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

THE INCOMPLETENESS FILE: Incompleteness meets incoherence!

TUESDAY, SEPTEMBER 18, 2018

Lucid writer intones:
Subscribers to The New Yorker had a major treat in store.

Or at least, so it seemed.

Their February 28, 2005 issue had arrived in the mail. It featured a lengthy essay in which a writer named Jim Holt discussed a pair of new books.

One of the books concerned Albert Einstein, an extremely famous theoretical physicist. The other new book concerned Kurt Godel, a "logician" who isn't well-known by the average shlub at all.

This May, Holt's New Yorker essay, lightly edited, reappeared as the title essay in his own new book, When Einstein Walked with Godel: Excursions to the Edge of Thought.

As seems to be required by law, Holt's new book was praised by major reviewers—was praised for its lucidity. In the thirteen years since that essay appeared, Holt had become a "made man" in New York publishing circles.

Within those circles, Holt is now reflexively praised for the clarity of his writing about difficult science and math. As you can see, Wikipedia even headlines him as a "philosopher!" That's how silly and mandatory this sort of thing has become.

Holt's book of essays was praised this year for its brilliant lucidity. As we noted last Friday, the same was true of the new book about Kurt Godel which he discussed in The New Yorker back in 2005.

Hurrah! That new book, by Rebecca Goldstein, had been described as "accessible"—but also as "surprisingly accessible," even as "remarkably accessible."

It had been praised as a "lucid expression" of Godel's ideas—but it had also been hailed as "eminently lucid." So it goes within our tightly scripted academic journalistical complex.

Goldstein's treatment of Godel's ideas had been widely praised. Now, a writer at The New Yorker was going to boil matters down even further! Subscribers would finally get a chance to understand Godel's "incompleteness theorems," on the basis of which, Holt now said, Godel has often been called "the greatest logician since Aristotle."

Truth to tell, nothing dimly resembling that occurred in Holt's piece. In fairness, Goldstein hadn't been especially lucid when it came to explaining Godel's theorems either.

For the general reader, Goldstein's treatment of Godel's theorems would almost surely have been extremely hard to follow. When Holt took his turn in The New Yorker, his attempt to describe those "incompleteness theorems" was almost comically incoherent—incoherent all the way down.

Today, the book by Holt which features that essay is being praised by major journalists for its brilliant clarity. In this way, a comical aspect of our journalism—indeed, of our upper-end culture's most basic attempt at rationality—has once again been put on display, for perhaps the ten millionth time.

Holt's essay appeared in early 2005. Its author discussed Einstein's theory of relativity, then turned to Godel's theorems. In this week's reports, we'll speak of Godel alone.

Back in May, readers of the New York Times and the Wall Street Journal were told that Holt's rather obvious incoherence is an example of brilliant lucidity.

In this way, we've been given another look at the classic misassessment we've now christened as "Aristotle's error." We've been given another look at the way we humans, at least in the west, keep "seeing ourselves from afar."

Thanks to his incompleteness theorems, Godel has often been described as the greatest logician since Aristotle. But what did Godel actually say in his theorems? What was he trying to show?

In his essay for The New Yorker, Holt addressed those basic questions in two enormously long paragraphs. Today, we'll examine the first of those paragraphs, transcribing it as it appears in Holt's current book.

We'll start with an apology. In the past two weeks, we've already posted the start of the first paragraph in question. Before we show you Holt's full paragraph, we'll revisit that part, for review:
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...
According to Holt, Godel had come to believe that "abstractions like numbers and circles had a perfect, timeless existence independent of the human mind." As it turns out, Godel was "seduced" by this "doctrine" as a mere freshman in college.

Already, a perceptive reader might suspect that she's being directed by "a guide...who only has at heart [her] getting lost." What in the world does a person believe when he believes that "numbers and circles have a perfect, timeless existence independent of the human mind?" What does it mean to "believe," to be seduced by, such a peculiar notion?

Already, a perceptive reader should be asking such questions. But Holt just kept plowing ahead.

In the passage we've posted above, Holt said that's what Godel believed. He didn't try to explain what that peculiar formulation might possibly mean. Instead, he moved on to describe a major dispute within the intellectual world of Godel's Vienna.

What follows is the first of the two lengthy paragraphs in which Holt explains, or attempts or pretends to explain, Godel's "incompleteness theorems." Warning! The second graf, which we'll review tomorrow, is almost twice as long as the first.

According to recent reviews in the Times and the Journal, this paragraph appears within the title essay of a book in which the writing is brilliantly incisive and clear. Additional warning! By the end of this paragraph, the greatest logician since Aristotle is asking himself how we can know that 2 + 2 equals 4!

That's what Godel is asking himself! People, we're just saying:
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.
What makes a proposition like “2 + 2 = 4” true?

As the people of Europe struggled and groaned between two deeply destructive world wars, that's the type of question the western world's most brilliant thinkers were laboring to resolve!

You've now seen the first of the two paragraphs in which Holt explains, or attempts to explain, Godel's "incompleteness theorems." In this first paragraph, Holt basically sets the stage for his ultimate explanation. That will come in the second paragraph, which is roughly twice as long.

What makes a proposition like “2 + 2 = 4” true? In this, the first of his two paragraphs, Holt—the brilliant, incisive writer—sets the stage for spelling it out. This is what he has said:

According to one group of thinkers in Godel's Vienna, "2 + 2 = 4" is true because this rather familiar arithmetical proposition "can be derived in a logical system according to certain rules."

Tell the truth, dear general reader: Do you have the slightest idea what that statement means?

Tell the truth, New Yorker subscriber: Do you understand what it means, even in a general sense, to "derive [an arithmetical proposition] in a logical system according to certain rules?"

Friend, of course you don't! But if that's what one group of thinkers were thinking, at least one other brilliant thinker was brilliantly thinking this:

According to the greatest logician since Aristotle, "2 + 2 = 4" is true because it "correctly describes [an] abstract world of numbers." Perhaps more precisely, it correctly describes one aspect of the perfect, timeless existence enjoyed by numbers and circles outside the human mind!

Europe was struggling between two wars. Tearing their hair in the loftiest circles, the western world's most brilliant "thinkers" were laboring over this.

That being said, did Holt go on to make Godel's approach to this matter brilliantly clear? More specifically, was he able to explain Godel's "incompleteness theorems" in a way the general reader might find wonderfully clear?

