Lesson 15-6: Angles Formed by Tangents (Day 156)

Today I subbed in an eighth grade science class. Three of the classes have a co-teacher in charge of the special ed students, and thus there is no "Day in the Life" today. Before I describe how the day went, let me get it out of the way -- ixnay on the arterchay iencescay. (Every time I sub for science this year, I can't help but think about the ixnay year.)

The students are currently learning about Mendelian genetics. They have a "packet" of four worksheets to fill out. The first page has a passage to read and the second page asks the students to take notes on the passage in outline form, which confuses the students for some reason. With the pure period rotation at this school starting with second, I must write down many names on the bad list.

I finally crack down on outline failures in fifth period. This time, I write down the steps for filling out an outline on the board. I also find another excuse to sing a song as an incentive to complete the packet, the "Packet Song." Actually, I'm changing it to "Packet Rap" as this song actually sounds better as a rap. I call it "packet" even though they ultimately them into interactive notebooks.

The only other change I made to the song is "four-page packet" instead of "ten-page packet," since the students have only four pages to work on today. The outline hints and rap work for the most part, since I only need to write one name on the bad list in fifth period and none in sixth -- after I coax one girl to finish enough questions to earn the reward for the class.

Dialogue 15 of Douglas Hofstadter's Godel, Escher, Bach is called "Birthday Cantatatata ..." Here's how the Dialogue begins:

"One fine May day, the Tortoise and Achilles meet, wandering in the woods. The latter, all decked out handsomely, is doing a jiggish sort of thing to a tune which he himself is humming. On his vest he is wearing a great big button with the words 'Today is my Birthday!'"

Tortoise: Hello there, Achilles. What makes you so joyful today? Is it your birthday, by any chance?

Achilles: Yes, yes! Yes it is, today is my birthday!

Tortoise: That is what I had suspected, on account of that button which you are wearing, and also because unless I am mistaken, you are singing a tune from a Birthday Cantata by Bach, one written in 1727 for the fifty-seventh birthday of Augustus, King of Saxony.

Achilles: You're right. And Augustus' birthday coincides with mine, so THIS Birthday Cantata has double meaning. However, I shan't tell you my age.

Well, Zeno first wrote about Achilles over 2500 years ago, so I assume that Achilles must be celebrating his 2500th birthday or more. Then again, Homer described Achilles several centuries before that, which would make the Greek hero's age closer to 3000.

Meanwhile, considering that it's almost May, I was considering delaying this chapter until his actual birthday that's coming up soon. But the king who shares his special day with Achilles was born on May 12th -- and unfortunately, that's a Sunday this year, hence a non-posting day. (Indeed, Sunday, May 12th is Mother's Day this year.)

Let's proceed a few lines in the Dialogue. In the Tortoise's usual fashion, he doesn't conclude it's truly his friend's birthday unless Achilles answers many more questions. This is just as we saw back in the Lewis Carroll Dialogue -- just because A and B are true, the Tortoise doesn't necessarily accept that Z is true. Here Z is "Today is the birthday of Achilles":

Tortoise: Why, who would ever have thought you to be gullible? Quite to the contrary, I regard you as an authority in the science of valid deductions, a fount of knowledge about correct methods of reasoning ... To tell the truth, Achilles, you are, in my opinion, a veritable titan in the art of rational cogitation. And it is only for that reason that I would ask you, "Do the foregoing sentences present enough evidence that I should conclude without further puzzlement that today is your birthday?"

So Achilles suddenly decides to answer "yes" infinitely many times. The reptile refers to this as "Answer Schema omega." Yet the Tortoise still demands one more thing before he finally accepts that it's his friend's birthday:

Achilles: What? That I ask for no present?
Tortoise: Not at all. In fact, Achilles, I am looking forward to treating you to a fine birthday dinner, provided merely that I am convinced that knowledge of all those yes answers at once (as supplied by Answer Schema omega) allows me to proceed directly and without any further detours to the conclusion that today is your birthday. That's the case, isn't it?
Achilles: Yes, of course it is.

But the Tortoise just labels this as Answer Schema omega + 1 and then continues. Subsequent answers from Achilles are labeled omega + 2, omega + 3, omega + omega, omega + omega + omega, and finally omega * omega, which we can write as omega^2.

Tortoise: Well, after Answer Schema omega^2 there's answer omega^2 + 1. And then answer omega^2 + 2. And so forth. But you can wrap those all together into a packet, being Answer Schema omega^2 + omega. And then there are quite a few other answer-packets, such as omega^2 + 2omega, and omega^2 + 3omega ... Eventually, you come to Answer Schema 2omega^2, and after a whie, Answer Schemas 3omega^2 and 4omega^2. Beyond them there are yet further Answer Schemas, such as omega^3, omega^4. omega^5, and so on. It goes on quite a ways, you know.

Achilles realizes that subsequent answers can be labeled omega^omega, omega^omega^omega, and so on. The Tortoise tells him that these can be covered by Answer Schema epsilon_0. But then, not only does the Tortoise still not accept that it's his friend's birthday, the reptile reveals that it's actually his uncle's birthday. Thus he, the Tortoise, deserves a present, even though Achilles is the one whose birthday it really is:

Tortoise: Ah, but you never did succeed in convincing me of the veracity of that remark. You kept on beating around the bush with answers, Answer Schemas, and whatnot. All I wanted to know was if it was your birthday or not, but you managed to befuddle me entirely. Oh, well, too bad. In any case, I'll be happy to let you treat me to a birthday dinner this evening.
Achilles: Very well. I know just the place. They have a variety of delicious soups. And I know exactly what kind we should have ...

