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Godel, Escher, Bach Chapter 9. Mumon and Godel
Author: Douglas R. Hofstadter Publisher: New York, NY: Basic Books. Publish Date: 1979 Review Date: Status:⌛️
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What Is Zen? I’M NOT SURE I know what Zen is. In a way, I think I understand it very well; but in a way, I also think I can never understand it at all. Ever since my freshman English teacher in college read Joshu’s MU out loud to our class, I have struggled with Zen aspects of life, and probably I will never cease doing so. To me, Zen is intellectual quicksand-anarchy, darkness, meaninglessness, chaos. It is tantalizing and infuriating. And yet it is humorous, refreshing, enticing. Zen has its own special kind of meaning, brightness, and clarity. I hope that in this Chapter, I can get some of this cluster of reactions across to you. And then, strange though it may seem, that will lead us directly to Godelian matters.
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One of the basic tenets of Zen Buddhism is that there is no way to characterize what Zen is. No matter what verbal space you try to enclose Zen in, it resists, and spills over. It might seem, then, that all efforts to explain Zen are complete wastes of time. But that is not the attitude of Zen masters and students. For instance, Zen koans are a central part of Zen study, verbal though they are. Koans are supposed to be “triggers” which, though they do not contain enough information in themselves to impart enlightenment, may possibly be sufficient to unlock the mechanisms inside one’s mind that lead to enlightenment. But in general, the Zen attitude is that words and truth are incompatible, or at least that no words can capture truth.
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Possibly in order to point this out in an extreme way, the monk Mumon (“No-gate”), in the thirteenth century, compiled forty-eight koans, following each with a commentary and a small “poem”. This work is called “The Gateless Gate” or the Mumonkan (“No-gate barrier”). It is interesting to note that the lives of Mumon and Fibonacci coincided almost exactly: Mumon living from 1183 to 1260 in China, Fibonacci from 1180 to 1250 in Italy. To those who would look to the Mumonkan in hopes of making sense of, or “understanding”, the koans, the Mumonkan may come as a rude shock, for the comments and poems are entirely as opaque as the koans
which they are supposed to clarify. Take this, for example:’ -
Koan:
Hogen of Seiryo monastery was about to lecture before dinner when he noticed that the bamboo screen, lowered for meditation, had not been rolled up. He pointed to it. Two monks arose wordlessly from the audience and rolled it up. Hogen, observing the physical moment, said, “The state of the first monk is good,
not that of the second.”
Mumon’s Commentary:
I want to ask you: which of those two monks gained and which lost? If any of you has one eye, he will see the failure on the teacher’s part. However, I am not discussing gain and loss.
Mumon’s Poem:
When the screen is rolled up the great sky opens, Yet the sky is not attuned to Zen. It is best to forget the great sky And to retire from every wind.
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Or then again, there is this one:2
Koan:
Goso said: “When a buffalo goes out of his enclosure to the edge of the abyss, his horns and his head and his hoofs all pass through, but why can’t the tail also pass?”
Mumon’s Commentary:
If anyone can open one eye at this point and say a word of Zen, he is qualified to repay the four gratifications, and, not only that, he can save all sentient beings under him. But if he cannot say such a word of Zen, he should turn back to his tail.
Mumon’s Poem:
If the buffalo runs, he will fall into the trench;
If he returns, he will be butchered.
That little tail
Is a very strange thing.
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I think you will have to admit that Mumon does not exactly clear everything up. One might say that the metalanguage (in which Mumon writes) is not very different from the object language (the language of the koan). According to some, Mumon’s comments are intentionally idiotic, perhaps meant to show how useless it is to spend one’s time in chattering about Zen. How ever, Mumon’s comments can be taken on more than one level. For instance, consider this :3
Koan:
A monk asked Nansen: “Is there a teaching no master ever taught before?” Nansen said: “Yes, there is.” “What is it?” asked the monk. Nansen replied: “It is not mind, it is not Buddha, it is not things.”
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Mumon’s Commentary:
Old Nansen gave away his treasure-words. He must have been greatly upset
Mumon’s Poem:
Nansen was too kind and lost his treasure. Truly, words have no power. Even though the mountain becomes the sea, Words cannot open another’s mind.
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In this poem Mumon seems to be saying something very central to Zen, not making
idiotic statements. Curiously, however, the poem is referential, and thus it is a comment not only on Nansen’s words, but on its own ineffectiveness. This type of paradox is quite characteristic of Zen. It is an attempt to “break the mind of logic”. You see this paradox quality in the koan, as well. Concerning Mumon’s commentary, do think that Nansen was really so sure of his answer? Or did the “correctness” of his answer matter at all? Or does correctness play any role in Zen? What is the difference between correctness and truth, or is there any? What if Nansen had said, “No, there is not any such teaching”? Would it have made any difference? Would his remark have been immortalized in a koan?
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Here is another koan which aims to break the mind of logic:’
The student Doko came to a Zen master, and said: “I am seeking the truth. In what state of mind should I train myself, so as to find it?” Said the master, “There is no mind, so you cannot put it in any state. There is no truth, so you cannot train yourself for it.” “If there is no mind to train, and no truth to find, why do you have these monks gather before you every day to study Zen and train themselves for this study?” “But I haven’t an inch of room here,” said the master, “so how could the monks gather? I have no tongue, so how could I call them together or teach them?” “Oh, how can you lie like this?” asked Doko. “But if I have no tongue to talk to others, how can I lie to you?” asked the master. Then Doko said sadly, “I cannot follow you. I cannot understand you""I cannot understand myself,” said the master.