That's what major reviewers have suggested. But at this point, does any of this seem especially clear?

Tomorrow, we're going to ask you to strap yourselves into your seats. We'll quickly revisit this first paragraph, into which a substantial amount of incoherence has already been poured.

Then, we'll look at Godel's endless succeeding paragraph, in which he attempts to describe the working of Godel's theorems in a way the general reader will be able to comprehend.

We humans! Seeing ourselves from afar once again, major reviewers have seemed to say that Holt did a wonderful job!

Tomorrow: "Beginning with a logical system for mathematics..."

THE PLATONIST FILE: Digest of reports!

MONDAY, SEPTEMBER 17, 2018

New chapter starts tomorrow:
Friend, do you know what a Platonist is? How well were Rebecca Goldstein and Jim Holt able to explain the puzzling "doctrine" of Platonism?

Tomorrow, we start our "incompleteness file"—our reports on the efforts by Goldstein and Holt to explain Kurt Godel's "incompleteness theorems."

Those reports start tomorrow. For today, we offer these links to last week's reports from the Platonist file:
Tuesday, September 11: You might be a Platonist if...! A timeless lack of clarity.

Wednesday, September 12:: Attempts to explain 2 + 2! The world of our greatest logicians.

Thursday, September 13: Goldstein tries to explain what Godel believed. Professor takes Platonist challenge!

Friday, September 14: What makes 17 a prime? When mathematicians wander.
For links to our earlier reports from the Godel file, you can just click here.

Tomorrow, we start the incompleteness file. With what degree of clarity did Holt explain Godel's theorems?

BREAKING: How many pundits watched the match?

SATURDAY, SEPTEMBER 15, 2018

Pseudolibs and dittoheads together:
At something approaching the speed of light, the Washington Post's Sally Jenkins was able to spot the sexism.

The events in question transpired late Saturday afternoon. By Sunday morning, this hard-hitting headline graced the front page of the hard-copy Washington Post:
Sexist power play ruins powerful final
The headline appeared, on page A1, atop an opinion column by Jenkins. According to the hard-hitting headline, Jenkins had spotted a "sexist power play."

At issue was the conduct of Carlos Ramos, the tennis official who umpired last Saturday's match between Serena Williams and Naomi Osaka. That headline captured Jenkins' assessment of two, or possibly even three, decisions Ramos made.

Jenkins offered an instant assessment. Another part of her fiery piece made us wonder if she had actually watched the match.

We'll quote that passage below. First, let's consider a fiery assessment which appeared in Tuesday's New York Times.

Jenkins is a major sports columnist at the Washington Post. Wesley Morris is a major "performance critic" at the New York Times.

Jenkins had quickly spotted the sexism which caused Ramos to behave as he did. As part of a lengthy assessment, Morris worked in the racism too:
MORRIS (9/11/18): ''This is unbelievable. Every time I play here I have problems,'' she told Ramos, justifying the question I whisper to myself before she starts any U.S. Open: Which of the bad old times would she draw upon if things go awry?

You remember Serena Williams's temper for how it singes but also for its aberration. Actresses might win Oscars for emotional combustion, but there's little tolerance for a nonfictional black woman undamming herself. Black female rage is an incarcerating stereotype whose social costs remain absurdly high.
This is the way we pseudo-liberals now play the game. But as we kept reading, that basic question recurred:

Had Morris watched the match?

In part, we wondered because, by sheer happenstance, we had watched the (rather lengthy) part of the match in question. We'd accidentally flipped to the tennis match just as the (lengthy) dispute was beginning. We sat and watched the (lengthy, multi-game) discussion, tirade or colloquy which followed.

We've often been struck by the smart, sane, sensible interviews Williams conducts on TV. (So too with her sister, Venus Williams.) But on this occasion, we thought she behaved extremely poorly, as almost everyone does at some point along the way.

In part for that reason, we found ourselves wondering if pundits like Jenkins and Morris (and quite a few others) had actually watched the match. Their accounts of what had occurred struck us as almost comically selective, except in the way they toyed with the subjects of gender and race, topics which shouldn't be toyed with.

That said, we modern liberals sometimes seem to live for the joy of toying with gender and race. We drop our bombs with lightning speed and with stunning certainty. Three days after the fact, we may compose groaners like this:
MORRIS: I've always found Williams's eruptions at the U.S. Open acutely depressing. As someone who's watched her in awe, suspense and pride, I find what's particularly awful is the way that pride—in her excellence, in her improbable historicism, in her grit—has compelled me to make excuses for her descents into viciousness. It's just ... Serena.

We're uneasy about how to criticize Williams's behavior without that criticism seeming racist or sexist, given the racism and sexism that Williams and her sister Venus continue to endure. You see something like an Australian op-ed cartoonist caricaturing Williams as a kind of Jim Crow-era savage and Osaka as a faceless blonde (she's the daughter of a Japanese mother and Haitian father) and have just a glimpse of what else Williams has been lugging with her onto the tennis court these many years.
Did Morris watch the match? Even as he seems to say that Williams may sometimes "descend into viciousness," he spots the racism in a cartoon in which Osaka was portrayed as, of all things, a blonde!

Osaka's parents are Japanese and Haitian! Morris seemed to know what that just had to mean about her hair! That said:

Whatever a person may think of that Australian cartoon on the whole, it's fairly obvious why it portrayed Osaka's hair as it did. The sheaf of hair protruding from the back of Osaka's cap that day was indeed curly and blonde, just as it appeared in the racist cartoon which Morris diagnosed as he did.

Had Morris actually watched the match? Did he have the slightest idea what Osaka had looked like that day? Even as he tossed his claims and insinuations around, did he have even the first idea what he was talking about?

Morris had had several days to get his reactions together. Jenkins had spotted the sexism right away—but had she watched the match?

We wondered about that Sunday morning because her general account of what had occurred seemed cartoonishly selective. But also because she offered the highlighted claim about the way the fiendish Ramos refuses to "take it" from women:
JENKINS (9/9/18): The controversy should have ended there. At that moment, it was up to Ramos to de-escalate the situation, to stop inserting himself into the match and to let things play out on the court. In front of him were two players in a sweltering state, who were giving their everything, while he sat at a lordly height above them. Below him, Williams vented, "You stole a point from me. You're a thief."