Chapter 15 of Douglas Hofstadter's Godel, Escher, Bach is called "Jumping out of the System." Here is how it begins:

"One of the things which a thoughtful critic of Godel's proof might do would be to examine its generality. Such a critic might, for example, suspect that Godel has just cleverly taken advantage of a hidden defect in one particular formal system, TNT."

The author reminds us that TNT is incomplete because G is true yet unprovable:

"I Cannot Be Proven in Formal System TNT"

or, expanded a bit,

"There does not exist a natural number which forms a TNT-proof-pair with the Godel number of this string."

So now Hofstadter asks, if we add G to TNT, is the resulting system TNT+G still incomplete?

"Since Godel's proof relies primarily on the expressive power of a formal system, we should not be surprised to see our new system succumb, too. The trick would be to find a string which expresses the statement:"

"I Cannot Be Proven in Formal System TNT+G."

Once again, the author likes to return to the MIU-system. We know that MU is unprovable in MIU, and so we can add MU to form the MIU+MU system. Here's a proof in MIU+MU:

Statements  Reasons
1. MU         1. axiom
2. MUU      2. rule 2

Thus m = 30300, n = 300 is a MIU+MU-proof-pair, though not a MIU-proof-pair.

So Hofstadter returns to TNT and extends it to TNT+G. He begins by defining the formula:

(TNT+G)-PROOF-PAIR{a,a'}

and then we give it an "uncle":

~Ea: Ea': <(TNT+G)-PROOF-PAIR(a,a') ^ ARITHMOQUINE(a",a')>

assign the uncle the Godel number u', and then arithmoquine it:

~Ea: Ea': <(TNT+G)-PROOF-PAIR(a,a') ^ ARITHMOQUINE(u'S0/a",a')>

Its interpretation is

"There is no number a that forms a TNT+G-proof-pair with the arithmoquinification of u'."

More concisely,

"I Cannot Be Proven in Formal System TNT+G."

And of course there's no point in calling this new Godel statement G' and adding it to TNT+G, since TNT+G+G' will be just as incomplete. In fact, the author explains that we can add Godel statements to TNT+~G and its descendants as well:

"Each time we make a new extension of TNT, its vulnerability to the Tortoise's method -- pardon me, I mean Godel's method -- allows a new string to be devised, having the interpretation:"

"I Cannot Be Proven in Formal System X."

And he asks, what if we try to add all the possible G's to TNT at once, as in TNT+G_omega? Once again, we return to the Tortoise in the Dialogue, and define a new symbol:

(TNT+G_omega)-PROOF-PAIR{a,a'}

Arithmoquine its uncle, and voila -- we have another new undecidable uncle. Hofstadter asks, why wasn't G_omega already included as one of the added G's?

"The answer is that G_omega was not clever enough to foresee its own embeddability inside number theory. In the Contracrostipunctus, one of the essential steps in the Tortoise's making an 'unplayable record' was to get a hold of a manufacturers blueprint of the record play which he was out to destroy."

The author now returns to Cantor's diagonal argument along with a very commonly given objection -- why can't we just add the diagonal number to the list?

(1a) Take list L, and construct its diagonal number d.

(1b) Throw d somewhere into list L, making a new list L+d.

(2a) Take list L+d, and construct its diagonal number d'.

(2b) Throw d' somewhere into list L+d, making a new list L+d+d'.

...

But even if we add all of the d numbers into the list somehow, we'd just obtain a new list which we can diagonalize yet again. The same is true of TNT:

"TNT is therefore said to suffer from essential incompleteness because the incompleteness here is part and parcel of TNT; it is an essential part of the nature of TNT and cannot be eradicated in any way, whether simple-minded or ingenious."

The author tells us that there are three basic conditions that must be satisfied in order for there to be an undecidable string (or an unplayable record):

(1) That the system should be rich enough so that all desired statements about numbers, whether true or false, can be expressed in it. (Otherwise you have a refrigerator, not a phonograph.)

(2) That all the general recursive relations should be represented by formulas in the system. (Or otherwise you have a record player of low fidelity.)

(3) That the axioms and typographical patterns defined by its rules be recognizable by some terminating decision procedure. (Otherwise you have a partially-designed phonograph.)

"The threshold seems to be roughly when a system attains the three properties listed above. Once this ability for self-reference is attained, the system has a hole which is tailor-made for itself; the hole takes the features of the system into account and uses them against the system."

Now Hofstadter quotes author J.R. Lucas, who attempts to use a Godelian argument to show that Artificial Intelligence can never reproduce human intelligence -- that is, that Turing's test will always fail and the singularity will never occur. Here is the gist of it:

Rigid internal codes entirely rule computers and robots; ergo ...

Computers are isomorphic to formal systems. Now ...

Any computer which wants to be as smart as we are has got to be able to do number theory as well as we can, so ...

Among other things, it has to be able to do primitive recursive arithmetic. But for this very reason ...

It is vulnerable to the Godelian "hook," which implies that ...

We, with our human intelligence, can concoct a certain statement of number theory which is true, but the computer is blind to that statement's truth (i.e., will never print out), precisely because of Godel's boomeranging argument.