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If any koan serves to bewilder, this one does. And most likely, can bewilderment is its precise purpose, for when one is in a bewildered s one’s mind does begin to operate nonlogically, to some extent. Only by stepping outside of logic, so the theory goes, can one make the leap to enlightenment. But what is so bad about logic? Why does it prevent the leap to enlightenment?
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To answer that, one needs to understand something about what enlightenment is. Perhaps the most concise summary of enlightenment would be: transcending dualism. Now what is dualism? Dualism is the conceptual division of the world into categories. Is it possible to transcend this natural tendency? By prefixing the word “division” by the word “conceptual”, I may have made it seem that this is an intellectual or conscious effort, and perhaps thereby given the impression that dualism could overcome simply by suppressing thought (as if to suppress thinking actually were simple!). But the breaking of the world into categories takes place far below the upper strata of thought; in fact, dualism is just as a perceptual division of the world into categories as it is a conceptual division In other words, human perception is by nature a dualistic phenomenon which makes the quest for
enlightenment an uphill struggle, to say the least.
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At the core of dualism, according to Zen, are words—just plain words. The use of words is inherently dualistic, since each word represents, obviously, a conceptual category. Therefore, a major part of Zen is the against reliance on words. To combat the use of words, one of the devices is the koan, where words are so deeply abused that one’s mind is practically left reeling, if one takes the koans seriously. Therefore perhaps wrong to say that the enemy of enlightenment is logic; rather dualistic, verbal thinking. In fact, it is even more basic than that: it is perception. As soon as you perceive an object, you draw a line between it and the rest of the world; you divide the world, artificially, into parts you thereby miss the Way.
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Here is a koan which demonstrates the struggle against words:
Koan: Shuzan held out his short staff and said: “If you call this a short staff, you oppose its reality. If you do not call it a short staff, you ignore the fact. Now what do you wish to call this?
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Mumon’s Commentary:
If you call this a short staff, you oppose its reality. If you do not call it a short staff, you ignore the fact. It cannot be expressed with words and it cannot be expressed without words. Now say quickly what it is. Mumon’s Poem:
Holding out the short staff, He gave an order of life or death. Positive and negative interwoven, Even Buddhas and Patriarchs cannot escape this attack.
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(“Patriarchs” refers to six venerated founders of Zen Buddhism, of whom Bodhidharma is the first, and Eno is the sixth.) Why is calling it a short staff opposing its reality? Probably because such a categorization gives the appearance of capturing reality, whereas the surface has not even been scratched by such a statement. It could be compared to saying “5 is a prime number”. There is so much more-an infinity of facts-that has been omitted. On the other hand, not to call it a staff is, indeed, to ignore that particular fact, minuscule as it may be. Thus words lead to some truth-some falsehood, perhaps, as well-but certainly not to all truth. Relying on words to lead you to the truth is like relying on an incomplete formal system to lead you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system-no matter how powerful-cannot lead to all truths. The dilemma of mathematicians is: what else is there to rely on, but formal systems? And the dilemma of Zen people is, what else is there to rely on, but words? Mumon states the dilemma very clearly:
“It cannot be expressed with words and it cannot expressed without words.”
Here is Nansen, once again:
‘Joshu asked the teacher Nansen, “What is the true Way?” Nansen answered, “Everyday way is the true Way.’ Joshu asked, “Can I study it?” Nansen answered, “The more you study, the further from the Way.” Joshu asked, “If I don’t study it, how can I know it?” Nansen answered, “The Way does not belong to things seen: nor to things unseen. It does not belong to things known: nor to things unknown. Do not seek it, study it, or name it. To find yourself on it, open yourself wide as the sky.” [See Fig. 50.]
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This curious statement seems to abound with paradox. It is a little reminiscent of this surefire cure for hiccups: “Run around the house three times without thinking of the word ‘wolf’.” Zen is a philosophy which seems to have embraced the notion that the road to ultimate truth, like the only surefire cure for hiccups, may bristle with paradoxes.
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If words are bad, and thinking is bad, what is good? Of course, to ask this is already horribly dualistic, but we are making no pretense of being faithful to Zen in discussing Zen-so we can try to answer the question seriously. I have a name for what Zen strives for: ism. Ism is an antiphilosophy, a way of being without thinking. The masters of ism are rocks, trees, clams; but it is the fate of higher animal species to have to strive for ism, without ever being able to attain it fully. Still, one is occasionally granted glimpses of ism. Perhaps the following koan offers such a glimpse:7
Hyakujo wished to send a monk to open a new monastery. He told his pupils that whoever answered a question most ably would be appointed. Placing a water vase on the ground, he asked: “Who can say what this is without calling its name?” The chief monk said: “No one can call it a wooden shoe.” Isan, the cooking monk, tipped over the vase with his foot and went out. Hyakujo smiled and said: “The chief monk loses.” And Isan became the master of the new monastery.
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To suppress perception, to suppress logical, verbal, dualistic thinking-this is the essence of Zen, the essence of ism. This is the Unmode-not Intelligent, not Mechanical, just “Un”. Joshu was in the Unmode, and that is why his ‘MU’ unasks the question. The Un-mode came naturally to Zen Master Unmon:8
One day Unmon said to his disciples, “This staff of mine has transformed itself into a dragon and has swallowed up the universe! Oh, where are the rivers and mountains and the great earth?“
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Zen is holism, carried to its logical extreme. If holism claims that things can only be understood as wholes, not as sums of their parts, Zen goes one further, in maintaining that the world cannot be broken into parts at all. To divide the world into parts is to be deluded, and to miss enlightenment.
A master was asked the question, “What is the Way?” by a curious monk.
