"The Tortoise and Achilles are spending a day at Coney Island. After buying a couple of cotton candies, they decide to take a ride on the Ferris wheel."
And of course, there's no point in asking how an ancient Greek reptile and athlete can spend a day at Coney Island. I warn you that this is the longest Dialogue in the book -- indeed, it's almost as long as the accompanying chapter.
Tortoise: This is my favorite ride. One seems to move so far, and yet in reality one gets nowhere.
Achilles: I can see why it would appeal to you.
At this point, they grab a hook and find themselves drawn into a helicopter:
Voice: Welcome aboard -- Suckers.
Achilles: Wh -- who are you?
Voice: Allow me to introduce myself. I am Hexachlorophene J. Goodfortune, Kidnapper-at-Large, and Devourer of Tortoises par Excellence, at your service.
Tortoise: Gulp!
While waiting for Goodfortune to cook his half-shelled friend, Achilles spots a book in the kitchen:
Achilles: Indeed -- and the adventure it's opened to looks provocative. It's called "Djinn and Tonic."
Tortoise: Hmm ... I wonder why. Shall we try reading it? I could take the Tortoise's part, and you could take that of Achilles.
Achilles: I'm game. Here goes nothing ...
(They begin reading "Djinn and Tonic.")
So now Achilles and the Tortoise are reading a book about Achilles and the Tortoise. Hofstadter distinguishes the readers and the characters by indenting. This is difficult to show correctly on the blog, so I'll indicate it as follows: the Level 1 Achilles and Tortoise are reading about the Level 2 Achilles and Tortoise.)
(Level 2. Achilles has invited the Tortoise over to see his collection of prints by his favorite artist, M.C. Escher.)
Tortoise: These are wonderful prints, Achilles.
Achilles: I knew you would enjoy seeing them. Do you have any particular favorite.
Tortoise: One of my favorites is Convex and Concave, where two internally consistent worlds, when juxtaposed, make a completely inconsistent composite world.
At this point, the Level 2 Tortoise describes a Pushing-Potion -- one that allows them to visit pictures such as the Escher drawing mentioned above. Achilles asks whether it's impossible to get out of the pictures once they've entered them.
Tortoise: In certain cases, that's not so bad a fate. But there is, in fact, another potion -- well, not a potion, actually, but an elixir -- no, not an elixir, but a -- a --
(Level 1.)
Tortoise: He probably means "tonic."
(Level 2.)
Achilles: Tonic?
Tortoise: That's the word I'm looking for! "Popping-Tonic" is what's it's called ...
Achilles: That sounds very interesting. What would happen if you took some popping-tonic without having previously pushed yourself into a picture?
Tortoise: I don't precisely know, Achilles ...
At this point, Achilles and the Tortoise agree to drink the pushing-potion and enter the Escher work:
Tortoise: Bottoms up!
(They swallow and enter Level 3, the Escher painting Convex and Concave.)
Achilles: That's an exceeding strange taste.
Tortoise: One gets used to it.
Achilles: Does taking the tonic feel this strange?
They enter the picture and find themselves in a gondola. They try speaking to the gondolier:
Tortoise: He doesn't speak English. If we want to get out of here, we'd better just clamber out quickly before he enters the sinister "Tunnel of Love," just ahead of us.
They try to leave, but Achilles just has to try grabbing a magic copper lamp first. Lizards grab him.
Achilles: He-e-e-elp!
A young boy saves Achilles by having some trumpeters lull the lizards to sleep. Then he pulls the athlete out of danger, but Achilles still has the magic copper lamp.
Achilles: Oh, Mr. T, how can I repay them?
The Tortoise invites the boy and his friends to a cup of coffee. Then Achilles gets ready to rub the magic copper lamp, which is marked with the letter 'L':
Achilles: Well, this is fantastic! I can have any wish I want, eh? I've always wished this would happen to me ...
Genie: Hello, my friends -- and thanks ever so much for rescuing my Lamp from the evil Lizard-Duo.
For his first wish, Achilles tries wishing for 100 wishes. But the Genie informs him that wishing for more wishes is a meta-wish that GOD won't let him grant.
Achilles: GOD? Who is GOD? And why won't he let you grant meta-wishes? That seems like such a puny thing compared to the others you mentioned.
Genie: Well, it's a complicated matter, you see. Why don't you just go ahead and make your three wishes? Or at least make one of them. I don't have all the time in the world, you know ...
Achilles insists that the Genie try to grant his meta-wish. And so the Genie takes out another lamp, labeled "ML" and colored silver:
Achilles: And what is that?