This implies that there is one thing which computers just cannot be programmed to do, but which we can do. So we are smarter.

Before Hofstadter attempts to rebut Lucas here, he shows us an Escher drawing -- Dragon. It's all about a dragon biting its own tail. Now the author quotes the artist:

I. Our three-dimensional space is the only true reality we know.

II. However much this dragon tries to be spatial, he remains completely flat. Two incisions are made in the paper on which he is printed. Then it is folded in such as way as to leave two square openings. But this dragon is an obstinate beast, and in spite of his two dimensions he persists in assuming that he has three; so he sticks his head through one of the holes and his tail through the other.

Hoftstadter explains:

"But the dragon cannot step out of his two-dimensional space, and cannot know it as we do. We could, in fact, carry the Escher picture any number of steps further."

And indeed:

"Each time at the instant that it becomes two-dimensional -- no matter how cleverly we seem to have simulated three dimensions inside two -- it becomes vulnerable to being cut and folded again. Now with this wonderful Escherian metaphor, let us return to the program versus the human."

He tells us that just as computers have a limit as revealed when the human Godelizes, so do humans have a limit -- we can't Godelize at will. Just ask Achilles. The author returns to the birthday story, where the Tortoise runs out of names such as omega, omega^omega, omega^omega^omega:

"Thus a new name must be supplied ad hoc. Now offhand you might think that these irregularities in the progression from ordinal to ordinal (as these names of infinity are called) could be handled by a computer program."

But as mathematicians Alonzo Church and Stephen C. Kleene proved, it can't:

There is no recursively related notion-system which gives a name to every constructive ordinal.

Speaking of ordinals and their names, I've discussed ordinals on the blog before. It was in the context of the Common Core Standards. Recall that there was much ado concerning whether:

5 * 3 = 3 + 3 + 3 + 3 + 3

or

5 * 3 = 5 + 5 + 5

The first method is recommended by some Common Core texts. But the second method is actually consistent with how ordinals are defined, because:

omega * 3 = omega + omega + omega

Ordinal multiplication is not commutative. In fact:

3 * omega = 3 + 3 + 3 + 3 + ...

                  = omega 

since the limit of the natural numbers 3, 3 + 3, 3 + 3 + 3, is just omega. (In terms of Achilles and his birthday, if he said "yes yes yes" three times at each step, then he skips to steps 3, 6, 9, ..., which are still all covered by Answer Schema omega.)

But unfortunately, Hofstadter in his Dialogue gives omega + omega the name 2omega. This is not correct -- omega * 2 is omega + omega, but 2omega (or 2 * omega) is just omega.

Indeed, we don't even need infinite ordinals to refute Hofstadter and the Common Core -- Axiom 5 of TNT states that:

Axiom 5: Aa: Ab: (a * Sb) = ((a * b) + a)

So we write:

Statements

1. Aa: Ab: (a * Sb) = ((a * b) + a) [axiom 5]

2. (SSSSS0 * SSS0) = ((SSSSS0 * SS0) + SSSSS0

3. (SSSSS0 * SS0) = ((SSSSS0 * S0) + SSSSS0

4. (SSSSS0 * SSS0) = (((SSSSS0 * SS0) + SSSSS0) + SSSSS0

5. (SSSSS0 * S0) = (SSSSS0 * 0) + SSSSS0

6. (SSSSS0 * SSS0) = ((((SSSSS0 * 0) + SSSSS0) + SSSSS0) + SSSSS0)

7. (SSSSS0 * 0) = 0 [axiom 4]

8. (SSSSS0 * SSS0) = (((0 + SSSSS0) + SSSSS0) + SSSSS0)

which states that 5 * 3 = 0 + 5 + 5 + 5.

But let's get back to refuting Lucas. Humans are limited by what they can decide -- after all, we can't decide the Epimenides paradox:

"This was cleverly pointed out by C.H. Whitely, when he proposed the sentence 'Lucas cannot consistently assert this sentence.'"

The author asks, is it possible for either a human or a computer to be self-transcendent? For humans, it might seem to be possible:

"Similarly, it is entirely conceivable that a partial ability to 'step outside of itself' could be embodied in a computer program. However, it is important to see the distinction between perceiving oneself, and transcending oneself."

Now Hofstadter makes one last reference -- Jaich's four Dialogues in Are Quanta Real? which is based on a work by Galileo. There are three characters in these Dialogues:

"Why wouldn't two have sufficed: Simplicio, the educated simpleton, and Salviati, the knowledgeable thinker?"

Well, Sagredo, the third character, is a neutral third party, yet also helps prove Simplicio wrong.

Hofstadter concludes the chapter as follows:

"Somewhere along this elusive path may come enlightenment. In any case (as I see it), the hope is that by gradually deepening one's self-awareness, by gradually widening the scope of 'the system,' one will in the end come to a feeling of being at one with the entire universe."

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

The perimeter of rectangle ABCD is 32. Its length is 2 less than twice its width. If point E can be located anywhere along side AB between points A and B, then what is the area of Triangle DEC?

We can use algebra to find the length and width of the rectangle:

w + 2w - 2 + w + 2w - 2 = 32
6w - 4 = 32
6w = 36
w = 6
2w - 2 = 10

Thus the area of the rectangle is 60. Notice that the area of the triangle is half that of the rectangle (no matter where E is), which is 30. Therefore the desired area is 30 square units -- and of course, today's date is the thirtieth.