“It is right before your eyes,” said the master.
“Why do I not see it for myself?”
“Because you are thinking of yourself.”
“What about you: do you see it?”
“So long as you see double, saying ‘I don’t’, and ‘you do’, and so on, your eyes are clouded,” said the master.
“When there is neither ‘I’ nor ‘You’, can one see it?”
“When there is neither ‘I’ nor ‘You’, who is the one that wants to see it?“9
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Apparently the master wants to get across the idea that an enlightened state is one where the borderlines between the self and the rest of universe are dissolved. This would truly be the end of dualism, for as he says, there is no system left which has any desire for perception. But what is that state, if not death? How can a live human being dissolve the borderlines between himself and the outside world?
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The Zen monk Bassui wrote a letter to one of his disciples who was about to die, and in it he said:
“Your end which is endless is as a snowflake dissolving in the pure air.” The snowflake, which was once very much a discernible subsystem of the universe, now dissolves into the larger system which once held it. Though it is no longer present as a distinct subsystem, its essence is somehow still present, and will remain so. It floats in Tumbolia, along hiccups that are not being hiccupped and characters in stories that are being read … That is how I understand Bassui’s message.
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Zen recognizes its own limitations, just as mathematicians have learned to recognize the limitations of the axiomatic method as a method attaining truth. This does not mean that Zen has an answer to what lies beyond Zen any more than mathematicians have a clear understanding the forms of valid reasoning which lie outside of formalization. One of the clearest Zen statements about the borderlines of Zen is given in the following strange koan, very much in the spirit of Nansen:10
Tozan said to his monks, “You monks should know there is an even higher understanding in Buddhism.” A monk stepped forward and asked, “What the higher Buddhism?” Tozan answered, “It is not Buddha.”
There is always further to go; enlightenment is not the end-all of Zen. And there is no recipe which tells how to transcend Zen; the only thing can rely on for sure is that Buddha is not the way. Zen is a system and cannot be its own metasystem; there is always something outside of Zen which cannot be fully understood or described in Zen.
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In questioning perception and posing absurd answerless riddles, Zen has company, in the person of M. C. Escher. Consider Day and Night (Fig. 4 masterpiece of “positive and negative interwoven” (in the words of Mumoni). One might ask, “Are those really birds, or are they really fields? Is it really night or day?” Yet we all know there is no point to such questions. The picture, like a Zen koan, is trying to break the mind of logic. Escher also delights in setting up contradictory pictures, such as Another World
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Figure 15
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(Fig. 48)-pictures that play with reality and unreality the same way as Zen plays with reality and unreality. Should one take Escher seriously? Should one take Zen seriously?
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There is a delicate, haiku-like study of reflections in Dewdrop (Fig. 47); and then there are two tranquil images of the moon reflected in still waters: Puddle (Fig. 51), and Rippled Surface (Fig. 52). The reflected moon is a theme which recurs in various koans. Here is an example:’
Chiyono studied Zen for many years under Bukko of Engaku. Still, she could not attain the fruits of meditation. At last one moonlit night she was carrying water in an old wooden pail girded with bamboo. The bamboo broke, and the bottom fell out of the pail. At that moment, she was set free. Chiyono said, “No more water in the pail, no more moon in the water.”
Three Worlds: an Escher picture (Fig. 46), and the subject of a Zen koan:12
A monk asked Ganto, “When the three worlds threaten me, what shall I do?” Ganto
answered, “Sit down.” “I do not understand,” said the monk. Canto said, “Pick up the mountain and bring it to me. Then I will tell you.”
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In Verbum (Fig. 149), oppositions are made into unities on several levels. Going around we see gradual transitions from black birds to white birds to black fish to white fish to black frogs to white frogs to black birds … six steps, back where we started! Is this a reconciliation of the dichotomy of black and white? Or of the trichotomy of birds, fish, and frogs? Or sixfold unity made from the opposition of the evenness of 2 an oddness of 3? In music, six notes of equal time value create a rhythmic ambiguity-are they 2 groups of 3, or 3 groups of 2? This ambiguity has a name: hemiolia. Chopin was a master of hemiolia: see his Waltz op. 42, or his Etude op. 25 , no. 2. In Bach, there is the Tempo di Menuetto from the keyboard Partita no. 5, or the incredible Finale of the first Sonata unaccompanied violin, in G Minor.
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As one glides inward toward the center of Verbum, the distinctions gradually blur, so that in the end there remains not three, not two, but one single essence: “VERBUM”, which glows with brilliancy-perhaps a symbol of enlightenment. Ironically, ‘verbum” not only is a word, but means “word”-not exactly the most compatible notion with Zen. On the other hand, “verbum” is the only word in the picture. And Zen master 1 once said, “The complete Tripitaka can be expressed in one character (“Tripitaka”, meaning “three baskets”, refers to the complete texts of the original Buddhist writings.) What kind of decoding-mechanism, I wonder would it take to suck the three baskets out of one character? Perhaps one with two hemispheres.
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Finally, consider Three Spheres II (Fig. 53), in which every part of the world seems to contain, and be contained in, every other part : the writing table reflects the spheres on top of it, the spheres reflect each other, as well as the writing table, the drawing of them, and the artist drawing it. The endless connections which all things have to each other is only hinted at here, yet the hint is enough. The Buddhist allegory of “Indra’s Net” tells of an endless net of threads throughout the universe, the horizontal threads running through space, the vertical ones through time. At every crossing of threads is an individual, and every individual is a crystal bead. The great light of “Absolute Being” illuminates and penetrates every crystal bead; moreover, every crystal bead reflects not only the light from every other crystal in the net-but also every reflection of every reflection throughout the universe.