Genie: This is my Meta-Lamp ...
Meta-Genie: I am the Meta-Genie. You summoned me, O Genie? What is your wish?
Genie: I have a special wish to make of you, O Djinn, and of GOD.
Notice that Hofstadter indents, but he doesn't consider this to be another level of the story. The entire genie section is labeled as "Tumbolina."
Meta-Genie: This is my Meta-Meta-Lamp ...
Meta-Meta-Genie: I am the Meta-Meta-Genie? You summoned me, O Meta-Genie? What is your wish?
Eventually, the Meta-Meta-Genie takes out another lamp. Hofstadter avoids giving the color of this new lamp, the MMML, but it's possible to assume it's the next element in order on the Periodic Table after copper, silver, and gold -- Roentgenium.
Each Meta-Genie takes half as long to appear as the previous Genie, and so we are able to reach Level Infinity -- GOD -- in finite time.
Genie: Thank you, O Djinn, and GOD. Your wish is granted, Achilles.
Achilles: Thank you, O Djinn, and GOD.
Achilles has been granted a Typeless wish, which can be a meta-wish, meta-meta-wish, etc. Now he asks what GOD stands for:
Genie: Not at all. "GOD" is an acronym which stands for "GOD Over Djinn." The word "Djinn" is used to designate Genies, Meta-Genies, Meta-Meta-Genies, etc. It is a Typeless word.
Achilles protests -- how can the G in GOD stand for GOD? (This recursive acronym reminds me of what the B stands for in Benoit B. Mandlebrot. It stands for -- Benoit B. Mandlebrot.)
Achilles: I see. You mean GOD sits up at the top of the ladder of djinns?
Genie: No, no, no! There is nothing "at the top," for there is no top.
Tortoise: It seems to me that each and every djinn would have a different concept of what GOD is, then, since to any djinn, GOD is the set of djinns above him or her, and no two djinns share that set.
Eventually, Achilles finally makes his Typeless Wish:
Achilles: I wish my wish would not be granted!
(At that moment, an event -- or is "event" the word for it? -- takes place which cannot be described, and hence no attempt will be made to describe it.)
(Level 1.)
Achilles: What on earth does that cryptic comment mean?
Tortoise: It refers to the Typeless Wish Achilles made.
Achilles: But he hadn't yet made it.
Tortoise: Yes, he had. He said, "I wish that wish would not be granted," and the Genie took THAT to be his wish.
Achilles: Oh my! That sounds ominous. Turn the page and let's see.
(Level 3.)
Achilles: Hey! What happened? Where is my Genie?
Tortoise: I'm afraid our context got restored incorrectly, Achilles.
Achilles: What on earth does that cryptic comment mean?
Tortoise: It refers to the Typeless Wish Achilles made.
Achilles: But he hadn't yet made it.
Tortoise: Yes, he had. He said, "I wish that wish would not be granted," and the Genie took THAT to be his wish.
Achilles: Oh my! That sounds ominous.
Tortoise: It spells PARADOX. For that Typeless Wish to be granted, it had to be denied -- yet not to grant it would be to grant it.
So instead, the "System" lands our Level 3 heroes in another Escher picture -- Reptiles. Achilles starts to drink the Popping-Tonic, but the Tortoise grabs a book. Achilles drops the tonic down a stairwell, but instead of saving it, the Tortoise opens the book.
Tortoise: Provocative Adventures of the Tortoise and Achilles Taking Place in Sundry Parts of the Globe. It sounds like an interesting book to read out of.
Achilles: We're losing our tonic! It's just fallen down the stairwell. There's only one thing to do! We'll have to go down one story!
Tortoise: Go down one story? My pleasure. Won't you join me?
(Level 4. This is the story that Level 3 Achilles and Tortoise are reading now.)
Achilles. It's very dark here, Mr. T. I can't see a thing. Oof! I bumped into a wall. Watch out!
Tortoise: Here -- I have a couple of walking sticks. Why don't you take one of them.
It takes some time for the Level 4 pair to figure out where they are.
Achilles: Have you made me shrink? So that this labyrinth we're in is actually some teeny tiny thing that someone could STEP on?
Tortoise: Labyrinth? Labyrinth? Could it be? Are we in the notorious Little Harmonic Labyrinth of the dreaded Majotaur?
Achilles: Yiikes! What is that?
It takes some time, but the Tortoise eventually figures it out.
Tortoise: We are walking down a spiral groove of a record in its jacket. Your stick scraping against the strange shapes in the wall acts like a needle running down the groove, allowing us to hear the music.
