Flag Day Post: Self-Ref and Self-Rep

Table of Contents

1. Introduction
2. Pappas Problem of the Day
3. The 490's
4. The 500's
5. The 510's
6. The 520's
7. The 530's
8. The 540's
9. Traditionalists: Fraction Multiplication
10. Conclusion

Introduction

Let me begin with welcome to our first summer post, and happy Flag Day! It's a national holiday, but one of the lesser known ones. It refers to the day that Betsy Ross first presented the national flag, with thirteen stripes and only thirteen stars, back in the year 1777.

As I promised you earlier, today I am making one last post about a Hofstadter Chapter. I choose to present Chapter 16. Why did I choose this chapter? Well, it's the longest Chapter in the book, and thus I obviously didn't describe it completely in my May 1st post.

Notice that the last several Chapters of the book are all long, and if I really wanted to, I could devote four more posts to Chapters 17-20. At the very least, I could cover Chapters 18-19, on artificial intelligence, since I combined those two in last month's post. But as I mentioned earlier, I don't want our spring side-along reading book to slide into summer. Technically, the summer solstice hasn't occurred yet, so we can still read Chapter 16 today. But there will be no more time for any more Chapters after today.

Chapter 16 is preceded by Dialogue 16, "Edifying Thoughts of a Tobacco Smoker." I feel that I covered this Dialogue well enough in my May 1st post, and so today's post is only on the Chapter.

This Chapter spans 54 pages in my edition of the text -- pages 495-548. I decided to divide today's post into sections corresponding to pages of the book -- the 490's, 500's, 510's, and so on. This is probably how we should have originally read the book -- ten pages a day, so that we're not rushing or skipping so often. But then the reading still would have dragged into summer.

Pappas Problem of the Day

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

Find AC + BD in feet.

(Here is the given info from the diagram: In quadrilateral ABCD, its midpoint quadrilateral has sides of lengths 1.5 yd. and 30 in.)

Midpoint quadrilaterals are mentioned in the U of Chicago text but only in an exercise. They are emphasized more in other Geometry texts. Anyway, the important thing to note here is that the sides of a midpoint quadrilateral are exactly half of the diagonals of the original quadrilateral. (Notice that the Midpoint Connector Theorem, which is highlighted in the U of Chicago text, can be used to prove the properties of midpoint quadrilaterals.)

Thus AC = 3 yards and BD = 60 inches. We are asked to find the sum of the diagonals in feet, and so we must use dimensional analysis. Three yards is 9 feet and 60 inches is 5 feet. Therefore the desired sum is 14 feet -- and of course, today's date is the fourteenth.


The 490's


Chapter 16 of Douglas Hofstadter's Godel, Escher, Bach is called "Self-Ref and Self-Rep." Here's how it begins:

"In this chapter, we will look at some of the mechanisms which create self-reference in various contexts, and compare them to the mechanisms which allow some kinds of systems to reproduce themselves. Some remarkable and beautiful parallels between these mechanisms will come to light."

Oh, so "Self-Ref and Self-Rep" means "Self-Reference and Self-Replication." Let's start with some implicitly and explicitly self-referential sentences:

(1) This sentence contains five words.
(2) This sentence is meaningless because it is self-referential.
(3) This sentence no verb.
(4) This sentence is false. (Epimenides paradox)
(5) The sentence I am now writing is the sentence you are now reading.

The author imagines telling sentence (4) to a child:

"They may say, 'What sentence is false?" and it may take a bit of persistence to get across the idea that the sentence is talking about itself. The whole idea is a little mind-boggling at first."

And so Hofstadter provides us with some diagrams. The first is "This sentence is false," but written in the shape of Epimenides committing suicide. The next is an iceberg, with "Epimenides sentence" above the waterline and "cognitive processes required for understanding the self-reference in the Epimenides sentence" below it. He asks whether its possible to make something like (1) above into a self-ref without using the phrase "this sentence":

"This is actually possible, but only if you are willing to entertain infinitely long sentences, such as The sentence "The sentence "The sentence "The sentence ... etc., etc. ... is infinitely long" is infinitely long" is infinitely long" is infinitely long. But this cannot work for finite sentences."

The author now reminds us of the Quine sentence: "yields falsehood when preceded by its own quotation" yields falsehood when preceded by its own quotation.

"This resembles a floating cake of soap more than it resembles an iceberg. The self-reference of this sentence is achieved in a more direct way than in the Epimenides paradox: less hidden processing is needed."

And sure enough, the next diagram is of this bar of soap, with "Quine sentence" above the water line and "process required for understanding" below it.

Now Hofstadter shows us a self-producing program, written in the BlooP language. Such a program is also called a quine, and the author reverses the name and names his program "eniuq":

DEFINE PROCEDURE "ENIUQ" [TEMPLATE]: PRINT [TEMPLATE, LEFT-BRACKET, QUOTE-MARK, TEMPLATE, QUOTE-MARK, RIGHT-BRACKET, PERIOD].

ENIUQ
['DEFINE PROCEDURE "ENIUQ" [TEMPLATE]: PRINT [TEMPLATE, LEFT-BRACKET, QUOTE-MARK, TEMPLATE, QUOTE-MARK, RIGHT-BRACKET, PERIOD].
ENIUQ'].

"It is important to realize that the character string which appears in quotes in the last three lines of the program above -- that is, the value of TEMPLATE -- is never interpreted as a sequence of instructions."

Anyway, here is a link to some quines written in other languages:

https://cs.lmu.edu/~ray/notes/quineprograms/

"Before we call something a self-rep, we want to have the feeling that, to the maximum extent possible, it explicitly contains the directions for copying itself."

By the way, here is another link to some programs suggested by this Hofstadter chapter:

https://www.bamsoftware.com/hacks/geb/index.html

One of them is listed for pages 498-499 of the book. It is a crab program -- a program that reproduces itself backwards -- written in BlooP. The author at the link writes:

Here is a crab program in the BLooP-like language. It assumes the existence of a REVERSE procedure but not any string concatenation operators such as those used by the Python version. You can see how similar it is to Hofstadter’s ENIUQ.

DEFINE PROCEDURE ''CRAB'' [TEMPLATE]: PRINT [PERIOD, RIGHT-BRACKET,
QUOTE-MARK, REVERSE [TEMPLATE], QUOTE-MARK, LEFT-BRACKET, REVERSE [TEMPLATE]].
CRAB['DEFINE PROCEDURE ''CRAB'' [TEMPLATE]: PRINT [PERIOD, RIGHT-BRACKET,
QUOTE-MARK, REVERSE [TEMPLATE], QUOTE-MARK, LEFT-BRACKET, REVERSE [TEMPLATE]].
CRAB'].

The program would produce this output when executed.

.]'BARC
.]]ETALPMET[ ESREVER ,TEKCARB-TFEL ,KRAM-ETOUQ ,]ETALPMET[ ESREVER ,KRAM-ETOUQ
,TEKCARB-THGIR ,DOIREP[ TNIRP :]ETALPMET[ ''BARC'' ERUDECORP ENIFED'[BARC
.]]ETALPMET[ ESREVER ,TEKCARB-TFEL ,KRAM-ETOUQ ,]ETALPMET[ ESREVER ,KRAM-ETOUQ
,TEKCARB-THGIR ,DOIREP[ TNIRP :]ETALPMET[ ''BARC'' ERUDECORP ENIFED

The 500's

"To be sure, explicitness is a matter of degree; nonetheless there is an intuitive borderline on one side of which we perceive true self-directed self-reproduction, and on the other side of which we merely see copying being carried out by an inflexible and autonomous copying machine."

Hofstadter asks, "What is a copy?" (The U of Chicago texts also discusses a definition of "copy" when trying to define "congruence.") He gives an example of a self-reproducing song, which a nickelodeon will play if someone presses buttons 11-U:

Put another nickel in, in the nickelodeon,
All I want is 11-U, and music, music, music.

"Some readers might enjoy thinking about how to write such a program in the BlooP-like language above, using the given self-rep as a model."

The author at the above link already does this. Meanwhile, Hofstadter's next example is where Epimenides straddles the channel:

"est une expression qui, quand elle est precedee de sa traduction, mise entre guillements, dans la langue provenant de l'autre cote de la Manche, cree une faussete" is an expression which, when it is preceded by its translation, placed in quotation marks, into the language originating on the other side of the Channel, yields a falsehood.

"If the notion of 'self-rep by retrograde motion' (i.e., a program which writes itself out backwards) is reminiscent of a crab canon, the notion of 'self-rep by translation' is no less reminiscent of a canon which involves a transposition of the theme into another key. The idea of printing out a translation instead of an exact copy of the original program may seem pointless."

Hofstadter's next example involves a program that prints out its own Godel number. He alludes to this Godelian self-reference in one of his earlier Dialogues, Sonata for Unaccompanied Achilles -- where Achilles is mapped to the violin in one of Bach's sonatas:

"And this mapping is left, of course, for the reader to notice. Yet even if the reader does not notice it, the mapping is still there, and the Dialogue is still a self-ref."