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To my mind, this brings forth an image of renormalized particles: in every
electron, there are virtual photons, positrons, neutrinos, muons … ; in every photon, there are virtual electrons, protons, neutrons, pions … ; in every pion, there are … But then another image rises: that of people, each one reflected in the minds of many others, who in turn are mirrored in yet others, and so on. Both of these images could be represented in a concise, elegant way by using Augmented Transition Networks. In the case of particles, there would be one network for each category of particle; in the case of people, one for each person. Each one would contain calls to many others, thus creating a virtual cloud of ATN’s around each ATN. Calling one we create calls on others, and this process might cascade arbitrarily far, until bottomed out.
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Let us conclude this brief excursion into Zen by returning to Mumon. Here is his comment on Joshu’s MU:13
To realize Zen one has to pass through the barrier of the patriarchs. Enlightenment always comes after the road of thinking is blocked. If you do not pass the barrier of the patriarchs or if your thinking road is not blocked, whatever you think, whatever you do, is like a tangling ghost. You may ask “What is a barrier of a patriarch?” This one word, ‘MU’, is it.
This is the barrier of Zen. If you pass through it, you will see Joshu face to face. Then you can work hand in hand with the whole line of patriarchs. I this not a pleasant thing to do?
If you want to pass this barrier, you must work through every bone in you body, through every pore of your skin, filled with this question: “What ‘MU’?” and carry it day and night. Do not believe it is the common negative symbol meaning nothing. It is not nothingness, the opposite of existence. I you really want to pass this barrier, you should feel like drinking a hot iron ball that you can neither swallow nor spit out.
Then your previous lesser knowledge disappears. As a fruit ripening in season, your subjectivity and objectivity naturally become one. It is like dumb man who has had a dream. He knows about it but he cannot tell it. When he enters this condition his ego-shell is crushed and he can shake the heavens and move the earth. He is like a great warrior with a sharp sword. If Buddha stands in his way, he will cut him down; if a patriarch offers him an obstacle, he will kill him; and he will be free in his way of birth and death. He can enter any world as if it were his own playground. I will tell you how to d this with this koan:
Just concentrate your whole energy into this MU, and do not allow any discontinuation. When you enter this MU and there is no discontinuation — your attainment will be as a candle burning and illuminating the whole universe.
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From the ethereal heights of Joshu’s MU, we now descend to the private lowlinesses of Hofstadter’s MU … I know that you have already concentrated your whole energy into this MU (when you read Chapter 1). So now I wish to answer the question which was posed there:
Has MU theorum-nature, or not?
The answer to this question is not an evasive MU; rather, it is a resounding NO. In order to show
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We made two crucial observations in Chapter I:
(1) that the MU-puzzle has depth largely because it involves the interplay of lengthening and shortening rules; (2) that hope nevertheless exists for cracking the problem by employing a tool which is in some sense of adequate depth to handle matters of that complexity: the theory of numbers.
We did not analyze the MU-puzzle in those terms very carefully in Chapter I; we shall do so now. And we will see how the second observation (when generalized beyond the insignificant MIU-system) is one of the most fruitful realizations of all mathematics, and how it changed mathematicians’ view of their own discipline. For your ease of reference, here is a recapitulation of the MIU-system:
SYMBOLS: M, I, U
Axiom: MI
RULES:
I. If xI is a theorem, so is xIU.
II. If Mx is a theorem, so is Mxx.
III. In any theorem, III can be replaced by U.
IV. UU can be dropped from any theorem.
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According to the observations above, then, the MU-puzzle is merely a puzzle about natural numbers in typographical disguise. If we could only find a way to transfer it to the domain of number theory, we might be able to solve it. Let us ponder the words of Mumon, who said, “If any of you has one eye, he will see the failure on the teacher’s part.” But why should it matter to have one eye?
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If you try counting the number of I’s contained in theorems, you will soon notice that it seems never to be 0. In other words, it seems that no matter how much lengthening and shortening is involved, we can never work in such a way that all I’s are eliminated. Let us call the number of I’s in any string the I-count of that string. Note that the I-count of the axiom MI is 1. We can do more than show that the I-count can’t be 0-we can show that the I-count can never be any multiple of 3.
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To begin with, notice that rules I and IV leave the I-count totally undisturbed. Therefore we need only think about rules II and III. As far as rule III is concerned, it diminishes the I-count by exactly 3. After an application of this rule, the I-count of the output might conceivably be a multiple of 3-but only if the I-count of the input was also. Rule III, in short, never creates a multiple of 3 from scratch. It can only create one when it began with one. The same holds for rule II, which doubles the I-count. The reason is that if 3 divides 2n, then-because 3 does not divide 2-it must divide n (a simple fact from the theory of numbers). Neither rule II nor rule III can create a multiple of 3 from scratch. But this is the key to the MU-puzzle! Here is what we know:
(1) The I-count begins at 1 (not a multiple of 3); (2) Two of the rules do not affect the I-count at all; (3) The two remaining rules which do affect the I-count do so in such a way as never to create a multiple of 3 unless given one initially.
The conclusion-and a typically hereditary one it is, too-is that the I-count can never become any multiple of 3. In particular, 0 is a forbidden value of the I-count. Hence, MU is not a theorem of the MIU-system. Notice that, even as a puzzle about I-counts, this problem was plagued by the crossfire of lengthening and shortening rules. Zero became the goal; I-counts could increase (rule II), could decrease (rule III). Until we analyzed the situation, we might have thought that, with enough switching back and forth between the rules, we might eventually hit 0. Now, thanks to a simple number-theoretical argument, we know that is the impossible.
