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.
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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

THE PLATONIST FILE: You might be a Platonist if...!

TUESDAY, SEPTEMBER 11, 2018

Timeless lack of clarity: Friend, are you a Platonist? You might be a Platonist if you adhere to certain "beliefs."

Maybe you've found yourself thinking that "numbers and circles have a perfect, timeless existence independent of the human mind."

Perhaps you've come to believe that "the truths of mathematics are determined by the reality of mathematics"—more specifically, "by the nature of the entities that make up that reality."

Granted, those are fuzzy claims, at least as presented. But such fleeting thoughts may serve as signs of incipient Platonism.

Borrowing from Professor Foxworthy, You might be a Platonist if...—if you believe such puzzling and/or apparently vapid claims. Beyond that, you may be part of a slightly embarrassing, mildly endearing human story—a story which has been unfolding for several millennia now.

We mention Professor Foxworthy, a resident scholar at Humorist U., because this is, in large part, an ongoing comical story. Indeed, if it weren't for all the warfare and killing which accompany the intellectual dysfunction of our upper-end intellectual elites, we could call it a comical story and leave the matter right there.

Who is Professor Foxworthy? This professor, searching for fun, has often employed this formulation: "You might be a redneck if..."

After presenting that formulation, the professor has told a series of jokes. Like his forerunner, Professor Irwin Corey, the professor has done this for fun.

That's what one resident scholar has done, though only in search of amusement. Concerning the loftier question—the question of whether you might be a Platonist—well, that largely ridiculous question emerges from a more consequential tale.

Friend, are you a Platonist? Depending on how you want to score it, it's possible that no one is—but some of our leading intellectual lights apparently say they are. We base that claim upon this passage from Jim Holt's well-reviewed new collection of essays, When Einstein Walked With Godel: Excursions to the Edge of Thought:

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 Holt who was willing to define Platonism as "the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind." To us, that formulation seems to exhibit a perfect, timeless lack of clarity. But there it sits, right near the start of Holt's robotically-praised new book!

Friend, you might be a Platonist if you find yourself advancing that fuzzy claim or quite a few others much like it. And if you are, it seems that you'll never walk alone!

According to Holt, this supposed doctrine "has always been popular among mathematicians." This claim also appears in Goldstein's earlier book, the book Holt was reviewing back in 2005, when he penned this version of the title essay from his current book.

Warning! "Philosophy" done by mathematicians may be a bit like shortstop play as done by 350-pound left tackles. That is to say, even the most brilliant mathematician may not be especially skilled at the types of rumination which fall under the traditional rubric of "philosophy"—and judgments about this fuzzy "doctrine" would certainly land on that pile.

Alas! Each Thanksgiving, your Uncle Charlie, who you otherwise love, may prove to you, at the dinner table, that he may not be the most expert political analyst in the land. That doesn't necessarily mean that he isn't a wonderful uncle.

A similar type of situation may imaginably obtain when those perfervid mathematicians succumb to a bout of philosophizin', however brilliant they may be in their principal field.

As we saw last week, Holt is the fellow who offered that first, remarkably fuzzy account of Platonism. The second account we've cited above comes from Goldstein, a highly-regarded philosophy professor who wrote the 2005 general interest book, Incompleteness: The Proof and Paradox of Kurt Godel.

That second, tautological-sounding definiton seems to be lurking here::

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...

Friend, do you believe in "the objective existence of mathematical reality," whatever that perfect, timeless bowl of salad is supposed to mean? You might be a Platonist if you adhere to such murky ideas!

Now that we've had our transient fun, let's get down to brass tacks. Let's go where the rubber meets the road, where the reader will surely think this:

The reader may be inclined to assume that we're being unfair to Goldstein and Holt, each of whom is praised as a highly skilled authority figure by members of our lightly-skilled upper-end mainstream press corps.

The reader may be inclined to assume that Holt and Goldstein went on, in their respective works, to offer fuller, clearer accounts of this "doctrine. A decent, fair-minded reader may be inclined to make that assumption. We'll explore that assumption all week.

Above, we've offered shards of explication by Goldstein and Holt. Sadly, you might be an Aristotelian if—if you assume that these widely praised writers went on to explain "Platonism" in a capable manner.

More precisely, you may be another victim of the misconception we'll describe as Aristotle's error—of the self-admiring supposition that "man [sic] is the rational animal."

Stating the obvious, no rational person would leave those fuzzy accounts of Platonism laying there all by their lonesome, with no further attempt at clarification. You may assume that star writers like Goldstein and Holt, proceeding onward from there, produced explications of this alleged doctrine which are amazingly clear.

It would be natural to assume such a thing. It would also be wrong.

To what extent are we humans really "the rational animal?" To what extent do our rational abilities, such as they are, really define our essence?

In our own studied view, that timeless doctrine from Aristotle "surrounds the actual working of [human nature] with a haze which makes clear vision impossible." It may disperse the fog to spend a few days studying the efforts made by Goldstein and Holt to make the Platonist doctrine more clear, with mathematician G. H. Hardy thrown in.

