Flying spaghetti monsters: Are Godel's "incompleteness theorems" actually "important?"
Do they carry any social significance? In the end, do they even make sense?
We'll admit to being doubters on the last of those points. Consider a part of Rebecca Goldstein's book which we'll explore in more detail at some later point.
Goldstein's book, designed for general readers, appeared in 2005. It bore this title: Incompleteness: The Proof and Paradox of Kurt Godel.
In our view, the general reader won't likely emerge from this book with the ability to discuss these supposedly transplendent theorems. For ourselves, we were surprised by the way Goldstein, a philosophy professor, leaned on the concept of "paradox" in her discussions, not excluding this rumintaion on a famous "abstract object:"
GOLDSTEIN (page 91): Russell's paradox concerns the set of all sets that are not members of themselves. Sets are abstract objects that contain members, and some sets can be members of themselves. For example, the set of all abstract objects is a member of itself, since it is an abstract object. Some sets (most) are not members of themselves. For example, the set of all mathematicians is not itself a mathematician—it's an abstract object—and so is not a member of itself. Now we form the concept of the set of all sets that aren't members of themselves and we ask of ourselves: is it a member of itself?...Now we form the concept of the set of all sets that aren't members of themselves? But why in the world would we do that?
The paragraph continues from there. The reference to "Russell" is a reference to Lord Russell, eventual husband of Lady Ottoline—that is to say, to Bertrand Russell—who came up with this world-class groaner back in 1901.
When we first encountered Goldstein's book, it surprised us to think that a capable philosophy professor would still be trafficking in this antique hocus-pocus about these "abstract objects"—about "abstract objects" which may or may not be "members of themselves."
We were even more surprised to see her marveling about this pseudo-paradox, which is even more simple-minded:
"This very sentence is false."Good God! The later Wittgenstein returned to England hoping to remove these flying spaghetti monsters from the pseudo-discourse in which he himself had trafficked as the early Wittgenstein. We were surprised to see a ranking professor still shoveling these snowstorms around.
We'll discuss these matters in the weeks ahead, possibly next week. For ourselves, if Godel's theorems turn on piddle like this, we'll float the shocking possibility that they may not make any real sense.
We know it's shocking to hear such claims about the genius theorems Goldstein gushes about. Then again, this "greatest logicians since Aristotle" seems to have been mentally ill his entire life; eventually died of self-starvation; and believed all sorts of crazy idea, perhaps including the crazy idea that numbers and circles live "a perfect, timeless existence" somewhere, apparently in an "abstract" realm we can access through something resembling ESP.
Do the theorems of this unfortunate man actually make any sense? For now, we'll vote with the doubters. Meanwhile, when Jordan Ellenberg discussed Goldstein's book for Slate, he offered these remarks, among others:
ELLENBERG (3/10/05): In his recent New York Times review of Incompleteness, Edward Rothstein wrote that it’s “difficult to overstate the impact of Gödel’s theorem.” But actually, it’s easy to overstate it: Goldstein does it when she likens the impact of Gödel’s incompleteness theorem to that of relativity and quantum mechanics and calls him “the most famous mathematician that you have most likely never heard of.” But what’s most startling about Gödel’s theorem, given its conceptual importance, is not how much it’s changed mathematics, but how little. No theoretical physicist could start a career today without a thorough understanding of Einstein’s and Heisenberg’s contributions. But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel’s work. So far, I’ve done it myself.You can read the rest of what Ellenberg wrote. For now, we're just saying!
When our greatest logicians devote their lives to the antics of spaghetti monsters, should we be surprised by the sheer stupidity which obtains all over the national discourse engineered by corporate journalists? We'll be focusing on that question next week. For today, let's visit an early part of Goldstein's book, where she starts to get something right.
When Holt summarized Goldstein's book, he profiled the strangeness of Godel. Again, we ask you to marvel at the highlighted part of this pile:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind...[T]he members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.Sadly, strangely, possibly dumbly, the greatest minds in Europe were puzzling hard over this:
What makes a proposition like "2 + 2 = 4" true?Seriously though, folks! From 1901 right up through Godel's arrival at college, that's what our allegedly greatest minds were struggling to figure out!