There was absolutely nothing worthy of penalizing in the statement.
It was pure vapor release. She said it in a tone of wrath, but it was compressed and controlled. All Ramos had to do was to continue to sit coolly above it, and Williams would have channeled herself back into the match. But he couldn't take it. He wasn't going to let a woman talk to him that way. A man, sure. Ramos has put up with worse from a man. At the French Open in 2017, Ramos leveled Rafael Nadal with a ticky-tacky penalty over a time delay, and Nadal told him he would see to it that Ramos never refereed one of his matches again.

But he wasn't going to take it from a woman pointing a finger at him and speaking in a tone of aggression. So he gave Williams that third violation for "verbal abuse"
and a whole game penalty, and now it was 5-3, and we will never know whether young Osaka really won the 2018 U.S. Open or had it handed to her by a man who was going to make Serena Williams feel his power. It was an offense far worse than any that Williams committed.
That was the proof of the sexism! When Nadal "told him he would see to it that Ramos never refereed one of his matches again," Ramos just sat there and took it. It was worse than what Williams said!

He'd tolerate that crap from Nadal. He just wasn't willing to "take it" from Williams. Except he did exactly that. In fact, he did it two times!

Sad! As Jenkins would have known if she watched the match, Williams specifically told Ramos, at two separate points, that she would never let him referee another one of her matches.

She told him this after the fifth game of the second set, then again after the seventh game of the set. The second time she made this statement, she went so far as to tell Ramos this:

"You will never, ever, ever be on another court of mine as long as you live.''

Williams dropped this bomb on Ramos at two separate point this day. And as with Nadal, so too here: Ramos simply "took it" each time!

He did, in fact, "allow a woman to talk to him that way;" he did so two separate times. It was only when Williams continued ranting that he charged her with "verbal abuse," as he certainly could have done long before that.

Did Williams stage a "meltdown," as some have now said? That word came to mind for us as we watched her go on, and on, and on and on, berating Ramos over the course of five games during this second set.

For ourselves, we were mainly impressed by the rudeness and disrespect Williams was exhibiting toward her 20-year-old (female) opponent, who was forced to endure a storm of shouting and booing from the crowd when Williams stopped her ranting long enough to let the match continue.

Everybody can have a bad day. As we watched the match, it seemed to us that Williams was having a corker. She said several things which made no earthly sense, and even as she insisted, over and over, that she would never cheat, her coach was telling a TV reporter that yes, as a matter of fact, he had been coaching when Ramos made that initial call. If Williams never accepts any coaching, why does her coach provide it?

We thought Williams had a very bad day. That said, Jenkins and Morris had strange days too. Had they watched the match?

No law requires the modern pundit to evaluate events in a balanced, intelligent way, but a few still manage to do so. On September 10, the New York Times' Juliet Macur reviewed the events at issue in this basically fair and balanced column. She added many of the points of complexity which most pseudo-liberal pundits quickly erased from view.

Macur engaged in something resembling traditional rational conduct. While presenting a range of possibilities about Williams' extremely long harangue, she even went so far as to perhaps suggest a possibility:
MACUR (9/10/18): [I]nstead of a match for the ages, the heralding of a young and deserving talent, it will probably be remembered for Williams's calling the umpire a sexist liar and later saying her complaints were made for the equal rights of all women. But on closer examination, it's also true that this umpire has been tough on top male players, too. The difference is that the men didn't belabor their arguments with him.

[...]

Ramos officiated with his usual exacting eye. He gave Williams a warning for receiving coaching in the second set. His action was warranted because Williams's coach, Patrick Mouratoglou, admitted to coaching her.

But Williams exploded into a tantrum that included her shouting that she would never cheat because she is a mother now and wants to be a good example for her daughter. She pointed her finger and demanded an apology from Ramos.

You can argue the nuances. Lots of coaches coach and lots of players are coached from off the court. And lots of umpires don't call them on it. You also have to wonder if Williams would have gone after Ramos so relentlessly—and with such conviction to stand up for women's rights—if she were winning.
Is Ramos equally "tough" and "exacting" with women and men? We have no idea, and very few of our legion of pseudo-liberal pundits seemed to worry about such niceties as they scattered their bombs about and delivered their scripted views.

Late in that passage, Macur might even have seemed to suggest that Williams might have staged her multi-game rant as a way to stir up the crowd against Ramos (and against Osaka). As we watched the events that day, it didn't seem that Williams was trying to do that—but she did produce those showers of catcalls and boos, and she did, in the process, show gross disrespect toward her younger (female) opponent.

Osaka was able to tough it out and win the match when Williams finally let it proceed. But to our eye, Williams had a terrible day, as we humans sometimes do, in ways which were often disappeared by impassioned scribes like Jenkins and Morris.

Are we famous "rational animals" able to reason at all? As we swith the focus of this site, we're trying to explore this eternal question.

Again and again and again and again, we contemporary pseudo-liberals give our answer: no. Especially when gender and race are involved, it tends to be narrative all the way down within our impassioned ranks.

Tending toward narrative all the way down: Also from Macur's column:
MACUR: Billie Jean King, a pioneer for women's equality in sports, weighed in on Twitter.

''When a woman is emotional, she's 'hysterical' and she's penalized for it,' '' King wrote. ''When a man does the same, he's 'outspoken' and there are no such repercussions. Thank you, Serena Williams, for calling out this double standard. More voices are needed to do the same.''

Hard to argue with that. But it was disappointing that King said nothing about the poor timing of Williams's powerful voice. It made me think back to last year's Open, when the Italian player Fabio Fognini unleashed a barrage of Italian curses upon a female umpire and was kicked out of the tournament.

So sometimes, there are repercussions.
When men do that, they're called "outspoken?" We'd love to see the cite for that from within the world of tennis. The cite may exist, but no one seemed inclined to present it. There were too many bombs to drop!

That said, many members of our tribe have followed King down that road. Inevitably, the Times felt the need to print this ridiculous letter:
LETTER TO THE NEW YORK TIMES (9/12/18): We should apply the same standard of sportsmanship for men and women. Women currently have much less leeway when it comes to what's considered good sportsmanship...

