"The telephone rings; Achilles picks it up."
Hmm, so Achilles apparently telephones in his day? I assume it's a landline, since we know that cell phones didn't exist in ancient Greece.
Achilles: Hello, this is Achilles.
Achilles: Oh, hello, Mr. T. How are you?
OK, I assume that Mr. T is the Tortoise. But we can't hear the reptile's side of the conversation. And that combined with the fact that I don't post the entire dialogue on the blog means that we'll really have to guess what's going on here.
Skipping down a few lines, we read:
Achilles: Wondrous many of them, eh? What kinds, for example?
Achilles: What do you mean, "phantasmagorical beasts"?
Achilles: Wasn't it terrifying to see so many of them at the same time?
Achilles: A guitar?! Of all things to be in the midst of all those weird creatures. Say, don't you play the guitar?
Let's skip a few more lines. Music segues into the pair's favorite composer (Bach, who else?) and then to some puzzles.
Achilles: A word with the letters 'A,' 'D,' 'A,' 'C' consecutively inside it. Hmm...What about "abracadabra"?
Apparently, this doesn't count since the letters are backwards, and we never hear the answer. So let's skip to the next puzzle:
Achilles: I agree -- can't do any harm. Here it is: What's a word that begins with the letters "HE" and also ends with "HE"?
Again, the answer is never revealed, though Achilles rejects the Tortoise's "degenerate solution" (which I assume is "HE" itself).
Next, the pair moves on to their favorite artist's painting, Mosaic II.
Achilles: Yes, I see all the black animals.
Achilles: Yes, I also see how their "negative space" -- what's left out -- defines the white animals.
Apparently, the Tortoise makes some connection between the Escher painting and "ADAC." Soon, the conversation ends as follows:
Achilles: Right. Well, I'll see you shortly.
Achilles: Good-bye, Mr. T.
Chapter 3 of Douglas Hofstadter's Godel, Escher, Bach is called "Figure and Ground." Here's how the chapter begins:
"There is a strangeness to the idea that concepts can be captured by simple typographical manipulations. The one concept so far captured is that of addition, and it may not have appeared very strange."
In this chapter, Hofstadter's goal is to create a formal system such that Px means "x is prime" -- that is, P with a prime number of hyphens is a theorem, but not a composite number of hyphens. Here are the six things we're allowed to do:
(1) reading and recognizing any of a finite set of symbols;
(2) writing down any symbol belonging to that set;
(3) copying any of those symbols from one place to another;
(4) erasing any of those symbols;
(5) checking to see whether one symbol is the same as another;
(6) keeping and using a list of previously generated theorems.
To accomplish this goal, the author begins with the tq-system, which is just like the pq-system, except that t represents "times."
Axiom Schema: xt-qx is an axiom, whenever x is a hyphen-string.
Rule of Inference: Suppose that x, y, and z are all hyphen-strings. And suppose that xtyqz is an old theorem. Then, xty-qzx is a new theorem.
Below is the derivation of the theorem --t---q------:
Statements Reasons
1. --t-q-- 1. axiom
2. --t--q---- 2. by rule of inference, using line 1 as the old theorem
3. --t---q------ 3. by rule of inference, using line 2 as the old theorem
The author writes:
"Multiplication, a slightly trickier concept than addition, has now been 'captured' typographically, like the birds in Escher's Liberation."
In the tq-system, we can capture compositeness as follows:
Rule: Suppose x, y, and z are hyphen-strings. If x-ty-qz is a theorem, then Cz is a theorem.
The author explains that in the I-mode ("intelligent mode"), the justification for this rule is that z is composite if (x + 1)(y + 1) = z for some x and y. But in the M-mode ("mechanical mode"), no such justification is necessary, since it's all about just following the rules:
"It is the essential thing which keeps you from mixing up the I-mode with the M-mode; or said another way, it keeps you from mixing up arithmetical facts with typographical theorems."
Proposed Rule: Suppose x is a hyphen-string. If Cx is not a theorem, then Px is a theorem.
But this doesn't quite work -- we're not allowed to say "is not a theorem," since it's difficult to tell what is and isn't a theorem. We can't check all theorems within the six things we're allowed to do, in items (1)-(6) above.
Hofstadter lists some theorems of the C-type:
C---- (4)
C------ (6)
C-------- (8)
C--------- (9)
C---------- (10)
C------------ (12)
C-------------- (14)
C--------------- (15)
C---------------- (16)
C------------------ (18)
"The 'holes' in this list are the nontheorems. To repeat the earlier question: Do the holes also have some 'form' in common?"
