"Achilles and the Tortoise have just finished a delicious Chinese banquet for two, at the best Chinese restaurant in town."
Hmm, so our ancient Greek pair is eating at a Chinese restaurant. Fortunately, today's Dialogue is much shorter than yesterday's, which should leave more time for discussing the Chapter proper.
Achilles: You wield a mean chopstick, Mr. T.
Tortoise: I ought to.
Then the Tortoise explains that he's been eating Chinese food for years. He must explain to Achilles why there is paper inside his cookie:
Tortoise: That's your fortune, Achilles. Many Chinese restaurants give out fortune cookies with their bills, as a way of softening the blow. If you frequent Chinese restaurants, you come to think of fortune cookies less as cookies than as message bearers. Unfortunately you seem to have swallowed some of your fortune. What does the rest say?
Achilles reads his fortune as "ONE WAR TWO EAR EWE" -- or "O NEW ART WOE ARE WE." A haiku is in the reptile's cookie: "Fortune lies as much/in the hand of the eater/as in the cookie." Then the Tortoise starts talking about a jukebox belonging to his friend, the Crab.
Tortoise: As a matter of fact, it has exactly one record.
Achilles: What? A jukebox with only one record? That's a contradiction in terms. Why is the jukebox so big, then? Is its single record gigantic -- twenty feet in diameter?
Tortoise: No, it's just a regular jukebox-style record.
The Tortoise plays the record and presses B-1. The song it plays is -- what else? -- B-A-C-H. Recall that this is German notation -- in American notation, we'd write it as Bb-A-C-B. Then the Tortoise presses C-3 instead, and a different melody plays. Achilles asks his friend how this is possible -- perhaps there's another song on the other side.
Tortoise: No, the record has grooves only on one side, and has only a single band.
Achilles: I don't understand that at all. You CAN"T pull different songs off the same record!
But a different melody does play -- C-A-G-E. Achilles notices that this is the name of another modern composer, John Cage. The Tortoise explains that when he enters the code, the letter stands for the starting note (B for BACH, C for CAGE). The number is sort of like the scale factor of a dilation -- after all, we've already seen musical translations, rotations, and reflections. For some reason, the 3 in C-3 means take the intervals in BACH (-1 semitone, +3, -1) and multiply then by 3 1/3 (instead of just 3) and rounded to obtain the intervals in CAGE (-3 semitones, +10, -3).
Achilles: Well, blow me down and pick me up! So does that mean that only some sort of skeletal code is present in the grooves, and that the various records add their own interpretations to that code?
Tortoise: I don't know, for sure. The cagey Crab wouldn't fill me in on all the details. But I did get to hear a third song, when record player B-10 swiveled in place.
The third song is B-C-A-H, in semitones -10, +33, -10. Once again, the intervals multiplied by 3 1/3.
Hofstadter concludes the dialogue as follows:
Achilles: I find it curious that when you augment BACH you get CAGE, and when you augment CAGE over again, you get BACH back, except jumbled up inside, as if BACH had an upset stomach after passing through the intermediate stage of CAGE.
Tortoise: That sounds like an insightful commentary on the new art from of Cage.
Before we move on to the Chapter, this BACH/CAGE record sounds exactly like the sort of song that we can program in Mocha. There will be a few differences, though:
- The Crab's record player is based on semitones (12EDO), but Mocha uses EDL instead.
- I assume that the strange number 3 1/3 was chosen in order to make it come out as CAGE, but since Mocha is a computer, it will use exact multiplication.
But let's try it anyway -- let's see how close we can get to BACH and CAGE using Mocha EDL's and exact multiplication.
Here is a list of Mocha notes ranging from C down to A:
(S=Sound, D=Degree)
S D Note
182 79 79u C
181 80 gu C
180 81 wa C
179 82 41u B#
178 83 83u B
177 84 ru B
176 85 17ugu B
175 86 43u B
174 87 29u B
173 88 lu Bb
172 89 89u Bb
171 90 gu Bb
170 91 thuru A#
169 92 23u A#
168 93 31u A#
167 94 47u A
166 95 19ugu A
165 96 wa A
164 97 97u A
These names come from the Mocha note naming program that we coded 2 1/2 weeks ago. It includes the naming conventions that I explained that day (Degree 11=lu Bb, Degree 13=thu G#). As we know, the names for primes 11 and 13, as well as 41, are controversial.