Lesson 15-6 of the U of Chicago text is called "Angles Formed by Tangents." In the modern Third Edition of the text, angles formed by tangents appear in Lesson 14-5.

This is what I wrote last year about today's lesson:

The theorems in this lesson are similar to those in yesterday's lesson.

Tangent-Chord Theorem:
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Given: AB chord of Circle O, Line BC tangent to Circle O
Prove: Angle ABC = Arc AB/2

Proof:
Statements                                     Reasons
1. Draw diameter BD.                    1. Through any two points there is exactly one line.
2. Arc AD = 180 - AB                    2. Arc Addition Postulate
3. CB perpendicular BD                 3. Radius-Tangent Theorem
4. Angle ABC = 90 - ABD             4. Angle Addition Postulate
5. Angle ABC = 180/2 - Arc AD/2 5. Inscribed Angle Theorem
6. Angle ABC = (180 - Arc AD)/2 6. Distributive Property
7. Angle ABC = Arc AB/2              7. Substitution Property of Equality

Tangent-Secant Theorem:
The measure of the angle between two tangents, or between a tangent and a secant, is half the difference of the intercepted arcs.

Given: Line AB secant, Ray EC tangent at point C, forming Angle E,
Arc AC = x, Arc BC = y
Prove: Angle E = (x - y)/2

Proof ("between a tangent and a secant"):
Statements                                     Reasons
1. Draw AC.                                   1. Through any two points there is exactly one line.
2. Angle DCA = x/2, EAC = y/2     2. Inscribed Angle Theorem
3. Angle DCA = EAC + E              3. Exterior Angle Theorem
4. Angle E = DCA - EAC               4. Subtraction Property of Equality
5. Angle E = x/2 - y/2                     5. Substitution Property of Equality
6. Angle E = (x - y)/2                     6. Distributive Property

In the text, the "between two tangents" is given as an exercise. The Given part of this proof with the way the points are labeled is completely different from the first part.

Given: Ray PV tangent at Q, Ray PU tangent at R
S on Circle O (same side of QR as P), T on Circle O (opposite side of QR as P)
Prove: Angle P = (Arc QTR - QSR)/2

Proof ("between two tangents"):
Statements                                     Reasons
1. Draw QR.                                   1. Through any two points there is exactly one line.
2. Angle VQR = Arc QTR/2,          2. Inscribed Angle Theorem
    Angle PQR = Arc QSR/2
3. Angle VQR = PQR + P              3. Exterior Angle Theorem
4. Angle P = VQR - PQR               4. Subtraction Property of Equality
5. Angle P = Arc QTR/2 - QSR/2  5. Substitution Property of Equality
6. Angle P = (Arc QTR - QSR)/2   6. Distributive Property

In some ways, the Tangent-Chord Theorem is just like yesterday's Angle-Chord Theorem, except that one of the intercepted arcs is 0 degrees. The bonus question concerns a solar eclipse.

Lesson 15-5: Angles Formed by Chords or Secants (Day 155)

Dialogue 14 of Douglas Hofstadter's Godel, Escher, Bach is called "Air on G's String." Here's how the Dialogue begins:

"The Tortoise and Achilles have just completed a tour of a porridge factory."

Achilles: You don't mind if I change the subject, do you?
Tortoise: Be my guest.
Achilles: Very well, then. It concerns an obscene phone call I received a few days ago.

According to our Greek athlete hero, here's what the caller said:

Caller (shouting wildly): Yields falsehood when preceded by its quotation! Yields falsehood when preceded by its quotation!

The two friends enter a strange tower:

Achilles: We can just step inside right here. Follow me.

(Achilles opens the door. They enter, and begin climbing the steep helical staircase inside the tower.)

Tortoise (puffing slightly): I'm a little out of shape for this sort of exercise, Achilles. How much further do we have to go/?

It turns out that the tower resembles the Escher picture, Above and Below. It is a two-story building such that the ceiling of one level becomes the floor of the other level. The two friends go here to discuss the obscene phone call:

Tortoise: A most imaginative example. But suppose we restrict the word "preceded" to the idea of precedence on a printed sheet, rather than elaborate entries into a banquet room.
Achilles: All right. But what exactly do you mean by "quotation" here?

The reptile explains that when we discuss a word or phrase, we put it in quotes. Thus here are some examples of phrases preceded by their own quotations:

"HUBBA" HUBBA
"'PLOP' IS NOT THE TITLE OF ANY BOOK, SO FAR AS I KNOW" 'PLOP' IS NOT THE TITLE OF ANY BOOK, SO FAR AS I KNOW.
"IS NOT THE TITLE OF ANY BOOK, SO FAR AS I KNOW" IS NOT THE TITLE OF ANY BOOK, SO FAR AS I KNOW.

Actually, on Amazon I found a comic book called "Plop," dated 1973 -- about a decade before Hofstadter wrote his book. On the other hand. "Is not the title of any book, so far as I know" still is not the title of any book, so far as Amazon knows. Achilles thinks this whole thing is silly.

Tortoise: Why silly?
Achilles: Because it's so pointless. Here's another one for you:
"WILL BE BOYS" WILL BE BOYS.

The Tortoise decides to give a name to the operation of preceding a phrase by its own quotation -- "to quine a phrase." He tells his friend that this name refers to its inventor, philosopher William Van Orman Quine.