And it goes without saying whose music is being played on this record:
Tortoise: Very simple. When I heard the melody B-A-C-H in the top voice, I immediately realized that the grooves that we're walking through could only be the Little Harmonic Labyrinth, one of Bach's lesser known organ pieces. It is so named because of its dizzyingly frequent modulations.
Achilles: Wh-what are they?
The reptile explains that the song changes key many times throughout. It starts in the key of C major.
Tortoise: Yes, C acts like a home base, in a way. Actually, the usual word is "tonic."
Oh -- so the "tonic" mentioned in the title of the story actually refers to a musical tonic! The Tortoise explains that the song is set up to resolve, or end, on the tonic.
Tortoise: Exactly. The composer has used his knowledge of harmonic progressions to manipulate your emotions, and to build up hopes in you to hear that tonic.
He explains that the first modulation in this song is from C major to G major. Thus the song seems to end on G, while keeping in mind that's only a false ending, as the ultimate true ending is on C.
Achilles: Why should you have to keep anything in mind when listening to a piece of music? Is music only an intellectual exercise?
After discussing keys and chords for a while, the pair refocuses on leaving the record:
Achilles: Oh, Mr. T, how will we ever get out of here, if we can't just retrace our steps?
Tortoise: That's a very good question. Achilles -- are you all right?
Achilles: Just a bit shaken up but otherwise fine. I fell into some big hole.
Tortoise: You've fallen into the pit of the Evil Majotaur.
Oh -- I get it now. It's not a "minotaur" but a "Majotaur," since we're dealing with major keys. Inside the pit, the pair smells something tasty.
Achilles: Oh, boy -- popcorn! I'm going to munch my head off!
(Level 1.)
Tortoise: Let's hope it isn't pushcorn! Pushcorn and popcorn are so extraordinarily difficult to tell apart.
(Level 4.)
Achilles: What's this about Pushkin?
Tortoise: I didn't say a thing. You must be hearing things.
Achilles: Go-golly! I hope not. Well, let's dig in.
(And the two friends begin munching the popcorn (or pushcorn?) -- and all at once -- POP! I guess it was popcorn, after all.)
(Level 3.)
Tortoise: What an amusing story. Did you enjoy it?
Achilles: Mildly.
The Level 3 pair sees the lizards leaving the Escher picture that they're in. The Tortoise realizes that even though Achilles lost the popping-tonic, they can leave their picture by doing the same -- moving perpendicularly to its plane.
Tortoise: Of course! We just need to go UP one story. Do you want to try it?
Achilles: Anything to get back to my house! I'm tired of all these provocative adventures.
Tortoise: Follow me, then, up this way.
(And they go up one story.)
(Level 2.)
Achilles: It's good to be back. But something seems wrong. This isn't my house! This is YOUR house, Mr. Tortoise.
Tortoise: Well, so it is -- and am I glad for that! I wasn't looking forward one whit to the long walk back from your house. I am bushed, and doubt if I could have made it.
Achilles: I don't mind walking home, so I guess it's lucky we ended up here, after all.
Tortoise: I'll say! This certainly is a piece of Good Fortune!
(Level 0.)
David Walker: And as I eat from this bag of kettle corn here -- oops, this must be popcorn, because now I've just moved up one story too far-- out of Dialogue 5 and fully out of Godel, Escher, Bach!
Let's get back to the book now, so we can start Chapter 5. I must put that kettle popcorn away now!
Chapter 5 of Douglas Hofstadter's Godel, Escher, Bach, called "Recursive Structures and Processes," begins as follows:
"What is recursion? It is what was illustrated in the Dialogue Little Harmonic Labyrinth: nesting, and variations on nesting."
This is a complex chapter, since recursion isn't a simple concept. I've already spent so much of this post on the Dialogue, so I don't want to write too much about this Chapter. But in this Chapter, Hofstadter begins with an example (someone calls, you put them on hold when someone else calls, you put them on hold when a third person calls, and so on) and then writes:
"This executive is hopelessly mechanical, to be sure -- but we are illustrating recursion in its most precise form. In the preceding example, I have introduced some basic terminology of recursion -- at least as seen through the eyes of computer scientists."
Therefore, I will link to some computer science websites -- specifically the Brian Harvey Logo website from Chapter 13 -- to save some space and time. Here's a link to the chapter on recursion:
https://people.eecs.berkeley.edu/~bh/v1ch7/recur1.html
And Hofstadter defines the terms "push," "pop," and "stack." Brian Harvey defines these here:
https://people.eecs.berkeley.edu/~bh/v2ch4/solitaire.html
A stack (also called a pushdown list) is a data structure that is used to remember things and recall them later. A stack uses the rule "Last In, First Out." That is, when you take something out of a stack, the one you get is the one you put in most recently. The name "stack" is based on the metaphor of the spring-loaded stack of trays you find in a self-service cafeteria. You add a tray to the stack by pushing down the trays that were already there, adding the new tray at the top of the pile. When you remove a tray, you take the one at the top of the pile--the one most recently added to the stack.