We need to start with a B (German B=Bb), and so we can choose Degrees 88, 89, or 90. Then there are four possible A's, three C's, and five H's (B's).
Let's assume we want the lowest possible limit, so we choose the 5-limit Bb (gu/green), 3-limit A and C (wa/white), and 7-limit B (ru):
90-96-81-84 = BACH
Now we assume that CAGE starts with the same C and A as in BACH -- 81-96. The number of Degrees has jumped from 6 to 15, so we might choose 2 1/2 as our scale factor:
81-96-58.5-66 = CAF#Eb
Well, that's close to CAGE, but not quite. Let's try choosing a different C and H in BACH -- we'll move up to Degree 79 (for C) and Degree 83 (for H/B):
90-96-79-83 = BACH
81-96-53.5-63.5 = CAGE
Can we obtain the final BCAH now? This is trickier, since EDL isn't the same as EDO. The intervals between Degrees are narrower for bass notes than for treble notes, so our simple dilations don't have the same effect as they would for EDO.
For example, we might choose Degree 162 for the middle C in BCAH. The Degree distance from 90 to 162 is now 72 -- it has now increased by a factor of 12. But now the distance from 96-79 must increase twelvefold as well, from 17 to 208 -- yet 208 Degrees from 162 would be negative!
We were able to dilate from BACH to CAGE in Mocha EDL since in this range, EDL is close enough to EDO to make it work out. But it's impossible to dilate from BACH to BCAH in EDL, where the differences between step sizes is too noticeable.
Chapter 6 of Douglas Hofstadter's Godel, Escher, Bach is called "The Location of Meaning." It begins as follows:
"Last chapter, we came upon the question, 'When are two things the same?' In this Chapter, we will deal with the flip side of this question, 'When is one thing not always the same?'"
The author distinguishes between information-bearers and information-revealers. Records are the former, while record players are the latter. Theorems are the former, and interpretations are the latter, and some information can be pulled out from the structures but not others.
"But what does this phrase 'pull out' really mean? How hard are you allowed to pull? There are cases where by investing sufficient effort, you can pull very recondite pieces of information out of certain structures."
Hofstadter gives another example from biology -- genotype is an information-bearer found in DNA, while phenotype is an information-revealer found as part of a physical organism. In the last chapter (which I skipped since it involved diagrams that I can't post), Gplot is an information-bearer while the butterfly graph is an information-revealer. He writes:
"The isomorphism between DNA structure and phenotype structure is anything but prosaic, and the mechanism which carries it out physically is awesomely complicated."
That's why there's a Human Genome Project. The author continues:
"And one says that the phenotype is the revelation -- the 'pulling out' -- of the information that was present in the DNA to start with, latently. (The term "revelation" in this context is due to Jacques Monod, one of the deepest and most original of twentieth-century molecular biologists.)"
He tells us that playing a record counts as a revelation of information, because:
(1) the music does not seem to be concealed in the mechanism of the record player;
(2) it is possible to match pieces of the input (the record) with pieces of the output (the music) to an arbitrary degree of accuracy;
(3) it is possible to play other records on the same record player and get other sounds out;
(4) the record and the record player are easily separated from one another.
Indeed, we can ask does DNA by itself contain any information?
"One view says that the DNA is quite meaningless out of context; the other says that even if it were taken out of context, a molecule of DNA from a living being has such a compelling inner logic to its structure that its message could be deduced anyway."
There are different levels of understanding of a message. Hofstadter asks us to consider what would happen if a record were to travel through space and land on a distant planet:
"Now suppose that an alien civilization hit upon the idea that the appropriate mechanism for translation of the record is a machine which converts the groove-patterns into sounds."