Achilles: That should be easy ... I'd say the quining gives this:
"WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG" WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG.
Hmm ... There's something just a little peculiar here. Oh, I see what it is! The sentence is talking about itself! Do you see that?
Tortoise: What do you mean? Sentences can't talk.

Achilles explains that he can consider Sentence P to be "_____ WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG" and Sentence Q to be the sentence formed by quining the blank. Thus Sentence P claims that Sentence Q is a reptile's love song. But if the phrase that we place in the blank is "WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG," then Sentence Q -- the quine of that phrase -- becomes Sentence P itself:

Achilles: And since Sentence Q is always the topic of Sentence P, there is a loop, so now P points back to itself. But you see, the self-reference is a sort of accident. Usually Sentences Q and P are entirely unlike each other, but with the right choice for the blank in Sentence P, quining will do this magic trick for you.

And so the pair eventually figure out that the "obscene phone call" is a similarly quined phrase:

"YIELDS FALSEHOOD WHEN PRECEDED BY ITS QUOTATION" YIELDS FALSEHOOD WHEN PRECEDED BY ITS QUOTATION."

Achilles: Thank you, Mr. T., for your lucid clarification of that obscene telephone call.
Tortoise: And that you, Achilles, for the pleasant promenade. I hope we meet again soon.

Chapter 14 of Douglas Hofstadter's Godel, Escher, Bach is called "On Formally Undecidable Propositions of TNT and Related Systems." It begins as follows:

"This Chapter's title is an adaptation of the title of Godel's 1931 paper -- 'TNT' having been substituted for 'Principia Mathematica.' Godel's paper was a technical one, concentrating on making his paper watertight and rigorous; this Chapter will be more intuitive, and in it I will stress the two key ideas which are at the core of the proof."

The author begins with the first idea -- that of proof-pairs:

"A proof-pair is a pair of natural numbers related in a particular way. Here is the idea:"

Two natural numbers, m and n respectively, form a TNT-proof-pair if and only if m is the Godel number of a TNT-derivation whose bottom line is the string with Godel number n.

He gives us a simpler example, using MIU instead of TNT. Recall that MIU is the system mentioned all the way back in Chapter 1, where we started with MI and were supposed to derive MU (which turned out to be impossible). He tells us that m = 3131131111301 and n = 301 form a MIU-proof-pair, because m is the Godel number of the MIU-derivation:

MI
MII
MIIII
MUI

(Recall that in Godel-numbering MIU-proofs, M=3, I=1, and U=0.) On the other hand, we see that the values m = 31311311130 and n = 30 do not form a MIU-proof pair, because:

MI
MII
MIII
MU

is not a valid derivation -- from MII we can't derive MIII. Now Hofstadter returns to TNT:

"Here are two parallel examples, one being merely an alleged TNT-proof-pair, the other being a valid TNT-proof-pair. Can you spot which is which?"

(1) m = 626,262,636,223,123,262,111,666,611,223,123,666,111,666
      n = 123,666,111,666

(2) m = 626,262,636,223,123,262,111,666,611,223,333,262,636,123,262,111,666
      n = 223,333,262,636,123,262,111,666

I briefly listed the Godel-numbering for some of the TNT symbols previously with Chapter 9, but not all of them. In particular, we now need the symbol 611, which is a line separator. With that plus a few others, we can now translate the above Godel numbers into proofs:

(1) Aa: ~Sa=0
      ~S0=0
      is a proof of S0=0?

(2) Aa: ~Sa=0
     ~Ea: Sa=0
     is a proof of ~Ea: Sa=0?

The author asks us to check:

(1) whether the alleged derivation coded for by m is actually a legitimate derivation;
(2) if so, whether the last line of the derivation coincides with the string which n codes for.

Both of the proposed derivations satisfy the first criterion. Derivation (1) is using the specification rule (from "for all a, 0 is not the successor of a," derive "0 is not the successor of 0") and Derivation (2) is using the interchange rule (from "for all a, 0 is not the successor of a," derive "there does not exist a such that 0 is the successor of a").

But derivation (1) is a proof of ~S0=0, instead of S0=0 as claimed. (We proved that "0 is not the successor of 0," instead of "0 is the successor of 0.") Thus (1) fails the second criterion. Therefore only (m, n) for (2) is a valid TNT-proof-pair.

The author asserts the following facts:

Fundamental Fact 1: The property of being a proof-pair is a primitive recursive number-theoretical property, and can therefore be tested for by a BlooP program.

Here BlooP is the coding language mentioned in my last post. But Hofstadter warns:

"There is a notable contrast to be made here with that other closely related number-theoretical property: that of being a theorem-number. To assert that n is a theorem number is to assert that some value of m exists which forms a proof-pair with n."

In other words, it's easier to check a proof than it is to find one. The author continues:

Fundamental Fact 2: The property of forming a proof-pair is testable in BlooP, and consequently, it is represented in TNT by some formula having two free variables.

This formula is very, very complex -- but all that matters is that it exists. The same is true of MIU:

"Let us abbreviate that formula, whose exists we are assured of by Fundamental Fact 2, this way:"

MIU-PROOF-PAIR{a,a'}

"Since it is a property of two numbers, it is represented by a formula with two free variables."

Then the statement that there exists a derivation of MU becomes:

Ea: MIU-PROOF-PAIR{a,SSSSSSSSSSSSSSSSSSSSSSSSSSSSSS0/a'}

where "/a'" means that a' has been replaced by a numeral -- the thirtieth successor of 0 (since 30 is the Godel number of MU).

