Believe it or not, I find yet another excuse to sing in class today. Since it's Fast Friday, students with good grades get to leave early (with green passes), while those with D's or F's have an extra tutorial (red passes). I end up with three students -- one D+, one D, and one F. So I promise to sing a song to make sure that these students are working on assignments, not playing games on computers/phones.
But one student -- the D-student -- claims that he has just forgot to submit an assignment, so he does so in seconds and then plays games the remaining 35 minutes. Unfortunately, there's no way for me to confirm whether this is true or not. The girl with an F works for half of tutorial -- it's actually the girl with a D+ who works hard the entire time.
Figuring that I've extracted as much work as possible, I sing "Meet Me in Pomona, Mona." Actually, I also sing it to the fifth period class (since I'd already written the song announcement on the board in anticipating tutorial). All of those students have green passes, and so in both fifth and tutorial, the Pomona song is the last thing they hear during the Big March.
That's right -- today is the last day before spring break in this district. Last year, I wrote that this district takes the week after Easter ("Bright Week") for spring break. Well, many districts that normally take the week after Easter off take the week before the holiday ("Holy Week") instead in years when Easter is especially late. What qualifies as a "late Easter" varies by district, but since Easter falls in the March 22nd-April 25th range, we expect Easters during the last week of that range (April 19th or later) to count as late.
This is especially noticeable in the UK and other European countries. These countries normally take both Holy Week and Bright Week off, for a two-week Easter holiday. But this year, most of them are taking both weeks before Easter off instead. (In years when Easter is especially early, they take both weeks off after Easter instead.)
On the other hand, I know of at least two Southern California school districts where spring break is always Good Friday plus the week after Easter, even when the holiday is late as it is this year. Those districts have a nine-week Big March this year! One of those districts is the one from which I graduated as a young student -- the high school where I ran for the Cross Country and Track teams.
One thing that makes the late Easter a bit weird this year is its effect on sports seasons. In California, the football season now starts and ends a week earlier (due to complaints that the State Finals games were played too close to Christmas), and many other sports followed suit. Cross Country didn't change, since the Saturday after Thanksgiving is so convenient for the State Meet. On the other hand, Track did change -- the State Meet for Track used to be the first Saturday in June, but now it's the last Saturday in May.
Normally, my old district held its League Prelims the first Wednesday in May (and League Finals two days later, on Friday). But with the change in the schedule, the league meets need to be held a week earlier -- but that week is exactly spring break this year, due to the late Easter! Thus the league meets are held two weeks earlier than last year -- and on a Tuesday and Thursday, to avoid Good Friday!
It also means that the entire track season proper falls during the Big March. The first meet of the season was the Saturday of President's Day Weekend and League Finals is on Maundy Thursday. (All meets after that are considered the postseason.)
Sometimes I wish that CIF didn't choose the 2018-2019 to implement the earlier schedule -- a year with both an early Thanksgiving and a late Easter. It would have wreaked less havoc if the CIF had waited until 2019-2020 to do so instead.
Dialogue 4 of Douglas Hofstadter's Godel, Escher, Bach is called "Contracrostipunctus." Here's how it begins:
Achilles has come to visit his friend and jogging companion, the Tortoise, at his home.
Achilles: Heavens, you certainly have an admirable boomerang collection!
Tortoise: Oh, pshaw. No better than that of any other Tortoise. And now, would you like to step into the parlor?
Achilles: Fine. (Walks to the corner of the room.) I see you also have a large collection of records. What sort of music do you enjoy?
We might as well skip a few lines, since we know the answer's going to be Bach. The Tortoise also mentions a record player that he received from his friend the Crab -- a Perfect phonograph:
Achilles: Naturally, I suppose you disagreed.
And of course the reptile disagrees -- he proved his crustacean friend wrong by playing a song called "I Cannot Be Played on Record Player 1," which caused the phonograph to break.
Achilles: Nasty fellow! You needn't spell out for me the last details: that you recorded those sounds yourself, and offered the dastardly item as a gift...
And the Tortoise continues to create more songs -- "I Cannot Be Played on Record Player 2," 3, and so on -- which cause more record players to break. Hence none are perfect. The only way for a player not to break is it to be low fidelity -- but a low fidelity player can hardly be called perfect:
Achilles: Compassion for the Crab overwhelms me. High fidelity or low fidelity, he loses either way.
But the Tortoise informs his friend that the Crab built a "Record Player Omega" -- one that rebuilds itself whenever a record tries to break it. And in a way, Godel's Theorem -- as we'll soon find out in today's Chapter -- is just like Record Player Omega.
Tortoise: What! It's almost midnight! I'm afraid it's my bedtime. I'd love to talk some more, but really, I am growing quite sleepy.
The conversation now turns back towards Bach. Achilles owns a goblet -- Goblet G -- in which the composer has inscribed his name. The Tortoise explains that Bach's name, B-A-C-H, consists of all musical notes, using the German convention where H=American B and B=American Bb. These are notes in one of Bach's songs -- his Contrapunctus.