The author now mentions another Cage song -- "Imaginary Landscape no. 4" -- which consists of random sounds coming from random radios:
"Cage's attitude is expressed in his own words: 'to let sounds be themselves, rather than vehicles for man-made theories or expressions of human sentiments.' Now imagine that this is the piece on the record sent out into space."
The aliens would be so confused when they tried to decipher the Cage song. Hofstadter now writes about a group of archaeologists attempting to read an ancient text:
"This intuition comes mainly from one fact: I feel that the result was inevitable; that, had the text not been deciphered by this group at this time, it would have been deciphered by that group at that time -- and it would have come out the same way."
Now there's a picture of the one of the most famous translations -- the Rosetta stone:
"The inscription on this basalt stele was first deciphered in 1921 by Jean Francois Champollion, the 'father of Egyptology'; it is a decree of priests assembled at Memphis in favor of Ptolemy V Epiphanes. In these examples of decipherment of out-of-context messages, we can separate out fairly clearly three levels of information: (1) the frame message; (2) the outer message; (3) the inner message."
Here are the differences among the three layers of any message:
Hofstadter quotes After Babel, by George Steiner:
"We normally use a shorthand beneath which there lies a wealth of subconscious, deliberately concealed or declared associations so extensive and intricate that they probably equal the sum and uniqueness of our status as an individual person."
For example, suppose we were to find a bottle washed up on the beach:
"Even without seeing writing, one recognizes this type of artifact as an information-bearer, and at this point it would take an extraordinary -- almost inhuman -- lack of curiosity, to drop the bottle and not look further. Next, one opens the bottle and examines the marks on the paper."
Here the outer message would be the name of the language -- for example, Japanese -- in which the inner message is written:
"It is in the nature of outer messages that they are not conveyed in any explicit language. To find an explicit language in which to convey outer messages would not be a breakthrough -- it would be a contradiction in terms!"
Hofstadter now mentions the "jukebox" theory of meaning -- no message contains inherent meaning, because, before any message can be understood, it has to be used as the input to some "jukebox":
"The inborn hardware is like a jukebox: it supplies the additional information which turns mere triggers into complete messages. Now if different people's 'jukeboxes' had different 'songs' in them, and responded to given triggers in completely idiosyncratic ways, then we would have no inclination to attribute intrinsic meaning to those triggers."
According to the author, meaning is intrinsic if intelligence is natural. He warns us that if intelligent alien life exists, they might think differently from us -- beware of "Earth chauvinism":
"In the same way, we might imagine that there could exist other kinds of 'jukeboxes' -- intelligences -- which communicate among each other via messages which we would never recognize as messages, and who also would never recognize our messages as messages."
And indeed:
"This would serve to lessen the feeling of having formulated an anthropocentric concept. And of course, if contact were established with an alien civilization from another star system, we would feel supported in our belief that our own type of intelligence is not just a fluke, but an example of a basic form which reappears in nature in nature in diverse contexts, like stars and uranium nuclei."
At this point Hofstadter presents two plaques in space -- one with 1, 1 dots, and the other with the dots arranged in rows with the following numbers of dots:
1, 1, 2, 3, 5, 8, 13, 21, 34.
This, of course, is the Fibonacci sequence. The author compares this to DNA -- the first two numbers are the "genotype" and the other numbers are the "phenotype":
"Thus, the genotype does not contain the full specification of the phenotype. On the other hand, if we consider the second version of the plaque to be the genotype, then there is much better case to suppose that the phenotype could actually be reconstituted."
And for the following sequence:
1, 3, 4, 7, 11, 18, 29, 47, ...
Here 1, 3 is the genotype and the remaining numbers -- the Lucas sequence -- the phenotype:
"And this 'jukebox,' unlike pure intelligence, is not at all universal; it is highly earthbound, depending on idiosyncratic sequences of events all over our globe for long periods of time."
The author now asks, how universal is DNA's message?