Aristotle is said to have said that we humans are "the rational animal." At least over here in the western world, this doctrine lies at the heart of the way we've tended to define ourselves, rather plainly seeing ourselves from afar.

By way of contrast, Professor Harari has said that we came to rule the world through our chance acquisition of a less lofty set of skills. Friend, you may be seeing things Harari's way by the end of our current file.

Tomorrow: As cited by Holt, 2 + 2 = 4!

THE GODEL FILE: Digest of reports!

MONDAY, SEPTEMBER 10, 2018

New chapter starts tomorrow: Below, you find links to the four reports from last week which constitute the Godel file.

Tomorrow, we start a new report, the Platonist file. Internationally, experts agree—somebody has to say it!

Tuesday, September 4: Who the heck was Godel? The greatest since Aristotle!

Wednesday, September 5: In what way was Godel "strange?" Let's start with the self-starvation.

Thursday, September 6: A Platonist was hiding among them. But was that a crazy idea?

Friday, September 7: We've been seeing ourselves from afar. Bushmen meet Aristotle!

Starting tomorrow: Holt, Goldstein and G. H. Hardy try to explain Platonism. With side trips to such curious places as 2 + 2 = 4!

(But nobody cares about this, you complain. In a leopard-skin pill-box nutshell, that's precisely our point!)

BREAKING: How drunk was "Mr. Drunky Pants?"

SATURDAY, SEPTEMBER 8, 2018

Rational animal's horseplay: According to the New York Times, "a night of heavy drinking" had been involved.

This claim could be true, of course. That said, it also could be false. At the end of December, this is way the front-page report began:

LAFRANIERE, MAZZETTI AND APUZZO (12/31/17): During a night of heavy drinking at an upscale London bar in May 2016, George Papadopoulos, a young foreign policy adviser to the Trump campaign, made a startling revelation to Australia’s top diplomat in Britain: Russia had political dirt on Hillary Clinton.

About three weeks earlier, Mr. Papadopoulos had been told that Moscow had thousands of emails that would embarrass Mrs. Clinton, apparently stolen in an effort to try to damage her campaign.

Exactly how much Mr. Papadopoulos said that night at the Kensington Wine Rooms with the Australian, Alexander Downer, is unclear. But two months later, when leaked Democratic emails began appearing online, Australian officials passed the information about Mr. Papadopoulos to their American counterparts, according to four current and former American and foreign officials with direct knowledge of the Australians’ role.

You'll note the lack of specific sourcing concerning the amount of drinking. That said, the Times led with the "heavy drinking," which of course made the story more fun.

Especially in the realm of corporate cable news, rational animals often just want to have fun. A few months later, a major star of the clownish medium entertained us humans as shown below.

As this corporate channel explicitly says in its ads, this is why we watch:

ANONYMOUS HUGE CABLE STAR (3/1/18): The New York Times reported late last year that it all started because of a demon drink. The Times reported that the FBI's counterintelligence investigation on the Russia matter began during the campaign because of some drunken talk by Trump adviser George Papadopoulos.

Quote: "During a night of heavy drinking at an upscale London bar in May 2016, George Papadopoulos made a startling revelation to Australia's top diplomat in Britain. He said that Russia had political dirt on Hillary Clinton." That is what triggered the FBI investigation.

When the special counsel charged George Papadopoulos with lying in October, we got more about where that drunken talk had come from. Quote: On or about April 26th, 2016, the professor, a professor from Malta named Joseph Mifsud, mysterious character, told defendant George Papadopoulos that they, the Russians, have dirt on her. The Russians had e-mails of Clinton. They have thousands of e-mails.

George Papadopoulos bragged about that to a guy in a bar drunkenly in May 2016. But he really had been told about it the month before, in April 2016. And it turned out to be true.

I mean, it was drunken smack talk when he bragged about it to this Australian guy, but it was true information. The Russians did have thousands of e-mails they stole from the Democrats and the Clinton campaign.

Thanks to Mr. Drunky Pants and his eventual guilty plea, we know that before those stolen Democratic e-mails began appearing online in July 2016, the Trump campaign, at least one member of the Trump campaign, knew that Russia had them—knew, was aware, that Russia had those e-mails, aware about it enough to brag about it in a bar over a few drinks.

Good lord, that was fun! On this occasion, Ms. Cable News Stupid Ass entertained us rubes with highly entertaining talk about Mr. Drunky Pants!

Just to be clear, Drunky Pants hadn't just been drunk on that drunken evening! Drunky Pants had been bragging too, back when he engaged in all that drunken smack talk.

On Olympus, the gods refer to this type of clowning as "the horseplay of the animals." They laugh, indeed they laugh very hard, as they delightedly say this.

This particular cable clown had similarly entertained us on January 2. On that occasion, she had described the way the "drunk braggart" in question "got drunk in London last May and, on a drinking binge...told a top Australian diplomat that Russia" blah blah blah blah blah.

That's what the drunk braggart had done "on this drunk evening in London."

For all our famous rationality, we famously rational rational animals love this type of amusement. But how odd! After March, this particular clowning cable news clown stopped amusing us in this manner.