The author does the same for TNT:

TNT-PROOF-PAIR{a,a'}

Then "0=0 is a theorem of TNT" becomes:

Ea: TNT-PROOF-PAIR{a,SSSSS ......... SSSS0/a'}

where a' has been replaced with the 666,111,666th successor of 0. For reasons that will soon be apparent, the author abbreviates this as "JOSHU." Hofstadter then asserts that all of the following statements can be written in TNT:

(1) 0=0 is not a theorem of TNT.
(2) ~0=0 is a theorem of TNT.
(3) ~0=0 is not a theorem of TNT.

"How do the solutions differ from the example done above, and from each other? Here are a few more translation exercises:"

(4) JOSHU is a theorem of TNT. (Call the TNT-string which expresses this "META-JOSHU.")
(5) META-JOSHU is a theorem of TNT. (Call the TNT-string which expresses this "META-META-JOSHU.")
(6) META-META-JOSHU is a theorem of TNT.
(7) META-META-META-JOSHU is a theorem of TNT.
(etc., etc.)

The author warns us that we can't write the formula:

Ea: TNT-PROOF-PAIR{a,a'}

to mean "a' is a TNT-theorem-number," because this property is not primitive-recursive, and so no such formula can be written in TNT.

Hofstadter now wants to consider substitution as:

replacement of all free variables by a specific numeral.

"Below are shown a couple of examples of this operation, which are followed by the parallel changes in Godel numbers."

Formula: a=a
Godel Number: 262,111,262

We now replaces all free variables by the numeral for 2:

Formula: SS0=SS0
Godel Number: 123,123,666,111,123,123,666

Formula: ~Ea: Ea': a"=(SSa*SSa')
Godel Number: 223,333,262,636,333,262,163,636,262,163,163,111,362,123,123,262,236,123,123,262,163,323

We now replaces all free variables by the numeral for 4:

Formula: ~Ea: Ea': SSSS0=(SSa*SSa') (We don't substitute a or a' because these aren't free variables.)
Godel number: 223,333,262,636,333,262,163,636,123,123,123,123,666,111,362,123,123,262,236,123,123,262,163,323

The author gives us some more exercises -- check to see that if we start with the formula represented by the first Godel number and substitute in the second numeral for all free variables, then we obtain the formula represented by the last Godel number:

(1) 362,262,112,262,163,323,111,123,123,123,123,666;
      2;
      362,123,123,666,112,123,123,666,323,111,123,123,123,123,666.

(2) 223,362,262,236,262,323,111,262,163;
      1;
      223,362,123,666,236,123,666,323,111,262,163.

OK, let's do the exercises:

(1) (a+a')=SSSS0;
      2;
      (SS0+SS0)=SSSS0.

(2) ~(a*a)=a'
      1;
      ~(S0*S0)=a'

Only (1) is a valid substitution. In both examples both a and a' are free (since neither one is bound by an A or E quantifier), but in (2) only a is substituted for. It's easy for us to check which substitution is correct because substitution is primitive-recursive. Thus TNT can check for us as well:

"Let us abbreviate that TNT-formula by the following notation:"

SUB{a,a',a"}

(Here a and a" are Godel numbers, while a' is the numeral to be substituted into a to obtain a".)

"Because this formula represents the substitution relationship, the formula shown below must be a TNT-theorem":

SUB{262,111,262S0/a, SS0/a', 123,123,666,111,123,123,666S0/a"}

where the number before S tells how many S's there are.

At this point the author introduces arithmoquining -- that's "quine" is just like the quines we see earlier in the Dialogue. We take a formula:

a=S0 (Godel number 262,111,123,666)

and arithmoquine it by substituting in its own Godel number for a:

262,111,123,666S0 = S0.

This is clearly false since 262,111,123,666 does not equal 1. Hofstadter tells us that if we were to take ~a=S0 and arithmoquine it, the resulting statement would be true:

"When you arithmoquine, you are of course performing a special case of the substitution operation we defined earlier. If we wanted to speak about arithmoquining inside TNT, we would use the formula:"

SUB{a",a",a'}

which he abbreviates as:

ARITHMOQUINE{a",a'}

He also writes it as:

a' is the arithmoquinification of a"

and states that the arithmoquinification of 262,111,123,666 is:

123,123,123, ....., 123,123,123,666,111,123,666

where there are 262,111,123,666 copies of  '123.'

Hofstadter now uses arithmoquining to define a new formula, which he calls G's uncle:

~Ea: Ea': <TNT-PROOF-PAIR{a,a'} ^ ARITHMOQUINE{a",a'}>

"You can see explicitly how arithmoquinification is thickly involved in the plot. Now this "uncle" has a Godel number, of course, which we'll call 'u.' The head and tail of u's decimal expansion, and even a teeny bit of its midsection, can be read off directly:"

u = 223,333,262,636,333,262,163,636,212, ..., 161, ..., 213

And now the authors takes G's uncle and arithmoquines it:

~Ea: Ea': <TNT-PROOF-PAIR{a,a'} ^ ARITHMOQUINE{uS0/a", a'}>

He tells us that this is Godel's famous string, or 'G.' Then he asks:

(1) What is G's Godel number?
(2) What is the interpretation of G?