"Successive more refined attempts along these lines might eventually lead to a full restoration of the chemical context necessary for the revelation of DNA's phenotypical meaning."
I know, all of this discussion of DNA makes me think about my past failures -- oops, ixnay on the arterchay iencescay! Hofstadter concludes the chapter as follows:
"Lest you think this all sounds hopelessly abstract and philosophical, consider the exact moment when phenotype can be said to be 'available,' or 'implied,' by genotype, is a highly charged issue in our day: it is the issue of abortion."
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The circumference of one of the great circles of this sphere is 4sqrt(pi). The sphere's surface area is ____pi.
Since we're given the circumference, it's easy to find the radius:
C = 2pi r
4sqrt(pi) = 2pi r
2sqrt(pi) = pi r
2/sqrt(pi) = r
Now we can use the formula for surface area of a sphere, Lesson 10-9:
SA = 4pi r^2
= 4pi (2/sqrt(pi))^2
= 16pi/pi
= 16
Therefore the desired area is 16 square units -- and of course, today's date is the sixteenth.
But wait a minute -- Pappas gives a factor of pi in the answer: ____pi. This is an error! Indeed, the whole purpose of having a factor of sqrt(pi) in the circumference is to make all factors of pi vanish and the surface area come out a whole number. She could have given the circumference as 4pi rather than 4sqrt(pi) -- this would make the radius 2 and the surface area 16pi square units. Then the factor of pi would be correct.
This is what I wrote last year about today's activity:
Today I finally post the vector activity that I've been planning this week.
There is also a page to cut out with 36 given vectors -- since as I mentioned earlier, I don't want the students to choose the vectors. Of course, cutting out the vectors is time consuming -- even if the teacher does it before the class -- and those tiny slips of paper are easily lost. Then teachers will have to cut them out several times throughout the day.
Another way to have the students choose vectors would be to have something larger represent the vectors, such as playing cards. The playing cards can be converted to vectors, as follows:
For the horizontal component:
Ace through 10 -- valued 1 through 10
Jack -- valued -2
Queen -- valued -1
King -- valued 0
For the vertical component, use the suit:
Clubs -- valued -1
Diamonds -- valued 0
Hearts -- valued 1
Spades -- valued 2
Examples:
Eight of Diamonds -- (8, 0)
Ace of Clubs -- (1, -1)
Jack of Hearts -- (-2, 1)
Because they are larger, playing cards are less likely to be lost than the little slips of paper that I provide for this activity. But there are problems with using playing cards. First of all, the conversion from playing card to vector is another step that the first partner can get wrong -- and once again this may frustrate the second partner. (I've heard that some people don't even know the difference between clubs and spades!) Furthermore, vectors with large components, such as (8, 0) for the eight of diamonds above, become (30, -2) after performing the steps in Task Three -- and then they are asked to graph that vector (30, -2) in Task Four. So I leave it up to individual teachers whether to use playing cards or the slips of paper that I provide.
"Last chapter, we came upon the question, 'When are two things the same?' In this Chapter, we will deal with the flip side of this question, 'When is one thing not always the same?'"
The author distinguishes between information-bearers and information-revealers. Records are the former, while record players are the latter. Theorems are the former, and interpretations are the latter, and some information can be pulled out from the structures but not others.
"But what does this phrase 'pull out' really mean? How hard are you allowed to pull? There are cases where by investing sufficient effort, you can pull very recondite pieces of information out of certain structures."
Hofstadter gives another example from biology -- genotype is an information-bearer found in DNA, while phenotype is an information-revealer found as part of a physical organism. In the last chapter (which I skipped since it involved diagrams that I can't post), Gplot is an information-bearer while the butterfly graph is an information-revealer. He writes:
"The isomorphism between DNA structure and phenotype structure is anything but prosaic, and the mechanism which carries it out physically is awesomely complicated."
That's why there's a Human Genome Project. The author continues:
"And one says that the phenotype is the revelation -- the 'pulling out' -- of the information that was present in the DNA to start with, latently. (The term "revelation" in this context is due to Jacques Monod, one of the deepest and most original of twentieth-century molecular biologists.)"