John McEnroe challenges a call and smashes his racket, and he's praised as a competitor. Serena Williams does the same and she's disrespecting the sport? Please. Not allowing female athletes to be hotheaded, fallible and unsportsmanlike fails to recognize female athletes as having a competitive spirit equal to that of their male counterparts.

S— M—, Los Angeles
McEnroe played long ago. He was routinely called a jerk, which is what he routinely was.

Pundits branded him "McBrat." In service to current pseudolib scripts, such history must disappear.

Are we able to reason at all? Again and again and again and again, we pseudolibs join our dittohead pals. We give a loud answer:

No.

THE PLATONIST FILE: What makes 17 a prime?

FRIDAY, SEPTEMBER 14, 2018

When mathematicians wander:
Jim Holt got off easy.

Back in 2005, he wrote a slightly-disguised review of Rebecca Goldstein's new general interest book, Incompleteness: The Proof and Paradox of Kurt Godel. Holt's review appeared in The New Yorker, an upper-end general interest magazine.

In her general interest book, Goldstein had told the story of Godel's life. She'd also tried to explain his "incompletenesss theorems," on the basis of which he's often been called the greatest logician since Aristotle.

First, though, Goldstein tried to explain the doctrine which, she said, lay at the heart of Godel's intellectual life from the time of his first year in college. Because he was only writing a review, Holt described the doctrine very briefly, and then quickly moved on.

We'd have to say he got off easy! Here's his key passage again:
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...
Platonism is built around the belief that "abstractions like numbers and circles have a perfect, timeless existence independent of the human mind." So said Holt, in The New Yorker, and then he quickly moved on.

Do numbers and circles have a perfect, timeless existence independent of the human mind? Friend, do you have even the slightest idea what that word sequence might possibly mean? We'll go ahead and answer for you:

No, you don't have the slightest idea. And neither does anyone else!

Because he was only writing a review, Holt got to leave things right there. Today, Holt's review, edited to remove most references to Goldstein, is the title essay of his own new book, When Einstein Walked with Godel: Excursions to the Edge of Thought.

Just this once, let's be honest! Holt's account of Godel's beloved "doctrine" is straight outta Jabberwocky. You don't know what the heck it means, and neither does anyone else.

But so what? When Holt's new book appeared, fronted by that title essay, major reviewers stood in line to exclaim, for the ten millionth, about how amazingly lucid and clear Holt's science/math writing is. Reviewers swore that Holt's writing was brutally lucid and clear.

In effect, reviewers swore that Holt had made Godel and Einstein easy. This has been standard journalistic practice dating at least to Einstein's own general interest book about relativity—to the brilliant physicist's failed attempt to make his own theories clear.

(More on that effort below.)

This is standard journalistic behavior—and Goldstein benefited from this practice when her book about Godel appeared. Her own account of Platonism is so murky that it seems to have come from the third planet beyond Jabberwocky. That said, the usual suspects stood in line to say how lucid her writing was.

Three major academic stars blurbed Goldstein's book on its jacket. You already know what they said:

In his dust jacket blurb, Alan Lightman praised Goldstein for her "penetrating, accessible, and beautifully written book."

Brian Greene went one step further. He said Goldstein's account of Godel was "remarkably accessible."

In a New York Times review, Polly Shulman said that Goldstein's writing was "surprisingly accessible." Meanwhile, back on the book's dust jacket, Stephen Pinker said this:
This book is a gem...Rebecca Goldstein, the gifted novelist and philosopher, offers us not just a lucid expression of Godel's brainchild but a satisfying and original narrative of the ideas and people it touched. Written with grace and passion, Incompletenesss is an unforgettable account of one of the great moments in the history of human thought.
"Lucid" was Pinker's word of choice. At Salon, Laura Miller stepped in to top him on this part of the color wheel, calling the book "eminently lucid."

This constitutes a familiar practice within several modern guilds. (Goldstein praises Holt's new book on that book's dust jacket!) Within the burgeoning publishing world of modern science-and-math-made-easy, the professors praise each other in these ways, as do the major reviewers.

Are these blurbs ever accurate? Back in 2005, Miller said that Goldstein's "masterful" book provided "an eminently lucid explanation of Gödel’s theorem and its implications.”

Does anyone think that Miller, a general interest reviewer, could string two coherent words together about Godel's highly abstruse theorems? We'd be very surprised if she could, but if she can, it's hard to believe that her ability stems from Goldstein's widely praised book.

At this point in our explorations, we haven't examined Goldstein and Holt's attempts to explain, elucidate, unpack or describe Godel's actual theorems. This week, we've been trying to see if either writer could explain, elucidate or describe the alleged "doctrine" called Platonism, which is said by Goldstein to lie at the heart of all Godel's ruminations.

What the heck is Platonism? We've already seen what Holt said. According to Goldstein's first bite at this apple, Platonism involves the belief that "the truths of mathematics are determined by the reality of mathematics"—and as we showed you yesterday, the project goes downhill from there, all the way to an impossibly strange rumination about the way the Platonist would evaluate the claim that Santa Claus exists.

Please understand—Goldstein isn't some second-rate shlub who got hauled in from the cold. As we explained in an earlier post, she lives a perfect timeless existence at or near the very top of modern academic elites.

She's a ranking philosophy professor, and a highly-regarded novelist. This helps explain why her book was blurbed so favorably by other elites—unless you think that writing like this really does deserve to be praised as transplendently lucid:
GOLDSTEIN (page 87): For a Platonist, mathematical truth is the same sort of truth as that prevailing in lesser realms. A proposition p is true if and only if p. "Santa Claus exists" is true if and only if Santa Claus exists.
"Santa Claus exists" is true if and only if Santa Claus exists? If that explains the doctrine with which Godel "fell in love" as a teen, then the whole world is a Platonist, including you and yours.

Casey Stengel is said to have said it when he managed the 1968 Mets: "Can't anybody here play this game?" the gent is said to have asked.

The later Wittgenstein said the same thing about a wide range of major "philosophers," not excluding himself in his own earlier phase. In time, we'll be perusing this major jailbreak which, according to Professor Horwich, is being strategically ignored.

That will come at a later date! For today, we're going to see what can happen when brilliant people decide to play out of position.

The greatest shortstop would probably make an extremely poor tight end. In 1968, Rod Laver was the world's top-rated male tennis player. There's little reason to think he could have helped Stengel's hapless Mets.

