THE INCOMPLETENESS FILE: Do Godel's theorems even make sense?

FRIDAY, SEPTEMBER 21, 2018

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?

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?

[Pregnant pause]

PUZZLED FRESHMAN: Same answer?

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.

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:
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.

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!

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.

THE INCOMPLETENESS FILE: What the heck is a "formal system?"

THURSDAY, SEPTEMBER 20, 2018

Once again, Joe Average won't know: Friend, are you a general reader? That is to say, are you a non-specialist in the fields of mathematics, mathematical logic, theoretical physics and the like?

Friend, if you're a general reader, let's consider the title essay of Jim Holt's new book.

The new book is called When Einstein Walked with Godel: Excursions to the Edge of Thought. The title essay is called When Einstein Walked with Godel—and friend, we're telling you this:

Ignore the various things you read about how "readable" Holt's essays are! Friend, if you're a general reader, there is exactly zero chance that you'll emerge from that title essay with even the slightest idea what Kurt Godel's "incompleteness theorems" are alleged to be all about.

As we showed you yesterday, there's zero chance you'll have any idea! Ignore what reviewers have said!

Alas! Several layers of academia, journalism and the publishing world are involved in the creation of this strange state of affairs. Before we look at Rebecca Goldstein's first attempt at explaining Godel's theorems—it's Goldstein's "Godel made easy" book which Holt reviewed in his title essay—let's take a minute to consider, once again, who these high-ranking players are.

We'll start with Godel himself, the man "who has often been called the greatest logician since Aristotle."

Who the Sam Hill was Kurt Godel? As it turns out, he seems to have been mentally ill throughout the whole course of life. (At age 72, he died of self-starvation.)

During his adult years in Princeton, he was famous for believing all sorts of crazy ideas. Among them, perhaps, was his foundational belief in "Platonism"—his ardent belief, in Holt's formulation, that "numbers and circles have a perfect, timeless existence" somewhere. (We're able to access this perfect world through some form of ESP.)

Should it seem strange that our greatest logician can be described in this way? We'll examine that question in more detail next week.

For now, let's continue assembling our list of players. Let's consider the circle of thinkers among whom Godel was moving when he devised his iconic theorems, when he was just 23.

According to the profiles offered by Holt and Goldstein, Godel was moving among the Vienna Circle, a group which is said to have included some of the western world's greatest thinkers. As Europe suffered between two wars, these thinkers were puzzling over how we can know that 2 + 2 = 4. They were also puzzling over how we can know that 4 is an even number.

Later, one of their descendants was puzzling over the question of how we can know that 317 (or 17, or 7) is a prime. Godel, our second greatest logician, was apparently puzzling out these crucial topics too.

Friend, do you find it odd to think that our greatest thinkers were puzzling over such questions? We find that odd (and unimpressive) too, just the way you do!

We find that unimpressive, a point we'll discuss next week. But at this point, we must consider the role in this story which gets played by the publishing industry. We must also consider the work of our own modern-day professors and upper-end journalists.

Our publishing business is awash in "Einstein made easy" books (and the like). None of these books has ever managed to make Einstein easy, including the 1916 "Einstein made easy" book written by Einstein himself.

(Einstein, our greatest theoretical physicist, was not a skilled popular writer.)

No one can understand these books, but professors keep turning them out. They take turns blurbing each other's books, telling us rubes how "lucid" these "accessible" books really are. In response, major reviewers stand in line to say beautifully readable these amazingly easy books are!

As any lover of humor would, we've found this fandango fascinating for a great many years. Next week, we'll consider the real-world problems our savants ignore as they produce unreadable books about 2 + 2 equaling 4 and about how we can know such facts.

Quick question! When our ranking professors behave in these ways, should we really be surprised by the intellectual chaos which characterizes our journalism? When our greatest thinkers behaved (and behave) in these ways, should we really be surprised by the low-IQ mugging and clowning which gets presented on corporate cable each night, as our nation slides into the sea?

(And each morning, on Morning Joe, whose entire panel flipped today concerning the need for an FBI probe of what the accuser has said. The panel moved from yesterday's "no" to today's full-throated "yes." We'd use the accuser's name, except the Times is calling her "Blasey" and the Post is still calling her "Ford.")

When the title essay to his new book first appeared, Holt was reviewing Professor Goldstein's 2005 "Godel made easy" book. Because Goldstein is a highly regarded novelist as well as a ranking philosophy prof, it may have seemed like a great idea to have her write a book about the life and the work of this puzzling, disordered man.