His answer is that the holes are negatively defined, since they are the things that are left out of a list which is positively defined. Naturally, he compares this to Escher's paintings -- a recursive figure is one whose ground can be seen as a figure in its own right, as opposed to a cursively drawable figure whose ground isn't its own figure (that is, it's just there).
"M.C. Escher was a master at drawing recursive figures -- see, for instance, his beautiful recursive drawing of birds. Our distinction is now as rigorous as one in mathematics, for who can definitely say that a particular ground is not a figure?"
The painting to which the author refers here is Tiling of the plane using birds. Notice that another word for "tiling of the plane" is tessellation -- a Common Core word also used in Lesson 8-2 of the U of Chicago text.
Hofstadter also gives an Escher-like painting by Scott Kim, called his "FIGURE-FIGURE" figure:
"If you read both black and white, you will see 'FIGURE' everywhere, but 'GROUND' nowhere! It is a paragon of recursive figures."
He tells us that the in this figure, we can see black regions:
(1) as the negative space to the white regions;
(2) as altered copies of the white regions (produced by coloring and shifting -- that is, a translation -- each white region.
Now we move from art to music -- that is, from Escher to Bach:
"But in baroque music -- Bach above all -- the district lines, whether high or low or in between, all act as 'figures.' In this sense, pieces by Bach can be called 'recursive.'"
The composer accomplished this by taking advantage of the rhythm:
"Other times, however, he puts one voice on the on-beats, and the other voice on the off-beats, so the ear separates them and hears two distinct melodies weaving in and out, and harmonizing with each other."
Returning to formal systems, Hofstadter writes:
"In our example, the role of positive space is played by the C-type theorems, and the role of negative space is played by strings with a prime number of hyphens."
The goal is to find rules that characterize the primes -- that is, to make the primes the positive space or "figure" rather than the negative space or "ground." But there's a problem here:
There exist formal systems whose negative space (set of nontheorems) is not the positive space (set of theorems) of any formal system.
And this is essentially Godel's Theorem. More formally, Godel would say:
There exist recursively enumerable sets which are not recursive.
or
There exist formal systems for which there is no typographical decision procedure.
As Hofstadter explains, if we had a such a decision procedure, we could list all strings, apply the decision procedure to them, and that will tell us which ones are theorems, which we cross out:
"This amounts to a typographical method for generating the set of nontheorems. But according to the earlier statement (which we here accept on faith), for some systems this is not possible."
Thus for those systems, the original decision procedure can't exist.
Here is a puzzle to think about in connection with the above matters. Can you characterize the following set of integers (or its negative space)?
1, 3, 7, 12, 18, 26, 35, 45, 56, 69, ...
As usual, the author doesn't reveal the answer.
As it turns out, there is a way to generate primes using a formal system after all. Instead of tq-system, Hofstadter introduces "DND" for "does not divide":
Axiom Schema: xyDNDx where x and y are hyphen strings.
For example, -----DND--, where x has been replaced by '--' and y by '---'.
Rule: If xDNDy is a theorem, then so is xDNDxy.
For example: -----DND------------ (5 does not divide 12) is derived from the above axiom using the rule twice.
Next, the authors adds another symbol, DF, where zDFx means "Z is divisor-free up to X" (that is, it has no divisors from 2 to X):
Rule: If --DNDz is a theorem, so is zDF--.
Rule: If zDFx is a theorem and also x-DNDz is a theorem, then zDFx- is a theorem.
Then the primes are numbers which are divisor-free up to 1 less than themselves:
Rule: If z-DFz is a theorem, then Pz- is a theorem.
Using these rules, P- isn't a theorem (1 isn't prime), but then P-- (2 is prime) is also a special case (since the simplest case of DF is defined for z = 3). So the author adds it as an axiom:
Axiom: P--
And there you have it. Hofstadter concludes the chapter as follows:
"And it is this potential complexity of formal systems to involve arbitrary amounts of backwards-forwards interference that is responsible for such limitative results as Godel's Theorem, Turing's Halting Problem, and the fact that not all recursively enumerable sets are recursive."
Lesson 14-5 of the U of Chicago text is called "Vectors." In the modern Third Edition of the text, vectors appear only in connection with translations in Lesson 4-6. In other words, Lessons 14-6 and 14-7 of the Second Edition have no counterpart in the Third Edition.
This is what I wrote last year about today's lesson:
Lesson 14-5 of the U of Chicago text is on vectors. Much of physics deals with vectors. Force is a vector quantity, as Einstein knew all too well.