So too in the worlds of mathematics and physics! Consider what happened when Goldstein quoted G. H. Hardy, who was, by all accounts, a brilliant mathematician.

Who the heck was G. H. Hardy? The leading authority on his life answers your question here.

By all accounts, Hardy was a brilliant mathematician. For better or worse, he also crossed over to do some "philosophizing" in his iconic 1940 essay, A Mathematician's Apology.

According to Goldstein, Hardy, "an English mathematician of great distinction, expressed his own Platonist convictions" in this "classic" text. She seems to think that the passage she quotes in her book will help us understand this alleged doctrine.

Below, you see the passage Goldstein quotes, on page 46 of her book. We'll especially focus on the way Hardy puzzles over how we humans can know that 317 is a prime:
HARDY (1940): I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it...

[T]his realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more than what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but "2" or "317" has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy—I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
In our view, that passage is the work of a brilliant mathematician who's playing way out of position. In effect, Hardy—a brilliant mathematician—becomes your Uncle Charlie at Thanksgiving dinner, going on and on.

Step by step, a brilliant mathematician leads us away from clarity in that jumpy passage. This is the sort of thing which can happen when mathematicians wander far afield.

Hardy muddles his thinking in that passage at an array of points. At one point, he refers to the number 2 as a "mathematical object."

Do you have any idea why a person would want to do that?

That passage starts with Hardy saying that "mathematical reality" (whatever that is) "lies outside us." Consider:

You surely know what Homer meant when he said the battle between Achilles and Hector took place "outside the walls of Troy." But are you sure you understand what Hardy means when he says that "mathematical reality" (whatever that is) is somehow found "outside us?"

That's a rather unusual formulation. Are you sure you know what it means? Can you think of any conceivable way to disagree with that peculiar statement?

In this passage, Hardy plays with dueling "isms"—with "realism" and "idealism." This will almost surely work to further confuse the general reader. Indeed, we'd advise you to check your wallet even when full-fledged "logicians" start burying you in such jargon.

Eventually, the rubber meets the road. Hardy, a brilliant mathematician, chooses to tell us this:

"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."

What makes 317 a prime? According to Hardy, 317 is a prime because it is so! (Hardy's emphasis), not because our minds are shaped in some way.

Friend, would you have any idea how to disagree with that? Has it ever occurred to you to think that 317 (or, more simply, 7 or 17) is a prime because "your mind is shaped in some way," whatever that might mean?

Do you understand what you're being told when you're told that 317 (or, more simply, 17) is a prime because it is so? Are you really completely sure that you're being told anything at all? Does that sound a bit like Uncle Charlie arguing some favorite political point?

In this passage, you see a brilliant mathematician making little clear sense. Is the number 17 a prime just because we think it is? Do you have any idea why anyone would ever make such a claim? In the absence of any such idea, do you understand why Hardy seems to be aggressively "refuting" this claim?

Why is 317 a prime? Now that you've asked, we can explain it amazingly simply. The number 317 is a prime because it can't be divided evenly by the number 2, or by any other "natural number," as you will quickly be able to see if you just give it a try.

317 can't be divided evenly by any other number! Go ahead—you can try them all, though if you're arithmetically slick, you'll know that you only have to try these numbers: 2, 3, 5, 7, 11, 13, 17, 19.

After you try 19 and fail, you don't have to try any more. (Reason: 19 x 19 is larger than 317.) But go ahead—try them all! No other number will divide evenly into 317. That's the most straightforward, simple-minded answer to the (rather imprecise) question Hardy semi-poses in that peculiar passage.

Playing out of position, Hardy almost seems to fashion a tautology: 317 is a prime because it is so! Writing a general interest book, Goldstein presents this passage as if it will help us understand Hardy's "Platonist convictions" and the doctrine of Platonism as a whole.

We humans! If we weren't so inclined to defer to authority, we'd respond to this in the manner of the child who saw that the emperor forgot to put on his clothes. We'd marvel at the lunacy involved in Holt's lucid but ludicrous statement:
"[Gödel was] seduced 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...has always been popular among mathematicians."
If we weren't so strongly inclined to defer to academic (and journalistic) authority, we'd react as a sensible person might. We'd marvel at the peculiar claim that mathematicians are inclined to think that numbers and circles "have a perfect, timeless existence" of some undisclosed kind. We'd wonder why a man like Hardy was throwing various "isms" around as he heatedly seemed to explain how we can know that 317 is a prime.

What was Hardy trying to say in that passage? We have no idea.

That said, the burden of clarity falls on the person who's making the lofty claim. It isn't your task, as Hardy's reader, to pretend to make sense of something he's said. Nor should you ever simply assume that something makes sense just because it's being said by a ranking academic.

It was Hardy's job to make his statement lucid! If Goldstein is going to quote him, it's Goldstein's job to let us know why "317 is a prime because it is so" isn't simply the holiday raving of a type of Uncle Charlie.

This brings us back to Einstein's general interest book, even as it points us toward the work of the later Wittgenstein.

Einstein is widely viewed as the most brilliant physicist at least since Newton. After he fashioned his theories of relativity, a publisher asked him to write a general interest book to explain what he had done to the general reader.

The book appeared in German in 1916, in English in 1920. Manifestly, it didn't "make Einstein easy." In his recent biography of Einstein, Walter Isaacson told the comical story which explains how this happened.

Einstein, the world's most brilliant physicist, wasn't a general interest writer! In effect, he was the greatest athlete of all time, but not real good at cooking.

According to Isaacson, as Einstein tried to make Einstein easy, he selected his cousin Elsa's teen-aged daughter as his focus group. "He read every page" to her, Isaacson writes, "pausing frequently to ask whether she indeed got it."

She kept saying she understood, "even though (as she confided to others), she found the whole thing totally baffling." So it went when the planet's most brilliant physicist briefly played out of position.

Friend, do you have the slightest idea what it means to believe in the doctrine of Platonism? Was this really some deep philosophical view? Or was it possibly one of the first of Godel's crazy ideas?

We ask because Godel is described as the greatest logician since Aristotle. What might it mean if our greatest logicians was in thrall to crazy ideas? What does it our highest ranking professors can't make out this fact?

Could it mean that we in the west, like the Bushmen of the Kalahari, have been "seeing ourselves from afar?" Could it light the way toward the work of the later Wittgenstein, which we plan to discuss in coming weeks, if we get there before Mr. Trump decides to start his war.

