"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.
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.
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