In my earlier blog on this year's Miami Book Fair, I promised to report back after I read Rebecca Goldstein's book, Incompleteness: the Proof and Paradox of Kurt Gödel, and let you know if it could actually be comprehended by a non-mathematician. Well, I've read it, and the answer is ... kind of.
Even though my mind rebelled at the limited amount of logical notation that was included, I did grasp at least the general outline of what Gödel did. And even that was enough to amaze me with its wizardry, and to give me some small appreciation of the import of his accomplishment.
The best way for me to present my take on it is to start with another author and another interesting mind. In his book, Hackers--Heroes of the Computer Revolution, Steven Levy quotes the early MIT pioneer of computer programming, Bill Gosper, as saying, "Data is just a dumb kind of programming."
This opaque proclamation, seeming to make pretensions of being deep, is really just a concise way of saying that there is no difference in a computer between a byte of program code or a byte of data, except for the context.
A computer that attempts to "run" data, executing it as a series of instructions, will certainly "crash," since the data will not conform to the precise requirements of the codes and sequences which are the syntax of the logical language of the program. Equally, a program trying to read a stream of data will likely reach the same impasse if it suddenly encounters program codes instead--the codes would not conform to the expected type of data.
Programmers know that both of these situations occur frequently as a result of logical glitches that cause the program to look in the wrong memory location for its next instruction or next piece of data.
But now imagine a computer in which data is also programming, and vice versa. This analogy is as close as I can come to understanding the method that Kurt Gödel used in his proof of "incompleteness." He devised an ingenious system of formal logic in which the statements are simultaneously logical and arithmetical--they have logical meanings and also numerical identities.
Using this system, he proceeded to show how any logically provable statement in it had a certain mathematical characteristic. So he could mathematically analyze any statement and determine if it was provable or not.
Finally, he was able to show that certain statements which are demonstrably true (because their arithmetic works) can also be proven to be unprovable (because they don't have that telltale mathematic signature of the provable ones).
If this isn't enough to bend your mind, you must already have a mind with a mathematical bent (!) and be able to imagine this with greater perfection than I can. In fact, you may have studied Gödel himself, in which case you should write to explain how I have got it all wrong. But I think, because of what Gosper said about data and programming, that I can get a glimpse of how this would work.
What it means is that Gödel created a logical system in which numbers are simultaneously logic. Code and data are one and the same, and function together to measure what is true and what is provable.
Note that truth and provability are two separate issues here. That is both the nature of the tool that Gödel used in his proof, and also what he set about to prove: that truths will always exist that we cannot logically prove. Like Plato, he believed that Truth exists quite apart from whether we can prove it, or whether we even know about it. Truth is a priori, before experience.
Most amazing about all this is that the implications of his proof (which, not being mathematicians, we will have to take on their word) reach beyond the "sandbox" of the formal system he created, beyond mathematics and logic itself, into the realm of philosophy and metaphysics. It says, provably and conclusively, that no matter what we do there will always be truths that we cannot prove are true. This cuts to our most fundamental experience as living, conscious beings--the abundant obviousness that Something exists, that we are part of it, somehow identical with it, though we will be forever unable to prove the What and Why of it, or even that we did not imagine the whole thing.
Some trick, huh? The idea that mathematics can have something so profound to tell us about our lives is incredible and almost unprecedented.
Near the end of the book, in a wonderful sidelight on Gödel's personality, Goldstein relates what happened as he was preparing for his U.S. citizenship examination. Ever the diligent student, Gödel made a thorough study of the Constitution and was startled to discover a logical flaw in it which, it seemed to him, would allow the democracy to degenerate into tyranny!
Alas, the details of this insight, like the legendary Fermat's Proof, were never recorded even by the people who told the story, and so it has been lost to posterity. Since we will be unable to patch the error with legislation, it remains for us to live out the proof. Only time will tell if Gödel left us with one more evidence of his genius.
Monday, December 26, 2005
Saturday, December 24, 2005
At each holiday season for the past few years I've found myself thinking of the words, "for unto us a child is born," and of the wonderful musical setting given to them by Handel in his Messiah.
First, the words lead me to muse on how this particular holiday has become a celebration of the child. Even though we adults give things to one another, we know it's really about that shower of presents we rain on our children (and in my case, grandchildren). And even though there was one particular child in the past whose birth this is meant to commemorate, we give our attention, appropriately, to those who are among us now.
With wisdom, the birth of each child should be taken as the great gift that it is, the miraculous appearance on earth of a new being, a new consciousness. Each one is a new blank slate on which a future may be written, each one a new hope that the future will be an improvement on the past as we grow toward a state of perfection that we glimpse as possible.
It's as if any child might be our savior--or maybe all of them, maybe each one born is one six-billionth of a savior, each contributing to the construction of the new year, the new beginning, that is always upon us. And why not? If, as Quakers believe, "there is that of God in everyone," why should we not celebrate this universal divinity by worshiping our own children?
In light of all this, the joy expressed in Handel's music, particularly in that one chorus, seems even more meaningful. I've been listening to the excellent recording of it by Christopher Hogwood and the Academy of Ancient Music, who, despite the antique aura of their name, manage to make each note breathe with fresh life.
And if that one isn't exciting enough for you, try to find the Roche Sisters Christmas album. Their version has all the vitality of their legendary a capella performance of the Halleluia Chorus, and adds the contemporary touch of tasteful electric bass and synthesizers, with voice doubling effects to sweeten their angelic sound even further. On top of that, they came up with an inspired concluding sequence of descending tones that nails down the message with magnificent finality.