As we noted yesterday, Holt's treatment of the "incompleteness theorems" will be totally incoherent for the general reader. For our money, the general reader won't likely be able to make hide nor hair of Goldstein's treatment either.

Holt wrote a book review for The New Yorker; by way of contrast, Goldstein had written a complete book. In our view, the general reader will have little chance of understanding Godel's theorems from reading that book, but for obvious reasons, we can't reproduce Goldstein's full presentation in the way we could do with Holt.

(We also think the professor went places which we found astounding. "This very sentence is false?" It's stunning to think that ranking professors can still find meaning in places like that. More on that starting tomorrow.)

Where Holt wrote an incoherent essay, Goldstein wrote a hard-to-read book. For our money, the general reader will almost surely emerge from that book with no idea what those "incompleteness theorems" are actually all about.

For today, we'll only show you the way Goldstein introduced the theorems. A person might claim that this is unfair, although we aren't sure it is.

On page 23 of Goldstein's book, she stops discussing Albert Einstein and turns to the young Kurt Godel. As she introduces Godel, she marvels at how young he was when he devised his iconic theorems. She almost seems to say that the theorems are easy to state:

GOLDSTEIN (page 23): He is Kurt Godel, and in 1930, when he was 23, he had produced an extraordinary proof in mathematical logic for something called the incompleteness theorem—actually two logically related incompleteness theorems.

Unlike most mathematical results, Godel’s incompleteness theorems are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof’s overall strategy, delightfully, is not. The two conclusions that emerge at the end of all the formal pyrotechnics are rendered in more or less plain English. The Encyclopedia of Philosophy’s article “Godel’s Theorem” opens with a crisp statement of the two theorems:

Tell the truth! Reading that passage, it sounds like it won't be hard to make Godel easy!

The two conclusions Godel reached "are rendered in more or less plain English," Goldstein writes. "Delightfully," the overall strategy of his proof isn't formidably technical!

Goldstein makes it sound like Godel and his theorems won't be all that hard! Then, she quotes the Encyclopedia of Philosophy's "crisp statement of the two theorems." The passage she quotes goes like this:

GOLDSTEIN (continuing directly): "By Godel's theorem, the following statement is generally meant:

"In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable. (This statement is occasionally referred to as Godel’s first theorem.)

"A corollary to the theorem is that the consistency of a formal system adequate for number theory cannot be proved within the system. (Sometimes it is this corollary that is referred to as Godel’s theorem; it is also referred to as Godel’s second theorem.)"

That quoted passage is attributed the Encyclopedia of Philosophy. We'll suggest you consider this:

According to Goldstein, this account of Godel's theorems has been "rendered in more or less plain English." We trust and believe that you, a general reader, can see that this just isn't so.

How does the Encyclopedia define or describe the first theorem? In plain English, it goes like this:

In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable.

Friend, that passage simply isn't written in plain English. We hope you could already see that.

Citizens, can we talk? The general reader will have no idea what a "formal system" is! Beyond that, this general reader will have little idea what "number theory" is.

In part for these reasons, this general reader won't be able to imagine what a formal system "adequate for" number theory is. The general reader will have no idea what that passage is talking about.

However "crisp" this statement may be, this statement will be clear as mud to the general reader. It contains the kind of technical language which may not look like technical language. But this language is guaranteed to leave the general reader on the outside, haplessly looking in.

Briefly, let's be fair. This passage represents Goldstein's first attempt at describing these iconic theorems. This strikes us as a strange first attempt but, at least in theory, Goldstein could have continued on from there to unpack these theorems in a way the average Joe could actually understand.

For our money, that doesn't happen in Goldstein's book. Along came Holt, to offer the crazily incoherent summary we perused in full in yesterday's report.

On page 26, Goldstein reassures the general reader. She does so in this passage, in which she once again plays the "plain English" card:

GOLDSTEIN (page 26): [Godel’s theorems] are the most prolix theorems in the history of mathematics. Though there is disagreement about precisely how much, and precisely what, they say, there is no doubt that they say an awful lot and that what they say extends beyond mathematics, certainly into metamathematics and perhaps even beyond. In fact, the mathematical nature of the theorems is intimately linked with the fact that the Encyclopedia of Philosophy stated them in (more or less) plain English. The concepts of “formal system,” “undecidable,” and “consistency” might be semi-technical and require explication (which is why the reader should not worry if the succinct statement of the theorems yielded little understanding); but they are metamathematical concepts whose explication (which will eventually come) is not rendered in the language of mathematics.

Finally! Three pages later, Goldstein notes that the general reader has no idea what a "formal system" is. For the record, she offers her first definition of the term on page 129 [sic].

