"...then, one by one, the four voices of the fugue chime in."
Achilles: I know the rest of you won't believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word -- but what an important one: "MU."
This Dialogue is a continuation of yesterday's. But here we go again with "Hofstadter's Law" again -- I was hoping to have more time to discuss the entire Chapter today, except that today's Dialogue is much longer than yesterday's. That will leave me less time to type about the Chapter, despite there being no subbing or traditionalists to discuss today.
At this point the author finally reveals the picture from yesterday's Dialogue. It consists of a word "MU," but the letters M and U are composed of -- uhh, let's just have Achilles tell us:
Achilles: I know what is going on here. Each of you has seen letters which compose, or are composed of, other letters. In the left-hand piece, there are indeed three "HOLISM"s, but each one of them is composed out of smaller copies of the word "REDUCTIONISM." And in complementary fashion, in the right-hand piece, there is indeed one "REDUCTIONISM," but it is composed out of smaller copies of the word "HOLISM." Now this is all fine and good, but in your silly squabble, the two of you have actually missed the forest for the trees. You see, what good is it to argue over whether "HOLISM" or "REDUCTIONISM" is right, when the proper way to understand the matter is to transcend the question, bu answering "MU"?
But the squabblers -- the Anteater and Crab -- keep on squabbling:
Anteater: Absurd! Your "MU" is as silly as a cow's moo. I'll have none of this Zen wishy-washiness.
Crab: Ridiculous! Your "MU" is as silly as a kitten's mew. I'll have none of this Zen wishy-washiness.
Soon, the Tortoise finally speaks:
Tortoise: I know the rest of you won't believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word -- but what an important one: "MU."
(Just as he says this, the fourth voice in the fugue being played enters, exactly one octave below the first entry.)
That's right -- he just repeats the first line from Achilles! Now the Anteater starts discussing what he knows best -- ant colonies:
Anteater: Most of my clients suffer from some sort of speech impairment. You know, colonies which have to grope for words in everyday situations. It can be quite tragic. I attempt to remedy the situation by, uhh -- removing -- the defective part of the colony. These operations are sometimes quite involved, and of course years of study are required before one can perform them.
The Dialogue continues with discussing the ant colony. Here are some key lines:
Achilles: Well, I can vaguely see how it might be possible for a limited and regulated amount of ant consumption to improve the overall health of a colony -- but what is far more perplexing is all this talk about having conversations with ant colonies. That's impossible. An ant colony is simply a bunch of individual ants running around at random looking for food and making a nest.
...
Achilles: Oh, no, clearly not. With brain cells, I see your point completely. Only ... ants are a horse of another color. I mean, ants just roman about at will, completely randomly, chancing now and then upon a morsel of food ... They are free to do what they want to do, and with that freedom, I don't see at all how their behavior, seen as a whole, can amount to something coherent -- especially something so coherent as the brain behavior necessary for conversing.
As you can see, the direction of the conversation is headed towards the human brain:
Achilles: Normally, I think that I'M in control of what I think -- but the way you put it turns it all inside out, so that is sounds as though "I" am just what comes out of all this neural structure, and natural law. It makes what I consider my SELF sound at best like a by-product of an organism governed by natural law, and at worst, an artificial notion produced by my distorted perspective. In other words, you make me feel like I don't know who -- or what -- I am, if anything.
The Anteater continues with this analogy -- ants in a colony vs. cells in the brain, working together:
Anteater: That's correct. Aside from the queen, there are males, who do practically nothing towards the upkeep of the nest, and then --
...
Anteater: In an ant colony, the situation is quite the contrary. In fact, it is just the exactly the constant to-ing and fro-ing of ants inside the colony which adapts the caste distribution to varying situations, and thereby preserves the delicate caste distribution. You see, the caste distribution cannot remain as one single rigid pattern; rather, it must constantly be changing so as to reflect, in some manner, the real-world situation with which the colony is dealing, and it is precisely the motion inside the colony which updates the caste distribution, so as to keep it in line with the present circumstances facing the colony.
...
Anteater: They are equivalent statements. For instance, in food-gathering, if there is an inconsequential amount of food somewhere which goes discovered by some wandering ant who then attempts to communicate its enthusiasm to other ants, the number of ants who respond will be proportional to the size of the food sample -- and an inconsequential amount will not attract enough ants to surpass the threshold. Which is exactly what I meant by saying there is nothing to do -- too little food ought to be ignored.
...
