sounds from one's center, (in Chinese, known as your TAN-DIEN, or in Japanese, I-TEN ... or, one-point ... this is where your energy/life-force/"CHI/KI" is said to be stored, for what its worth), it is located approximately 2 inches below your navel, hence the western translation of the proverbial zen practice of "navel gazing" really has nothing at all to do with actually physically looking at your navel, but rather, looking at, or rather, looking from, your navel, (or CENTER) with your mind's eye, and, equally importantly, breathing from this region of your body, but in reality, one should breathe with one's entire body, ... but note, here, the tan-dien(i-ten) is not on the surface of your belly, but rather, it is really located at the epicenter of your body, ... if you draw a line from your perineum to your cranium, you will get the picture, and you should picture this, because vision, ie, mental imagery, is, yes, a core part of this ... a core part of the power of this practice ... the term in Japanese for the (trinity) key to this, is, san-mit-su ... which means, THOUGHT-WORD-DEED, which is where I am heading with this dissertation ... you see, one might, I believe, readily accept that certain words, like yelling the word "FIRE" in a crowded theater, do have a certain measure of power, and as we, in this dear kossack community, must acknowledge, the words and speeches of Obama do have power, (have had power) .... and further to this, "thoughts" have power, too, an inspirational thought can move/motivate a man to exert great effort to achieve earth changing accomplishments, ... which leads to the third, which is, of course, "deed" ... I think it is undeniable to all that "deeds" have power, a measure of power ... and herein lies the heart of the axiom .... each of these, three, alone, (thought-word-deed) have power, they have a certain measure of power, but together, combined, they have an exponentially greater amount of power, ... ie, when used in tandem, they awaken a fourth thing, a living thing, .... the whole is greater than the sum of all its parts ....

    Consider, a car, which has four wheels, rubber tires, a steering wheel, a chassis, brakes, an engine, seats, doors, windows, axles, ... etc ... it has a whole host of things, but until it is a fully working cohesive whole, it is merely a very heavy piece of metal, that maybe you can roll down a hill, or not even that if the wheels are off, but otherwise, it is simply a place to sit, and sleep uncomfortably,

    but, if you complete the car, and add gas, well, it now becomes a vehicle of an orders-of-magnitude greater power,

    ... it, all of a sudden, becomes another whole thing entirely ...

    the completion of the car, quite literally, has a transformative resulting affect that is profoundly greater than the mere sum of all its parts,

    and, similarly, it is said, thought-word-deed, combined, with certain archetypal sounds, has the same affect on the nature of our existence and what we are capable of, as human beings,

    ie, that this will awaken certain dormant aspects of humanity ...

    it is said, within the ancient tantric teachings, that certain words, do not MEAN-A-THING, rather, that they ARE-A-THING, in and of themselves,

    consider, what does a HIGH-C NOTE mean?

    remember, if you will, that this is a note, if sung correctly, by a soprano singer, can shatter glass, right?

    okay, so, what does that HIGH-C NOTE sound mean?

    well, it doesn't matter what meaning we ascribe to that sound, because that sound is a thing, in and of its own self, irrespective of whatever meaning we ascribe to it,

    and this is, I assert, a transcendent axiom,

    why?

    well, because, although the destruction of a thing, like the shattering of glass, may be impressive, no doubt, what is far more important and relevant, AND POWER is the creating/building of a thing,

    As Obama recently stated in his message to Iran ... "the measure of that greatness is not the capacity to destroy, it is your demonstrated ability to build and create" ... which he also happened to mention in his victory speech, on my birthday, November 4th.