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Not all problems of the the type which the MU-puzzle symbolizes are so easy to solve as this one. But we have seen that at least one such puzzle could be embedded within, and solved within, number theory. We are going to see that there is a way to embed all problems about any formal system, in number theory. This can happen thanks to the discovery, by Gödel, of a special kind of isomorphism. To illustrate it, I will use MIU-system.
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We begin by considering the notation of the MIU-system. We map each symbol onto a new symbol:
M ⇐ ⇒ 3
I ⇐ ⇒ 1
U ⇐ ⇒ 0
The correspondence was chosen arbitrarily; the only rhyme or reason is that each symbol looks a little like the one it is mapped onto. Each number is called the Gödel number of the corresponding letter. Now I sure you can guess what the Gödel number of a multiletter string will be:
MU ⇐ ⇒ 30
MIIU ⇐ ⇒ 3110
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It is easy. Clearly this mapping between notations is an information preserving transformation; it is like playing the same melody on two different instruments.
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Let us now take a look at a typical derivation in the MIU-system, written simultaneously in both notations:
(1) MI --- axiom --- 31
(2) MII --- rule 2 --- 311
(3) MIIII --- rule 2 --- 31111
(4) MUI --- rule 3 --- 301
(5) MUIU --- rule 1 --- 3010
(6) MUIUUIU --- rule 2 --- 3010010
(7) MUIIU --- rule 4 --- 30110
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The left-hand column is obtained by applying our four familiar typographical rules. The right-hand column, too, could be thought of as having been generated by a similar set of typographical rules. Yet the right-hand column has a dual nature. Let me explain what this means.
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We could say of the fifth string (‘3010’) that it was made from the fourth, by appending a ‘0’ on the right; on the other hand we could equally well view the transition as caused by an arithmetical operation-multiplication by 10, to be exact. When natural numbers are written in the decimal system, multiplication by 10 and putting a `0’ on the right are indistinguishable operations. We can take advantage of this to write an arithmetical rule which corresponds to typographical rule I:
ARITHMETICAL RULE Ia: A number whose decimal expansion ends on the right in ‘1’
can be multiplied by 10.
We can eliminate the reference to the symbols in the decimal expansion by arithmetically describing the rightmost digit:
ARITHMETICAL RULE Ib: A number whose remainder when divided by 10 is 1, can
be multiplied by 10.
Now we could have stuck with a purely typographical rule, such as the following one:
TYPOGRAPHICAL RULE I: From any theorem whose rightmost symbol is ‘1’ a new
theorem can be made, by appending ‘0’ to the right of that ‘1’.
They would have the same effect. This is why the right-hand column has a “dual nature”. It can be viewed either as a series of typographical operations changing one pattern of symbols into another, or as a series arithmetical operations changing one magnitude into another. But there are powerful reasons for being more interested in the arithmetical version. Stepping out of one purely typographical system into another isomorphic typographical system is not a very exciting thing to do; whereas stepping clear out of the typographical domain into an isomorphic part of number theory has some kind of unexplored potential. It is as if somebody had known musical scores all his life, but purely visually-and then, all of a sudden, someone introduced him to the mapping between sounds a musical scores. What a rich, new world! Then again, it is as if somebody had been familiar with string figures all his life, but purely as string figures, devoid of meaning-and then, all of a sudden, someone introduced him the mapping between stories and strings. What a revelation! The discovery of Gödel-numbering has been likened to the discovery, by Descartes, of the isomorphism between curves in a plane and
equations in two variables; incredibly simple, once you see it-and opening onto a vast new world.
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Before we jump to conclusions, though, perhaps you would like to a more complete rendering of this higher level of the isomorphism. It is a very good exercise. The idea is to give an arithmetical rule whose action is indistinguishable from that of each typographical rule of the MIU-system. A solution is given below. In the rules, m and k are arbitrary natural numbers, and n is any natural number which is less than 10 to the m power.
|
Note: Fix these V
RULE 1: If we have made 10m + 1, then we can make 10 x (10m + 1)
Example: Going from line 4 to line 5. Here, m = 30.
RULE 2: If we have made 3 x 10m + n, then we can make 10m x (3 x 10m+n)+n.
Example: Going from line 1 to line 2, where both m and n equal 1.
RULE 3: If we have made k x 10m+ 111 x 10’+n, then we can make k x 10”+` + n.
Example: Going from line 3 to line 4. Here, m and n are 1, and k is 3.
RULE 4: If we have made k x 10rn+z + n, k x 10” +n. then we can make k x 10m + n
Example: Going from line 6 to line 7. Here, m = 2, n = 10, and k = 301.
Let us not forget our axiom! Without it we can go nowhere. Therefore, let us postulate that:
We can make 31.
Now the right-hand column can be seen as a full-fledged arithmetic process, in a new arithmetical system which we might call the 310-system
(1) 31 given
(2) 311 rule 2 (m=1, n=1)
(3) 31111 rule 2 (m=2, n=11)
(4) 301 rule 3 (m=1, n=1, k=3)
(5) 3010 rule 1 (m=30)
(6) 3010010 rule 2 (m=3, n=10)
(7) 30110 rule 4 (m=2, n=10, k=301)
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Notice once again that the lengthening and shortening rules are ever with us in this “310-system”; they have merely been transposed into the domain of numbers, so that the Godel numbers go up and down. If you look carefully at what is going on, you will discover that the rules are based on nothing more profound than the idea that shifting digits to left and right in decimal representations of integers is related to multiplications and divisions by powers of 10. This simple observation finds its generalization in the following
CENTRAL PROPOSITION: If there is a typographical rule which tells how certain digits are to be shifted, changed, dropped, or inserted in any number represented decimally, then this rule can be represented equally well by an arithmetical counterpart which involves arithmetical operations with powers of 10 as well as additions, subtractions, and so forth.