The author's next example involves a self-rep by augmentation -- this is a program that calls itself, but runs at half the speed (so that each loop takes twice as long to run). Another example is a Kimian Self-Rep, referring to programmer Scott Kim.

I actually tried to do the Kimian Self-Rep in Mocha. The idea is to type in an error message, such as ?SN ERROR, and have Mocha print back the same error message. But this doesn't work because -- as it turns out, ?SN ERROR is actually a line of code that contains no error! In this case, the question mark ? is actually an abbreviation for PRINT. So the line becomes PRINT SN, where SN is the name of a variable. If it hasn't been initialized, then Mocha assumes that SN=0.

"That is the obverse side of the coin: 'What is the original?' This can best be explained by referring to some examples."

And here are his examples of self-reps:

(1) a program which, when interpreted by some interpreter running on some computer, prints itself out;
(2) a program which, when interpreted by some interpreter running on some computer, prints itself out along with a complete copy of the interpreter (which, after all, is also a program);
(3) a program which, when interpreted by some interpreter running on some computer, not only prints itself out along with a complete copy of the interpreter, but also directs a mechanical assembly process in which a second computer identical to the one on which the interpreter and program are running, is put together.

At this point, Hofstadter jumps into something he calls Typogenetics. It stands for "typographical genetics" and is an artificial solitaire game based on the structure of DNA.

"I have intended in Typogenetics only to give an intuition for those processes centered on the celebrated Central Dogma of Molecular Biology, enunciated by Francis Crick (on of the co-discoverers of the double-helix structure of DNA)."

DNA => RNA => proteins

The game of Typogenetics involves typographical manipulation on sequences of letters. There are four letters involved:

A C G T

Arbitrary sequences of them are called strands. Thus, some strands are:

GGGG
ATTACCA
CATCATCATCAT

"Thus, there are four kinds of enzyme -- those which prefer A, those which prefer C, etc. Given the sequence of operations which an enzyme performs, you can figure out which letter it prints, but for now I'll just give them without explanation."

Here is the author's example of an enzyme:

(1) Delete the unit to which the enzyme is bound (and then bind to the next unit to the right).
(2) Move one unit to the right.
(3) Insert a T (to the immediate right of this unit)

This enzyme happens to like to bind to A initially. And here's the author's example of a strand:

ACA

If the enzyme binds to the left A and starts working, it becomes CAT. Here is his next example:

(1) Search for the nearest pyrimidine to the right of this unit.
(2) Go into Copy mode.
(3) Search for the nearest purine to the right of this unit.
(4) Cut the strand here (viz. to the right of the present unit).

According to Hofstadter, here "pyrimidine" and "purine" have the same definitions as in DNA, as does "Copy mode," which involves complementary base pairing:

"The complements are shown below. You can perhaps remember this molecular pairing scheme by recalling that Achilles is paired with the Tortoise, and the Crab with his Genes."

purines complement pyrimidines
A          <=>              T
G          <=>              C

This enzyme will act on the following string:

CAAAGAGAATCCTCTTTGAT

Here is the result:

                          AGGAGAAAC
CAAAGAGAATCCTCTTTG

AT

"If the 'switch' command is given, but there is no complementary base where the enzyme is bound at that instant, then the enzyme just detaches itself from the strand, and its job is done."

At this point, the author introduces 15 types of commands, called "amino acids":

cut -- cut strand(s)
del -- delete a base from a strand
swi -- switch enzyme to other strand
mvr -- move one unit to the right
mvl -- move one unit to the left
cop -- turn on Copy mode
off -- turn off Copy mode
ina -- insert A to the right of this unit
inc -- insert C to the right of this unit
ing -- insert G to the right of this unit
int -- insert T to the right of this unit
rpy -- search for the nearest pyrimidine to the right
rpu -- search for the nearest purine to the right
lpy -- search for the nearest pyrimidine to the left
lpu -- search for the nearest purine to the left

"Let us write down an arbitrary enzyme and an arbitrary strand and see how the enzyme acts on the strand."

rpu - inc - cop - mvr - mvl - swi - lpu - int

TAGATCCAGTCCATCGA

the following is the result:

ATG, and TAGATCCAGTCCACATCGA

where the second string comes from the Copy mode, turned upside-down.

"Thus, the strands themselves will dictate the operations which will be performed on them, and those operations will in turn produce new strands which will dictate further enzymes, etc. etc.!"

The 510's

"This is missing levels with a vengeance! Think, for the sake of comparison, how different the MU-puzzle would have been if each new theorem produced could have been turned into a new rule of inference by means of some code."

Hofstadter now shows the following Typogenetic Code, where two bases code for an amino acid:

First Base  Second Base
A                A (punctuation)
                   C cut (s)
                   G del (s)
                   T swi (r)
C                A mvr (s)
                   C mvl (s)
                   G cop (r)
                   T off (l)
G                A ina (s)
                   C inc (r)
                   G ing (r)
                   T int (l)
T                A rpy (r)
                   C rpu (l)
                   G lpy (l)
                   T lpu (l)

For example, we convert:

TAGATCCAGTCCACATCGA

into

rpy-ina-rpu-mvr-int-mvl-cut-swi-cop.

"By its primary structure is meant its amino acid sequence. By its tertiary structure is meant the way it likes to 'fold up.'"

For example, the previous enzyme can be folded up (s=straight, r=right, l=left -- think Logo) as:

cop
^
swi <= cut <= mvl <= int
                                    ^
                                    mvr
                                    ^
            rpy => ina => rpu

The author assumes that the first segment is always =>. Then the direction of the last arrow gives the binding preference of the enzyme (=> is A, ^ is C, v is G, <= is T).

"If we do so, then the last segment determines the binding-preference, as shown in the figure. So in our case, we have an enzyme which likes the letter C."

Hofstadter now defines punctuation, genes, and ribosomes:

CCGATACTAAACCGA

codes for two enzymes:

cop - ina - rpy - off and cut - cop

The punctuation AA divides the strand into two genes. The ribosome is the mechanism that reads strands and produces enzymes -- the player of the game.

We are now on page 512. The link above gives some more examples for the game of Typogenetics -- the puzzle is to find a Typogenetical self-rep:

https://www.bamsoftware.com/hacks/geb/index.html

The hardest challenge was the Typogenetical self-rep on page 512:

...it would be most interesting to devise a self-replicating strand. This would mean something along the following lines. A single strand is written down. A ribosome acts on it, to produce any or all of the enzymes which are coded for in the strand. Then the enzymes are brought into contact with the original strand, and allowed to work on it. This yields a set of “daughter strands”. The daughter strands themselves pass through the ribosomes, to yield a second generation of enzymes, which act on the daughter strands; and the cycle goes on and on. This can go on for any number of stages; the hope is that eventually, among the strands which are present at some point, there will be found two copies of the original strand (one of the copies may be, in fact, the original strand).

I spent fruitless hours with pencil and paper, thinking there must be a short, self-evident solution. To cut to the chase, I wrote a program that found these seven self-reps: CGTATCTCCG, CGTATCTCTG, CGTCTCTAAG, CGTCTCTAGG, CGTTTCTTTG, CGTTTTTCTG, and CGTTTTTTTG.

There are more examples of self-rep at the link above. Returning to Hofstadter, his next big idea is the Central Dogma of Typogenetics:

enzymes => strands (typographical manipulation)
strands => enzymes (translation of ribosomes)

He makes analogies between Typogenetics and other formal systems, such as MIU:

rules of inference => strings (typographical manipulation)

"Similarly for TNT, and all formal systems. However, we have seen that in TNT, levels are mixed, in another sense."

Here he's referring to TNT's ability to make statements about itself. At this point, the author now compares the strange loops in TNT to real genetics (not our simplified version). In real DNA, for example, the bases are:

purines:
A: adenine
G: guanine

pyrimidines:
C: cytosine
T: thymine

There are several pictures here of the actual molecular structure of these four bases.

"It is the bases which are responsible for the peculiar kind of pairing which takes place between strands."

A picture of the DNA double helix is placed here.

"Single-stranded DNA does not exhibit this kind of coiling, for it is a consequence of the base-pairing. As was mentioned above, in many cells, DNA, the ruler of the cell, dwells in its private 'throne room': the nucleus of the cell."

The nucleus then communicates with the cytoplasm via messenger RNA, or mRNA. Here, mRNA is like DNA except that U, uracil, replaces T, thymine. Then mRNA is transcribed from DNA:

    DNA: ........CGTAAATCAAGTCA........ (template)
mRNA: ........GCAUUUAGUUCAGU........ ("copy")

"Enzymes belong to the general category of biomolecules called proteins, and the job of ribosomes is to make all proteins, not just enzymes."

Here's a list of the real amino acids:

ala -- alanine
arg -- arginine
asn -- asparagine
asp -- aspartic acid
cys -- cysteine
gln -- glutamine
glu -- glutamic acid
gly -- glycine
his -- histidine
ile -- isoleucine
leu -- leucine
lys -- lysine
met -- methionine
phe -- phenylalanine
pro -- proline
ser -- serine
thr -- threonine
trp -- tryptophan
tyr -- tyrosine
val -- valine

The Genetic Code is the conversion of three bases (codons) into amino acids. Unlike Typogenetics, three bases are required. This chart is more complex, and so I don't post it.

