On Thursday of this week we will begin our group discussion of Hofstadter’s book, Gödel, Escher, Bach (see here for more details). I’m posting this a little early since it is the first assignment/discussion. The plan is to post on Thursdays and Mondays so that participants can start commenting (this means you will have to do the reading BEFOREHAND, so check and follow the schedule!). We’re still working out the best day/format for a live group chat. Stay tuned.
Assignment for Thursday, September 10 (Part I: GEB begins)
Read: Introduction
Listen: Brandenburg Concerto No. 5 (BWV 1050)
You still have time to order the book from Amazon.com (or pick one up at a used book store, mine was 25¢). This is a no guilt discussion group. If you miss a week, try to jump back in. You can also read/listen ahead by looking at the schedule posted here.
Douglas Hofstadter wrote a very helpful preface to the 20th anniversary edition of the book. The actual content of the book remains unchanged. However, if you read the preface, you will get a good idea of what Hofstadter was hoping readers would “get” in his book. He wrote it because so many people over the years have not understood what he was really trying to say. I guess you could say there was a real chasm between reader response and authorial intention! You can read a fair bit of it by creatively using the Amazon “Look Inside” feature. Actually, you can read all of it if you try hard enough (first puzzle of the course to solve).
Feel free to start posting your thoughts and/or questions on the Introduction!
Up next (For September 14):
Read: Three-Part Invention and Chapter I: The MU-puzzle
Listen: The Three-Part Ricercar, from the Musical Offering (BWV 1079), introduces the Kings theme (which appears in nearly every piece of the Musical Offering) and the fugue style in general.
A few questions to get us started (pick whichever one you want to comment to):
Whats the connection between Bachs endlessly rising canon and Eschers Waterfall?
What is Hofstadters main objection to the Hierarchical Theory of Types?
A quote from GEB (p15): Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way? Are there things about infinity you find paradoxical? What are they? What is the essential quality of infinity?
Describe an example of a strange loop (other than the ones we’ve read about).
I’m not sure if I’ll have time after all to follow this to the end, but I’ll try. I just read the intro so I’ll go ahead and chime in now.
To me, “infinite” describes any set that has a proper subset which maps onto it in a 1-1 correspondence. The strange loops that H describes seem to present “infinity” to the viewer/reader in such a way that it is not always easy to distinguish between the original set and its subset. This raises the question of whether the subset is actually “proper.” I can see how these points could become relevant to thinking about philosophy of mind.
Not only that, but the hierarchy he finds so interesting have a compelling quality about them because they contain so many “puzzles,” as H calls them. For example, the subsets themselves can have their own proper, 1-1 corresponding subsets and they, too, can qualify as infinite sets, making it potentially difficult for us to decide upon appopriate frames of reference. Add to this the consideration that a set (or canon) could be made to rise one more time before looping back to the beginning suggests to me that infinity can be transcended, but his could never happen from within–you’d have to go outside of it.
1) What’s the connection between Bach’s endlessly rising canon and Escher’s Waterfall?
Short? They both are good examples of strange loops. Bach’s piece, followed through some number of times, will only loop back to the same key you started in. Escher, after 6 twists and turns from the initial fall, will be right back with it.
2) What’s H’s main objection to the Hierarchical Theory of Types?
The HToT just falls down. The act of judging any sentence as ‘nonsensical’ is itself outside of the system of hierarchy of the language.
3) Is infinity paradoxical?
Maybe. The system of mathematics is robust enough to deal with several kinds of infinity. CBOVELL: this relates to your point of transcending infinity. Math describes 2 special sets: Aleph(0) and Aleph(1). Aleph(0) is the set of all ‘countable’ infinities. The set of all whole, positive numbers ( {1,2,3…} ) would be infinitely long, but still countable. As would the set of all positive multiples of 3 ( {3,6,9,12…} ). Both would be infinitely long, yet the second would be one third the size of the first. And both are considered as part of Aleph(0)
Aleph(1) however is the set of uncountable infinities. A set of all ‘real’ numbers is in Aleph(1). More info can be found in wikipedia: http://en.wikipedia.org/wiki/Aleph_number
4) Describe an example of a strange loop
10 PRINT “Climpsun is a redneck school”
20 GOTO 10
It’s an infinite loop (chosen because GaTech is playing Clemson tonight). The interesting thing about computers is that it’s always VERY easy to write an infinite loop. But there’s a famous problem called ‘The Halting Problem’. From within, a computer can never detect that it’s inside an infinite loop. An external entity has to come in and break it. If you typed the above into a BASIC interpreter, it’d repeat “Climpsun is a redneck school” forever.
–Jason
Just stopping back to see if there was any discussion. I see that I forgot to try for my own example of a strange loop.