More briefly:
Typographical rules for manipulating numerals are actually arithmetical rules for operating on numbers.
This simple observation is at the heart of Gödel’s method, and it will have an absolutely shattering effect. It tells us that once we have a Gödel numbering for any formal system, we can straightaway form a set of arithmetical rules which complete the Gödel isomorphism. The upshot is that we can transfer the study of any formal system-in fact the study of all formal systems-into number theory.
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Just as any set of typographical rules generates a set of theorems, a corresponding set of natural numbers will be generated by repeated applications of arithmetical rules. These producible numbers play the same role inside number theory as theorems do inside any formal system. Of course, different numbers will be producible, depending on which rules are adopted. “Producible numbers” are only producible relative to a system of arithmetical rules. For example, such numbers as 31, 3010010, 3111, and so forth could be called MIU-producible numbers-an ungainly name, which might be shortened to MIU-numbers, symbolizing the fact that those numbers are the ones that result when you transcribe the MIU-system into number theory, via Gödel-numbering. If we were to Gödel-number the pq-system and then “arithmetize” its rules, we could call the producible numbers “pq-numbers”-and so on
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Note that the producible numbers (in any given system) are defined by a recursive method: given numbers which are known to be producible, we have rules telling how to make more producible numbers. Thus, the class of numbers known to be producible is constantly extending itself, in much the same way that the list of Fibonacci numbers, or Q-numbers, does. The set of producible numbers of any system is a recursively enumerable set. What about its complement-the set of nonproducible numbers? Is that set always recursively enumerable? Do numbers which are nonproducible share some common arithmetical feature?
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This is the sort of issue which arises when you transpose the study of formal
systems into number theory. For each system which is arithmetized, one can ask, “Can we characterize producible numbers in a simple way?” “Can we characterize
nonproducible numbers in a recursively enumerable way?” These are difficult questions of number theory. Depending on the system which has been arithmetized, such questions might prove too hard for us to resolve. But if there is any hope for solving such problems, it would have to reside in the usual kind of step-by-step reasoning as it applies to natural numbers. And that, of course, was put in its quintessential form in the previous Chapter. TNT seemed, to all appearances, to have captured all valid mathematical thinking processes in one single, compact system.
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Could it be, therefore, that the means with which to answer any question about any formal system lies within just a single formal system-TNT? It seems plausible. Take, for instance, this question: Is MU a theorem of the MIU-system?
Finding the answer is equivalent to determining whether 30 is a MIU number or not. Because it is a statement of number theory, we should expect that, with some hard work, we could figure out how to translate the sentence “30 is a MIU-number” into TNT-notation, in somewhat the same way as we figured out how to translate other number-theoretical sentences into TNT-notation. I should immediately caution the reader that such a translation, though it does exist, is immensely complex. If you recall, I pointed out in Chapter VIII that even such a simple arithmetical predicate as “b is a power of 10” is very tricky to code into TNT-notation-and the predicate “b is a MIU-number” is a lot more complicated than that! Still, it can be found; and the numeral SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSO can be substituted for every b. This will
result in a MONstrous string of TNT, a string of TNT which speaks about the MU-puzzle. Let us therefore call that string “MUMON”. Through MUMON
and strings like it, TNT is now capable of speaking “in code” about the MIU-system.
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In order to gain some benefit from this peculiar transformation of the original question, we would have to seek the answer to a new question: Is MUMON a theorem of TNT? All we have done is replace one relatively short string (MU) by another (the monstrous MUMON), and a simple formal system (the MIU-system) by a complicated one (TNT). It isn’t likely that the answer will be any more forthcoming even though the question has been reshaped. In fact, TNT has a full complement of both lengthening and shortening rules, and the reformulation of the question is likely to be far harder than the original. One might even say that looking at MU via MUMON is an intentionally idiotic way of doing things. However, MUMON can be looked at on more than one level.
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In fact, this is an intriguing point: MUMON has two different passive meanings.
Firstly, it has the one which was given before: 30 is a MIU-number. But secondly, we know that this statement is tied (via isomorphism) to the statement MU is a theorem of the MIU-system. So we can legitimately quote this latter as the second passive meaning of MUMON. It may seem very strange because, after all, MUMON contains nothing but plus signs, parentheses, and so forth-symbols of TNT. How can it possibly express any statement with other than arithmetical content?
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The fact is, it can. Just as a single musical line may serve as both harmony and
melody in a single piece; just as “BACH” may be interpreted as both a name and a melody; just as a single sentence may be an accurate structural description of a picture by Escher, of a section of DNA, of a piece by Bach, and of the dialogue in which the sentence is embedded, so MUMON can be taken in (at least) two entirely different ways. This state of affairs comes about because of two facts:
Fact 1. Statements such as “MU is a theorem” can be coded into number theory
via Gödel’s isomorphism.
Fact 2. Statements of number theory can be translated into TNT. It could be said that MUMON is, by Fact 1, a coded message, where the symbols of the code are, by Fact 2, just symbols of TNT.
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Now it could be objected here that a coded message, unlike an uncoded message, does not express anything on its own-it requires knowledge of the code. But in reality there is no such thing as an uncoded message. There are only messages written in more familiar codes, and messages written in less familiar codes. If the meaning of a message is to be revealed it must be pulled out of the code by some sort of mechanism, or isomorphism. It may be difficult to discover the method by which the decoding should be done; but once that method has been discovered, the message becomes transparent as water. When a code is familiar enough, it ceases appearing like a code; one forgets that there is a decoding mechanism. The message is identified with its meaning.