Anteater: Let me explain my way of seeing things, and then see if you agree. Once a signal is formed, there is no awareness on its part that it should head off in any particular direction. But here, the delicate caste distribution plays a crucial role. It is what determines the motion of signals through the colony, and also how long a signal will remain stable, and where it will "dissolve."
...
Anteater: Ant colonies have been subjected to the rigors of evolution for billions of years. A few mechanisms were selected for, and most were selected against. The end result was a set of mechanisms which make ant colonies work as we have been describing. If you could watch the whole process in a movie -- running a billion or so times faster than life, of course -- the emergence of various mechanisms would be seen as natural responses to external pressures, just as bubbles in boiling water are natural responses to an external heat source. I don't suppose you see "meaning" and "purpose" in the bubbles in boiling water -- or do you?
...
Achilles: I know how hard it is for me --
Tortoise: Not for me, thank you.
Achilles: -- to track a single voice --
Anteater: I'd like some tea, too, Mr. Crab --
Achilles: -- in a musical fugue --
Anteater: -- if it isn't too much trouble.
Achilles: -- when all of them --
Crab: Not at all. Four cups of tea --
Tortoise: Three!
Achilles: -- are going at once.
Crab: -- coming right up!
Yes, all four voices are indeed going at once. By the way, there is so much discussions of ants and fugues, but "Ant Fugue" is actually the name of a woodcut by M.C. Escher, which Hofstadter shows us here. Apparently, it's not the name of a fugue by Bach.
Anteater: As a matter of fact, the individuals in a signal sometimes break off and get replaced by others of the same caste, if there are a few in the area. Most often, signals arrive at their disintegration points with nary an ant in common with their starting lineup.
...
Anteater: Oh, I but dabble in brains, and therefore couldn't set up the image in its glorious detail. But -- and correct me if I'm wrong, Mr. Crab -- I would surmise that the brain counterpart to an ant colony's signal is the firing of a neuron; or perhaps it is a larger-scale event, such as a patter of neural firings.
...
Achilles: What about a description on the level of signals, or teams?
By the way, my summary of today's Dialogue is sort of on the level of signals or teams. Here I'm describing snippets of the Dialogue, but you must put the entire Dialogue (colony) together yourself.
Anteater: Exactly. There is no natural mapping from the individual letters into the real world. The natural mapping occurs on a higher level -- between words, and parts of the real world. If you wanted to describe the book, therefore, you would make no mention of the letter level.
Achilles: Of course not! I'd describe the plot and the characters, and so forth.
...
Anteater: This gets back to the question which you earlier raised about purpose. You're right that symbols themselves are active, but the activities which they follow are nevertheless not absolutely free. The activities of all symbols are strictly determined by the state of the full system in which they reside. Therefore, the full system is responsible for how its symbols trigger each other, and so it is quite reasonable to speak of the full system as the "agent." As the symbols operate, the state of the system gets slowly transformed, or updated. But there are many features which remain over time. It is this partially constant, partially varying system which is the agent. One can give a name to the full system. For example, Aunt Hillary is the "who" who can be said to manipulate her symbols; and you are similar, Achilles.
...
Crab: But precisely that sort of thing DID happen when you read "MU," without perceiving the lower levels "HOLISM" and "REDUCTIONISM."
...
Tortoise: An organ point is the sustaining of a single note by one of the voices in a polyphonic piece (often the lowest voice), while the other voices continue their own independent lines. This organ point is on the note of G. Listen carefully, and you'll hear it.
By the way, the quartet is now continuing to listen to Bach's Well-Tempered Clavier on the record. Of course, now Achilles compares Bach's fugue to the ant colony, where the "organ point" at the note of G is just like ant Aunt Hillary:
Achilles: How does that jibe with the communistic nature of ant colonies which you earlier described to us? It sounds quite inconsistent, to me, to preach communism and to live in a fancy estate!
Note: This book was written over thirty years ago. Thus Aunt "Hillary" has nothing to do here with the 2016 presidential candidate. Please don't try to read politics into this Dialogue (despite the mention of "communism" and "libertarianism" here).
Tortoise: Do they spell anything?
Achilles: Ah ... "J.S. BACH." Oh! I understand now. It's Bach's name!
By the way, they are now discussing another picture which combines the names "J.S. BACH" in capitals with "fermat" in lowercase letters.
Anteater: But it was all to no avail. J.S.F was gone, beyond reconstitution. However, something very strange then began to take place: over the next few months, the ants which had been components of J.S.F. slowly regrouped, and built up a new organization. And thus was Aunt Hillary born.