We mention this for a reason. Near the start of her book, Goldstein gives a weirdly decent explanation of this potent conundrum. She speaks about a different fact—the fact that 5 + 7 = 12—but, as you can probably see, the basic logic of all such statements is pretty much the same.
Citizens, we encountered this same traditional groaner as college freshmen ourselves! How can we know that 7 + 5 = 12? Professor Nozick raised this "problem" in the introductory course, Phil 3: Problems in philosophy.
How do we know that 7 + 5 = 12? One wag in the back of the class dared to ask himself this:
Who is this "problem in philosophy" a problem for?Or words to that effect! On the world's most exalted comedy stages, we've occasionally recalled one subsequent discussion. We did so just a few years ago, with comedy-loving Clarence Page as an opening act:
PHILOSOPHICALLY TORTURED TEACHING ASSISTANT: Students, how can we know that 7 + 5 equals 12?Did that exchange really take place? Memory sometimes plays tricks. But we're fairly sure that we remember the paper we finally wrote on this topic, and it resembled the explanation Goldstein supplies early in her book.
INNOCENT FRESHMAN: Miss Cummings told us? In second grade?
FRUSTRATED TEACHING ASSISTANT (tearing his hair as he stares out the window, seeming to contemplate the abyss): No, no, students, you're missing my point! How do we know that 7 + 5 equals 12?
PUZZLED FRESHMAN: Same answer?
What makes a proposition like “2 + 2 = 4” true? Using a slightly tougher example, Goldstein offers this:
GOLDSTEIN (page 17): The rigor and certainty of the mathematician is arrived at a priori, meaning that the mathematician neither resorts to any observations in arriving at his or her mathematical insights nor do these mathematical insights, in and of themselves, entail observations, so that nothing we experience can undermine the grounds we have for knowing them. No experience would count as grounds for revising, for example, that 5 + 7 = 12. Were we to add up 5 things and 7 things, and get 13 things, we would recount. Should we still, after repeated recountings, get 13 things we would assume that one of the 12 things had split or that we were seeing double or dreaming or even going mad. The truth that 5 + 7 = 12 is used to evaluate counting experiences, not the other way around.Goldstein is on the right track. That said, and stating the obvious, it makes more sense to explore the logic of this conundrum through the simplest possible example: 1 + 1 = 2.
How do we know that 1 + 1 = 2? Simple! Among other factors, we would be strongly disinclined to accept alleged counterexamples! Let's think in terms of marbles.
"Two" is simply the name we give to the number of marbles you'll typically have if you start with one marble, then receive one additional marble. If you counted your marbles at that point and found you had three marbles, we would assume that you hadn't noticed the addition of the third marble. Beyond that, we wouldn't accept such counterexamples as these:
Haystack Calhoun does the math:What would we say to such counterexamples? We would say they aren't what we mean! In each case, that simply isn't what we mean when we say 1 + 1 = 2!
A farmer has one haystack. He adds to it a second haystack. He sees that he still has one (larger) haystack. The farmer declares that, at least on the farm, 1 + 1 = 1.
Porky Pig adds to the wealth:
A farmer has one (male) pig. He adds one (female) pig. Months later, he finds that he has eight pigs. The farmer declares that, at least on the farm, 1 + 1 = 8.
The evaporation monologues:
A chemist has one beaker of a chemical. He adds a second beaker of a different chemical. The beaker's contents go "poof" and all the liquid disappears. When he added the second beaker, he ended up with no beakers. The chemist declares that, at least in the lab, 1 + 1 = 0.
How do we know that 1 + 1 = 2? We know it because we know what we mean when we make the familiar statement. All other addition facts follow from there. No flying spaghetti monsters, abstract or not, need apply!
Goldstein made a decent play on page 17. In our view, her book goes downhill from there, biographical writing excluded.
People, one plus one equals two! As our greatest thinkers argued this point, war came to Europe again.
Next week: The guardians file
Now for the rest of the story: After we freshmen took Phil 3, we all decided to abandon philosophy as a major. Nozick, who was only 26 at the time, went on to become a huge star. (He was always very nice to us pitiful freshmen.)
We switched back after sophomore year. Historical inevitability seemed to take over from there.