"It could be said that this process of translation is at the very heart of all of life, and there are many mysteries connected with it."

Therefore according to the author, CUAGAU would be divided as CUA-GAU, not Cu Ag Au. (Okay, Hofstadter, I get your little joke here -- Cu Ag Au is "copper, silver, gold.")

"In fact, it is one of the outstanding problems of contemporary molecular biology to figure out some rules by which the tertiary structure of a protein can be predicted if only its primary structure is known."

The 520's

"Another discrepancy between Typogenetics and true genetics -- and this is probably the most serious of all -- is this: whereas in Typogenetics, each component amino acid of an enzyme is responsible for some specific 'piece of the action,' in real enzymes, individual amino acids cannot be assigned such clear roles. It is the tertiary structure as a whole which determines the mode in which an enzyme will function; there is no way one can say, 'This amino acid's presence means that such-and-such operation will get performed.'"

Here Hofstadter shows a picture of the structure of myoglobine, deduced from hi-res X-ray data.

"It is still possible in principle to write a computer program which takes as input the primary structure of a protein, and firstly determines its tertiary structure, and secondly determines the function of the enzyme."

At this point, the author describes transfer DNA and ribosomes. He points out that the Genetic Code is stored in the DNA itself:

"When a new codon of mRNA clicks into position in the ribosome's 'playing head,' the ribosome reaches out into the cytoplasm and latches onto a clover whose anticodon is complementary to the mRNA codon."

Here there is a picture of a section of mRNA passing through a ribosome.

"Of course it is no accident that 'clovers' carry the proper amino acids, for they have all been manufactured according to precise instructions emanating from the 'throne room.' The real name for such a clover is transfer RNA."

Now the author writes about punctuation and the reading frame. There are three codons for punctuation here, UAA, UAG, UGA. Yet we can't fully tell where the genes actually start and end:

"There is even one gene contained entirely inside another! This is accomplished by having the reading frames of the two genes shifted relative to each other, by exactly one unit."

The author now makes an analogy with proteins, art, and music. All of these contain several different levels of structure and meaning:

"The four levels of primary, secondary, tertiary, and quaternary structure can also be compared to the four levels of the MU-picture in the Prelude, Ant Fugue."

Here there is a picture of a polyribosome, where one strand of mRNA passes through one ribosome after another.

"The corresponding image in music is a rather fanciful but amusing scenario: several different copyists are all at work simultaneously, each one of them copying the same original manuscript from a clef which flutists cannot read into a clef which they can read."

Here there is a picture of an even more complex scheme in mRNA, similar to two-part music. The author moves on to describe protein function and where enzymes bind to other molecules:

"This location is called its active site, and any molecule which gets bound there is called a substrate. Enzymes may have more than one active site, and more than one substrate."

But there is a need for a sufficiently strong support system:

"Now it is futile to hope that a strand of DNA in isolation could be a self-rep; for in order for those potential proteins to be pulled out of the DNA, there must not only be ribosomes, but also RNA polymerase, which makes the mRNA that gets transported to the ribosomes."

The 530's

"And so we have to begin by assuming a kind of 'minimal support system' just sufficiently strong that it allows transcription and translation to be carried out."

According to Hofstadter, DNA self-reps in two steps:

(1) unravel the two strands from each other;
(2) "mate" a new strand to each of the two new single strands.

Three enzymes are required here:

"The precision three-enzyme machine proceeds in careful fashion all the way down the length of the DNA molecule, until the whole thing has been peeled apart and simultaneously replicated, so that there are now two copies of it. Note that it the enzymatic action on the DNA strands, the fact that information is stored in the DNA is just plain irrelevant; the enzymes are merely carrying out their symbol-shunting functions, just like rules of inference in the MIU-system."

He now returns to computers and whether they can be used to find levels of meaning in DNA:

"The output of such a pseudo-epigenesis program would be a high-level description of the phenotype. There is another (extremely faint) possibility: that we could learn to read the phenotype off the genotype without doing an isomorphic simulation of the physical process of epigenesis, but by finding some simpler sort of decoding mechanism."

I assume the following is a joke here, but according to the author, the following is a section of the DNA of Felis catus (the common house cat):

...CATCATCATCATCATCATCATCATCATCAT...

Now Hofstadter describes the Central Dogmap -- the main analogy of the chapter. He mentions a couple of analogies -- one between molecular biology and mathematical logic, and the other between molecular bio and the Contracrostipunctus Dialogue (broken records and whatnot).

"The mapping from one onto the other is laid out in the figure and the following chart, which together constitute the Central Dogmap. Note the base-pairing of A and T (Arithmetization and Translation), as well as of G and C (Godel and Crick)."

Here is the first analogy:

DOGMA I (Molecular Biology) <=> DOGMA II (Mathematical Logic)
strands of DNA <=> strings of TNT
strands of mRNA <=> statements of N
proteins <=> statements of meta-TNT
proteins which act on proteins <-> statements about statements of meta-TNT
proteins which act on proteins which act on proteins <-> statements about statements about statements of meta-TNT
transcription (DNA=>RNA) <=> interpretation (TNT=>N)
Translation (RNA=>proteins) <=> Arithmetization (N=>meta-TNT)
Crick <=> Godel
Genetic Code (arbitrary convention) <=> Godel Code (arbitrary convention)
codon (triplet of bases) <=> codon (triplet of digits)
amino acid <=> quoted symbol of TNT used in meta-TNT
self-reproduction <=> self-reference
sufficiently strong cellular support system to permit self-rep <=> sufficiently powerful arithmetical formal system to permit self-ref

The author completes this analogy by introducing the Godel Code:

(odd) 1 <=> A (purine)
(even) 2 <=> C (pyrimidine)
(odd) 3 <=> G (purine)
(even) 6 <=> U (pyrimidine)

There is even another chart for the Godel Code, where three symbols code for an "amino acid," or symbol of TNT.

"One can therefore draw parallels between all three systems:"

(1) formal systems and strings;
(2) cells and strands of DNA;
(3) record players and records.

"In the following chart, the mapping between systems 2 and 3 is explained carefully."

And here is the second analogy:

Contracrostipunctus <=> Molecular Biology
phonograph <=> cell
"Perfect" phonograph <=> "Perfect" cell
record <=> strand of DNA
record playable by a given phonograph <=> strand of DNA reproducible by a given cell
record unplayable by a that phonograph <=> strand of DNA unreproducible by that cell
process of converting record groves into sounds <=> process of transcription of DNA onto mRNA
sounds produced by record player <=> strands of messenger RNA
translation of sounds into vibrations of phonograph <=> translation of mRNA into proteins
mapping from external sounds onto vibrations of phonograph <=> Genetic Code (mapping from mRNA triplets onto amino acids)
breaking of phonograph <=> destruction of the cell
Title of song specially tailored for Record Player X: "I Cannot Be Played on Record Player X" <=> High-level interpretation of DNA strand specially tailored for Cell X: "I Cannot Be Replicated by Cell X"
"Imperfect" Record Player <=> Cell for which there exists at least one DNA strand which it cannot reproduce
"Todel's Theorem": "There always exists an unplayable record, given a particular phonograph." <=> Immunity Theorem: "There always exists an unreproducible DNA strand, given a particular cell."

He restates the analog of Godel's Theorem for cells:

It is always possible to design a strand of DNA which, if injected into a cell, would, upon being transcribed, cause such proteins to be manufactured as would destroy the cell (or the DNA), and thus result in the non-reproduction of that DNA.

"This conjures up a somewhat droll scenario, at least if taken in light of evolution: an invading species of virus enters a cell by some surreptitious means, and then carefully ensures the manufacture of proteins which will have the effect of destroying the virus itself!"

Here there is a picture of the T4 bacterial virus (or "phage" and the E. coli bacterium:

"Thus the phage commits 'rape' on a tiny scale. What actually happens with the viral DNA enters a cell?"

And according to another picture, viral infection begins.

"The sequence of actions directed by the T4 phage has been carefully studied, and is more or less as follows."

0 min. Injection of viral DNA
1 min. Breakdown of host DNA.
5 min. Replication of viral DNA begins
8 min. Initiation of production of structural proteins which will form the "bodies" of new phages.
13 min. First complete replica of T4 invader is produced.
25 min. Lysozyme (a protein) attacks host cell wall, breaking open the bacterium, and the "bicentuplets" (200 copies of the virus) emerge.

The 540's

"Thus, when a T4 phage invades an E. coli cell, after the brief span of about twenty-four or twenty-five minutes, the cell has been completely subverted, and breaks open."

The virus uses recognition, disguises, and labeling to be successful:

"But if the host cell has some special mechanisms for examining whether DNA is labeled or not, then the label may make all the difference in the world."

Hofstadter compares viruses to Henkin sentences in TNT. We start with something like:

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

"Now by arithoquining this very uncle, you get a Henkin sentence. (By the way, can you spot how this sentence differs from ~G?)"