The Dialogue ends as the Tortoise takes out his violin:
The Tortoise begins to play: B-A-C- -- but as he bows the final H, suddenly, without warning, a shattering sound rudely interrupts his performance. Both he and Achilles spin around, just in time to catch a glimpse of myriad fragments of glass tinkling to the floor from the shelf where Goblet G had stood, only moments before. And then...dead silence.
Chapter 4 of Douglas Hofstadter's Godel, Escher, Bach, called "Consistency, Completeness, and Geometry," begins as follows:
"In Chapter II, we saw how meaning -- at least in the relative simple context of formal systems -- arises when there is an isomorphism between rule-governed symbols, and things in the real world."
In this Chapter, Hofstadter explains the symbolism inherent in the previous dialogue:
"All this is by way of preparation of a discussion of the Contracrostipunctus -- a study in levels of meaning. The Dialogue has both explicit and implicit meanings. Its most explicit meaning is simply the story which was related."
The author explains that the grooves in the records have two levels of meaning. Level One is that playing the grooves causes the air to vibrate. The Level Two meaning depends upon a chain of two isomorphisms:
(1) isomorphism between arbitrary groove patterns and air vibrations;
(2) isomorphism between arbitrary air vibrations and phonograph vibrations.
And these vibrations ultimately cause the goblet to break:
"Thereafter, the vibrations act back on the goblet just as they did on the escalating series of phonographs."
Now Hofstadter moves on to the implicit meanings. The story is about backfiring on two levels, as follows...
Level One: Goblets and records which backfire'
Level Two: The Tortoise's devilish method of exploiting implicit meaning to cause backfires -- which backfires.
The author now charts out a mapping between the Contracrostipunctus and Godel's Theorem:
phonograph <=> axiomatic system for number theory
low-fidelity phonograph <=> "weak" axiomatic system
high-fidelity phonograph <=> "strong" axiomatic system
"Perfect" phonograph <=> complete system for number theory
"blueprint" of phonograph <=> axioms and rules of formal system
record <=> string of the formal system
playable record <=> theorem of the axiomatic system
unplayable record <=> nontheorem of the axiomatic system
sound <=> true statement of number theory
reproducible sound <=> interpreted theorem of the system
unreproducible sound <=> true statement which isn't a theorem
song title: "I Cannot Be Played on Record Player X" <=> implicit meaning of Godel's string: "I Cannot Be Derived in Formal System X"
Hofstadter now goes back to Bach:
"A few words on the Art of the Fugue...Composed in the last year of Bach's life, it is a collection of eighteen fugues all based on one theme."
To make a long story short, soon after he included the notes B-A-C-H in the Contrapunctus -- one of the fugues included in the Art of the Fugue -- Bach died.
The author returns to Godel's Theorem. Just as no sufficiently powerful record player is perfect, no sufficiently powerful formal system is perfect. It must contain true yet unproveable statements:
"A most puzzling fact about Godel's method of proof is that he uses reasoning methods which seemingly cannot be 'encapsulated' -- they resist being incorporated into any formal system."
The author now explains what it means for a formal system to be inconsistent. He tries adding the following schema to the pq-system:
Axiom Schema II: If x is a hyphen-string, then xp-qx is an axiom.
Then -p-q- is a theorem (1 + 1 = 1), even though we've previously proved -p-q-- (1 + 1 = 2). But this isn't an inconsistency, since "p" doesn't have to mean plus and "q" doesn't have to mean "equals":
"Notice that I said 'some'; not necessarily all symbols will have to be mapped onto new notions. Some may very well retain their 'meanings,' while others change."
For example, we can keep "p" as plus but reinterpret "q" as "is greater than or equal to." Then we can have both -p-q- (1 + 1 > 1) and -p-q-- (1 + 1 > 2) as theorems.
At this point, Hofstadter gives another example of a formal system -- Euclidean geometry. Since this is a Geometry blog, we already know all about Euclidean and non-Euclidean geometry. So I don't have to quote everything the author writes about Geometry. He describes Euclid's Elements:
"Thus there was a definite plan to the work, an architecture which made it strong and sturdy. Nevertheless, the architecture was of a different type from that of, say, a skyscraper."
At this point the author presents a drawing of a skyscraper, Tower of Babel. It goes without saying that the artist is Escher.
Meanings of words are important to Euclidean geometry, as Hofstadter explains:
"Therefore, when someone gives a definition for a common word in the hopes that we will abide by that definition, it is a foregone conclusion that we will not do so but will instead be guided, largely unconsciously, by what our minds find in their associative stories."
The author now states the five postulates of Euclid. I don't need to repeat them, since we already know what they are. Here what he says about the fifth postulate:
"Though he never explicitly said so, Euclid considered this postulate to be somehow inferior to the others, since he managed to avoid using it in the proofs of the first twenty-eight propositions."