Here J.S.F. is J.S. Fermant, an ant farmer. The Anteater explains that he also once proved a famous conjecture by mathematiciant Fourmi -- though 2^a + 2^b = 2^c has infinitely many solutions, the similar equation n^a + n^b = n^c for n > 2 has no solutions.
At this point, the Tortoise shows the Crab another picture, which combines the words "HOLISM" and "REDUCTIONISM" yet again. Hofstadter concludes the Dialogue with the Tortoise's own interpretation of this last picture:
Tortoise: I know the rest of you won't believe this, but in fact this picture consists of the word "REDUCTIONISM" written once, with the letters continually growing as they proceed from left to right.
Achilles: At last -- I heard the new twist on the theme this time! I am so glad that you pointed it out to me, Mr. Tortoise. Finally, I think I am beginning to grasp the art of listening to fugues.
Chapter 11 of Douglas Hofstadter's Godel, Escher, Bach is called "Brains and Thoughts." Here's how it begins:
"It was only with the advent of computers that people actually tried to create 'thinking' machines, and witnessed bizarre variations on the theme of thought. Programs were devised whose 'thinking' was to human thinking as a slinky flipping end over end down a staircase is to human locomotion."
Both the Dialogue and the Chapter are ultimately about the human brain. The author compares symbols in the brain to symbols in a formal system:
"In the brain, the rules are mixed right in with the symbols themselves, whereas on paper, the symbols are static entities, and the rules are in our heads."
The author writes that in TNT, the following expressions represent the same number, "fifty":
((SSSSSSS0 * SSSSSSS0) + (S0 * S0))
((SSSSS0 * SSSSS0) + (SSSSS0 * SSSSS0))
And likewise, in the brain, the following expressions might represent the same person:
"The person whose book I sent to a friend in Poland a while back."
"The stranger who started talking with me and my friends tonight in this coffee house."
Hofstadter now describes a friend to informed you that she crashed your car off a cliff:
"Then she tells you that it's all been an April Fool's joke, and that both she and the car are fine! Later, you may even think of her as an unsafe driver because of the strength of the first impression, which should have been wiped out when you learned it was all untrue."
He writes about the brain's "ants" -- the nerve cells or neurons -- and how they work:
"In between the entry ports of a neuron and its output channel is its cell body, where 'decisions' are made. The type of decision which a neuron faces -- and this can take place up to a thousand times per second -- is this: whether or not to fire -- that is, to release ions down its axon, which eventually will cross over into the entry ports of one or more other neurons, thus causeing them to make the same sort of decision."
The author tells us that there are "teams," or suborgans, in the brain, but:
"We will not describe any of the brain's suborgans in detail because, as it turns out, only the roughest mapping can at this time be made between such large-scale suborgans and the activities, mental or physical, which they are responsible for."
The author now refers David Hubel, a neurophysiologist, who considers what worms' brains might look like to an alien:
"Earthworms have isomorphic brains! One could say, 'There is only one earthworm.' But such one-to-one mappability between individuals' brains disappears very soon as you ascend in the thinking-hierarchy and the number of neurons increases -- confirming one's suspicions that there is not just one human!"
Hofstadter describes the findings of the 1940's neurosurgeon Wilder Penfield:
"What Penfield found was that stimulation of certain neurons would reliably create specific images or sensations in the patient."
There are two types of neuron:
"The on-center neurons are those whose firing rate increases whenever, in the small circular retinal area to which they are sensitive, the center is bright but the outskirts are dark; the off-center neurons are those which fire faster when there is darkness in the center and brightness in the outer ring."
Hofstadter asks, is there a "grandmother cell" in the brain:
"Just when we seem to be approaching the threshold where 'symbol' might emerge from 'signal,' the trail gets lost -- a tantalizingly unfinished story."
He compares the visual processing of cats, monkeys, and humans:
"First of all, all three species have 'dedicated' areas of cortex at the back of the brains where visual processing is done: the visual cortex. Secondly, in each of hem, the visual cortex breaks up into three subregions, called areas 17, 18, and 19 of the cortex."
According to the author, information in the brain is possibly funneled into neural modules:
"This means that no one knows where or how the output from complex and hypercomplex cells gets transformed into conscious recognition of shapes, rooms, pictures, faces, and so on."
Still, the assumption in this book is that this is indeed how the brain works:
"So a funneling process must take place at some point after the reception of the visual signature and before the words are uttered."