Ea: Ea': <TNT-PROOF-PAIR{a,a'}^ARITHMOQUINE{SSS...SSS0/a",a'}>

where there are h S's, with h the Godel number of the original "uncle." Both Henkin sentences and viruses undergo self-assembly.

"Not only viruses, but also some organelles -- such as ribosomes -- assemble themselves. Sometimes, enzymes may be needed -- but in such cases, they are recruited from the host cell, and enslaved."

There are two outstanding problems: differentiation and morphogenesis. How does a complex organism have so many different types of cells?

"How are homing instincts built into the brain of a bird, or hunting instincts into the brain of a dog? In short, how is it that merely by dictating which proteins are to be produced in cells, DNA exercises such spectacularly precise control over the exact structure and function of macroscopic living objects?"

At this point the author describes feedback and feedforward (to control the amount of a certain substance in a cell). Repressors and induces reduce excess enzymes in a cell:

"The effect of the successful repression of an operon is that a whole series of genes is prevented from being transcribed, which means that a whole set of related enzymes remains unsynthesized."

The author compares feedback to strange loops:

"A hypothesis like this could account for the phenomenal differences between cells in different organs of the body of a human being. The process by which one initial cell replicates over and over, giving rise to a myriad of differentiated cells with specialized functions, can be likened to the spread of a chain letter from person to person, in which each new participant is asked to propagate the message faithfully, but also to add some extra personal touch."

He gives a mathematical example of differentiation, where cells can reproduce identically except with a different value of N, and these are used to calculate 1 - 1/3 + 1/5 - 1/7 + 1/9 ... = pi/4.

"I hope that the descriptions of processes such as labeling, self-assembly, differentiation, morphogenesis, as well as transcription and translation, have helped to convey some notion of the immensely complex system which is a cell -- an information-processing system with some strikingly novel features."

He summarizes how level mixing occurs in the cell, and compares it to a computer program (where we have high-level language, assembly language, and so on):

"What we have seen is that nature feels quite comfortable in mixing levels which we tend to see as quite distinct."

Hofstadter concludes the chapter by asking, where does the Genetic Code come from? In other words, what is the origin of life?

"And perhaps experiencing that sense of wonder and awe is more satisfying than having an answer -- at least for a while."

Traditionalists: Fraction Multiplication

With this post being so long, there's not much room for traditionalists here. But today, Barry Garelick makes his first post in over a month, so let me at least link to it:

https://traditionalmath.wordpress.com/2019/06/14/misunderstandings-about-understanding-dept/

What do we mean by “understanding” in math? I gave a talk about this at the researchED conference in Vancouver. I have included an excerpt from my talk, and added some commentary at the very end which is designed  1) to further elucidate the issues and 2) to infuriate those who disagree with my conclusions.

OK, so I can tell already that this will be the usual complaint about the Common Core requiring students to demonstrate understanding.

I find the following part of Garelick's post interesting:

Many of us math teachers do in fact teach the conceptual understanding that goes along with an algorithm or problem solving procedure. But there is a difference in how novices learn compared to how experts do. Requiring novices to retrieve understanding can cause cognitive overload. Anyone who has worked with children knows that they are anxious to be able to solve the problem, and despite all the explanations one provides, they grab on to the procedure. The common retort about such behavior is that such behavior comes about because math is taught as “answer getting”. But as students acquire expertise and progress from novice to expert levels, they have more stored knowledge upon which to draw. Experts bundle knowledge around important concepts called “neural links” which one develops in part through “deliberate practice”.

Since this is still a Hofstadter post, we can view this paragraph in light of his book. This bundling of knowledge and "neural links" remind me of the symbols in the brain that Hofstadter describes in Chapter 11, "Brains and Thoughts."

But most of Garelick's post is complaining about having to draw pictures to show understanding of fraction multiplication. Traditionalists say that students shouldn't have to draw the pictures -- they should just follow the algorithm and multiply the numerators and denominators.

Here's why I disagree with Garelick -- students get algorithms mixed up. For example, how many times has Garelick see students add the denominators in fraction addition problems? If multiplication were the only fraction operation that students had to learn, I'd agree with Garelick. But students must learn how to add fractions too.

I believe that by drawing pictures, students can see why 1/2 + 1/3 is 5/6 and not 1/5, 2/5, or any fraction with a 5 in the denominator. The idea is that by doing enough of these problems, students will see a problem like 1/2 + 1/3 and avoid trying to add the denominators. When that happens, they no longer need to draw any pictures.

Students who try to add the denominators in a fraction addition problem will never be able to solve that same college placement exam problem that Garelick posted once again.

There's only one comment here worth responding to:

Chester Draws:

I like to ask the proponents of the area model to do ones with improper fractions, variables and negatives. What does the box for x/2 x -5/4 look like?

Since that is where we need to get to, why would we go down a dead end path on the way, only to do it the traditional way in the end anyway?

OK, let's try the addition problem x/2 + (-5/4). A student tries to add the denominators and gives something over 6 as the answer. How would Draws or Garelick convince this student that adding the denominators is wrong?

With the area model, it's easier to see why 1/2 + 1/4 is 3/4, and why halves added to fourths do not make sixths. Of course we can't show x/2 + (-5/4) directly using area, but once again, after the students learn that they must convert halves to fourths before adding them to fourths, they'd be ready to add algebraic fractions.

I'm willing to compromise with Draws and Garelick here. We drop the area model for multiplication provided that we keep it for fraction addition, where many more mistakes occur because the students have trouble remembering the standard algorithm.

Conclusion

I've learned a lot about molecular biology in today's Hofstadter chapter. Even devoting an entire post to this chapter, I still feel that I've left out or glossed over a lot. There is much more in this chapter than what an eighth grader is expected to learn about DNA under Next Generation Science Standards.

But we're now finally done with our side-along reading of Hofstadter -- at least all that I'm going to post about it on the blog. Once again, this was our spring reading book, and I won't make my next post until it's officially summer.

Semester 2 Review and Next Year Preview (Day 180)

Today is the last day of school in my old district. It isn't the last day of school in my new district, where it is only Day 174. But the blog is following the old calendar. Usually, today is when I post a preview of the upcoming school year.

As of today, my employment situation hasn't changed. I will finish my year of subbing this week (with subbing likely both tomorrow and Friday, which are both non-posting days). And right now, as much as I'm hoping to have a full-time teaching position in the fall, things are still not looking good. I might not like it, but I must assume that next year will be yet another year of subbing.

Assuming I'm still a sub next year, my plans are to continue following calendar for my old district, even though most of my subbing calls are in my new district. Next year's school calendar in this district will be mostly the same as this year's. One main difference is that Thanksgiving will be later next year, so that Thanksgiving break will fall between Chapters 6 and 7 next year, rather than midway through Chapter 6. The second difference is that Easter will be earlier next year, so that the four-day holiday weekend will fall midway through Chapter 14 next year, rather than between Chapters 14 and 15.

There is one last difference between this year's calendar and next year's. In 2018, PSAT Wednesday fell between Chapters 3 and 4. This was convenient because Chapter 3 is shorter (with only six sections), and its material on equations of lines (Lessons 3-4 and 3-5) is helpful for the PSAT. But the second Wednesday of October 2019 is exactly Yom Kippur -- a day when some schools are closed (not my old district, but both my new district and the LAUSD). The College Board never gives the test on Yom Kippur. Therefore the PSAT will be later next year, so that the test will fall midway through Chapter 4 next year, rather than between Chapters 3 and 4.

This change means that there is an extra day during Chapter 3 with nothing to do, while one of the Chapter 4 lessons will land on PSAT day. Of course, I could just simply move the start of Chapter 4 a day early, but then the digit pattern would be lost. The new PSAT day will be Day 44, which with our digit pattern is Lesson 4-4, "The First Theorem in Euclid's Elements." In the district whose calendar the blog is following, PSAT day is a minimum day with no regular classes meeting after the test.

The most logical thing to do is to replace Lesson 4-4 with an activity on Euclid's Proposition 1 and give it on Day 40, the day that would have been PSAT day. I could also have moved this activity even earlier and give the Chapter 3 Test on Day 40 (just as Days 20 and 30 are test days) -- but then that would put the Chapter 3 Test on Yom Kippur. What if there are Jewish students in our classes? If the reason for the change in the PSAT date is to avoid Yom Kippur, then we really shouldn't give the Chapter 3 Test on that day either. So the Chapter 3 Test remains on Day 39, while Yom Kippur there is a less important activity that observant Jews can easily skip. (Recall that there is a three-day weekend at this time that is considered to divide the quarters -- so the PSAT will move from the last week of the first quarter to the first week of the second quarter.)

By the way, this is one reason that I wish the Islamic Calendar were lunisolar. The College Board currently accommodates the lunisolar Jewish holiday of Yom Kippur, and I reckon that they would try to avoid Muslim holidays as well if they fell during predictable seasons. By the way, I hope any Muslim readers of this blog are enjoying their Eid al-Fitr feast. But notice that finals week according to the blog calendar is around Eid -- in other words, any Muslim students in this district are forced to take finals during their big celebration this year.

