Are elite humans conscious? Trust us! No one reading Rebecca Goldstein's 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel, will come away with even the slightest understanding of Godel's famous alleged theorems and proofs.
Goldstein is a well-regarded novelist as well as a ranking philosophy professor. In her well-received book, she spins a web of interesting tales about Godel's intensely difficult life, which ended in suicide by self-starvation at age 71.
That said, Goldstein's book was aimed at general readers, and no general reader on the face of the earth will have any idea how Godel's incompleteness theorems actually worked after reading Goldstein's attempts to explain them.
Those readers won't know what Godel is said to have proven, or how he is said to have proved it. We'll be proving this point below.
Tomorrow, we'll return to the paradox wars, moving from the liar's paradox onward to Lord Russell's. For today, if only for comic relief, let's focus on an intriguing part of the modern publishing industry. As we do, let's continue to ask our latest gloomy question:
Are we humans even conscious? Or are we, in many cases, simply misfiring machines? We were forced to ask that gloomy question after reading the New York Times review of Goldstein's well-received book, a review which insists—here we go again!—that Rebecca Goldstein "magically" managed to make Kurt Godel easy.
Books which claim to make difficult science easy are now a basic part of the publishing business. This culture began with the Einstein-made-easy books, especially with Stephen Hawkings' incomprehensible best-sellers which claimed to accomplish this task.
Einstein-made-easy books created this publishing niche. But the practice expanded from there, even to Professor Goldstein's Godel-made-easy effort.
Goldstein's book concerns Godel's incompleteness theorems—theorems which established Godel, according to Goldstein and others, as "the greatest logician since Aristotle." These theorems are plainly hard (or impossible) to explain, but on the dust jacket of Goldstein's book, three major academic stars blurb the lucidity of her work, as is required by law.
(Alan Lightman: Incompleteness is "a penetrating, accessible, and beautifully written book." Stephen Pinker: Goldstein "offers us a lucid exposition of Godel's brainchild." Brian Greene: "a remarkably lucid account of Godel's most stunning breakthrough—a proof that there are true but unprovable statements.")
According to the blurbs on the back, Goldstein's book isn't simply accessible and lucid; the book is remarkably lucid. And then, along came the New York Times review, written by Polly Shulman.
The Times identified Shulman as "a contributing editor for Science magazine [who] has written about mathematics for many other publications." According to Shulman's current web site, "she majored in math at Yale."
We're going to guess that Shulman knows a whole lot of math! We'll also guess that she understands the modern journalistic convention in which insiders insist that wholly incomprehensible books make various abstruse science/math topics amazingly easy to grasp.
We don't know why major newspapers like the Times enjoy this game so much. Is it possible that its editors aren't even conscious—that they're simply misfiring machines?
We'll leave that one to the experts! For today, if only as comic relief, let's walk through Shulman's upbeat assessment of Goldstein's work. After that, we'll look at part of the way Goldstein explains Godel's proof.
As far as we know, Shulman understands Godel's theorems, whatever that might entail. That said, her review is already working the lucidity beat by the end of its first paragraph:
SHULMAN (5/1/05): Rebecca Goldstein, as anyone knows who has read her novels—particularly "The Mind-Body Problem"—understands that people are thinking beings, and the mind's loves matter at least as much as the heart's. After all, she's not just a novelist, but a philosophy professor. She casts "Incompleteness," her brief life of the logician Kurt Gödel (1906-78), as a touching intellectual love story. Though Gödel was married, his wife barely appears here; as Goldstein tells it, his romance was with mathematical Platonism, the idea that the glories of mathematics exist eternally beyond our grasp. Gödel's Platonism inspired him to deeds as daring as any knight's: he proved his famous incompleteness theorem for its sake. His Platonism also set him apart from his intellectual contemporaries. Only Einstein shared it, and could solace Gödel's loneliness, Goldstein argues. A biography with two focuses—a man and an idea—Incompleteness" unfolds its surprisingly accessible story with dignity, tenderness and awe.Goldstein's book is "surprisingly accessible," Shulman says. (Also, it's a "love story!") By rule of law, elite reviewers are required to voice this judgment concerning easy-to-understand books written by the elect.
By the third paragraph of her review, Shulman is discussing "the famous Liar's Paradox." ("The proofs of Godel's famous theorems rely on just that sort of twisty thinking: statements like the famous Liar's Paradox, 'This statement is false.' ") Shulman then moves to her basic account of what Godel is said to have proved.
