Yesterday's sub wrote that she had trouble with cell phones, and so today I write down the names of the offenders in first (zero) and third periods. In order to avoid writing so many names, I switch to an incentive where I sing a song if the students avoid getting on my list. (It seems as if I'm finding excuses to sing songs everyday now.) This time I sing Square One TV's "The Mathematics of Love."
It's been some time since I've subbed in a Grade 12 ERWC class. It reminds me of a possible math equivalent to this class -- a senior "Algebra III" class proposed by Governor Brown. Well, that was three years ago, and nothing ever came out of that idea -- and now Brown is no longer governor. So that's the end of that.
Dialogue 1 of Douglas Hofstadter's Godel, Escher, Bach is called "Three-Part Invention." (Again, there is a Dialogue before each Chapter.) He begins as follows:
"Achilles (a Greek warrior, the fleetest of foot of all mortals) and a Tortoise are standing together on a duty runway in the hot sun. Far down the runway, on a tall flagpole, there hangs a large rectangular flag. The flag is solid red, except where a thin ring-shaped hole has been cut out of it, through which one can see the sky."
So far, this story is recognizable. It's a parody of the short story "What the Tortoise Said to Achilles" by Lewis Carroll -- and this is itself an extension of a famous ancient Greek paradox. Here's a link to a full version of Carroll's story:
http://www.ditext.com/carroll/tortoise.html
But Hofstadter's version isn't the same as Carroll's. Indeed, here's how it begins:
Achilles: What is that strange flag down at the other end of the track? It reminds me somehow of a print by my favorite artist, M.C. Escher.
Tortoise: That is Zeno's flag.
Obviously, Carroll's version doesn't mention Escher, who was born six months after Carroll died. So this is Hofstadter's creation. And Zeno refers to Zeno's paradox -- the paradox of which Achilles and the tortoise are the main characters.
Let's skip down to when the tortoise talks about Zeno's paradox:
Tortoise: Not if things go according to Zeno's paradox, you won't. Zeno is hoping to use out footrace to show that motion is impossible, you see. It is only in the mind that motion seems possible, according to Zeno. In truth, Motion Is Inherently Impossible. He proves it quite elegantly.
Achilles: Oh yes, it comes back to me now: the famous Zen koan about Zen Master Zeno. As you say, it is very simple indeed.
Eventually, the two characters start arguing about the flag:
Achilles: I'm all confused. I remember vividly how I used to repeat over and over the name of the six patriarchs of Zen, and I always said, "The sixth patriarch is Zeno, the sixth patriarch is Zeno..." (Suddenly a soft warm breeze picks up.) Oh, look, Mr. Tortoise -- look at the flag waving! How I love to watch the ripples shimmer through its soft fabric. And the ring cut out of it is waving, too!
Tortoise: Don't be silly. The flag is impossible, hence it can't be waving. The wind is waving.
And soon, look who shows up. It's none other than Zeno himself:
Zeno: Thank you. You see, my Master, the fifth patriarch, taught me that reality is one, immutable, and unchanging: all plurality, change, and motion are mere illusions of the senses. Some have mocked his views; but I will show the absurdity of their mockery. My argument is quite simple. I will illustrate it with two characters of my own Invention: Achilles (a Greek warrior, the fleetest of foot of all mortals) and a Tortoise. In my tale, they are persuaded by a passerby to run a footrace down a runway towards a distance flag waving in the breeze. Let us assume that, since the Tortoise is a much slower runner, he gets a head start of, say, ten rods. Now the race begins. In a few bounds, Achilles has reached the spot where the Tortoise started.
Achilles: Hah!
Of course, we've heard this part of the paradox before -- by this time, the Tortoise has moved ahead of that spot, and so Achilles can never catch the Tortoise. After Zeno has discussed his paradox, the race is about to begin:
Tortoise and Achiiles: Ready!
Zeno: On your mark! Get set! Go!
Chapter 1 of Douglas Hofstadter's Godel, Escher, Bach is called "The MU-puzzle." It begins:
"One of the most central notions in this book is that of a formal system. The type of formal system I use was invented by the American logician Emil Post in the 1920's, and is often called a 'Post production system.'"
He begins with such a formal system -- the MIU-system -- in which every string contains only three letters of the alphabet: M, I, U. Below are some strings of the MIU-system:
MU
UIM
MUUMUU
UIIUMIUUIMUIIUMIUUIMUIIU
This should seem familiar to readers of this blog. Michael Serra does something similar in his Geometry text -- the Centauri challenge. I included it as an activity after Lesson 4-2. You may refer to my October 12th post for more information.
In October, we were given PQQRSS and had to derive QRQ. Today, we are given MI and our goal is to derive MU -- hence the name of the chapter, "The MU-puzzle."
Here are the rules that we're allowed to use in our derivation:
RULE I: If you possess a string whose last letter is I, you can add on a U at the end.