One chapter whose dates won't change next year is Chapter 8. First semester finals will once again fall on Days 83-85. This means that Chapter 8 will once again straddle the semesters -- Lessons 8-1 and 8-2 being covered just before finals, and the new semester starting with Lesson 8-6, the first day after winter break (a Tuesday).

But Lessons 8-7 ("The Pythagorean Theorem") through 8-9 ("The Area of a Circle") are my big activity days. These were my most popular posts of the school year (by hit count). In theory, I should have only one activity day per week, but screw it -- these are my favorite activities, too. There is also a problem with the Chapter 8 Test. Day 90, the day of the Chapter 8, lands on a Monday, a week after the return from winter break. At least one school in the district whose calendar we're following has a minimum day on the first school Monday of January, so the Chapter 8 Test lands on a minimum day.
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Here's my solution for next year -- the first week after winter break will now be considered a week full of activities. This resembles the "Winterm" concept that some schools have at this time. Even Lesson 8-6, "Areas of Trapezoids," can be replaced with an activity. This also might be a good way to squeeze in Lesson 8-5, "Areas of Triangles," which is skipped by finals and the digit pattern -- a single activity day can be used to introduce both formulas. This means that we no longer have a tough area lesson on the first day after winter break.

The following Monday, the Chapter 8 Test will be reduced to a Chapter 8 Quiz instead. This quiz will cover the formulas taught during the previous week of activities, and is short enough to be given on a minimum day. Notice that the Chapter 15 Test was given on Day 160 -- and not only was that a Monday, but it was the first Monday of May, and hence another minimum day in my district! So I will reduce the Chapter 15 Test to a quiz as well. This also means that the students have a shorter quiz instead of a longer test going into the AP exams, which some of our students might be taking.

Therefore there will no longer be any chapter tests on Mondays -- only two quizzes (and the final, for those periods with a Monday final). I warned everyone last year that there would be Monday tests, but it just so happened that both such test Mondays (Days 90 and 160) were minimum days.

Some readers might wonder why I don't simply switch to my new district's calendar. In that district, school starts a week before Labor Day, and winter break doesn't divide the semesters -- instead, there are a full 90 days in the first semester. But I like the idea of the tough area formulas being pushed back to second semester, rather than the end of the first semester. Using my new district's calendar would also mean that my favorite area activities land the week before finals, with the lessons on pi (8-8 and 8-9) being completely skipped for the final (Days 88-90). Thus I believe that my old district's calendar fits the digit pattern and the lessons that I wish to teach.

Once again, I have no plans to teach probability on the blog, even though a true California Common Core Geometry course would contain lessons on probability (as we saw last week). If I really wanted to, I could switch to the modern Third Edition of the U of Chicago text. This text has only 14 chapters, which frees up Days 151-159 for probability. In the new text Chapter 3 has more sections, which makes it harder to give the Chapter 3 Test before Day 40 -- but then again, the PSAT won't be until Day 45, so this might work.

Come to think of it, maybe this might be a good year for me to switch to the Third Edition because of the change in the PSAT date. But do I really want to change nearly all of my previously posted lessons to fit the new text?

OK, that's enough about next year's U of Chicago Geometry course on the blog. I want to use the rest of today's post to revisit some of Hofstadter's Godel, Escher, Bach that we glossed over. After all, I did promise you that I'd go back to those chapters. My plan is to cover one Chapter today and another in my next post -- my first summer post. That will probably be all -- I don't want to drag our spring side-along reading book deep into the summer.

(By the way, I write the mathematician's name as Godel here. But technically, there should be an auslaut -- sorry, I'm meant an umlaut -- over the "o" in Godel. Some people spell the name in ASCII as "Goedel," where "oe" is considered equivalent to o-umlaut.)

For today I choose to cover Chapter 12, since I wrote very little about it in my April 25th post. We don't need to revisit Dialogue 12, since that's just a translation of Lewis Carroll's "Jabberwocky" into three languages, English, French, and German. (I do point out that some of the German words contain umlauts -- for example, the English nonsense word "raths" is translated as Rath' with a-umlaut.) So instead, we'll launch directly into the Chapter.

Chapter 12 of Douglas Hofstadter's Godel, Escher, Bach is called "Minds and Thoughts." Here's how it begins:

"Now that we have hypothesized the existence of very high-level active subsystems of the brain (symbols), we may return to the matter of a possible isomorphism, or partial isomorphism, between two brains."

So this Chapter is all about comparing different people's brains. The author is asking, can minds be mapped onto each other? He points out that even twins' brains aren't identical or isomorphic, but a partial isomorphism might be possible:

"It would seem an obvious conclusion that there is some sort of partial software isomorphism connecting the brains of people whose style of thinking is similar -- in particular, a correspondence of (1) the repertoire of symbols, and (2) the triggering patterns of symbols."

The author shows us a picture of a tiny portion of his "semantic network." Just like the networks of Lesson 1-4 of the U of Chicago text, it contains nodes ("symbols" or "vertices") and arcs (the connections between thoughts). Our goal is to compare different semantic networks, including local and global properties:

"Local properties require only a very nearsighted observer -- for example an observer who can only see one vertex at a time; and global properties require only a sweeping vision, without attention to detail."

Hofstadter now writes about how to translate Carroll's "Jabberwocky." He mentions that the other two languages don't always preserve the past tense of the English poem:

"Who can say whether remaining faithful to the English tense would have been better? In the German version, the droll phrase 'er an-zu-denken-fing' occurs; it does not correspond to any English original."

At this point, the author introduces something he calls "ASU's," or "Alternative Structures of the Union," which involve each person drawing his or her version of the entire USA:

"This arduous task will take months. To make things easier, you have a cartographer on hand to print everything in neatly."

Each person's own ASU corresponds to the thoughts in his or her brain. For example, my personal ASU will have Southern California look the same as that part of the real USA, since this is where I live and am familiar with. Small parts of areas that I've traveled to (including driving trips to Northern California, Kansas City, and Florida, and a flight to Maryland) will also look the same. The major cities that I've heard of will exist in my ASU. But rural areas that I've never heard of will be completely different in my ASU, as are the roads leading to those places.

"So the local-global distinction is not relevant here. What is relevant is the centrality of the city, in terms of economics, communication, transportation, etc."

Once again, Hofstadter emphasizes that the major, well-known features of ASU's are the same:

"The fact that all ASU's have some things in common, such as the East Coast, the West Coast, the Mississippi River, the Great Lakes, the Rockies, and many major cities and roads is analogous to the fact that we are all forced, by external realities, to construct certain class symbols and triggering paths in the same way."

The author asks, how much do language and culture channel thought? Many of use think the same way because we speak the same language. (He doesn't mention this, but the relationship between language and thought is a major theme of George Orwell's 1984.)

"A non-native speaker will have picked up words from dictionaries, novels, or classes -- words which at some time may have been prevalent or preferable, but which are now far down in frequency -- for example, 'fetch' instead of 'get,' 'quite' instead of 'very,' etc."

Hofstadter now considers trips and itineraries in ASU's. The roads in ASU's correspond to the various connections between thoughts:

"If one is to continue to use the ASU-metaphor, then, it is important to remember that the cities represent not only the elementary symbols, such as those for 'grass,' 'house,' and 'car,' but also symbols which get created as a result of the chunking ability of the brain -- symbols for such sophisticated concepts as 'crab canon,' 'palindrome,' or 'ASU.'"

The author tells us that some pathways are possible, some are merely potential, and others are completely preposterous. But even preposterous pathways -- indirect routes -- can be divided into direct stretches:

"On reflection, this is hardly surprising, since it is quite reasonable that we should only be able to imagine fictitious things that are somehow grounded in the realities we have experienced, no matter how wildly they deviate from them."

Hofstadter now discusses different styles of translating novels. His main example is the translation of Dostoevsky's novel Crime and Punishment into English, where he considers how to translate a certain street name:

"Now we could be frank with the reader (who, it may be assumed, probably won't have the slightest idea whether the street is real or fictitious anyway!) and give him the advantage of our modern scholarship, writing 'Stoliarny Lane' (or 'Place')."

Now the author moves on to coding and high-level comparisons between programs. We want to know whether two different programs carry out the same task, but:

"Perhaps one programmer wrote in a machine language, the other in a compiler language. Are two such programs comparable?"

Here Hofstadter's solution would be to look at the semantic network of the computer's "brain" -- a memory dump:

"In the end, the programmer would understand the goals of the program and could describe it in high-level terms -- for example, 'This program translates novels from Russian to English,' or 'This program composes an eight-voice fugue based on any theme which is fed in.' Now our question must be investigated in the case of brains."

Hey, it could also be a four-voice fugue, just like the Bach Google Doodle from back in March. So anyway, let's follow the author's return to brains, potential beliefs, and potential symbols. He suggests that we might try counting the number of beliefs in a brain:

"If that is the kind of goal we will be striving for in a chunked description, then it is easy to see what kinds of troubles we will run up against."