In our view, things don't get a whole lot better for the general reader in what follows from there. Things seem technical all the way down. It seems to us that the general reader will likely be forced to quit.

Citizens, let's review:

Our greatest logician was mentally ill and possessed of crazy ideas. At the heart of his prolix theorems was his apparently crazy belief that numbers and circles live a perfect existence somewhere.

In turn, our philosophy professors seem to have no idea how to explain these prolix theorems (which "say an awful lot") to the general reader. But they produce books which claim to have done that anyway. When they do, journalists rush to say that they understood every word. And it all began with our greatest thinkers pondering 2 + 2.

When we see this cultural pattern unfold, are we surprised by the utter incoherence displayed by lesser thinkers on corporate cable? Are we surprised that our broken, pre-rational public discourse has now helped to place a Donald J. Trump in the White House?

Seeing ourselves from afar, we humans still tend to believe, say and suggest that we're the rational animal. In our view, this profoundly iconic notion qualifies as "Aristotle's [gigantic large howling] error."

Tomorrow, we'll debase Godel a tiny bit more, prepping a bit for next week. We'll also see Professor Goldstein do something amazingly rare.

Tomorrow: A (near) perfect statement by Goldstein

THE INCOMPLETENESS FILE: What the Sam Hill is a "logical system?"

WEDNESDAY, SEPTEMBER 19, 2018

No general reader will know: According to the headline on the New York Times review, Jim Holt's new book is a collection of essays which "make sense of the infinite and the infinitesimal."

It's Holt's "conviviality, and a crispness of style, that distinguish him as a popularizer of some very redoubtable mathematics and science,“ the gushing reviewer said, marching in upper-end lockstep.

Indeed, it wasn't just the New York Times making these mandated statements. According to the headline on the Christian Science Monitor review, "When Einstein Walked with Gödel"—that's the title of Holt's new book—"is science writing at its best."

The essays in Holt's new book "all wonderfully achieve [his] stated goal," which includes "enlighten[ing] the newcomer," the Monitor's reviewer said. "This is considerably more difficult than it sounds, and Holt does a beautifully readable job."

Holt's collection of essays wasn't reviewed by the Washington Post, but the reviewer for the Wall Street Journal completed the rule of three. Holt is "one of the very best modern science writers," this third reviewer opined. He specifically singled out Holt's "wonderful title essay."

That's the very essay we've been discussing—the essay in which Holt tries to explain Kurt Godel's "incompleteness theorems."

Reviewers seemed to agree. Holt's work is "beautifully readable," especially for "the newcomer"—for the general reader. But then we turn to that title essay, the one in which Holt attempts to explain Godel's theorems.

According to Holt, those theorems have established Godel, by widespread agreement, as "the greatest logician since Aristotle." An obvious question arises:

How "beautifully readable" is Holt's explanation of those iconic theorems? To what extent is Holt's account of those theorems "science writing at its best?"

As we noted yesterday, Holt explains those theorems in two extremely long paragraphs. As we showed you yesterday, the first of those paragraphs, by far the shorter of the two, reads as shown below in Holt's title essay, which first appeared in The New Yorker in 2005.

Below, you see the first of the two paragraphs in which Holt explains Godel's theorems. By the end of this paragraph, our greatest logician since Aristotle is, for reasons which don't quite get explained, pondering 2 + 2:

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. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of 1920s Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the 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.

As this first long paragraph ends, the greatest thinkers in Europe are puzzling over a knotty problem. According to Holt's own language, they're trying to explain "what makes a proposition like 2 + 2 = 4 true."

Without so much as chortling even once, Holt proceeds from there:

One group of Europe's most brilliant thinkers is said to have said that the truth of 2 + 2 = 4 lies in the fact that "it could be derived in a logical system according to certain rules." Without attempting to explain what that technical word salad means, Holt proceeds to say what the youthful Godel believed:

The youthful Godel is said to have thought that the truth of 2 + 2 = 4 lies in the fact that "it correctly describes some abstract world of numbers"—presumably, the world in which "abstractions like numbers and circles have a perfect, timeless existence independent of the human mind." So the greatest logician thought, as opposed to the other great thinkers.

Scotty, beam us down! Despite the gushing of those reviewers, no general reader will have any idea what that paragraph means.

What does it mean to say that 2 + 2 = 4 "can be derived in a logical system according to certain rules?" No general reader has the slightest idea, but Holt doesn't stop to decipher the claim. He merely compares it to what Godel is said to have thought—a belief which is said to involved the perfect existence of circles.

