Welcome to Digressions

You've arrived at the main page of Digressions, my personal blog. To leave a comment to a post (and please do!) click on the post title and you'll find a Comments box at the bottom of the post, or click on the Comments link at the bottom of posts on the main page. If you want to read about watches, clocks, and other mechanical diversions, I'm the US editor for Revolution Magazine, whose homepage is at www.revo-online.com.

Monday, May 23, 2011

Gödel, Incompleteness, and an Aging Brain

One of the interesting things about having kids is that they make you learn, re-learn, or realize that you have absolutely no understanding whatsoever of many things that you thought you knew, have forgotten, or prided yourself on understanding (erroneously.) Math, for me, is a biggy. Although I have managed to learn to read, when it comes to math I've always had a combination of genuine fascination with it, and almost totally functional illiteracy, at least when it comes to actually being able to calculate. One subject with which by which I was both intrigued and baffled for years was Gödel's Incompleteness Theorems, which were formulated by Kurt Gödel (left) as a refutation of the idea that mathematics could be put on a rigorously logical axiomatic foundation. He published his paper, which was a fatal torpedo in the side of the Good Ship Certainty, in 1931, under the rather diffident title "On Formally Undecidable Propositions of Principia Mathematica and Related Systems."

Like most people who can't do math but are fascinated by things for which an understanding of math is essential, I had lots of people and circumstances on which I blamed my own incompetence (the usual suspects: the emotional rigors of childhood; indifferent teachers; a disastrously disorganized educational system) but the truth is people who really want to understand math generally find a way to do it, external circumstances notwithstanding.

One of the most conspicuous examples is the Hungarian mathematician Paul Erdös (it is easy to think, considering their names, that an umlaut over a vowel in your name is a sign of predestination to mathematical genius) whose father was in a Russian POW camp for much of his childhood, and whose mother, who had lost Erdös' two older sisters to scarlet fever, kept him home until he was 10 years old, because she feared school was "full of germs" (which, in fact, happens to be true.) Erdös went on to become one of the most prolific mathematicians of all time (neck and neck with the Swiss mathematician Leonhard Euler, who also published around the same number of papers, and who had a considerably more socially developed childhood than Erdös) but in adulthood was also notoriously eccentric --Erdös lived entirely out of a suitcase, never owned a home (or even stayed in one place for very long) and spent his entire very long and productive life staying in a succession of the homes of his colleagues. (He was also an amphetamine addict, and once declared "a mathematician is a machine for turning coffee into theorems" though in his case substituting "dexedrine" for caffeine would have been more candid.)

Gödel was also notoriously eccentric --he was obsessed in his later years with a fear of being poisoned and when his beloved wife Adele died, he essentially starved to death as she was the only person he trusted to prepare his food. If people today know his name it is thanks to his famous Incompleteness Theorem. The Incompleteness Theorem is actually two theorems, which brought to an end a project which had preoccupied mathematicians for some time prior to Gödel's work: the attempt to develop a system of axioms which would form a perfect logical foundation for mathematics. Several such systems were developed, including Peano Arithmetic, something with the exotic name of Zermelo-Frankel Set Theory with Choice (you wouldn't want it without choice, would you?) and finally Bertrand Russel's Principia Mathematica. The idea that mathematics could be placed on firm epistemological ground was articulated probably with the most clarity and determination, at least among Gödel's contemporaries, by David Hilbert, who rejected the maxim ignoramus et ignoramibus ("we do not know, and we will not know") that essentially draws an outline around what is even possibly knowable and says there will always be something outside that line.

Gödel's Incompleteness Theorems are, I've read, very technical, and they have also had the interesting fate, along with the theory of relativity and quantum mechanics, of having had their meaning, or what many of us suppose to be their meaning, extended beyond the domain in which they were intended to function. I read about Gödel for the first time in Douglas Hofstadter's book Gödel, Escher, Bach: An Eternal Golden Braid, which I think I read when I was around twelve and most of which, of course, went rather over my head. Having the book around fed my desire to see myself as a deep thinker, though, and I do remember reading and understanding at least some of it. In particular I recall the bit about the Liar's Paradox ("This sentence is not true") staying with me, and spending a lot of middle school and high school with a constant sense of frustration that mathematical concepts could be so interesting but the actual study of the subject so tedious and humiliating. While Hofstadter, as far as I could, and can, tell, had a perfectly lucid grasp of Gödel's Incompleteness Theorems, I certainly didn't, and the idea that they showed that in general, the universe is a vast unknowable place where there are mysteries which science and reasoning can never unveil seemed to periodically rear its head as the years went by, and I gradually started to think of Gödel's work as implying exactly that --in a fuzzy, unexamined kind of way.