Are Goldstein and Holt amazingly lucid? It certainly isn't a moral failing, but no, we don't think they are.

That said, Goldstein and her partners in blurbing are among our highest academic elites. When you see the way our top professors perform, are you surprised that our journalists perform even worse? Are you surprised that our cultural breakdown has reached the point where Donald J. Trump holds such power?

Next week: The incompleteness file

THE PLATONIST FILE: Goldstein tries to explain what Godel believed!

THURSDAY, SEPTEMBER 13, 2018

Professor takes Platonist challenge:
Rebecca Goldstein is a ranking philosophy professor.

She's also a highly-regarded novelist. For our money, her inclination to tell the human story didn't serve her especially well when she wrote her 2005 general interest book, Incompleteness: The Proof and Paradox of Kurt Godel.

Goldstein goes into substantial internal detail about the way Godel "fell in love" when he was still a teen—about the "ecstatic transfiguration" produced by his love affair with Platonism.

She goes on and on, then on and on, about this "transfigurative intellectual love." At times, though, she also makes it sound like Godel had sworn allegiance, at this point in his life, to some revolutionary political group.

According to Goldstein, Godel "had become a Platonist in 1925," the year during which he turned 19. He "was already a committed Platonist in 1926," the year in which he began attending meetings of the super-elite discussion group called the Vienna Circle.

Godel was moving in lofty academic circles at a very young age. It's also true, according to Goldstein, that he didn't want the Circle to know about his commitment to Platonism. In effect, he was an Alger Hiss hiding within a circle of people all named Whittaker Chambers.

Back in 2005, Jim Holt reviewed Goldstein's book for The New Yorker. He told a shortened version of this slightly comical story in that review, which has now become the title essay of his own new book, When Einstein Walked with Godel:
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 nineteen-twenties 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...

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...
Members of the Vienna Circle had no idea that a Platonist was lurking among them! For better or worse, Goldstein tells the story of the "clandestine Platonist" at much greater length, and with much more dramatic flair.

At any rate, Godel was "a committed Platonist" by the time he was 20. He was lurking among the "logical positivists," unwilling to let them know about his deepest beliefs.

By the time he was 24, Godel had formulated and publicized his "incompleteness theorems," the theorems which are said to identify him as the greatest logician since Aristotle. Essentially, Godel saw these theorems as the proof of his beloved Platonism. Or so Goldstein says, we would assume correctly.

As of 1930, this silliness was consuming one of the western world's highest intellectual elites. Along the way, we may puzzle a bit at the stakes involved in this war of the worlds.

As we saw yesterday,
Godel, the second greatest logician, was trying to figure out how we can know that 2 + 2 equals 4. He'd fallen in love with Platonism, and wanted to prove that the doctrine was true.

Can major battles of this type really revolve around matters like 2 + 2? As we proceed today and tomorrow, we'll see other apparently vapid puzzlements move to center stage in this remarkable tale.

Before proceeding, we ought to admit it—this story of the "committed Platonist" strikes us as amazingly silly. Today, though, we ask a more challenging question:

In the course of writing her book, was Goldstein able to explain the nature of this "doctrine?" To what beliefs had Godel committed when he committed to Platonism, if in clandestine fashion?

As we saw yesterday, Holt made virtually no attempt to describe this powerful doctrine in his review of Goldstein's book. Was Goldstein, a ranking philosophy professor, able to clarify matters further, writing at much greater length?

In her first attempt at taking the Platonist challenge, Goldstein had, rather unhelpfully, offered the formulation shown below. We've shown you this passage before. Try to ignore the technical language:
GOLDSTEIN (page 44): Godel's commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher...

Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up that reality.
There it stands. In this, her first pass at the Platonist challenge, Goldstein tells us this:

The Platonist believes in "the objective existence of mathematical reality," whatever that is supposed to mean. But wait—there's more.

According to the Platonist, the truths of mathematics are determined by the reality of mathematics! More specifically, the truths of mathematics are determined by the nature of the entities which make up that reality.

So this ranking professor has said. Presumably, everyone can see how unhelpful this first attempt at explication was. That leaves us asking an obvious question:

Does Goldstein go on, in her book-length text, to clarify, unpack, elucidate or explain the essence of this alleged doctrine? Does she ever do a better job helping us understand the nature of the doctrine to which the second greatest logician in history is said to have committed his life?

What the heck is Platonism? Does Goldstein ever provide an answer which is, quoting from the blurbs on her book, lucid, accessible, clear?

We're going to say that she doesn't. To give you a sense of what we mean, let's take ourselves to page 87 of her general interest book.

Friend, let's start with a basic admission. There's no way to present an excerpt from this book without exciting a possible objection. The reader may suspect that a tiny shard of Goldstein's brilliantly lucid exposition has been taken out of some larger context.

Go ahead—keep that possible objection in mind! Then proceed to read this passage, in which Goldstein tells us what a possible fool believes:
GOLDSTEIN (page 87): For a Platonist, mathematical truth is the same sort of truth as that prevailing in lesser realms. A proposition p is true if and only if p. "Santa Claus exists" is true if and only if Santa Claus exists. "Every even number greater than 2 is the sum of two primes" is is true if and only if there is no even number greater than 2 that is the sum pf two primes (even if we can never prove it).
What does a Platonist believe? With what sorts of beliefs did our second greatest logician fall in transfigurative love?

According to Goldstein, a Platonist believes the following. A Platonist believes that the proposition "Santa Claus exists" is true if and only if Santa Claus does exist!

No, really. That's what it says!

Friend, you must be a Platonist if that's what the doctrine is! Every person on your block is a committed Platonist too.

"Santa Claus exists" is true if and only if Santa Claus exists? Everyone believes that statement—and everyone believes the other two examples provided in that peculiar passage.

To marvel further at the type of work which routinely emerges from our academic elites, we turn to the footnote on that same page. In that footnote, Goldstein further discusses the third example from the passage we've already posted, the example involving the sum of two primes.

According to that footnote, "the Prussian mathematician Christian Goldbach (1690-1764) had conjectured that every even number greater than 2 is the sum of two prime numbers."