He answers (1) by saying that it's "the arithmoquinification of u." He answers (2) in stages:

"There do not exist numbers a and a' such that both (1) they form a TNT-proof-pair, and (2) a' is the arithmoquinification of u."
"There is no number a that forms a TNT-proof-pair with the arithmoquinification of u."
"The formula whose Godel number is the arithmoquinifcation of u is not a theorem of TNT."
"G is not a theorem of TNT."

-- or, if you prefer:

"I am not a theorem of TNT."

And as Hofstadter reminds us, this is why TNT is incomplete -- there exists a statement G, which is true yet unprovable.

The above steps are confusing, so he gives a similar example:

~Ea: Ea': <TORTOISE-PAIR{a,a'} ^ TENTH-POWER{SS0,a",a'}>

where "Tortoise-pair" is as defined in my last post and "TENTH-POWER{a",a'}" is the statement that a' is the tenth power of a. So we translate:

"There do not exist numbers a and a' such that both (1) they form a Tortoise-pair, and (2) a' is the tenth power of 2."
"There is no number a that forms a Tortoise-pair with 1024"
"1024 does not have the Tortoise property":

As he explains:

"Here it was the number 1024 that was described as 'the tenth power of 2'; above, it was the number described as 'the arithmoquinification of u."

Hofstadter now takes time to remind us what Godel was trying to accomplish:

"Recall that this was a great philosophical dilemma of the day: how to prove a system consistent. Godel found a simple way to express the statement 'TNT is consistent' in a TNT-formula: and then he showed that this formula (and all others which express the same idea) are only theorems of TNT under one condition: that TNT is inconsistent."

And since we don't want TNT to be inconsistent, the only alternative is for it to be incomplete -- that there are true yet unprovable statements in TNT.

Hofstadter explains that TNT is omega-incomplete, as follows. He considers the statement:

Aa: ~Ea': <TNT-PROOF-PAIR{a,a'} ^ ARITHMOQUINE{uS0/a", a'}>

He tells us that from this statement we could derive G in one step (interchange) -- and since G is not a theorem, neither can this statement. Yet all the following are provable:

~Ea': <TNT-PROOF-PAIR{0/a, a'} ^ ARITHMOQUINE{uS0/a", a'}>
~Ea': <TNT-PROOF-PAIR{S0/a, a'} ^ ARITHMOQUINE{uS0/a", a'}>
~Ea': <TNT-PROOF-PAIR{SS0/a, a'} ^ ARITHMOQUINE{uS0/a", a'}>
~Ea': <TNT-PROOF-PAIR{SSS0/a, a'} ^ ARITHMOQUINE{uS0/a", a'}>

which translate as:

"0 and the arithmoquinification of u do not form a TNT-proof pair."
"1 and the arithmoquinification of u do not form a TNT-proof pair."
"2 and the arithmoquinification of u do not form a TNT-proof pair."
"3 and the arithmoquinification of u do not form a TNT-proof pair."

Not only are all of these true -- since G isn't a theorem, no integer forms a proof-pair with G -- but they are provable, since the proof-pair property is primitive recursive. But since:

"For all n, n and the arithmoquinification of u do not form a TNT-proof pair."

is not provable, this makes TNT omega-incomplete.

Hofstadter explains that there are two different ways to plug the hole:

"Now the standard type of extension would involve:"

adding G as a new axiom.

And the other way would be:

adding the negation of G as a new axiom.

The author compares this to the situation of non-Euclidean geometry. He explains:

"To put it bluntly, ~G says:"

"There exists some number which forms a TNT-proof-pair with the arithmoquinification of u"

"...but the various members of the pyramidal family successively assert:"

"0 is not that number"
"1 is not that number"
"2 is not that number"
...

Thus the Godel number of a proof of G is a new type of natural number -- a supernatural number. He explains that supernaturals look a lot like ordinary natural numbers:

"In fact, supernatural numbers share all of the properties of natural numbers, as long as those properties are given to us in theorems of TNT."

Thus supernatural numbers are not fractions or negative numbers, and so on. Instead, they are more like infinitely large natural numbers. Indeed, the proof of a supernatural theorem must be infinitely long in order to avoid contradictions:

"The supernatural number I won't cause any disaster. However, we will have to get used to the idea that ~G is now the one which asserts a truth ("G has a proof"), while G asserts a falsity ("G has no proof").

Hofstadter tells us that supernaturals are useful, but he asks, are they real? Again, he compares the situation to non-Euclidean geometry:

"In both instance, one is powerfully driven to ask, 'But which one of these rival theories is correct? Which is the truth?"

It's not necessarily correct to appeal to the physical would to answer the question about geometry:

"Therefore, one should be extremely cautious in saying that the brand of geometry which physicists might wish to use would represent 'the true geometry,' for in fact, physicists will always use a variety of different geometries, choosing in any given situation the one that seems simplest and most convenient."

Hofstadter considers how supernatural numbers might affect banks -- what would happen if someone has a supernatural amount of money? He also wonders about clouds -- sometimes one cloud plus one cloud equals one cloud (because the two clouds come together):

"This doesn't prove that 1 plus 1 equals 1; it just proves that our number-theoretical concept of 'one' is not applicable in its full power to cloud-counting. So bankers, cloud-counters, and most of the rest of us need not worry about the advent of supernatural numbers: they won't affect our everyday perception of the world in the slightest."