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Here we have a case where the identification of message and meaning is so strong that it is hard for us to conceive of an alternate meaning: residing in the same symbols. Namely, we are so prejudiced by the symbols of TNT towards seeing number-theoretical meaning (and only number theoretical meaning) in strings of TNT, that to conceive of certain string of TNT as statements about the MIU-system is quite difficult. But Gödel’s isomorphism compels us to recognize this second level of meaning in certain strings of TNT. Decoded in the more familiar way, MUMON bears the message:
30 is a MIU-number.
This is a statement of number theory, gotten by interpreting each sign the conventional way.
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But in discovering Gödel-numbering and the whole isomorphism built upon it, we
have in a sense broken a code in which messages about the MIU-system are written in strings of TNT. Gödel’s isomorphism is a new information-revealer, just as the
decipherments of ancient scripts were information-revealers. Decoded by this new and less familiar mechanism MUMON bears the message
MU is a theorem of the MIU-system.
The moral of the story is one we have heard before: that meaning is an automatic by-product of our recognition of any isomorphism; therefore there are at least two passive meanings of MUMON-maybe more!
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Of course things do not stop here. We have only begun realizing the potential of Gödel’s isomorphism. The natural trick would be to turn TNT’s capability of mirroring other formal systems back on itself, as the Tortoise turned the Crab’s phonographs against themselves, and as his Goblet G turned against itself, in destroying itself. In order to do this, we will have to Gödel-number TNT itself, just as we did the MIU-system, and then “arithmetize” its rules of inference. The Gödel-numbering is easy to do. For instance, we could make the following correspondence:
Symbol Codon Mnemonic Justification
O … 666 Number of the Beast for the Mysterious Zero
S … 123 successorship: 1, 2, 3, …
= .... 111 visual resemblance, turned sideways
- … 112 1+1=2
. … 236 2x3=6
( … 362 ends in 2 *
) … 323 ends in 3 *
< … 212 ends in 2 * these three pairs
… 213 ends in 3 * form a pattern
[ … 312 ends in 2 *
] … 313 ends in 3 *
a … 262 opposite to V (626)
´… 163 163 is prime
∧ … 161 ´∧´
is a “graph” of the sequence 1-6-1 ∨ … 616 ´∨’ is a “graph” of the sequence 6-1-6
⊃ … 6336 "implies" 3 and 3, in some sense . ~ ..... 223 . 2 + 2 is not 3 ℑ ..... 333ℑ’ looks like `3’
V … 626 opposite to a; also a “graph” of 6-2-6
.: … 636 two dots, two sixes
punc. … 611 special number, as on Bell system (411, 911)
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Each symbol of TNT is matched up with a triplet composed of the digits 1, 2, 3, and 6, in a manner chosen for mnemonic value. I shall call each such triplet of digits a Gödel codon, or codon for short. Notice that I have given no codon for b, c, d, or e; we are using austere TNT. There is a hidden motivation for this, which you will find out about in Chapter XVI. I will explain the bottom entry, “punctuation”, in Chapter XIV.
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Now we can rewrite any string or rule of TNT in the new garb. Here, for instance, is Axiom 1 in the two notations, the old below the new:
626,262,636,223,123,262,111,666 V a : ~ S a = 0
Conveniently, the standard convention of putting in a comma every third digit happens to coincide with our colons, setting them off for “easy” legibility.
Here is the Rule of Detachment, in the new notation:
RULE: If x and 212x633y213 are both theorems, then 1 is a theorem.
Finally, here is an entire derivation taken from the last Chapter, given in austere TNT and also transcribed into the new notation
Note: Image
Notice that I changed the name of the “Add S” rule to “Insert `123’ ”, since that is the typographical operation which it now legitimizes.
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This new notation has a pretty strange feel to it. You lose all sense of meaning; but if you had been brought up on it, you could read strings in this notation as easily as you do TNT. You would be able to look and, at glance, distinguish well-formed formulas from ill-formed ones. Naturally since it is so visual, you would think of this as a typographical operation but at the same time, picking out well-formed formulas in this notation i picking out a special class of integers, which have an arithmetical definition too.
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Now what about “arithmetizing” all the rules of inference? As matter stand, they
are all still typographical rules. But wait! According to the Central Proposition, a typographical rule is really equivalent to an arithmetical rule. Inserting and moving digits in decimally represented numbers is an arithmetical operation, which can be carried out typographically. Just as appending a ‘O’ on the end is exactly the same as multiplying by 10, so each rule is a condensed way of describing a messy arithmetical operation. Therefore, in a sense, we do not even need to look for equivalent arithmetical rules, because all of the rules are already arithmetical!
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Looked at this way, the preceding derivation of the theorem “362,123,666,112,123,666,323,111,123,123,666” is a sequence of highly convoluted number-theoretical transformations, each of which acts on one or more input numbers, and yields an output number, which is, as before, called a producible number, or, to be more specific, a TNT-number. Some the arithmetical rules take an old TNT-number and increase it in a particular way, to yield a new TNT-number; some take an old TNT-number a and decrease it; other rules take two TNT-numbers, operate on each of them some odd way, and then combine the results into a new TNT-number and so on and so forth. And instead of starting with just one known TNT-number, we have five initial TNT-numbers-one for each (austere) axiom, of course. Arithmetized TNT is actually extremely similar to the arithmetized MIU-system, only there are more rules and axioms, and to write out arithmetical equivalents explicitly would be a big bother-and quite unenlightening, incidentally. If you followed how it was done for the MIU-system, there ought to be no doubt on your part that its quite analogous here
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There is a new number-theoretical predicate brought into being by this “Godelization” of TNT: the predicate
a is a TNT-number.