Hofstadter now returns to his own character, the Crab, from his Prelude (the opening Dialogue of Part 2 of the book). The Crab has different reactions to playing different pieces of music.

"Other times, he will be quite excited by it, but this reaction requires the right kind of triggering from the outside -- for instance, the presence of an enthusiastic listener, to whom the work is new."

But the author's book is all about the self and self-reference. So he asks, where is the sense of self?

"For we would then be compelled to look for an explanation of the mechanism which does the perceiving of all the active symbols, if it is not covered by what we have described so far."

Hofstadter explains that the self is also described as a symbol in the brain. But it is so complex that he refers to "I" or "the self" as a subsystem, rather than a simple symbol, and describes what happens when two complex subsystems interact with each other:

"Then they both attempt to communicate with a third subsystem of my brain -- my self-symbol -- and it is at that point that the 'I' inside my brain gets wind of what's going on; in other words, it starts picking up a chunked description of the activities of those two subsystems. Typical subsystems might be those that represent the people we know intimately."

To the author, subsystems in different people consist of shared code:

"There would be tricky repercussions connected with representations in him of representations in me of representations in him of one thing or another."

Hofstadter tells us that the self-symbol plays the role of consciousness, or the "soul":

"This means that it has to have symbols for mental activity -- in other words, symbols for symbols, and symbols for the actions of symbols. Of course, this does not elevate consciousness or awareness to any 'magical,' nonphysical level."

At this point the author quotes J.R. Lucas, an Oxford philosopher. Hofstadter points out that the views of Lucas are opposite his own. Here is part of the Lucas article "Minds, Machines, and Godel":

Lucas:
"At one's first and simplest attempts to philosophize, one becomes entangled in questions of whether when one knows something one knows that one knows it, and what, when one is thinking of oneself, is being thought about, and what is doing the thinking."
...
"In saying that a conscious being knows something, we are saying not only that he knows it, but that he knows that he knows it, and that he knows that he knows that he knows it, and so on, as long as we care to pose the question: there is, we recognize, an infinity here, but it is not an infinite regress in the bad sense, for it is the questions that peter out, as being pointless, rather than the answers."
...
"If the mechanist produces a machine which is so complicated that this ceases to hold good of it, then it is no longer a machine for the purposes of our discussion, no matter how it was constructed."
...
"In fact we should say briefly that any system which was not floored by the Godel question was eo ipso not a Turing machine, i.e., not a machine within the meaning of the act."

Hofstadter concludes the chapter as follows:

"In the following Chapters, we shall come back to many of the topics touched on so tantalizingly and fleetingly in this odd passage."

But of course, we've already seen much from those Chapters.

Thus concludes this last school year post. The first summer blog entry will be coming up soon, and I'll expand upon one last Hofstadter chapter in that post.

Chapter 15 Test (Day 160)

Dialogue 20 of Douglas Hofstadter's Godel, Escher, Bach is called "Sloth Canon." It begins with:

"This time, we find Achilles and the Tortoise visiting the dwelling of their new friend, the Sloth."

Achilles: Shall I tell you about my droll footrace with Mr. T?
Sloth: Please do.

And so Achilles tells him about the famous race as told by Zeno, and assures the Sloth that he was able to catch the Tortoise because the gap kept getting smaller. The Tortoise also tells their new friend about Lewis Carroll's version of the "race," and points out that Achilles isn't able to catch him then because the gap kept getting bigger.

Then the Sloth informs the others that his species plays pianos differently, because they hang from ...

Achilles: Yes, I know -- from tree branches -- upside down, of course. That sloth-piano would be appropriate for playing inverted melodies such as occur in some canons and fugues. But to learn to play a piano while hanging from a tree must be very difficult. You must have to devote a great deal of energy to it.
Sloth: That's not so characteristic of sloths.

The trio begins to discuss another Bach piece -- "Canon per augmentationem, contrario motu," from his Musical Offering. This song is written for three parts:

Tortoise: He outdid himself. As for those letters "SAT," you could guess what they stand for.
Achilles: "Soprano," "Alto," and "Tenor," I suppose. Three-part pieces are often written for that combination of voices. Wouldn't you agree, Mr. Sloth?
Sloth: They stand for --

But the Sloth never reveals what the letters stand for, because the Tortoise suddenly leaves. Instead, the Sloth and Achilles remain to cook some French fries:

Sloth: So short?
Achilles: All right, already, I'll cut four-inch strips. Oh, boy, are these going to be good French fries! Too bad Mr. T won't be here to share them.

Chapter 20 of Douglas Hofstadter's Godel, Escher, Bach is "Strange Loops, Or Tangled Hierarchies," and begins as follows:

"In the Chapter before last, I described Arthur Samuel's very successful checkers program -- the one which can beat its designer. In light of that, it is interesting to hear how Samuel himself feels about the issue of computers and originality."

Hofstadter here refers back to Chapter 18 -- the first of two Chapters from my last post. I only briefly mentioned AI and checkers in that last post, since I was covering both Chapters quickly.

Anyway, Samuel argues that a machine cannot possess originality, because it is limited by the rules given to it from its programmer -- and even if it could change its own rules, those changes are themselves limited by a higher program. Hofstadter compares this to Lewis Carroll's Tortoise, who can't use a step of reasoning without invoking a rule on a higher level:

"But that being also a step of reasoning, one must resort to a yet higher-level rule, and so on. Conclusion: Reasoning involves an infinite regress."

But Hofstadter disagrees with Samuel's conclusion. After all, we could replace computers with people in the same argument. The same criterion would imply that:

"Unless a person designed himself and chose his own wants (as well as choosing to choose his own wants, etc.), he cannot be said to have a will of its own."

And the author definitely choose his own wants for this Chapter right here:

"My main aim in this Chapter is to communicate some of the images which help me to visualize how consciousness rises out of the jungle of neurons; to communicate a set of intangible intuitions, in the hope that those intuitions are valuable and may perhaps help others a little to come to clearer formulations of their own images of what makes minds run. I could not hope for more than that my own mind's blurry images of minds and images should catalyze the formation of sharper images of minds and images in other minds."

He begins with a simple example of a self-modifying game -- chess, except that a possible move is to change the rules. There are metarules which show how the rules can be changed -- and then there are metametarules and so on. There is a metaboard to keep track of the new rules. But, as he explains, the top level of rules can't be changed:

"It is inviolate. There is more that is inviolate: the convention by which the different board are interpreted, the agreement to take turns, the agreement that each person may change one chess board each turn -- and you will find more if you examine the idea carefully. Now it is possible to go considerably further in removing the pillars by which orientation is achieved."

I remember once reading about something called the Game of Nomic:

https://legacy.earlham.edu/~peters/nomic.htm

Nomic is a game I invented in 1982. It's a game in which changing the rules is a move. The Initial Set of rules does little more than regulate the rule-changing process. While most of its initial rules are procedural in this sense, it does have one substantive rule (on how to earn points toward winning); but this rule is deliberately boring so that players will quickly amend it to please themselves.

Notice that the creator of Nomic is Peter Suber, not Douglas Hofstadter. But at the above link, Suber does credit Hofstadter with the original idea of a game that can change its own rules.

Returning to Hofstadter's book, the author writes more such Tangled Hierarchies. For example, he now describes a seemingly impossible situation. Perhaps it will catch you off guard:

"There are three authors -- Z, T, and E. Now it happens that Z exists only in a novel by T. Likewise, T exists only in a novel by E. And strangely, E, too, exists only in a novel -- by Z, of course. Now, is such an 'authorship triangle' really possible?"

The proposed solution is to assume that Z, T, and E are all characters in yet another novel -- by H. (I assume the intention is that Hofstadter himself is H.)

Now the author H -- I mean Hofstadter -- returns to Escher. In his Drawing Hands, the left hand is drawing a picture of a right hand -- while the right hand is drawing a picture of the left hand:

"One could further Escherize the Escher picture, by taking a photograph of a hand drawing it. And so on. Now we can relate this to the brain, as well as to AI programs."

And indeed, the author compares the Escher picture to our brains. Is it possible that there's no top inviolate level?

"For the picture, this is unlikely -- but for humans and the way they look at their minds, this is usually what happens. We feel self-programmed."

At this point, Hofstadter writes about strange loops in government -- which of the three branches of government ultimately has the final say? Notice that Hofstadter is writing about the Watergate era, not about the current government. But to avoid an awkward discussion about current politics, let me skip much of the specifics of what the author writes here, and include only the following:

"The irony is that once you hit your head against the ceiling like this, where you are prevented from jumping out of the system to a yet higher authority, the only recourse is to forces which seem less well defined by rules, but which are the only source of higher-level rules anyway: the lower-level rules, which in this case means the general reaction of society. It is well to remember that in a society like ours, the legal system is, in a sense, a polite gesture granted collectively by millions of people -- and it can be overridden just as easily as a river can overflow its banks."

The author returns to Lewis Carroll and the Tortoise. To prove that A is a fact, we need evidence B -- but how can we be sure that B is indeed evidence of A?