    And, if certain sounds can destroy, it stands to reason that certain sounds can also stimulate growth, or awakening,

    Note ... it has been proven that children who listen to Mozart perform markedly ("statistically relevant") better, intellectually, than those who do not.

    in other words, certain sound patterns awaken our higher order brain functions,

    and, according to the Shao-Lin Taoist teachings, it is taught
    that certain patterns of thought/movement/sound will awaken
    a human being's greater self, an immortal self,

    ... if you believe in such notions ...

    and this pattern is, in fact, symbolized in the very
    term SHAO-LIN, itself,

    it is the heart of Chinese medicine,

    it is known as the FIVE-ELEMENT theory,

    or, the five elemental manifestations,

    *** shao-lin = little/young wood ****

    wood - the fifth elment,

    wood - fire - ash - metal - water

    wood = life
    fire = death
    ash = rejuvenation
    metal = cleansing
    water = rebirth

    ie, that there is a pattern (a progression) of five,

    correlating, of course, to our fingers,

    note, of course, that there are five vowels,

    hmmm, and though I realize that many today speak of the chakra system related to human body, as one that has seven charkra centers ... in certain tantric teachings they actually classify it as five "chakras" ... the main difference being the head region, which the seven-chakra system generally breaks into three(3) parts, but the five chakra system combines this region into one chakra ...

    ... (the three, as one) ...

    and what does the word chakra mean?

    chakra = wheel,
    (which is something that one must turn)
    (ie, a vortex of power/energy)
    (in which direction? both!)

    and what does TANTRIC/TANTRA mean?

    TANTRA = to weave,
    (as in, to weave a web)

    And so, my point here is this ...

    if thoughts do have power, and if we are speaking about a domain of thought (a set), called "mathematics," then the question arises, what is the nature of the power of thought in mathematics, ie, what is the isomorphic system wherein mathematical thoughts transmute/apply to other domains ... and to this question, I assert that the numeric pattern, itself, is a core part of this ... ie, a pattern of five, or seven, or nine, or three, or two ... or maybe, each of the numbers, from 1 to 9, each/all have their own unique power, as patterns ...

    ie, all numeric patterns have power,

    all are intrinsic to our humanity,

    and all are important, and all awaken a thing,

    all of them stimulate ... a thing ...

    all of them are catalysts to awaken a certain
    primal aspect of our humanity, (I would suggest),

    hmmm .... okay ...

    The second (related) topic is a somewhat purely mathematical comment, about these RULES, and the "COUNTING OF I" ... which, in effect, "sets",

    And, in set theory, in databases, we pay particularly close attention to the cardinality of data.

    Why?

    Well, because the cardinality of data actually represents sub-set, within the greater/superset,

    And further, the RULES delineated represent another set of sets.

    My point is this, (and I really wish I read the book, and in particular, I wish I had this passage to read ...)

    p 266, middle, the bit starting "the fact is..." is central to Godel's proof.

    In any case, my point is this, these super-sets and subsets represent a pattern, a system, a progression, and inherent to that patter/progression, is, I would suggest, a dormant pattern within our humanity, waiting to be awakened, (or already awakened) ... and this is the area wherein, I believe, the evolution of humanity is written in a certain eternal language, the language of our soul.

    Third ..

    p 251, middle: Are words inherently dualistic?

    Well, let me mention, to evolve this ~thought/pattern~ that if words are dualistic in nature, if we apply the Shao-Lin Taoist notion of what we believe is the true dualism, in nature, it is not a dualism of opposites, but rather, a dualism of compliments, ... ie, we are looking for sets of "two-things" (dualistic things) that complete each other ... and if we consider the intrinsic nature of the relationship between man and woman, it is NOT one of opposites (I would hope), but rather ... we say in shao-lin, it is (or should be) a relationship of compliments.

    In Shao-Lin we say, that it is not that YANG=STRONG and YIN=WEAK, ... rather, we say that if YANG=STRONG then YIN=SENSITIVE, because strength without sensitivity is worthless (ineffective, pointless, even dangerous) but strength added to sensitivity, now that is true power.

    And, of course, this mathematical pattern/thought, of two(2), being the numeric principle inherent to the isomorphic power of this two(2) archetype.

    And, of course, CENTER, (ie, breathing/thinking/moving from center (tan-dien/i-ten)) being the isomorphic power/pattern of the ONE(1) archetype ... a thunder lightening bold ... the VAJRA ... Thor's Hammer.

    (Hmmm, you know, I recently read somewhere that the heat of the thunder lightening bolt is about 20 times the heat of the surface of the sun ... this seemed important ... )

    And, thought-word-deed(mind-spirit-body) being the isomorphic power/pattern of the three(3) archetype.

    And what is the isomorphic power/pattern of the four(4) archetype?