I remember back when I was a high school senior taking AP Physics C, and our teacher wanted us to remember one thing about vectors:
Vectors operating at right angles are independent.
This means that if two vectors are perpendicular to each other -- most notably if one is parallel to the x-axis and the other to the y-axis -- then they are linearly independent. Motion along one vector has nothing to do with motion along the other. We notice this most clearly when one is throwing a ball, and we resolve the velocity into its horizontal and vertical components. The horizontal component of velocity is slowed down slightly by the force of air resistance, while the vertical component is slowed down -- and its direction is ultimately reversed -- by the force of gravity.
In fact, I remember learning about acceleration so many years ago, and I was confused as to why an object that is turning or slowing down is said to be "accelerating." After all, I thought that an object slowing down was decelerating. To find out the answer, we must look at vectors. If the velocity and acceleration vectors are perpendicular, then the object is turning. They must be perpendicular because of the theorem from Chapter 13 (Lesson 13-5) that the tangent and radius of a circle are perpendicular. So if the velocity and acceleration vectors are in opposite directions, then the object is slowing down. I was confused by the use of the word "acceleration" by non-technical English speakers, where it usually refers to an increase in speed without considering vectors at all.
As it turns out, we prove in Linear Algebra -- a college course beyond Calculus -- that any two vectors that aren't parallel are linearly independent. But the linear independence that we consider the most in physics involve right angles. We don't discuss this idea in Lesson 14-5, but we will look at both velocity and force vectors on the posted worksheet.
Vectors appear in the Common Core Standards, but not in the geometry section. Indeed, they appear in the "Vector & Matrix Quantities" section:
CCSS.MATH.CONTENT.HSN.VM.A.1
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
(The U of Chicago text uses the first option, a single boldface roman letter. So v is a vector.) The point is that the various domains of the standards don't line up exactly with courses -- not even for the traditional (as opposed to the integrated) pathway. So there are standards that appear in the geometry section that don't appear in a geometry course (i.e., conic sections), as well as vice versa (vectors).
The various Common Core Standards for vectors are spread out among the last three lessons of the chapter, 14-5 through 14-7. One standard that appears in today's Lesson 14-5 is:
CCSS.MATH.CONTENT.HSN.VM.B.4.A
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
This is only partly realized in Lesson 14-5 -- component-wise addition will appear in 14-6. But adding vectors end-to-end and by the parallelogram rule do appear in 14-5, and the U of Chicago does give an example where the magnitude of a sum isn't the sum of the magnitudes -- indeed, there's an example where the sum is shorter than one of the vectors being added! The text explains that this is to be expected since one vector is the velocity of the boat and the other is the velocity of the current (and the boat is moving nearly upstream).
So in a way, we are beginning this standard today as well:
CCSS.MATH.CONTENT.HSN.VM.A.3
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
The text points out that vectors are closely related to translations. Many texts tend to define the translation in terms of its vector. The U of Chicago does the opposite. Instead, translations are defined as compositions of reflections. The Two Reflection Theorem of Lesson 6-2 tells us how to find the magnitude and direction of a translation in terms of the two mirrors. Of course, some would say that the magnitude and direction of the translation really means the magnitude and direction of the translation vector. But the word "vector" can't appear in Lesson 6-2, since it isn't even defined until Lesson 14-5. Instead, we see the following theorem:
Theorem:
Two vectors are equal if and only if their initial and terminal points are preimages and images under the same translation.
This theorem, therefore, shows how to connect the U of Chicago definition of translation (two reflections) with the definition found in other texts (vectors). Notice that in both cases, "equal vectors" are defined to be vectors with the same magnitude and direction.
This problem is resolved in the modern Third Edition of the text. Translations appear in Lesson 4-4 and vectors appear in Lesson 4-6, rather than eight chapters apart as in the old text.
Finally, the text defines vector addition:
Definition:
The sum or resultant of two vectors AB and BC, written AB + BC, is the vector AC.
David Joyce criticizes the use of the word "resultant" to refer to vector sum:
The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?)
But then again, U of Chicago rarely uses the word "resultant" and mainly uses the word "sum" -- the word that we normally use when referring to addition. Joyce would prefer to omit the section on vectors, but they are included in the Common Core Standards -- and as I mentioned, the U of Chicago devotes three sections to vectors, not just one!
Joyce would be glad that the modern Third Edition has only one section on vectors -- but then again, he wouldn't be thrilled by the emphasis on translations and other transformations in both the U of Chicago text and Common Core.
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