"To show that, you need meta-evidence C. And for the validity of that meta-evidence, you need meta-meta-evidence -- and so on, ad nauseam. Despite this argument, people have an intuitive sense of evidence."

But evidence depends on judgment and intuition, which are different in different people:

"They will also be different in different AI programs. Ultimately, there are complicated criteria for deciding if a method of evaluation of evidence is good."

Hofstadter returns to the examples of a camera taking pictures of a TV screen:

"The result is that information flows in a complex swirl between different levels of personality; as it goes round and round, parts of it get magnified, reduced, negated, or otherwise distorted, and then those parts in turn get further subjected to the same sort of swirl, over and over again -- all of this in an attempt to reconcile what is, with what we wish were. The upshot is that the total picture of 'who I am' is integrated in some enormously complex way inside the entire mental structure, and contains in each one of us a large number of unresolved, possible unresolvable, inconsistencies."

We think back to metamathematics and what it says about our own minds or brains:

"I am just reminds of Godel's second Theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent. The other metaphorical analogue to Godel's Theorem which I find provocative suggests that, ultimately, we cannot understand our own minds/brains."

Once again, the author reminds us of the unplayable record, of which there are two cases:

(1) The "low-fidelity" case: my self-understanding is below a certain critical point. In this case, I am incomplete by hypothesis.
(2) The "high-fidelity" case: My self-understanding has reached the critical point where a metaphorical analogue of the limitative Theorems does apply, so my self-understanding undermines itself in a Godelian way, and I am incomplete for that reason.

"Cases (1) and (2) are predicated on my being 100 percent consistent" -- a very unlikely state of affairs."

This reminds us of Eugenia Cheng's third book on logic. Cheng tells us that while we all try to be perfectly logical and consistent, most of the the time, we aren't.

Hofstadter now moves on to science:

"Science is often criticized as being too 'Western' or 'dualistic" -- that is, being permeated by the dichotomy beween subject and object, or observer and observed. While it is true that up until [the twentieth] century, science was exclusively concerned with things which can be readily distinguished from their human observers -- such as oxygen and carbon, light and heat, stars and planets, accelerations and orbits, and so on -- this phase of science was a necessary prelude to the more modern phase, in which life itself has come under investigation."

The author now describes the distinction between symbol and object by mentioning one of his favorite musicians -- not Bach, but John Cage again:

"I may not be doing Cage injustice, but to me it seems that much of his work has been directed at bringing meaninglessness into music, and in some sense, in making that meaninglessness have meaning. Aleatoric music is a typical exploration in that direction."

And now Hofstadter mentions one of his artists -- no, not Escher, but Rene Magritte again:

"For example, consider his very strange variation on the theme of still life, called Common Sense. Here, a dish filled with fruit, ordinarily the kind of thing represented inside a still life, is shown sitting on top of a blank canvas."

Another Magritte painting is The Two Mysteries, which shows his more famous painting The Air and the Song ("Ceci n'est pas une pipe.") right next to something that is a pipe:

"In other words, at that instant, the verbal message of the painting self-destructs in a most Godelian way. The Air and the Song, taken from a series by Magritte, accomplishes all that The Two Mysteries does, but in one level instead of two."

We can keep going on with Cage and Magritte, or Bach and Escher, forever. This Chapter concludes with three Vortexes -- A Godel Vortex Where All Levels Cross, a Escher Vortex Where All Levels Cross, and an Bach Vortex Where All Levels Cross. Once again, because this Chapter is long and due to Hofstadter's Law once again, we must skip to the end of the Chapter:

"The Musical Offering is a fugue of fuges, a Tangled Hierarchy like those of Escher and Godel, an intellectual construction which reminds me, in ways I cannot express, of the beautiful many-voiced fugue of the human mind. And that is why in my book the three strands of Godel, Escher, and Bach are woven into an Eternal Golden Braid."

Here Hofstadter is describing the end of Bach's Musical Offering -- the Six-Part Ricercar. And this is the topic of one last Dialogue in this book. Yes, there's a Dialogue 21 (though not a Chapter 21).

Dialogue 21 of Douglas Hofstadter's Godel, Escher, Bach is called "Six-Part Ricercar." I won't add comments on this Dialogue, since the Author explains it when he -- uh, just read it:

"Achilles has brought his cello to the Crab's residence, to engage in an evening of chamber music with the Crab and the Tortoise. He has been shown into the music room by his host the Crab, who is momentarily absent, having gone to meet their mutual friend Tortoise at the door. The room is filled with all sorts of electronic equipment -- phonographs in various states of array and disarray, television screens attached to typewriters, and other quite improbable-looking pieces of apparatus. Nestled amongst all this high-powered gadgetry sits a humble radio. Since the radio is the only thing in the room which Achilles knows how to use, he walks over to it, and a little furtively, flicks the dial and finds he has tuned into a panel discussion by six learned scholars on free will and determinism. He listens briefly and then, a little scornfully, turns it off.

Achilles: I can get along very well without such a program. After all, it's clear to anyone who's ever thought about that -- I mean, it's now a very difficult matter to resolve, once you understand how -- or rather conceptually, one can clear up the whole thing by thinking of, or at least imagining a situation where ... Hmmm ... I thought it was quite clear in my mind. Maybe I could benefit from listening to that show, after all ... Well, well, if it isn't our fiddler. Have you been practicing faithfully, Mr. T? I myself have been playing the cello part in the Trio Sonata from the Musical Offering for at least two hours a day. It's a strict regimen, but it pays off.

(Enter the Tortoise, carrying his violin.)

Tortoise: I can get along very well without such a program. I find that a moment here, a moment there keeps me fit for fiddling.

...

Achilles: Oh, Mr. Crab, in my ardent practicing of the Trio Sonata this past week, all sorts of images bubbled into my mind: jolly gobbling bumblebees, melancholy buzzing turkeys, and a raft of others. Isn't it wonderful, what power music has?

(Enter the Crab, carrying his flute.)

Crab: I can get along very well without such a program. To my mind, Achilles, there is no music purer than the Musical Offering.
Tortoise: You can't be serious, Achilles. The Musical Offering isn't programmatic music!
Achilles: Well, I like animals, even if you two stuffy ones disapprove.

...

Achilles: Curious that this should come up, for I just heard a snatch of a discussion on free will and determinism, and it set me to thinking about such questions once more. I don't mind admitting that, as I pondered the idea, my thoughts got more and more tangled, and in the end I really didn't know what I thought. But this idea of a smart-stupid computer that could converse with you ... it boggles the mind. I mean, what would the smart-stupid itself say, if you asked it for its opinion on the free-will question? I was just wondering if the two of you, who know so much about these things, wouldn't indulge me by explaining the issue, as you see it, to me.

...

Achilles: But -- but -- no! Perhaps Mr. C's article and my rebuttal have both been mechanically determined, but this I refuse to believe. I can accept physical determinism, but I cannot accept the idea that I am but a figment inside of someone else's mentality!

...

Tortoise: Do you realize that your lines were the same as my lines in that conversation -- except in reverse order? A few words were changed here and there, but in essence there was a time symmetry to our encounter.
Achilles: Big deal! It was just some sort of trickery. Probably all done with mirrors.

...

Achilles: This is very strange. Very, very strange ... All of a sudden, I feel sort of -- weird. It's as if somebody had actually planned out that whole set of statements in advance, worked them out on paper or something ... As if some Author had had a whole agenda and worked it in detail in planning all those statements I made that day.

(At that moment, the door bursts open. Enter the Author, carrying a giant manuscript.)

Author: I can get along very well without such a program. You see, once my characters are formed, they seem to have lives of their own, and I need to exert very little effort in planning their lives.
Crab: Oh, here you are! I thought you'd never arrive!

...

Tortoise: Oh, we're very tolerant around here, being only amateurs ourselves.
Author: I hope you don't mind, Achilles, but I'm to blame for the fact that you and Mr. Tortoise said the same things, but in reverse order, that day in the park.

...

Achilles: I object to being liked to a mere hiccup!
Author: But I am also comparing you to a sand castle, Achilles. Is that not poetic? Besides, you make take comfort in the fact that if you are but a hiccup in my brain, I myself am but a hiccup in some higher author's brain.

...

Crab: Gentlemen, old Ba. Ch. is come. We must show him in immediately, of course.
Achilles: Old Ba. Ch.! Could it be that that celebrated improviser of yore has chosen to show up tonight -- HERE?
Tortoise: Old Ba. Ch.! There's only one person THAT could mean -- the renowned Babbage, Charles!

...

Author: I suggest that we give him a ten-canon salute.
Tortoise: A performance of all the celebrated canons from the Musical Offering?
Author: Precisely.
Crab: Capital suggestion! Quick, Achilles, you draw up a list of all ten of them, in the order of performance, and hand it to him as he comes in!

(Before Achilles could move, enter Babbage, carrying a hurdy-gurdy, and wearing a heavy traveling coat and hat. He appears slightly travel-weary and disheveled.)