Below, you see that basic account. We hate to be the killjoy here, but by the time this paragraph is done, the general reader will have zero idea what Shulman is talking about. Any such dream will have come to an end with the passage we're setting in bold:
SHULMAN: Gödel's work addresses the core of mathematics: finding proofs. Proofs are mathematicians' road to truth. To find them, mathematicians from the ancient Greeks on have set up systems consisting of three basic elements: axioms, true statements so intuitively obvious they are self-evident; rules of inference, logical principles indicating how to use axioms to prove new, less obviously true statements; and those new true statements, called theorems. (Many Americans met axioms and proofs for the first and last time in 10th-grade geometry.) A century ago, mathematicians began taking these systems to an extreme. Since mathematical intuition can be as unreliable as other kinds of intuitions—often things that seem obvious turn out to be just plain wrong—they tried to eliminate it from their axioms. They built new systems of arbitrary symbols and formal rules for manipulating them. Of course, they chose those particular symbols and rules because of their resemblance to mathematical systems we care about (such as arithmetic). But, by choosing rules and symbols that work whether or not there's any meaning behind them, the mathematicians kept the potential corruption of intuition at bay. The dream of these formalists was that their systems contained a proof for every true statement. Then all mathematics would unfurl from the arbitrary symbols, without any need to appeal to an external mathematical truth accessible only to our often faulty intuition.Uh-oh! In this passage, Shulman almost suggests that there may have been something a little bit odd in what these mathematicians (these elite logicians) began doing way back when. We applaud Shulman for this behavior. By the rules of deference to academic authority, hints and suggestions of this type are very rarely supplied.
At any rate, in the course of "taking these systems to an extreme," these mathematicians "built new systems of arbitrary symbols and formal rules for manipulating them." These mathematicians "chose those particular symbols and rules because of their resemblance to mathematical systems we care about (such as arithmetic)."
General reader, please! By now, you have exactly zero idea what Shulman is talking about.
You may not understand that fact, but it's a fact all the same. Apparently, Shulman's editors at the Times didn't understand this fact themselves, or perhaps they just didn't care.
Already, the general reader won't understand what Shulman is talking about. Despite this fact, she plows ahead, explaining what Godel proved, and even how he proved it.
The general reader won't understand a word of what's said in the passage below. But by the end of this passage, Shulman is insisting that Goldstein's explanation of Godel's "spectacular proof" is so amazingly easy to understand that even a house pet can grasp it:
SHULMAN (continuing directly): Gödel proved exactly the opposite, however. He showed that in any formal system complicated enough to describe the numbers and operations of arithmetic, as long as the axioms don't lead to contradictions there will always be some statement that is not provable—and the contradiction of it will not be provable either. He also showed that there's no way to prove from within the system that the system itself won't give rise to contradictions. So, any formal system worth bothering with will either sprout contradictions—which is bad news, since once you have a contradiction, you can prove anything at all, including 2 + 2 = 5—or there will be perfectly ordinary statements that may well be true but can never be proved.Hurrah! According to Shulman, Goldstein describes Godel's spectacular proof "with lucid discipline." She goes on to say that the essence of Godel's spectacular proof "fits magically into a few pages of a book for laypeople," by which she means Goldstein's book.
You can see why this result rocked mathematics. You can also see why positivists, existentialists and postmodernists had a field day with it, particularly since, once you find one of those unprovable statements, you're free to add it to your system as an axiom, or else to add its complete opposite. Either way, you'll get a new system that works fine. That makes math sound pretty subjective, doesn't it?
Well, Gödel didn't think so, and his reason grows beautifully from his spectacular proof itself, which Goldstein describes with lucid discipline. Though the proof relies on a meticulous, fiddly mechanism that took an entire semester to build up when I studied logic as a math major in college, its essence fits magically into a few pages of a book for laypeople. It can even, arguably, fit in a single paragraph of a book review—though that may be stretching.
Shulman goes on to offer her own one-paragraph account—an account no layperson will understand. But in obedience to the rules, Shulman says that Goldstein "magically" describes Godel's proof—indeed, that she does so with "lucid discipline," taking just a few pages to do it.
Are humans even conscious? We often find ourselves asking that question when we encounter claims like this—claims which are completely standard in the elite journalistic culture surrounding [WHOSOEVER]-made-easy books.