He writes about one of the first mathematicians to attempt to prove Euclid's fifth postulate -- Girolamo Saccheri, a contemporary of Bach. His proof attempt was indirect -- he assumed that the postulate was false in hopes of deriving a contradiction. But instead, there was no contradiction. In 1823, Janos Bolyai and Nikolay Lobachevsky independently discovered hyperbolic geometry:
"And ironically, in that same year, the great French mathematician Adrien-Marie Legendre came up with what he was sure was a proof of Euclid's fifth postulate, very much along the lines of Saccheri."
Recall that three summers ago, I wrote about Legendre's book on the blog.
In formal systems, there are often undefined terms, as Hofstadter explains:
"The symbol q is especially interesting, since its 'meaning' changed when a new axiom schema was added."
Likewise, in geometry "point," "line," and "plane" are undefined -- it's the postulates that tell us what they really are. For example, the author mentions elliptical (or spherical) geometry, where there are no parallel lines, since a "line" is actually a great circle of a sphere:
"This view would say that the postulates are implicit definitions of the undefined terms, all of the undefined terms being defined in terms of the others."
Hofstadter now attempts to define "consistent" and "inconsistent." He considers the three statements TbZ, ZbE, and EbT as follows:
The Tortoise always beats Zeno at chess.
Zeno always beats Egbert at chess.
Egbert always beats the Tortoise at chess.
It's possible, though not probable, that all three of these can be true. (Consider interpreting T as "rock," Z as "scissors," "E" as "paper," and "b" as "always beats" in Rock-Paper-Scissors.) Thus the three statements are internally consistent.
The author does show an inconsistency if we reinterpret "b" as "was invented by" (whether it's Tortoise-Egbert-Zeno or Rock-Paper-Scissors):
"Therefore, internal consistency depends on consistency with the external world -- only now, 'the external word' is allowed to be any imaginable world, instead of the one we live in. But this is an extremely vague, unsatisfactory conclusion."
If "b" is interpreted as "is bigger than," then we have an inconsistency regardless of what natural numbers T, Z, and E stand for:
"Once again we see a case where only some of the interpretation is needed, in order to recognize internal inconsistency."
It's Escher time again:
"It is particularly interesting in the case of understanding drawings by Escher, such as Relativity, in which there occur blatantly impossible images."
Hofstadter explains that the images are inconsistent no matter how we interpret them:
"There is, actually, one other out -- to leave all the lines of the picture totally uninterpreted, like the 'meaningless symbols' of a formal system. This ultimately escape route is an example of a 'U-mode' response -- a Zen attitude towards symbolism."
And no, Escher's drawings can't be interpreted as non-Euclidean geometry:
"The difference between an Escher drawing and non-Euclidean geometry is that in the latter, comprehensible interpretations can be found for the undefined terms, resulting in a comprehensible total system, whereas for the former, the end result is not reconcilable with one's conception of the world, no matter how long one stares at the pictures."
The author tells us that in every conceivable world, math -- or at the very least, logic -- would have to be the same as in our world.
To Hofstadter, completeness is the opposite direction from consistency -- the latter is "Everything produced by the system is true" while the former is "Every true statement is produced by the system."
But as we'll eventually find out, formalized number theory is incomplete. And yet if we try to add more axioms to it to "complete" it, the theory remains incomplete. Hofstadter concludes the chapter with a return to the analogy from above:
"If our record player passes its test, then we will say that the record is defective; contrariwise, if the record passes its test, then we will say that our record player is defective. What, however, can we conclude wen we find out that both pass their respective tests? That is the moment to remember the chain of two isomorphisms, and think carefully.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
"A 24 foot ladder leans again a brick wall as shown. How tall is the brick wall?"
[Here is the given info from the diagram -- the wall makes a 30-degree angle with the ground.]
This is the first 30-60-90 triangle problem since finally reaching Lesson 14-1 in the text. Here 24 is the hypotenuse and we need the leg opposite the 30-degree angle. This must be half of the hypotenuse, or 12. Therefore the desired length is 12 feet -- and of course, today's date is the twelfth.
Since I didn't write much about Lesson 14-1, this is a good problem for our students to consider as we continue Chapter 14 -- and it might even help them with today's activity, which is indirectly related.
Lesson 14-6 of the U of Chicago text is called "Properties of Vectors." In the modern Third Edition of the text, properties of vectors don't appear at all, as I explained yesterday.
I post my originally planned lesson for Lesson 14-6, which contains many of those properties of vector addition from the Common Core Standards that I mentioned yesterday.
But today is an activity day. Since I've rearranged last year's Chapter 14 activities, today I'm posting an activity from last year on the area of a regular hexagon.
In my new district, it is now spring break. But my old district already had its own spring break -- and that's the district whose calendar the blog follows. Therefore my next post will be Monday.
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