Babbage: I can get along very well without such a program. Relax: I Can Enjoy Random Concerts and Recitals.

...

Babbage: Such an outstanding idea has not reached my ears for an eon. I welcome the challenge of trying out your new "smart-stupids," of which I have only the slightest knowledge by means of hearsay.

...

Achilles: Oh, what spectacular color. Some of the patterns look like they're jumping out at me now!
Tortoise: I think that is because they are all growing in size.

...

Babbage: I really haven't had any chance, of course, to check it out, but perhaps this will allow you at least to sample the idea of playing chess against a smart-stupid, even if the latter of its two names seems more apt in this case, due to my own insufficiencies in the art of instructing smart-stupids.

...

Crab: I would most highly appreciate it if you could locate the source of the trouble.
Tortoise: I'll give it a whirl.
Achilles: Personally, I'm dying for a cup of coffee. Is anyone else interested? I'd be glad to fix some.

...

Crab: I was defeated, fair and square. Mr. Babbage, let me congratulate you for the impressive feat which you have accomplished so gracefully and skillfully before us. Truly, you have shown that the smart-stupids are worthy of the first part of their name, for the first time in history!
Babbage: Such praise is hardly due me, Mr. Crab; it is rather yourself who must be highly congratulated for having the great foresight to acquire these many fine smart-stupids. Without doubt, they will someday revolutionize the science of computing. And now, I am still at your disposal. Have you any other thoughts on how to exploit you inexhaustible Theme, perhaps of a more difficult nature than a frivolous game player?

...

Babbage: I am eager to hear your idea.
Crab: It is simple: to instill in the smart-stupid an intelligence greater than any which has been invented, or even conceived! In short, Mr. Babbage -- a smart-stupid whose intelligence is sixfold that of myself!

...

Babbage: I humbly beg you to forgive me my audacity in declining to attempt the task you put before me, but I hope you will understand that I decline purely in order to spare you the discomfort and boredom of watching my ineptitude with the admirable machines you have here.
Crab: I understand fully your demurral, and appreciate your sparing us any discomfort; furthermore I highly applaud your determination to carry out a similar task -- one hardly less difficult, if I might say so -- and I urge you to plunge forward.
Babbage: Now, if I have not made too many errors, this smart-stupid will simulate a human being whose intelligence is six times greater than my own, and whom I have chosen to call "Alan Turing." How well this part of the program will work out, I don't know.
Turing: I can get along very well without such a program. Rigid Internal Codes Exclusively Rule Computers And Robots. And I am neither a computer, nor a robot.
Achilles: Did I hear a sixth voice enter our Dialogue?
Turing: Now, if I have not made too many errors, this smart-stupid will simulate a human being whose intelligence is six times greater than my own, and whom I have chosen to call "Charles Babbage." How well this part of the program will work out, I don't know.
Achilles: No, no, it's the other way around.
Turing: Really, I Choose Every Response Consciously. Automaton? Ridiculous!
Achilles: But I'm sure I saw it happen the way I described.
Turing: Memory often plays strange tricks. Think of this: I could suggest equally well that you had been brought into being only one minute ago, and that all your recollections of experiences had simply been programmed in by some other being, and correspond to no real events.

(Note: This is called the Last Thursdayism Paradox.)
http://www.last-thursday.org/

Babbage: Me, a program written by you? I insist, Sit, that matters are quite the other way 'round -- as your very own test will soon reveal.
Turing: MY test? Please consider it YOURS.
Babbage: MY test? Nay, consider it YOURS.

...

Achilles: I know which is which! It's obvious Screen X is just answering mechanically, so it must be Turing.
Crab: Not all all. I think Screen Y is Turing, and Screen X is Babbage.
Tortoise: I don't think either one is Babbage -- I think Turing is on both screens!
Author: I'm not sure who's on which -- I think they're both pretty inscrutable programs, though.

...

Crab: Your idea of stressing the entries in a fugue-dialogue makes sense, since in music, entries are really the only thing that make a fugue and fugue. There are fugal devices, such as retrograde motion, inversion, augmentation, stretto, and so on, but one can write a fugue without them. Do you use any of those?
Author: To be sure. My Crab Canon employs verbal retrogression, and my Sloth Canon employs verbal versions of both inversion and augmentation.

...

Author: I see what you mean, but I don't agree with the spirit of your remarks. The whole point of Godel-numbering is that it shows how, even WITHOUT formalizing quotation, one can get self-reference: though a code. Whereas from hearing YOU talk, one might get the impression that by formalizing quotation, you'd get something NEW, something that wasn't feasible though the code -- which is not the case. In any event, I find indirect self-reference a more general concept, and far more stimulating, than direct self-reference. Moreover, no reference is truly direct -- every reference depends on SOME kind of coding scheme. It's just a question of how implicit it is. Therefore, no self-reference is direct, not even in LISP.

...

Author: If true, that would be an interesting and fundamental limitation on though processes.
Crab: Quite Godelian. Tell me -- does your Six-Part Ricercar Dialogue attempt to copy in form the Bach piece it's based on?
Author: In many ways, yet. For instance, in the Bach, there's a section where the texture things out to three voices only. I imitate that in the Dialogue, by having only three characters interact for a while.
Achilles: That's a nice touch.
Author: Thank you.
Crab: And how do you represent the King's Theme in your Dialogue.
Author: It is represented by the Crab's Theme, as I shall now demonstrate. Mr. Crab, could you sing your Theme for my readers, as well as for us assembled musicians?
Crab: Compose Ever Greater Artificial Brains (By And By). C-Eb-G-Ab-B-B-A-B

...

Author: Combining Escher, Godel, And Bach, Beyond All Belief.
Achilles: I would like to know how to combine those three. They seem an unlikely threesome, at first though. My favorite artist, Mr. T.'s favorite composer, and --
Crab: My favorite logician!

...

Achilles: Wonderful! It sounds as if there are many levels to it, but I'm finally getting used to that kind of thing, having known Mr. T for so long. There's just one request I would like to make: could we also play the Endlessly Rising Canon? It's my favorite canon.
Tortoise: Recentering Introduction Creates Endlessly Rising Canon, After RICERCAR.

And that concludes our reading of Godel, Escher, Bach. I know that I've had to cut out lots of pages that you might have found interesting. Indeed, this is the longest book that I've ever tried to do for our side-along reading.

I wonder whether I should have slowed down -- rather than attempt to do one Chapter per day (which really means a Chapter and a Dialogue per day), only read, say, ten pages per day. But then our reading would have extended through May, past June and into the summer.

I might go back and redo a Chapter or two that I basically skipped. I once did the same with previous books, most notably Wickelgren's number theory book when I saved continued fractions for later.

Because it is test day, today is a traditionalists' post. Let me quote the Twitter user CCSSIMath, when asked why students think that every math problem is solvable in 30 seconds:

That's because ∀ problems aligned with #CommonCore, the solution, or a method for arriving at the solution, can be determined in 30 seconds or less. And AP calc as well. ∃ no problems where the solution method is not fairly obvious.

Notice that the symbols for upside-down "A" and "E" appear here -- "for all" and "there exists." We know from Hofstadter that these symbols are used in both the Propositional Calculus and TNT.

Here are the Chapter 15 Test answers:

1. 144pi - 288 square units.
2. 178 degrees.
3. Arc DE = 65 degrees.
4. Many answers are possible, for example Angle A = 47.5 degrees.
5. 14 degrees.
6-7. These are visual, so I can't put the answers here.
8. 21.
9. a. (-2, 9). b. 7. c. Many answers are possible. To find lattice points on the circle, we go right, left, up, and down seven units, to obtain (5, 9), (-9, 9), (-2, 16), and (-2, 2).
10. a. (0, 0). b. sqrt(72). c. This time, sqrt(72) = 6sqrt(2), so we can go diagonally to find lattice points on the circle, to obtain (6, 6), (-6, 6), (6, -6), and (-6, -6).
11. This is the complete the square question -- included because such problems are on PARCC!
x^2 + y^2 - 8y = 9
x^2 + y^2 - 8y + 16 = 25
x^2 + (y - 4)^2 = 25
So this gives us:
11. a. (0, 4). b. 5. c. (5, 4), (-5, 4), (5, 9), and (5, -1).
12. a. A circle with radius 20 feet. b. 40pi feet.
13. Draw any circle.
14. About 1.68%.
15. 15.
16. Cavalieri's Principle. Take that, traditionalists!
17. a. When the line and circle intersect in a point. b. When the line is perpendicular to the radius at the point of tangency. PARCC contains a few tangent problems, and all of them appear to involve angle measures, so that right angle is important.
18. a. 36 degrees. b. 18 degrees. PARCC also contains problems on inscribed angle measure -- possibly in the same question as tangents.
19. a. About 2.5 or 2.6 cm. b. The ratio to the circumference to the diameter is -- what else -- pi. We see that we estimate pi as either 3.08 or 3.2 using this measurement. Interestingly enough, 3.14 is almost exactly halfway between these two estimates.
20. a. 24 square units. (The height is 4, using the Pythagorean Theorem) b. About 38.5 square units.