    And what is the isomorphic power/pattern of the five(5) archetype? (well, I already provided that above)

    And what is the isomorphic power/pattern of the six(6) archetype?

    And what is the isomorphic power/pattern of the seven(7) archetype?

    And what is the isomorphic power/pattern of the eight(8) archetype?

    And what is the isomorphic power/pattern of the nine(9) archetype?

    Hmmm ... I wonder ...

    Well, this could obviously become a lengthy discourse, (and I probably should write my own book on this subject one day, because it does most certainly warrent this, (oh wait, I already am, lol) but let me leave you with a final comment regarding the nine(9) archetype and .... ZEN .... because this commentary simply would not be complete without this.

    In the Shin-Den (spirit realm) teachings of the Shinto Mikkyo lineage, there is a body of knowledge/practices, known as the "KU-JI KIRI" (or, the "KU-JI NO IN") ... which roughly translates as "THE NINE LEVELS OF POWER" ... or "THE NINE SLASH METHOD" or "THE NINE PROTECTION METHOD" ...

    and, in these teachings, the ninth level of power, the most powerful of the levels, is known as "ZEN".

    Now, it should be noted that the word for NINE(9) in Japanese is "KU", and also, it should equally, if not more importantly/profoundly be noted, that the word "KU" also happens to mean "VOID" ... ie, that place from which all things spring, the primordial VOID ... or, maybe, one might refer to this as CHAOS, the first of the ancient greek prodigia gods, the god from which all other gods/goddesses sprung forth ... ie, what is creation, but the summoning from NO-THING("void", or "MU=EMPTINESS") into THING-NESS.

    (ie ... nine(9) is akin to zero(0) ...)

    And so, what is the power of ZEN/KU?

    What is the isomorphic power/pattern of the nine(9) archetype?

    Well, if these teachings and this mikkyo lore contains true keys to this puzzle, then I would suggest that the ninth level of power, which is, maybe, endemic of ZEN BUDDHISM, is the power of VOID, ie, to summon VOID, ie, the power "undo a thing".

    Peace,

    Namaste,
    Ubuntu,
    Satnam,

    Arthur,

    ~we study the old to understand the new~from one thing know ten thousand~to see things truly one must see what is in the light and what lies hidden in shadow~

    by ArthurPoet on Sun Mar 22, 2009 at 12:29:44 PM PDT

  •  Left and right spheres (2+ / 0-)
    Recommended by:
    Dave in RI, plf515

    Escher's drawing on p238 is a fascinating study in surface and bulk properties of materials, among other things. Hofstadter uses it in a way (and in a context) that doesn't really allow much reference to the left and right spheres, and glosses over the fact that the side spheres don't really reflect the other two spheres at all, despite the text.

    The middle sphere is reflective but the left hand sphere is transparent, and reacts in a very different way to the ambient light, as evidenced most obviously by the slightly reflective table top on which the three spheres rest. The same surface interestingly shows almost no difference between the mirror sphere and the matt sphere.

    An allegory might be the perception of various groupings of different phenomena. Looking at the top left reflection of the window in the glass sphere and the mirror sphere, one might regard them as similar; seeing the lack of reflected detail of the artist, one might regard the glass and matt spheres as similar; and seeing the dark shadows on the table, one could see the mirror and matt spheres as similar. All possible pairings can be justified.

    This is not a sig-line.

    by Joffan on Sun Mar 22, 2009 at 10:05:19 PM PDT

  •  Godel numbers ... (1+ / 0-)
    Recommended by:
    plf515

    ... Still trying to get my mind around them. One thing that bothers me -- just arbitrarily assigning a number to a symbol. Why should that give us anything arithmetically meaningful? How do we know it works for all numbers?

    Then in setting up the rules (page 263), why is it valid to assign different values to the variables from one rule to the other?

    Gotta go over it all again, I guess.

    The solution of the MU puzzle was so easy, though!!! Why didn't I think of that??? :-)

     

    •  I don't know if it works for ALL (0+ / 0-)

      numbers, but it seems to.  And I think (although I have no proof) that any set of substitutions could be made into arithmetical rules, with enough ingenuity.  

      The MU puzzle is one of those where a shift of perspective brings enlightenment.  

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