Berkeley Logo provides primitive procedures
push and pop to implement stacks. Each stack is represented as a list. To push something onto the stack, Logo uses fput; to pop something off the stack, it uses first.And indeed Hofstadter uses the same cafeteria metaphor in his book.
Now the author includes a recursive diagram of his Dialogue to show all the levels. He admits that he leaves Level 1 (Goodfortune's sky hideaway) unresolved. (No, I didn't skip that level when I ate the kettle corn.)
Hofstadter now explains that there can be stacks (of keys) in music as well:
"The listener can also distinguish (unlike Achilles) between a local easing a tension -- for example, a resolution into the pseudotonic -- and a global resolution."
After providing some examples of this in Bach's music, the author moves to recursion in language. I notice that his example is in German, but Brian Harvey gives an example in English:
https://people.eecs.berkeley.edu/~bh/v1ch8/recur2.html
Just as Logo programs can be iterative or recursive, so can English sentences. People are pretty good at understanding even rather long iterative sentences: "This is the farmer who kept the cock that waked the priest that married the man that kissed the maiden that milked the cow that tossed the dog that worried the cat that killed the rat that ate the malt that lay in the house that Jack built." But even a short recursive (nested) sentence is confusing: "This is the rat the cat the dog worried killed."
The author gives several more examples involving language. You might notice that this post has been labeled "traditionalists," because traditionalists posted over the weekend -- and indeed, I'm skipping several pages of Hofstadter to get to the traditionalists.
And what I'm skipping over includes diagrams that I can't post anyway. But what he's diagramming now is sentences -- and some traditionalists lament the lack of diagramming sentences in modern English classes. Thus these pages are relevant to the traditionalists' debate after all.
In the first diagram, the author writes about an ORNATE NOUN, which consists of at most one article, any number (zero or more) of adjectives, and a noun. A FANCY NOUN begins with an ornate noun, but it could be followed by a preposition and another fancy noun. A fancy noun can also start with a relative pronoun, followed by another fancy noun and a verb (or vice versa). This recursive structure describes the fancy noun "farmer" from the example above -- "that" is the relative pronoun that introduces the verb that leads to the next fancy noun.
The next example involves the Fibonacci sequence and two more diagrams, which the author calls "Diagram G" and "Diagram H." Once again, this is too pictorial to post on the blog, but suffice it to say that the diagram is recursive. Diagram G, for example, has a central point (node) with two branches coming out of it. The left branch leads to a copy of G, and the right branch leads to a node followed by another copy of G. Hofstadter refers to these as RTN's, "recursive transition networks."
Here is a recursive formula for how many nodes there are at each level of recursion:
G(n) = n - G(G(n - 1)), n > 0
G(0) = 0
H(n) = n - H(H(H(n - 1))), n > 0
H(0) = 0
It's also possible to have two functions F and M depend on each other:
F(n) = n - M(F(n - 1)), n > 0
M(n) = n - F(M(n - 1)), n > 0
F(0) = 1, M(0) = 0.
He doesn't explain how to calculate this, however. Instead, he lists a chaotic sequence:
Q(n) = Q(n - Q(n - 1)) + Q(n - Q(n - 2)), n > 2
Q(1) = Q(2) = 1.
1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, _____, ...
He writes the this is sort of, but not quite like, Fibonacci:
"Instead, the two immediately previous values tell how far to count back to obtain the numbers to be added to make the new value!"
For example, if we want to know how to fill in the blank after 10, we count back 9 and 10 spaces (starting from the blank) to reach 5 and 6, so the blank is 5 + 6 = 11.
At this point, Hofstadter describes a function from R to R called INT. It's a complicated function based on the continued fraction expansion of reals numbers (that I mentioned in previous posts). So he basically ends up showing us lots of graphs instead, including a related graph called Gplot that looks just like a butterfly.
Speaking of butterflies, he also mentioned Escher's Butterflies, as well as his Fish and Scales, as further examples of recursion.