Hofstadter asks many questions concerning these hypothesized neural complexes -- the modules which mediate thought processes:
Do they extend into the lower regions of the brain, such as the midbrain, the hypothalamus, etc.?
Can a single neuron belong to more than one such complex?
To how many such complexes can a single neuron belong?
By how many neurons can such complexes overlap?
Are these complexes pretty much the same for everybody?
Are corresponding ones found in corresponding places in different people's brains?
Do they overlap in the same way in everybody's brain?
Much of this Chapter is about the symbols in the brain. The author gives a definition of symbols as "these hypothetical neural complexes, neural modules, neural packets, neural networks, multineuron units -- call them what you will, whether they come in the form of pancakes, garden rakes, rattlesnakes, snowflakes, or even ripples on lakes." He proceeds:
"Now what does a symbol do, when awakened? A low-level description would say, 'Many of its neurons fire.'"
The author explains that this is a major theme of his dialogue -- symbols symbolize things, but mere neurons don't:
"About the 'size' of symbols, there are questions like this: Would there be a symbol for an entire story? Or for a melody? Or a joke?"
Once again, Hofstadter's Law must now kick in. We proceed to the end of this Chapter, which happens to be a preview:
"And next Chapter, we shall discuss some of the insights that our symbol-based model of brain activity give into the comparison of brains."
This is what I wrote last year about today's lesson:
Chapter 11 of Douglas Hofstadter's Godel, Escher, Bach is called "Brains and Thoughts." Here's how it begins:
"It was only with the advent of computers that people actually tried to create 'thinking' machines, and witnessed bizarre variations on the theme of thought. Programs were devised whose 'thinking' was to human thinking as a slinky flipping end over end down a staircase is to human locomotion."
Both the Dialogue and the Chapter are ultimately about the human brain. The author compares symbols in the brain to symbols in a formal system:
"In the brain, the rules are mixed right in with the symbols themselves, whereas on paper, the symbols are static entities, and the rules are in our heads."
The author writes that in TNT, the following expressions represent the same number, "fifty":
((SSSSSSS0 * SSSSSSS0) + (S0 * S0))
((SSSSS0 * SSSSS0) + (SSSSS0 * SSSSS0))
And likewise, in the brain, the following expressions might represent the same person:
"The person whose book I sent to a friend in Poland a while back."
"The stranger who started talking with me and my friends tonight in this coffee house."
Hofstadter now describes a friend to informed you that she crashed your car off a cliff:
"Then she tells you that it's all been an April Fool's joke, and that both she and the car are fine! Later, you may even think of her as an unsafe driver because of the strength of the first impression, which should have been wiped out when you learned it was all untrue."
He writes about the brain's "ants" -- the nerve cells or neurons -- and how they work:
"In between the entry ports of a neuron and its output channel is its cell body, where 'decisions' are made. The type of decision which a neuron faces -- and this can take place up to a thousand times per second -- is this: whether or not to fire -- that is, to release ions down its axon, which eventually will cross over into the entry ports of one or more other neurons, thus causeing them to make the same sort of decision."
The author tells us that there are "teams," or suborgans, in the brain, but:
"We will not describe any of the brain's suborgans in detail because, as it turns out, only the roughest mapping can at this time be made between such large-scale suborgans and the activities, mental or physical, which they are responsible for."
The author now refers David Hubel, a neurophysiologist, who considers what worms' brains might look like to an alien:
"Earthworms have isomorphic brains! One could say, 'There is only one earthworm.' But such one-to-one mappability between individuals' brains disappears very soon as you ascend in the thinking-hierarchy and the number of neurons increases -- confirming one's suspicions that there is not just one human!"
Hofstadter describes the findings of the 1940's neurosurgeon Wilder Penfield:
"What Penfield found was that stimulation of certain neurons would reliably create specific images or sensations in the patient."
There are two types of neuron:
"The on-center neurons are those whose firing rate increases whenever, in the small circular retinal area to which they are sensitive, the center is bright but the outskirts are dark; the off-center neurons are those which fire faster when there is darkness in the center and brightness in the outer ring."
Hofstadter asks, is there a "grandmother cell" in the brain:
"Just when we seem to be approaching the threshold where 'symbol' might emerge from 'signal,' the trail gets lost -- a tantalizingly unfinished story."