What comes to mind at the moment is a situation in “siamese chess” where four people are playing on two boards and the pieces captured by one teammate are passed over to the other teammate for immediate play. If both teams agree to mimic their teammates’ opponents’ moves then perhaps it might be possible to set up a strange loop after the first two moves are made.
That’s all for now. Until next week (hopefully)!
A strange loop (it’s not so strange, is it?):
1) What Carlos Bovell says in this paper is always right.
2) If what Carlos Bovell says in this paper turns out to be wrong, see #1.
Thanks, Carlos and Jason, for starting the conversation. I’m going to wait a bit to see if anyone else chimes in.
I’m no mathematician or logician, but I’m already seeing how what Hofstadter is after contains insights into areas of life that I do have some involvement in.
Here’s a summary of my thoughts so far, adapted from another forum where I’d posted them previously:
In the introduction, Hofstadter dares to take on one of the greatest intellectual works of the 20th century: Russell and Whitehead’s Principia Mathematica. Basically, R&W’s goal was to try to eliminate all paradox from mathematics. Until the 19th century, mathematicians and logicians had been very happy with the belief that Euclid had explained it all. Then in that century there came a quick burst of simultaneous discoveries that there were indeed many alternative geometries, and that while each geometry was pretty consistent within itself, it created unresolvable paradoxes when put side-by-side with another geometry. Some feared that mathematics, and even logic itself, were threatened with annihilation.
Then came Russell and Whitehead, the B. B. Warfield and Charles Hodge of mathematics. Their book was meant to devise a system that would rule out all the paradoxes. The only problem, as Hofstadter shows, is that everything they came up with violates what we know to be the case in the “real world.”Their attempt at eliminating self-contradiction is itself self-contradicting.
Without going any further into that, I was struck by the parallels to the Reformed theological world of which I was a part until recently. To vastly oversimplify, 19th century higher criticism upset the theological apple cart and threatened to throw what everyone had thought was Christianity into extinction. Along came the scientific systematic theologians. Each new work of ST produced is an attempt at a Principia Mathematica, a complete systematization of God that irons out all the paradoxes.
But now there are a whole contingent of post-liberal and post-conservative theologians who play a role similar to that of Douglas Hofstadter: if you attempt to iron out the paradoxes, you not only are doomed to failure, but you kill the very life and truth that the whole thing really is communicating.
Just some initial thoughts. I have a feeling where this book is going is going to be ripe with insights into our world, even though its about the world of mathematics.
One more thing. Loved this quotation I read this morning, from p. 22-23:
“ the drive to eliminate paradoxes at any cost, especially when it requires the creation of highly artificial formalisms, puts too much stress on bland consistency, and too little on the quirky and bizarre, which make life and mathematics interesting.”
God save us from bland consistency!
Joining the conversation late (root canal and work deadlines having intervened). I don’t have much to add to what has been said above, except to note that reading GEB also dovetails nicely with some of my other reading at the moment, namely, Plato’s Dialogues and Feser’s “The Last Superstition” (which discusses inter alia Aquinas’s proofs of God), in that all three apply logic and philosophy to the question of universals, of what is or is not real, of forms and formalisms, of infinity (of time/timelessness, of existence, etc.), and causality. It’s an interesting combination of (I think) mutually-reinforcing texts.
Oh, and here’s my example of a strange loop:
http://en.wikipedia.org/wiki/Shepard_tone
P.S. in corollary to Mark’s second post, I would add: God save us from the loss of mystery!
Yes, Carl!
I’ve become much more comfortable with the concept of mystery than my more fundamentalist past would have allowed. I’m excited that GEB may help us explore the intersection of mystery and “reality.”
Just ran across this, and I think it relates:
Turns out Dan Brown, author of the Da Vinci Code, is yet another casualty of inerrancy handled badly. From an interview in this week’s Parade:
“I was raised Episcopalian, and I was a very religious kid. Then in eighth or ninth grade, I studied astronomy, cosmology, and the origins of the universe. I remember saying to a minister, ‘I don’t get it. I read a book that said there was an explosion known as the Big Bang, but here it says God created heaven and Earth and the animals in six days. Which is right?’ Unfortunately, the response I got was, ‘Nice boys don’t ask that question.’ A light went off, and I said, ‘The Bible doesn’t make sense. Science makes much more sense to me.’ And I just gravitated away from religion.”
But then he continues…
“The irony is that I’ve really come full circle. The more science I studied, the more I saw that physics becomes metaphysics and numbers become imaginary numbers. The farther you go into science, the mushier the ground gets. You start to say, ‘Oh, there is an order and a spiritual aspect to science.'”
The first person to propose that the universe began with expansion from a primordial atom — what later was dubbed the “Big Bang” theory — was Georges Lemaitre, a Catholic priest. At the time his theory was dismissed by many scientists as a pious fantasy. Albert Einstein too initially rejected Lemaitre’s idea, but came eventually to applaud it. The rest is history.