He tells us that playing a record counts as a revelation of information, because:
(1) the music does not seem to be concealed in the mechanism of the record player;
(2) it is possible to match pieces of the input (the record) with pieces of the output (the music) to an arbitrary degree of accuracy;
(3) it is possible to play other records on the same record player and get other sounds out;
(4) the record and the record player are easily separated from one another.
Indeed, we can ask does DNA by itself contain any information?
"One view says that the DNA is quite meaningless out of context; the other says that even if it were taken out of context, a molecule of DNA from a living being has such a compelling inner logic to its structure that its message could be deduced anyway."
There are different levels of understanding of a message. Hofstadter asks us to consider what would happen if a record were to travel through space and land on a distant planet:
"Now suppose that an alien civilization hit upon the idea that the appropriate mechanism for translation of the record is a machine which converts the groove-patterns into sounds."
The author now mentions another Cage song -- "Imaginary Landscape no. 4" -- which consists of random sounds coming from random radios:
"Cage's attitude is expressed in his own words: 'to let sounds be themselves, rather than vehicles for man-made theories or expressions of human sentiments.' Now imagine that this is the piece on the record sent out into space."
The aliens would be so confused when they tried to decipher the Cage song. Hofstadter now writes about a group of archaeologists attempting to read an ancient text:
"This intuition comes mainly from one fact: I feel that the result was inevitable; that, had the text not been deciphered by this group at this time, it would have been deciphered by that group at that time -- and it would have come out the same way."
Now there's a picture of the one of the most famous translations -- the Rosetta stone:
"The inscription on this basalt stele was first deciphered in 1921 by Jean Francois Champollion, the 'father of Egyptology'; it is a decree of priests assembled at Memphis in favor of Ptolemy V Epiphanes. In these examples of decipherment of out-of-context messages, we can separate out fairly clearly three levels of information: (1) the frame message; (2) the outer message; (3) the inner message."
Here are the differences among the three layers of any message:
- To understand the inner message is to have extracted the meaning intended by the sender.
- To understand the frame message is to recognize the need for a decoding-mechanism.
- To understand the outer message is to build, or know how to build, the correct decoding mechanism for the inner message.
Hofstadter quotes After Babel, by George Steiner:
"We normally use a shorthand beneath which there lies a wealth of subconscious, deliberately concealed or declared associations so extensive and intricate that they probably equal the sum and uniqueness of our status as an individual person."
For example, suppose we were to find a bottle washed up on the beach:
"Even without seeing writing, one recognizes this type of artifact as an information-bearer, and at this point it would take an extraordinary -- almost inhuman -- lack of curiosity, to drop the bottle and not look further. Next, one opens the bottle and examines the marks on the paper."
Here the outer message would be the name of the language -- for example, Japanese -- in which the inner message is written:
"It is in the nature of outer messages that they are not conveyed in any explicit language. To find an explicit language in which to convey outer messages would not be a breakthrough -- it would be a contradiction in terms!"
Hofstadter now mentions the "jukebox" theory of meaning -- no message contains inherent meaning, because, before any message can be understood, it has to be used as the input to some "jukebox":
"The inborn hardware is like a jukebox: it supplies the additional information which turns mere triggers into complete messages. Now if different people's 'jukeboxes' had different 'songs' in them, and responded to given triggers in completely idiosyncratic ways, then we would have no inclination to attribute intrinsic meaning to those triggers."
According to the author, meaning is intrinsic if intelligence is natural. He warns us that if intelligent alien life exists, they might think differently from us -- beware of "Earth chauvinism":
"In the same way, we might imagine that there could exist other kinds of 'jukeboxes' -- intelligences -- which communicate among each other via messages which we would never recognize as messages, and who also would never recognize our messages as messages."
And indeed:
"This would serve to lessen the feeling of having formulated an anthropocentric concept. And of course, if contact were established with an alien civilization from another star system, we would feel supported in our belief that our own type of intelligence is not just a fluke, but an example of a basic form which reappears in nature in nature in diverse contexts, like stars and uranium nuclei."