Having a fourteen year old whose math homework I wanted to understand, and help with if I could, made me go back to mathematics after an extremely long hiatus, and not without considerable trepidation. Much to my surprise I found out that after a lapse of some thirty years things like the quadratic formula seemed rather transparently sensible --I wasn't sure quite how, but following a series of clearly articulated steps to find the solution to a math problem no longer evoked a feeling of literally physical anxiety, but instead seemed soothing and actually philosophically reassuring. I still can't calculate worth a damn, really, but I can understand, now, the attraction of mathematics aesthetically. There is something cooly urbane about the formalism of numbers, and I am beginning to suspect that my retreating into books about science as a form of recreation, which was a habit I developed very early in life, stemmed from a sense, even on the low rent linguistic fringes of the actual math that is the real natural language of science, that that stately formalism was there.

Thus, ironically, I got interested in Gödel's Incompleteness Theorem again, and after poking around the offerings on Amazon, bought a couple of e-books to read on the subject, one of which has been providing me not only with a lot of sweat inducing mental exercise but also a realization that Gödel's work tends to get shoehorned into a lot of philosophical discourse where it isn't a particularly good fit. The book is called Gödel's Theorem: An Incomplete Guide to its Use and Misuse, and while it is tough sledding at first (it contains sentences like, "Even if we have no idea whether or not S is consistent, we can prove the hypothetical statement, 'if S is consistent, the consistency of S is unprovable in S" which at first or even tenth reading are clear as mud) it does gradually put together the pieces of Gödel's Incompleteness Theorem in a way that makes both its structure and even the extent of the range of its meaning clear to a non-specialist reader. At least, to this non-specialist reader, after circling around it for thirty-plus years like a nervous fish around a shiny lure.

The Incompleteness Theorems, at least as far as I can summarize them (which is itself doomed to incompleteness --his original paper presented forty six preliminary definitions and several additional preliminary theorems before coming to the point) are as follows:

1. In any formal system S it is possible to make statements that can neither be proved nor disproved in S

2. For any formal system S, the consistency of S cannot be proved in S.

I am, as I mentioned, leaving out a lot --for instance, the stipulation that S must be a "formal system in which a certain amount of arithmetic can be carried out." The specificity of the theorems, though, is what both sets limits on what they themselves set limits on, and what gives them their beauty. To quote An Incomplete Guide:

"It is often said that Gödel demonstrated that there are truths that cannot be proved. This is incorrect, for there is nothing in the incompleteness theorem that tells us what might be meant by 'cannot be proved' in an absolute sense. 'Unprovable' in the context of the incompleteness theorem, means unprovable in some particular formal system."

Gödel rather famously found misunderstandings and minsinterpretations of his work irritating and he took particularly scathing exception to Wittgenstein's dismissal of the theorems as "trickery" (kunststücken) and his claim that the theorems could simply be bypassed; Gödel's response was (and he took a swipe at Bertrand Russel while he was at it):

"Russell evidently misinterprets my result; however, he does so in a very interesting manner. In contradistinction Wittgenstein . . . advances a completely trivial and uninteresting misinterpretation."

If persons of Russell's and Wittgenstein's apparent intelligence and lifelong engagement with the problems Gödel grappled with are capable of finding themselves vulnerable to such rough handling, maybe we can find our own difficulties in understanding his work forgiveable. For me it is just a pleasure --a reassuring pleasure --to find that there is, after all, in Gödel's work the same sort of exactness and splendidly beautiful formal specificity that I found reassuring about science in the first place as a kid . . . and to come a little closer to understanding it on its own terms. It's reassuring to find oneself able to learn, as one gets older. And I find myself delighted to have just a little taste of the same feeling expressed by a much finer mind than my own, that of Paul Erdös:

"I fell in love with numbers at a young age. They were my friends. I could depend on them to always be there and to always behave in the same way."

Perhaps that's exactly the reason I found mathematics so much more congenial the second time around, when life, as middle age looms, seems an increasingly dark business. Even if I'm still barred from citizenship in the metropolis of numbers, it's nice, in a violent and uncertain world, to know it's there.

PS. Erdös had a sense of humor, apparently. He once remarked, "The first sign of senility is that a man forgets his theorems. The second is that he forgets to zip up. The third is that he forgets to zip down."

PPS. One of the most fun things about number theory is how simple the formulations of some of its hardest unsolved problems are. Witness, O Best Beloved, the Collatz Conjecture. It says simply this: take any natural number n. If it is odd, multiply it by 3 and add 1 (n3+1) and if it is even, divide by two. Take the result, and apply the procedure again. No matter what natural number you start with, you will always eventually reach 1 (and, the last three numbers of the sequence are always 4,2,1.) Despite its simplicity, the Collatz Conjecture, proposed by Lothar Collatz in 1937, remains unproven, and no less an eminence than Paul Erdös declared, "Mathematics is not yet ready for such problems." But, you know, knock yourself out.