How magically lucid is Goldstein's account of Godel's spectacular proof? The task is undertaken in Part III of Goldstein's four-part book. (Part III, "The Proof of Incompleteness," starts on page 147.)
In fact, Goldstein's account stretches over more than just "a few pages." By page 174, the general reader is encountering the magical lucidity we invite you to gaze on below.
We're using plain numbers in place of subscripts. With the exception of that one change, Goldstein's magical lucidity looks exactly like this:
GOLDSTEIN (page 174): Now we're going to turn this wff into a number by just consecutively going along and replacing each symbol in the wff by its Godel number. Every symbol in the formula p1 has been assigned a number; the open parenthesis is a 7, the x a 3, the close parenthesis an 8, the tilda a 0. Replacing each element of the formula with the corresponding Godel number yields a large number, which is the Godel number for the wff. Abbreviating "the Godel number of the proposition p1" by (GN)p, we obtain:The lucidity is everywhere! On the next page, the magic continues:
GN(p1) = 738739877673846739882734398
In this way Godel numbers are assigned to wffs, which are sequences of symbols, and hence to propositions, which are just special wffs. In the same way, Godel numbers can be assigned to sequences of propositions, and in particular to potential proofs, which are, after all, just sequences of propositions, using the Godel numbers already assigned to propositions. Bookkeeping! In our simplified version, the Godel number of a sequence of propositions (a potential proof) is obtained basically by putting the Godel numbers of the sequences propositions together; however, since it is important to be able to unambiguously extract from the number the original sequence of propositions, we'll need some sort of signal that will indicate where one proposition ends and the next one begins—sort of like a carriage return on a typewriter. We'll let 0 function as our carriage return, indicating that we now go to a new line in the proof.
GOLDSTEIN (page 175): The Godel number that will correspond to the sequence of p1 followed by p2 is:"The metasyntactic and the arithmetic collapse into one another." To Goldstein, this is part of "the heart-stopping beauty" of this proof—the proof she is explaining in this magically lucid way.
GN(p1, p2) = 7387398776738467398827343980675846758
Through Godel's inspired contrivances, all of the logical relations that between propositions in the formal system become arithmetic relations expressible in the arithmetical language of the system itself. This is the essence of the heart-stopping beauty of the whole thing. So if, for example, wff1 logically entails wff2, then GN(wff1) will bear some purely arithmetical relation to GN(wff2). Suppose, say, that it can be shown that if wff1 logically entails wff2, then GN(wff2) is a factor of GN(wff1). We would then have two ways of showing that wff1 logically entails wff2; we could use the rules of the formal system to deduce wff2 from wff1; or we could show that GN(wff1) can be obtained from GN(wff2) by multiplying by an integer. Suppose that GN(wff1) = 195589 and GN(wff2) = 317. 317 is a factor of 195589, since 317 multiplied by 617 = 195589. So that wff1 logically entails wff2 could be demonstrated either by using the formal rules of proof to arrive at wff2 from wff1, or, alternatively, by using the rules of arithmetic to arrive at GN(wff1) = 195589 by multiplying 617 by 317 = GN(wff2). The metasyntactic and the arithmetic collapse into one another.
Are elite humans conscious? At present, it's part of elite journalistic culture to tell the general reader that explanations of this sort are lucid, accessible, surprisingly accessible, even remarkably lucid.
In the case of Shulman's review, New York Times subscribers were told that Goldstein describes Godel's proof "with lucid discipline," producing an account which "fits magically into a few pages of a book for laypeople."
All these claims—those from the blurbs, those from the review—are perfectly obvious nonsense. That said, they come to us from the highest realms of elite pseudo-culture, in which members of the elect praise the work of others.
Then again, Godel's spectacular theorems comes to us from the realm of paradox—more specifically, from the realm of "the liar's paradox," an utterly silly bit of nonsense which exists on the level of the ability to magically find a penny behind the ear of a 4-year-old child.
Have our most brilliant elite logicians been long-standing fools for paradox? Tomorrow, we'll move on to Goldstein's account of "Russell's paradox," the utterly silly fiddle-faddle which transfixed elite logicians like Godel as they ignored the affairs of the world while starving themselves to death.
We're told that they're our brightest lights. We end up with Trump in the White House.
Tomorrow: "Russell's paradox concerns the set of all sets that are members of themselves."