So far, so perfectly clear. But in her footnote, Goldstein further discusses Goldbach's conjecture. As she does, she tries again to let us know what a Platonist believes and asserts:
GOLDSTEIN (page 87): Goldbach's conjecture has been confirmed for every even number that has ever been checked; however, no proof has of yet been discovered for the universal conclusion that every even number greater than 2 is the sum of two primes. The fact that Goldbach's conjecture remains unproved means (at least according to the Platonist) that lurking out there beyond the point where mathematicians have checked there might be a counterexample: an even number that isn't the sum of two primes. Then again (according to the mathematical Platonist), there may not be a counterexample: every even number may be the sum of two primes, without there being a formal way of proving that this is so. A Platonist asserts that there either is or isn't a counterexample, irrespective of our having a proof one way or the other.
"A Platonist asserts that there either is or isn't a counterexample, irrespective of our having a proof one way or the other?"

Tell the truth! At least on its face, does that make any sense?

"A Platonist asserts that there either is or isn't a counterexample?" Who wouldn't make that "assertion?" And why would it take a Platonist to make the other assertions Goldstein lists in that passage about the conjecture?

Goldbach's conjecture remains unproved (and unrefuted)? Why would it take a Platonist to say that there might be a counterexample which hasn't yet been checked? Wouldn't anyone say the same thing? Ecstatic transfiguration to the side, what's love of Platonism got to do with it?

Goldstein's examples in this footnote bring us back to Santa Claus, who can only accurately be said to exist if he does exist. We flash on our second greatest logician trying to figure how we can know that 2 + 2 equals 4.

Misguided respect for academic authority will induce the trusting soul to assume that there must be some lofty explanation for these peculiar presentations. We'll strongly suggest that, in matters like these, such trust will be leading us wrong.

At substantial length, Goldstein tells a novelist's tale in her book—a tale of the powerful intellectual love which produced a committed Platonist. Respect for intellectual authority may incline us to believe that some actual great beliefs lie at the heart of this tale.

On the other hand, yesterday, in Holt's text, we saw our second greatest logician puzzling over 2 + 2 = 4. Today, we see a high-ranking professor offering what seems like perfect twaddle concerning the existence of Santa Claus.
.
In Goldstein's book, we also encounter the passage shown below. In this passage, Goldstein gives another example of what is, or was, at stake in this battle. Today, we'll highlight that passage:
GOLDSTEIN (page 44): Godel's commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher...

Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up that reality. The structure of, say, the natural numbers (which are the regular old counting numbers: 1, 2, 3, etc.) exists independent of us, according to the mathematical realist...and the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical properties of light and gravity.
According to the Platonist, the claim that the number 4 is even is an "objective" claim. So is the claim that 25 is a perfect square. (That is, that it's the product of 5 x 5.)

These claims are "objective," the committed Platonist cries. But so does everyone you've ever known, along with all those you've never met. This account of what a Platonist believes doesn't seem to make any sense—and yet, this is the way a ranking academic authority explain this ancient "doctrine," with three academic stars blurbing how lucid she is.

According to Goldstein and Holt, Platonism is the doctrine which defined the world of our second greatest logician, a man who seemed to be mentally ill throughout his life and who was famous for believing a long list of crazy ideas. According to Goldstein, it was on this peculiar plain outside Troy that a giant battle was waged involving our second greatest logician and a lofty academic Circle.

Does any of this make any sense? Respect for authority tells us it must.

Experience tells us something different. Then too, there's the passage from G. H. Hardy, another alleged Platonist, which Goldstein quotes early on.

Tomorrow: When mathematicians stray, or how can we humans possibly know that 317 is a prime?

THE PLATONIST FILE: Attempts to explain 2 + 2 = 4!

WEDNESDAY, SEPTEMBER 12, 2018

The perfect timeless world of our greatest logicians:
At this point, it's important to remember who we're talking about.

We're speaking here about Kurt Godel, "who has often been called the greatest logician since Aristotle." So says the widely praised science/math writer Jim Holt in the opening pages of his new book, When Einstein Walked With Godel: Excursions to the Edge of Thought.

Is it true? Was Godel really the greatest logician since Aristotle? Since Godel (1906-1978) lived in the twentieth century, that would make him the greatest logician in something like 2400 years!

You'd almost think that a person like that would have made some great, identifiable contribution to human thought. You'd almost think his name would be well known.

With Godel, the story is different. In Holt's telling, this second greatest logician seemed to struggle with mental illness since perhaps the age of 5—mental illness which became so extreme that he ended up dying of self-starvation.

Then too, there were the crazy ideas. Holt discusses them early on, contrasting Godel with his friend, Albert Einstein:
HOLT (page 4): Although Einstein’s private life was not without its complications, outwardly he was jolly and at home in the world. Gödel, by contrast, had a tendency toward paranoia. He believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases; he refused to go out when certain distinguished mathematicians were in town, apparently out of concern that they might try to kill him. “Every chaos is a wrong appearance,” he insisted—the paranoiac’s first axiom.
Our greatest logician believed in ghosts. He had a morbid fear of gases from the fridge.

Holt essay first appeared in 2005 as a somewhat disguised review of Rebecca Goldstein's book, Incompleteness: The Proof and Paradox of Kurt Godel. As such, almost all Holt's material seems to be drawn from Goldstein's book, in which she presented an amusing list of Godel's apparently crazy ideas—crazy ideas which didn't seem to spring directly from paranoia or other such illness.

Does it seem strange to think that our greatest logician was perhaps best known, in his adult years, for his various crazy ideas? We'll let you wrestle with that one. For ourselves, we'd be inclined to see this syndrome as perhaps being instructive, illustrative of the vast intellectual dysfunction at the heart of the human experience.

Godel became known as "the greatest logician since" because of his incompleteness theorems, which Holt and Goldstein struggle to explain in their respective texts. We'll examine those struggles next week, marveling at the kind of journalistic/academic work which is reflexively praised for its clarity by long lines of scripted nimrods within the upper-end press.

This week, we're puzzling about something different; we're puzzling about Godel's alleged "Platonism." At great length and in flowery language, Goldstein describes the murky "doctrine" as the enduring, rapturous love of Godel's life—as a doctrine with which he "fell in love" when he was only 19.