For example, we know from the preceding derivation that
362,123,666,112,123,666,323,111,123,123,666 is a TNT-number, while on the other hand, presumably 123,666,111,666 is not a TNT-number.
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Now it occurs to us that this new number-theoretical predicate is expressible
by some string of TNT with one free variable, say a. We could put a tilde in front, and that string would express the complementary notion
a is not a TNT-number.
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Now if we replaced all the occurrences of a in this second string by the TNT-numeral for 123,666,111,666-a numeral which would contain exactly 123,666,111,666 S’s, much too long to write out-we would have a TNT-string which, just like MUMON, is capable of being interpreted on two levels. In the first place, that string would say
123,666,111,666 is not a TNT-number.
But because of the isomorphism which links TNT-numbers to theorems of TNT,
there would be a second-level meaning of this string, which is:
S0=0 is not a theorem of TNT.
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This unexpected double-entendre demonstrates that TNT contains strings which talk about other strings of TNT. In other words, the metalanguage in which we, on the outside, can speak about TNT, is at least partially imitated inside TNT itself. And this is not an accidental feature of TNT; it happens because the architecture of any formal system can be mirrored inside N (number theory). It is just as inevitable a feature of TNT as are the vibrations induced in a record player when it plays a record. It seems as if vibrations should come from the outside world-for instance, from jumping children or bouncing balls; but a side effect of producing sounds-and an unavoidable one-is that they wrap around and shake the very mechanism which produces them. It is no accident; it is a side effect which cannot be helped. It is in the nature of record players. And it is in the nature of any formalization of number theory that its metalanguage is embedded within it.
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We can dignify this observation by calling it the Central Dogma of Mathematical Logic, and depicting it in a two-step diagram:
TNT ⇒ N ⇒ meta-TNT
In words: a string of TNT has an interpretation in N; and a statement of N may have a second meaning as a statement about TNT.
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G: A String Which Talks about Itself in Code
This much is intriguing yet it is only half the story. The rest of the story involves an intensification of the self-reference. We are now at the stage where the Tortoise was when he realized that a record could be made which would make the phonograph playing it break-but now the quest is: “Given a record player, how do you actually figure out what to put on the record?” That is a tricky matter. We want to find a string of TNT-which we’ll call ‘G’-which is about itself, in the sense that one of its passive meanings is a sentence about G. particular the passive meaning will turn out to be
“G is not a theorem of TNT.”
I should quickly add that G also has a passive meaning which is a statement of number theory; just like MUMON it is susceptible to being construed in least two different ways. The important thing is that each passive meaning is valid and useful and doesn’t cast doubt on the other passive meaning in any way. (The fact that a phonograph playing a record can induce vibrations in itself and in the record does not diminish in any way the fact that those vibrations are musical sounds!)
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The ingenious method of creating G, and some important concepts relating to TNT, will be developed in Chapters XIII and XIV; for now it is interesting to glance ahead, a bit superficially, at the consequences finding a self-referential piece of TNT. Who knows? It might blow up! In a sense it does. We focus down on the obvious question:
Is G a theorem of TNT, or not?
Let us be sure to form our own opinion on this matter, rather than rely G’s opinion about itself. After all, G may not understand itself any better than a Zen master understands himself. Like MUMON, G may express a falsity. Like MU, G may be a nontheorem. We don’t need to believe every possible string of TNT-only its theorems. Now let us use our power of reasoning to clarify the issue the best we can at this point
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We will make our usual assumption: that TNT incorporates valid methods of reasoning, and therefore that TNT never has falsities for theorems. In other words, anything which is a theorem of TNT expresses a truth. So if G were a theorem, it would express a truth, namely: “G is not a theorem”. The full force of its self-reference hits us. By being a theorem, G would have to be a falsity. Relying on our assumption that TNT never has falsities for theorems, we’d be forced to conclude that G is not a theorem. This is all right; it leaves us, however, with a lesser problem. Knowing that G is not a theorem, we’d have to concede that G expresses a truth. Here is a situation in which TNT doesn’t live up to our expectations-we have found a string which expresses a true statement yet the string is not a theorem. And in our amazement, we shouldn’t lose track of the fact that G has an arithmetical interpretation, too-which allows us to summarize our findings this way:
A string of TNT has been found; it expresses, unambiguously, a statement about certain arithmetical properties of natural numbers; moreover, by reasoning outside the system we can determine not only that the statement is a true one, but also that the string fails to be a theorem of TNT. And thus, if we ask TNT whether the statement is true, TNT says neither yes or no.
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Is the Tortoise’s string in the Mu Offering the analogue of G? Not quite. The analogue of the Tortoise’s string is ~G. Why is this so? Well, let us think a moment about what ~G says. It must say the opposite of what G says. G says, “G is not a theorem of TNT”, so ~G must say “G is a theorem”. We could rephrase both G and ~G this way:
G: “I am not a theorem (of TNT).” ~G: “My negation is a theorem (of TNT).”
It is ~G which is parallel to the Tortoise’s string, for that string spoke not about itself, but about the string which the Tortoise first proffered to Achilles — which had an extra knot on it (or one two few, how ever you want to look at it)
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Mumon penetrated into the Mystery of the Undecidable anyone, in his concise poem on Joshu’s MU:
Has a dog Buddha-nature?
This is the most serious question of all.
If you say yes or no,
You lose your own Buddha-nature