Each time they arrive at the part that says, "His name shall be called," it gives me chills as they seem to add the exclamation points that the text cries out for. Let me leave you with this:
And his name shall be called ...
The mighty God!
The everlasting Father!
The Prince of Peace.
Wednesday, December 21, 2005
To: Senators Bill Nelson and Mel Martinez,
Representative Illeana Ros-Lehtinen
I am writing to express my concern over the recent news that some members of a Quaker meeting in South Florida have been listed as a "threat" as a result of a domestic investigation by the military. (Story online at http://www.msnbc.msn.com/id/10454316/.)
For several years now I have attended the Quaker meeting in Miami. During that time I have met and got to know many Quakers. I can tell you that it would be hard to find, and almost impossible to imagine, any group whose members are more honest, kind, well intentioned, and --above all--open about their activities.
These are people who make a point of opening all their proceedings to the general public. The sign on the door always says, "All Are Welcome." Ironically, I am sure that even government investigators, had their identities been made known, would have been equally welcome to attend, listen, and participate in whatever discussions took place regarding military enlistment or other topics.
If there is any "threat" from such people, it is nothing other than the threat that truth always poses to lies and deceit. To subject any American citizens to covert investigation, and to stigmatize them for doing no more than exercising their Constitutional freedoms, would be shameful enough; but to do so to such a group of exemplary citizens, of exemplary human beings, goes beyond questions of legality, and should cause us to question the motives and intentions of those perpetrating the investigation.
I sincerely hope that, as our elected representative in Congress, you will investigate what has taken place and make every effort to see to it that our freedoms are not infinged by the ill-advised and overreaching actions of any governmental agency, and especially of the military, which has no Constitutional jurisdiction in domestic law enforcement.
We look to you to make the laws we live by, and likewise to see that they are upheld.
In accordance with the open nature of Quaker meetings, this letter is being published online at nortspews.blogspot.com so that nothing in it can be considered to be concealed. I would welcome the opportunity to publish your reply in the same spirit.
Thursday, December 08, 2005
I've just had my spiritual inspiration for the week from an unexpected source: Harper's Magazine. The December issue contains two articles I can highly recommend.
First is the cover story, "Jesus Without the Miracles," in which Eric Reese draws a fascinating parallel between Thomas Jefferson's Bible and the recently unearthed Gospel According to Thomas.
Jefferson, as a private amusement, literally took a pair of scissors to the Bible and over the course of "a few evenings" (amazing what there was time for before TV) extracted the actual teachings of Jesus from the encrustations of story and myth that grew up around them in later centuries. The result was something similar to the legendary source for the Gospels that was thought to have been lost. In it, according to Reese, "Jesus never performs a miracle and never claims that he will have to die for the sins of humankind." Instead we are left with the record of his teaching, a stream of exhortations for us to love one another, to do no harm, to "become as passers-by," to live gently on the earth.
Amazingly, even as Jefferson was working on his piece of clip-art, the actual source material was sleeping soundly in a clay jar beneath the sands of Egypt. When it was literally unearthed in the 20th century, after two thousand years, and the fragile scraps of papyrus were deciphered, the document revealed a remarkably similar teaching, full of parables and Zen koan-like puzzles, and equally devoid of any talk of miracles or salvation.
Right off the bat, for example, one of the first of these "sayings of Jesus" goes like this: "Let him who seeks continue seeking until he finds, and when he finds he will be troubled, and when he has been troubled he will marvel, and he will reign over the All." Heady stuff, seeming to describe the feelings of a modern reader trying to make sense of all this, as well as the condition of spiritual seekers of all ages, and pointing the way toward the eternal.
The companion piece in the same issue of Harper's is a wonderful memoir by Scott Korb titled "All That I Have is Yours," which is what his stepfather told him on his death bed, and also a quotation from the story of the prodigal son. That story is so well known it is easy to overlook its significance, but something about the immediacy of Korb's retelling brought it home to me in a new way.
As you may recall, the whole thing began with the younger son insisting on having his share of the inheritance right away, so his father obliged by dividing the estate and giving him half, which he promptly went away to squander. When he reappears later, destitute, the father not only welcomes him in great joy but throws him a big party and even kills the fatted calf in his honor. Seeing this, the elder brother says, "Um, excuse me, but I'm still here, I've never asked for anything, I've stayed by your side as a dutiful son, and now he comes back and you kill the fatted calf for him?"
The father replies that they have to celebrate their joy, that their son and brother who was as if dead has returned to them. But more than that, he reminds him, "everything I have is yours." In other words, you already have everything there is, what more could you want? In the larger context of the parable, in which God is the father, we are reminded that we already have everything there is, that we inherited it at birth--an entire world, a whole universe that grows in size and richness the more we learn of it--and that all we have to do in return is to share it with one another. What more could we want than everything there is?
The personal nature of this great gift appeals to the Quaker in me. It is something we all share simply by virtue of our humanity, with no need for salvation or the mediation of saints and priests on our behalf. Jesus was trying his best to tell us, "you're already saved, you already live in heaven, what more could you possibly want?" But, as he is also reported to have said, ruefully I suspect, "unless you see signs and wonders, you will not believe."
Personally, I think the survival of these words and the record of them is plenty in the way of miracles. And I'll be happy to take this day, this world, this one universe, as the plenty that it is, and say, "Thank you, thank you very much, this will do nicely."