Holt's essay condenses this portrait of Godel's affair—but what the heck is "Platonism" even supposed to be? As we've noted, the following passage includes Holt's first bite at that worm-infested apple. As he continues (see text below), some comical themes will emerge:
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...
It's just as we told you last week! According to Holt by way of Goldstein, the young Godel was "seduced by...the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind."

So says Holt, almost seeming to assume that this puzzling definition of Platonism makes some sort of earthly sense.

Does Holt go on to explicate this "doctrine?" We'll consider his fleeting effort below. First, let's consider the puzzling places to which we humans can be taken in explorations of modern "philosophy" at its highest levels.

Holt is halfway through a lengthy paragraph at the point where we've left off. From the rest of his graf, we can extract a minor attempt to flesh out the concept of "Platonism"—but we're also taken to a peculiar place, a place where we ponder 2 + 2 and the mysterious way in which 2 + 2 can be known to equal 4.

Are we humans the rational animal, or was that widely bruited assertion Aristotle's error? Is it possible that our highest intellectual elites have more often turned out ruminations which more closely resemble clown shows?

Have our highest academic elites sometimes resembled harlequins, clowns? We cut-and-paste, you decide:
HOLT (continuing directly): In the philosophical world of nineteen-twenties 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.
A significant name pops up in that passage—the name of Ludwig Wittgenstein. Before our post-philosophical explorations are done, we'll consider his admittedly murky, but conceptually simple, later work at some length.

Setting that matter aside for the present, let's look at Holt's attempt to explicate Platonism. Also, let's consider the fact that our greatest minds have struggled over the nature of the famous schoolboy proposition, 2 + 2 = 4.

In that extended passage, Holt seems to contrast young Godel's Platonism—his fervent belief that the number 3 lives a perfect timeless existence—with the hard-boiled views of a group of thinkers called the Vienna Circle.

Mercifully, Holt ascribes no particular "isms" to that particular group. But within that passage, a reader can discern a tiny attempt by Holt to explain Platonism in a bit more detail:

You might be a Platonist if! From that passage by Holt, it sounds like a Platonist believes that a proposition like 2 + 2 = 4 is true because "it correctly describes some abstract world of numbers."

Reader, tell the truth! Do you have even the slightest idea what that proposition might mean? Before you give a defensive answer in which you defer to academic authority, please remember that we've asked you to stick to the truth.

People, what makes a proposition like “2 + 2 = 4” true? According to Holt, our loftiest gangs of intellectuals were debating this chin-scratcher as recently as 1930, with Godel clinging to a belief which he was refusing to reveal:

According to Holt, Godel believed the proposition "2 + 2 = 4" is true because it correctly describes some abstract world of numbers. So believed our second greatest logician as he fell in love with the idea which would drive his work throughout his entire life.

Friend, do you have the slightest idea what any of that might mean? Do you feel sure that you can explain what it means to believe in the existence of "abstract worlds" at all? More specifically, do you know what it means to believe in the existence of the "abstract world of numbers"—which, presumably, is the place where the number 3 lives its perfect timeless existence, surrounded of course by the circles?

Do you have even the slightest idea what it means to believe in such things? We're going to guess that you do not—that you could never explicate, explain or unpack the essence of such alleged beliefs, which represent Holt's only attempts to explain what "Platonism" is.

Luckily, Holt briefly extends his explanation at one additional point. As it turns out, the truth of 2 + 2 = 4 was, for Godel, all about our ESP! For now, let's ignore the technical language and stick to the matter at hand:
HOLT (page 10): [Gödel] believed he had shown that mathematics has a robust reality that transcends any system of logic. But logic, he was convinced, is not the only route to knowledge of this reality; we also have something like an extrasensory perception of it, which he called “mathematical intuition.” It is this faculty of intuition that allows us to see, for example, that the formula saying “I am not provable” must be true, even though it defies proof within the system where it lives...
How can we know that 2 + 2 equals 4? According to Godel as told by Holt, we have "something like an extrasensory perception" which lets us discern such realities about the abstract world of numbers where the number 3, and all other numbers, live their perfect timeless existence in the company of circles.

Briefly, a word of praise. In her own book, Goldstein offered a perfect, simple explanation of how we can know that 2 + 2 equals 4.

Goldstein's perfect explanation doesn't involve ESP. You'll rarely see a philosophy professor make such a perfect statement. We'll get to that statement next week.

For today, let's leave matters at this. Back around 1930, the greatest logician in 2400 years was trying to determine how we can possibly know that 2 + 2 equals 4.

At Princeton, during his later years, he was known for his various crazy ideas. Starting around the age of 19, he developed the Platonistic idea that we can know about 2 + 2 because we have some faculty which resembles ESP.

This is the way our highest academic elites were playing during these not-so-distant years. According to Holt, even Lord Russell got dragged into this ridiculous mess. Again, ignore the technical language to focus on the sad absurdity of what is being said:
HOLT (page 9): Gödel’s incompleteness theorems did come as a surprise. In fact, when the fledgling logician presented them at a conference in the German city of Königsberg in 1930, almost no one was able to make any sense of them. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it? The very idea seemed absurd. Even the once great logician Bertrand Russell was baffled; he seems to have been under the misapprehension that Gödel had detected an inconsistency in mathematics. “Are we to think that 2 + 2 is not 4, but 4.001?” Russell asked decades later in dismay, adding that he was “glad [he] was no longer working at mathematical logic.”
Lord Russell, "the once great logician," worried that he was being asked to believe that 2 + 2 might not equal 4 after all! This made him glad that we was no longer mired in the world of (mathematical) logic.

Respect for academic authority tells us that we must believe that these ruminations, by these great intellectual figures, simply must have made sense. By the end of the 1940s, Wittgenstein had torn such suppositions to shreds in a self-admittedly poorly-written book, Philosophical Investigations.

Professor Horwich has said that our current professors have stopped teaching Wittgenstein because his demonstrations mean that they would pretty much have to stop teaching everyone else. Down the road, we'll return to that claim, concerning which we'd occasionally joked on the world's greatest comedy stages before first encountering Horwich.

We'll return to Horwich's conjecture at some later date! Tomorrow, we'll return to Professor Goldstein's attempts to explicate "Platonism." We'll start with a comical story mentioned by Holt in passing:

A Platonist was hiding among them! Yes, it actually gets that silly, that puny, that tiny, that dumb.

Tomorrow: What a Platonist believes