He compares the visual processing of cats, monkeys, and humans:
"First of all, all three species have 'dedicated' areas of cortex at the back of the brains where visual processing is done: the visual cortex. Secondly, in each of hem, the visual cortex breaks up into three subregions, called areas 17, 18, and 19 of the cortex."
According to the author, information in the brain is possibly funneled into neural modules:
"This means that no one knows where or how the output from complex and hypercomplex cells gets transformed into conscious recognition of shapes, rooms, pictures, faces, and so on."
Still, the assumption in this book is that this is indeed how the brain works:
"So a funneling process must take place at some point after the reception of the visual signature and before the words are uttered."
Hofstadter asks many questions concerning these hypothesized neural complexes -- the modules which mediate thought processes:
Do they extend into the lower regions of the brain, such as the midbrain, the hypothalamus, etc.?
Can a single neuron belong to more than one such complex?
To how many such complexes can a single neuron belong?
By how many neurons can such complexes overlap?
Are these complexes pretty much the same for everybody?
Are corresponding ones found in corresponding places in different people's brains?
Do they overlap in the same way in everybody's brain?
Much of this Chapter is about the symbols in the brain. The author gives a definition of symbols as "these hypothetical neural complexes, neural modules, neural packets, neural networks, multineuron units -- call them what you will, whether they come in the form of pancakes, garden rakes, rattlesnakes, snowflakes, or even ripples on lakes." He proceeds:
"Now what does a symbol do, when awakened? A low-level description would say, 'Many of its neurons fire.'"
The author explains that this is a major theme of his dialogue -- symbols symbolize things, but mere neurons don't:
"About the 'size' of symbols, there are questions like this: Would there be a symbol for an entire story? Or for a melody? Or a joke?"
Once again, Hofstadter's Law must now kick in. We proceed to the end of this Chapter, which happens to be a preview:
"And next Chapter, we shall discuss some of the insights that our symbol-based model of brain activity give into the comparison of brains."
This is what I wrote last year about today's lesson:
Lesson 15-2 of the U of Chicago text is called "Regular Polygons and Schedules." Just like Lesson 13-3 from last month, 15-2 naturally lends itself to an activity, so this will be considered the activity for this week.
The text defines a round-robin tournament as a tournament in which each competitor (or team) plays each other competitor exactly once. True round-robins are rare at the highest levels of sports. The first two rounds of the World Baseball Classic and first round of the World Cup (of soccer) divide the field into groups of four nations each, and each group is played as a round-robin tournament. After the first two baseball rounds or first soccer round, the tournaments become single-elimination instead.
On the other hand, round-robin regular seasons are fairly common in high school sports. During football season, every team in the league plays every other team. It might be easy to schedule the four-team round-robins by hand, but the school from which I graduated is in a seven-team league. So the U of Chicago text provides a method of complete a seven-team round-robin:
1. Let the 7 teams be vertices of an inscribed regular 7-gon (heptagon).
2. Draw a chord and all chords parallel to it. Because the polygon has an odd number of sides, no two chords have the same length. This is the first week's schedule.
3. Rotate the chords 1/7 of a revolution. This is the second week's schedule.
4. Continue rotating 1/7 of a revolution for each week. Because in a week no two chords have the same length, no pairing repeats. In a total of seven weeks, the schedule is complete.
Here is the resulting schedule as given by the U of Chicago text:
1st week: 7-2, 6-3, 5-4, 1 bye
2nd week: 1-3, 7-4, 6-5, 2 bye
3rd week: 2-4, 1-5, 7-6, 3 bye
4th week: 3-5, 2-6, 1-7, 4 bye
5th week: 4-6, 3-7, 2-1, 5 bye
6th week: 5-7, 4-1, 3-2, 6 bye
7th week: 6-1, 5-2, 4-3, 7 bye
In my alma mater's league, from one year to another, the home games become away games and vice versa, and then the schedule itself rotates so that the first week match-ups are contested the second week, the second week match-ups are contested the third week, and vice versa. Many of the stronger teams don't actually take the bye week -- instead they play special games against out-of-state teams.
If there are an even number of teams, then the extra team is placed at the center, and a radius is drawn in addition to the chords -- this replaces the bye week. For example, we can add an eighth team to the list above by having the eighth team play the team with the bye.
Here is the activity. Students should begin by cutting out the circle, drawing the parallel chords, and leaving the numbers where they are, so that the circle can be rotated to match up with the numbers.
(Yes, I know that today isn't Friday, but this is another one of those lessons that naturally lends itself to an activity, so this is it.)
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