Example: From MI, you may get MIU.
RULE II: Suppose you have Mx. Then you may add Mxx to your collection.
Examples: From MIU, you may get MIUIU.
From MUM, you may get MUMUM.
From MU, you may get MUU.
RULE III: If III occurs in one of the strings in you collection, you may make a new string with U in place of III.
Examples: From UMIIIMU, you could make UMUMU.
From MIIII, you could make MIU (also MUI).
From IIMII, you can't get anywhere using this rule. (The three I's have to be consecutive.)
From MIII, make MU.
From MU, you can't make MIII. This is wrong.
RULE IV: If UU occurs inside one of your strings, you can drop it.
From UUU, get U.
From MUUUIII, get MUIII.
Once again, our goal is to get from MI to MU. But Hofstadter warns us:
"The answer to the MU-puzzle appears later in this book."
That's "later in this book," as in "a later chapter," as in "a later blog post." So don't expect me to give the answer in this post.
Now the author tells us about theorems, axioms, and rules. Once again, an "axiom" is a postulate, or basic assumption. For today's puzzle, our only axiom is MI. The "theorems" are what we prove -- all the strings derivable from MI are theorems. The rules -- actually rules of production or inference -- are the four rules I-IV given above.
And a derivation is sort of like a proof of a theorem. Here is an example of a derivation:
Statements Reasons
1. MI 1. axiom
2. MII 2. from (1) by rule II
3. MIIII 3. from (2) by rule II
4. MIIIIU 4. from (3) by rule I
5. MUIU 5. from (4) by rule III
6. MUIUUIU 6. from (5) by rule II
7. MUIIU 7. from (6) by rule IV
Hofstadter tells us that it's possible for program a computer to search for solutions, but there is a key difference between people and machines:
"If you punch '1' into an adding machine, then add 1 to it, and then add 1 again, and again, and again, and continue doing so for hours and hours, the machine will never learn to anticipate you, and do it itself, although any person will pick up the repetitive behavior very quickly."
One of the earliest examples of artificial intelligence were the first chess-playing machines:
"It was not a very good chess player, but it at least had the redeeming quality of being able to spot a hopeless position, and to resign then and there, instead of waiting for the other program to go through the boring ritual of checkmating."
Now Hofstadter describes three modes of thinking -- M-mode, I-mode, U-mode. (Oh, so the three letters M, I, U in the puzzle aren't just random letters.) The M-mode is the Mechanical mode -- a sheet of paper filled with nothing M's, I's, and U's. As for the I-mode, or intelligent mode:
"One thing you might do is notice that the numbers 3 and 2 play an important role, since I's are gotten rid of in three's, and U's in two's -- and doubling of length (except for the M) is allowed by rule II. So the second sheet might also have some figuring on it."
The third mode, the Un-mode or U-mode, is the Zen way of approaching things. The author will explain this in a later chapter.
Now Hofstadter explains how a genie might solve this problem:
Step 1: Apply every applicable rule to the axiom MI. This yields two new theorems: MIU, MII.
Step 2: Apply every applicable rule to the theorems produced in step 1. This yields three new theorems: MIIU, MIUIU, MIIII.
Step 3: Apply every applicable rule to the theorems produced in step 2. This yields five new theorems: MIIIIU, MIIUIIU, MIUIUIUIU, MIIIIIIII, MUI.
And the author even draws a tree to represent this -- similar to the tree created by Logo to solve the pitcher problem (to which I alluded to in last week's Lesson 13-7 post). But it's not obvious that MU will eventually be reached, since the tree is infinite:
"If there is a test for theoremhood, a test which does always terminate in a finite amount of time, then that test is called a decision procedure for the given formal system."
But so far, we don't have such a decision procedure. Hofstadter concludes the chapter as follows:
"The lone axiom was known, the rules of inference were simple, so the theorems had been implicitly characterized -- and yet it was still quite unclear what the consequences of that characterization were.
In particular, it was still totally unclear whether MU is, or is not, a theorem."
Lesson 14-3 of the U of Chicago text is called "The Tangent Ratio." In the modern Third Edition of the text, the tangent ratio appears in Lesson 13-5. (By the way, Lesson 13-4 of the new edition is all about the Golden Ratio. This lesson doesn't appear in the old edition, but we did mention it back on January 6th, Phi Day.)
In the past I never created much of a worksheet for Lesson 14-3. I always ended up either replacing it with an activity or combining it with Lesson 14-4. So I'll finally create it this year.
By the way, during tutorial today I help an Algebra II student with her trig assignment. She is learning about unit circle, radian measure, and trig functions (including tangent, of course) of the special angles 30, 45, and 60. The only mistake she makes is when she misses negative signs when drawing some of the angles -- other than that, she has a strong understanding of it thus far.
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