In these ways, our greatest thinkers puzzled out 2 + 2. Last May, major journalists stood in line to say how "beautifully readable" Holt's new book is, especially the "wonderful title essay" in which this hodgepodge appears.

In such ways, we see a modern, high-end display of "Aristotle's error." These reviewers aren't reflecting the ancient claim that "man [sic] is the rational animal." Rather, they're acting out the "Harari heuristic," which holds that our warlike species, Homo sapiens, gained control of the planet when, through a set of chance mutations, our ancestors developed the ability to "gossip" and the ability to invent and affirm sweeping group "fictions."

As the weeks and months proceed, we'll return to Harari's account, reviewing his claims in more detail. For today, we'll only note an obvious fact—even by the end of this first paragraph, Holt's opaque, highly technical writing will have left any general reader several light-years behind.

Alas! Whether they know it or not, general readers will already be at sea by the end of that first paragraph. Most specifically, such readers will have no idea what a "logical system" is.

Nor will such readers have any idea what it means to say that a nursery school fact like 2 + 2 "can be derived in a logical system according to certain rules." Already, Holt may as well be writing in some form of ancient Etruscan.

In the paragraph which follows, Holt starts explaining those "incompleteness theorems." When he does, a large pile of Sandstorm arrives.

At the end of this pig-pile of abstruse phrases, Godel's two theorems get defined. The general reader will have zero idea what Holt is talking about:

HOLT (continuing directly): Gödel was introduced into the Vienna Circle by one of his professors, but he kept quiet about his Platonist views. Being both rigorous and averse to controversy, he did not like to argue his convictions unless he had an airtight way of demonstrating that they were valid. But how could one demonstrate that mathematics could not be reduced to the artifices of logic? Gödel’s strategy—one of preternatural cleverness and, in the words of philosopher Rebecca Goldstein, “heart-stopping beauty”—was to use logic against itself. Beginning with a logical system for mathematics, one presumed to be free of contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of doublespeak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another. In fact, as Gödel showed, a numerical formula could even be made to say something about itself. Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying when it asserts, "I am nor provable?" No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.

Ar the end of this, the world's longest paragraph, Holt defines, or pretends to define, Godel's two "incompleteness theorems." Despite the subsequent, mandated gushing of our journalistic elites, no general reader will have any idea what Holt is talking about.

Consider the various snares and traps that reader has encountered during this long forced march to the sea:

We're told that Godel wanted to demonstrate that "mathematics could not be reduced to the artifices of logic." The general reader will have no idea what such a reduction might look like.

In pursuit of this puzzling end, we're told that Godel "beg[an] with a logical system for mathematics, one presumed to be free of contradictions." The general reader won't know what "a logical system" is. He won't know what it means for such a creature to be "adequate for mathematics."

We're now told that Godel came up with "a formula that said something about numbers." On the pain of impending death, the general reader won't be able to imagine an example of any such formula making any such statement. Nor will she have any idea what it might mean to produce "a formula that, while ostensibly saying something about numbers, also says, 'I am not provable.' ”

A bit later on, the reader seems to be told that Godel produced "a self-referential formula" which generated a proposition whose truth "can be seen only from outside the logical system." No general reader has any idea what Holt is talking about.

At any rate, atthe end of this long harangue, Holt describes Godel's first incompleteness theorem. Excitement builds for the general reader. Then he's told that the theorem says this:

"No logical system can capture all the truths of mathematics."

That would be an exciting claim. Except, do you remember the problem with which the general reader started? He or she has no idea what a "logical system" is!

In these two paragraphs, Holt explains, or pretends or attempts to explain, Godel's two "incompleteness theorems." For unknown reasons, this Olympian hodgepodge was first offered to general readers in the pages of The New Yorker. Few subscribers could have had any idea what Holt was talking about.

Thirteen years later, Holt's piece was published as the title essay in a collection of his work. Mainstream reviewers stood in line to praise it for being readable, especially for newcomers to the subject matter.

What a long, strange journey it has been through those lengthy paragraphs! We started with Europe's greatest thinkers pondering the fact that 2 + 2 equals 4. We were told that the greatest logician since Aristotle believed that circles and numbers and other such critters have a perfect, timeless existence, an existence we can access through some version of ESP.

Eventually, an avalanche of technical language landed on our newcomer heads. But so what? An obedient line of upper-end journalists said this all made perfect sense.

We're going to say that all these groups are providing textbook illustrations of "Aristotle's error." Also this:

When our journalists behave in the manner described, they're helping us see how things fall apart when Plato's guardians fail.

Tomorrow: Goldstein's first attempt