At this point Hofstadter presents two plaques in space -- one with 1, 1 dots, and the other with the dots arranged in rows with the following numbers of dots:
1, 1, 2, 3, 5, 8, 13, 21, 34.
This, of course, is the Fibonacci sequence. The author compares this to DNA -- the first two numbers are the "genotype" and the other numbers are the "phenotype":
"Thus, the genotype does not contain the full specification of the phenotype. On the other hand, if we consider the second version of the plaque to be the genotype, then there is much better case to suppose that the phenotype could actually be reconstituted."
And for the following sequence:
1, 3, 4, 7, 11, 18, 29, 47, ...
Here 1, 3 is the genotype and the remaining numbers -- the Lucas sequence -- the phenotype:
"And this 'jukebox,' unlike pure intelligence, is not at all universal; it is highly earthbound, depending on idiosyncratic sequences of events all over our globe for long periods of time."
The author now asks, how universal is DNA's message?
"Successive more refined attempts along these lines might eventually lead to a full restoration of the chemical context necessary for the revelation of DNA's phenotypical meaning."
I know, all of this discussion of DNA makes me think about my past failures -- oops, ixnay on the arterchay iencescay! Hofstadter concludes the chapter as follows:
"Lest you think this all sounds hopelessly abstract and philosophical, consider the exact moment when phenotype can be said to be 'available,' or 'implied,' by genotype, is a highly charged issue in our day: it is the issue of abortion."
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The circumference of one of the great circles of this sphere is 4sqrt(pi). The sphere's surface area is ____pi.
Since we're given the circumference, it's easy to find the radius:
C = 2pi r
4sqrt(pi) = 2pi r
2sqrt(pi) = pi r
2/sqrt(pi) = r
Now we can use the formula for surface area of a sphere, Lesson 10-9:
SA = 4pi r^2
= 4pi (2/sqrt(pi))^2
= 16pi/pi
= 16
Therefore the desired area is 16 square units -- and of course, today's date is the sixteenth.
But wait a minute -- Pappas gives a factor of pi in the answer: ____pi. This is an error! Indeed, the whole purpose of having a factor of sqrt(pi) in the circumference is to make all factors of pi vanish and the surface area come out a whole number. She could have given the circumference as 4pi rather than 4sqrt(pi) -- this would make the radius 2 and the surface area 16pi square units. Then the factor of pi would be correct.
This is what I wrote last year about today's activity:
Today I finally post the vector activity that I've been planning this week.
There is also a page to cut out with 36 given vectors -- since as I mentioned earlier, I don't want the students to choose the vectors. Of course, cutting out the vectors is time consuming -- even if the teacher does it before the class -- and those tiny slips of paper are easily lost. Then teachers will have to cut them out several times throughout the day.
Another way to have the students choose vectors would be to have something larger represent the vectors, such as playing cards. The playing cards can be converted to vectors, as follows:
For the horizontal component:
Ace through 10 -- valued 1 through 10
Jack -- valued -2
Queen -- valued -1
King -- valued 0
For the vertical component, use the suit:
Clubs -- valued -1
Diamonds -- valued 0
Hearts -- valued 1
Spades -- valued 2
Examples:
Eight of Diamonds -- (8, 0)
Ace of Clubs -- (1, -1)
Jack of Hearts -- (-2, 1)
Because they are larger, playing cards are less likely to be lost than the little slips of paper that I provide for this activity. But there are problems with using playing cards. First of all, the conversion from playing card to vector is another step that the first partner can get wrong -- and once again this may frustrate the second partner. (I've heard that some people don't even know the difference between clubs and spades!) Furthermore, vectors with large components, such as (8, 0) for the eight of diamonds above, become (30, -2) after performing the steps in Task Three -- and then they are asked to graph that vector (30, -2) in Task Four. So I leave it up to individual teachers whether to use playing cards or the slips of paper that I provide.
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