He also refers to recursion in programs modeling chess, plus a simpler example, tic-tac-toe. We can go back to Brian Harvey who describes tic-tac-toe in further detail:
https://people.eecs.berkeley.edu/~bh/v1ch6/ttt.html
Actually, I like the XKCD version, which shows a recursive game of tic-tac-toe all at once:
https://xkcd.com/832/
I know that I'm whizzing through lots of examples now. Not only can't I post all of the diagrams, but I'm running out of time to type this post. Actually, there's a reason why I never seem to have enough time to type everything I want to say in this post:
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
This is my first real "spring break" post (as in spring break in the district where I get most of my calls, which isn't the district whose calendar the blog observes). Thus I had extra time to type up this post -- and I started early because I knew how much longer the Dialogue would be. Yet I still find myself rushing through typing this post anyway. This is Hofstadter's Law at work.
So instead, let's just jump to the end of the chapter:
"Instead of of just considering programs composed of procedures which can recursively call themselves, why not get really sophisticated, and invent programs which can modify themselves -- programs which can act on programs, extending them, improving them, generalizing them, fixing them, and so on? This kind of 'tangled recursion' probably lies at the heart of intelligence."
OK, now we can finally look at the traditionalists. Here is Barry Garelick's latest post:
https://traditionalmath.wordpress.com/2019/04/13/unintended-consequences-of-teaching-habits-of-mind-for-algebraic-thinking/
The idea of whether algebraic thinking can be taught outside of the context of algebra has attracted much attention over the past two decades. Interestingly, the idea was raised as a question and a subject for further research in an article appearing in American Mathematical Society Notices which asks, “Is there evidence that teaching sense making without algebra is more or less effective than teaching the same concepts with algebra?” I sincerely hope someone follows up on this question.
The main commenter here is Wayne Bishop:
The best teaching of algebraic thinking of which I am aware is the “unitary bar” approach of the Singapore Primary Math series. By Grade 6, you almost want to scream, “use algebra”; the underlying approach/thinking is exactly the same but the use of conventional unknowns simplifies and clarifies the process.
I assume that by "conventional unknowns," Bishop means variables like x. To him, variables simplify and clarify the problem-solving process. But to many students, just seeing that x scares them off. As usual, traditionalists don't take that into consideration.
By the way, Garelick's example of searching for patterns does not require recursion. That's all I have to say for now due to Hofstadter's Law.
Lesson 14-7 of the U of Chicago text is on adding vectors using trigonometry -- and we can't skip it because it appears in the following Common Core standard:
CCSS.MATH.CONTENT.HSN.VM.B.4.B
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
Notice that we are essentially converting the vectors from polar to rectangular form, adding their components, and then converting the sum back to polar form. And all of this is done without the students' even knowing what polar coordinates are.
In an actual trig course (as part of precalculus), we find out that we can avoid thinking about angles larger than 90 degrees by considering the reference angle -- the angle formed by the x-axis and the terminal side of the original angle. We see that even though the U of Chicago's Geometry text doesn't teach reference angles, the angles shown in the text are always formed using the x-axis -- that is, the west-east axis -- and never using the y-axis. So the U of Chicago, while not explicitly teaching reference angles, clearly has these reference angles in mind when writing this chapter.
The text states that one of the two components is found using the sine of the angle, while the other is using the cosine. But because of the way the angles are drawn, the horizontal component will always use the cosine and the vertical component will always use the sine -- just as they would be for polar coordinates in a trig class.
To convert from rectangular back to polar coordinates, we use the distance formula and the inverse tangent function. This is the only time that an inverse trigonometric function appears in the U of Chicago text -- although many other geometry texts discuss inverse trig in more detail.
CCSS.MATH.CONTENT.HSN.VM.B.4.B
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
Notice that we are essentially converting the vectors from polar to rectangular form, adding their components, and then converting the sum back to polar form. And all of this is done without the students' even knowing what polar coordinates are.
In an actual trig course (as part of precalculus), we find out that we can avoid thinking about angles larger than 90 degrees by considering the reference angle -- the angle formed by the x-axis and the terminal side of the original angle. We see that even though the U of Chicago's Geometry text doesn't teach reference angles, the angles shown in the text are always formed using the x-axis -- that is, the west-east axis -- and never using the y-axis. So the U of Chicago, while not explicitly teaching reference angles, clearly has these reference angles in mind when writing this chapter.
The text states that one of the two components is found using the sine of the angle, while the other is using the cosine. But because of the way the angles are drawn, the horizontal component will always use the cosine and the vertical component will always use the sine -- just as they would be for polar coordinates in a trig class.
To convert from rectangular back to polar coordinates, we use the distance formula and the inverse tangent function. This is the only time that an inverse trigonometric function appears in the U of Chicago text -- although many other geometry texts discuss inverse trig in more detail.
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