The only people who need to consider supernaturals are mathematicians and mathematical logicians:

"It is that intuition about reality which determines which 'fork' of number theory mathematicians will put their faith in, when the chips are down. But this faith may be wrong."

Hofstadter gives one final application of Godel's Theorem. He describes Diophantine equations, which I've mentioned on the blog before (most recently in December) -- polynomial equations whose solutions are integers:

a = 0
5x + 13y - 1 = 0
5p^5 + 17q^17 - 177 = 0
a^123,666,111,666 + b^123,666,111,666 - c^123,666,111,666 = 0

He tells us that Hilbert wants to find a general solution to all Diophantine equations:

"Little did he suspect that no such algorithm exists! Now for the simplification of G. It has been shown that whenever you have a sufficiently powerful formal number theory, and a Godel-numbering for it, there is a Diophantine equation which is equivalent to G."

And since G is unprovable, the equivalent Diophantine equation would both have a solution and have no solution -- thus making Hilbert's general solution impossible. Hofstadter concludes the chapter:

"This is what the Tortoise did in the Prelude, using Fermat's equation as his Diophantine equation. It is nice to know that when you do this, you can retrieve the sound of Old Bach from the molecules in the air!"

Of course, since Hofstadter wrote his book, FLT has been proved. All that means, of course, is that a different Diophantine equation must be used to reproduce Bach music!

For once, I've posted most of what I wanted to write about the Hofstadter Chapter. I'm glad it was this one, since this Chapter discusses Godel's proof in more detail.

This is what I wrote last year about today's lesson:

Lesson 15-5 of the U of Chicago text is on "Angles Formed by Chords or Secants." There is one vocabulary term as well as two theorems to learn.

The vocabulary word to learn is secant. The U of Chicago defines a secant as a line that intersects a circle in two points. This is in contrast with a tangent, a line that intersects the circle in one point.

At this point, I often wonder why we have tangent and secant lines as well as tangent and secant functions in trig. Well, here's an old (nearly 20 years!) Dr. Math post with the explanation:

http://mathforum.org/library/drmath/view/54053.html

Now, the tangent and the secant trigonometric functions are related to 
the tangent and secant of a circle in the following way.

Consider a UNIT circle centered at point O, and a point Q outside the 
unit circle. Construct a line tangent to the circle from point Q and 
call the intersection of the tangent line and the circle point P. Also 
construct a secant line that goes through the center O of the circle 
from point Q. The line segment OQ will intersect the circle at some 
point A. Next draw a line segment from the center O to point P. You 
should now have a right triangle OPQ.

A little thought will reveal that the length of line segment QP on the 
tangent line is nothing more but the tangent (trig function) of angle 
POQ (or POA, same thing). Also, the length of the line segment QO on 
the secant line is, not surprisingly, the secant (trig function) of 
angle POA.

And now let's look at the theorems:

Angle-Chord Theorem:
The measure of an angle formed by two intersecting chords is one-half the sum of the measures of the arcs intercepted by it and its vertical angle.

Given: Chords AB and CD intersect at E.
Prove: Angle CEB = (Arc AD + Arc BC)/2

Proof:
Statements                                            Reasons
1. Draw AC.                                          1. Through any two points there is exactly one segment.
2. Angle C = Arc AD/2,                        2. Inscribed Angle Theorem
    Angle A = Arc BC/2
3. Angle CEB = Angle C + Angle A    3. Exterior Angle Theorem
4. Angle CEB = Arc AD/2 + Arc BC/2 4. Substitution

Angle-Secant Theorem:
The measure of an angle formed by two secants intersecting outside the circle is half the difference of the arcs intercepted by it.

Given: Secants AB and CD intersect at E
Prove: Angle E = (Arc AC - Arc BD)/2

Proof:
Statements                                            Reasons
1. Draw AD.                                          1. Through any two points there is exactly one segment.
2. Angle ADC = Arc AC/2,                   2. Inscribed Angle Theorem
    Angle A = Arc BD/2
3. Angle A + Angle E = Angle ADC     3, Exterior Angle Theorem
4. Angle E = Angle ADC - Angle A      4. Subtraction Property of Equality
5. Angle E = Arc AC/2 - Arc BD/2       5. Substitution

In the end, I must admit that of all the theorems in the text, I have trouble recalling circle theorems the most.

I decided to include another Exploration Question as a bonus:

The sides of an inscribed pentagon ABCDE are extended to form a pentagram, or five-pointed star.
a. What is the sum of the measures of angles, F, G, H, I, and J, if the pentagon is regular?

Notice that each angle satisfies the Angle-Secant Theorem. So Angle F is half the difference between CE (which is two-fifths of the circle, Arc CD + Arc DE = Arc CE = 144) and AB (which is one-fifth of the circle, Arc AB = 72). So Angle F = (144 - 72)/2 = 36 degrees. All five angles are measured the same way, so their sum is 36(5) = 180 degrees.

b. What is the largest and smallest this sum can be if the inscribed polygon is not regular.

Well, let's write out the Angle-Secant Theorem in full:

Angle F + G + H + I + J
= Arc (CD + DE - AB + DE + EA - BC + EA + AB - BC + AB + BC - DE + BC + CD - EA)/2
= Arc (CD + DE + EA + AB + BC)/2
= (360)/2 (since the five arcs comprise the entire circle)
= 180

So the largest and smallest this sum can be is 180. The sum of the five angles is a constant.