Why did the Drunky Pants horseplay stop? It may have stopped because the Australian diplomat said, in an April interview, that there had been no heavy drinking on the drunken night in question. Not even by Drunky Pants!

This may or may not have been true, of course, but that's what the diplomat told a reporter from The Australian in a formal interview. The news report which resulted started off like this (no free link available):

MAGRAY (4/28/18): Alexander Downer is sitting in the comfy Australian high commissioner's office at the century-old Australia House that overlooks the Strand. As the bells of St. Martin-in-the-Fields ring in the background, he leans across the coffee table and insists conspiratorially: it was just the one gin and tonic.

And to stress the point, he says the gin measure was typically mean—just 25ml, which is normal for a London bar, not the sort of lavish pour one might offer friends at Stoke Lodge, the Australian government's residence near Hyde Park.

But it was over that single short drink, accompanied by Erika Thompson, a counsellor at the high commission, late one afternoon in May 2016 that Downer met George Papadopoulos, then an adviser in Donald Trump's campaign team.

Wouldn't you know it? All over the world, we famously rational animals made the amount of drinking the lede!

Downer may have been lying, of course, but that's what Downer said. As far as we know, Erika Thompson hasn't gone on the record about the amount of drinking involved, whether by Mr. Drunky Pants or by her diplomat boss.

Yesterday, Drunky Pants went on the record about the drinking himself. He did so in an interview with the New York Times.

The interview boasts a comical aspect. You can enjoy it here:

MAZZETTI (9/8/18): Going back to the meeting with [Alexander] Downer. What were you guys drinking?

PAPADOPOULOS: I think I had a gin and tonic.

MAZZETTI: Did you drink a number of gin and tonics? How much did you drink?

PAPADOPOULOS: No, I think I had one or two drinks. I think Downer himself, in numerous interviews, kept explaining that we had one drink or two drinks. No one was drunk, as some articles stated that we might have been. At least I don’t remember being drunk. I don’t think he was drunk. I don’t think his assistant was drunk. I think we had a couple drinks, that we were just talking.

This account may be untrue, of course. That said, Drunky Pants made a point of citing Downer's past statements on this topic—statements we'd never heard anyone mention, perhaps because they would have introduced uncertainty into this part of the tale, undermining our fun.

The humor occurs when Papadopoulos refers to "some articles [which] stated that we might have been [drunk]." He was speaking to one of the authors of the original news report which put that notion into play, but neither Mazzetti nor Drunky Pants chose to address this point.

Last night, the cable clown who likes to cavort quoted this part of the interview. She quoted Drunky's denial of drunkenness, and his reference to Downer's earlier denial of drunkenness.

Being a thoroughly rational animal, she didn't address the way these statements contradict the earlier fun she'd provided. Instead, she found a new way to destroy her viewers' brain cells, along with their sense of what actually matters:

ANONYMOUS CORPORATE-OWNED CLOWN (9/7/18): The judge ordered Papadopoulos to serve two weeks in jail. The judge said he had been inclined to give Mr. Papadopoulos a month in jail but he cut in it half because he was impressed by Papadopoulos` remorse over what he had done.

So, George Papadopoulos impressed the judge with his remorse. He will do 14 days in jail, plus a year of supervised release, plus 200 hours of community service, plus a fine of $9,500.

That sentence led to these ecstatic tweets from George's mom, Kiki Papadopoulos.

It is amazing the number of people I've learned about who I would never otherwise encounter because of the criminal cases involving people close to the president and his campaign.

Mrs. Papadopoulos said this today, after the sentence. Quote: "Just left courthouse. Amazing judge. Judge gave George two weeks of jail time! Amazing! God bless America."

And then, a few minutes later, "Judge Moss is a fair and good man! Everything went amazing! He is grrrrrreat."

Like Tony the Tiger. Grrrrreat!

So entertaining, and so insulting to the viewer's intelligence! To watch this idiot please you this way, you can just click here.

As usual, this cable star had found a way to talk about herself. It's amazing, the number of people she's learned about!

As usual, she was entertaining us the rubes with brainless, mindless drivel.

As we've told you in the past, she loves to drag in, and attempt to embarrass, any uninvolved parents and kids. In this case, she decided to go all Tony the Tiger on the mother of Drunky Pants.

This cable star clowned on other points, including a few basic matters. Back in April, for whatever it's worth, Downer had also undercut a few pleasing parts of the standard tale.

That said, we're going to let you search these points out. Ain't rational conduct grand?

THE GODEL FILE: We've been seeing ourselves from afar!

FRIDAY, SEPTEMBER 7, 2018

Bushmen meet Aristotle: Long ago and far away, even before Kurt Godel died, we were teaching delightful fifth graders in the Baltimore City Schools.

(For our previous report, click here.)

At one point, we came upon a rather dull textbook which included a memorable anecdote. If memory serves, it was the third or fourth grade social studies text of one of the major publishers.

The textbook focused on the planet's many peoples and cultures. The memorable anecdote concerned the Bushmen of the Kalahari. They were among the shortest people in the world, the lower grade textbook said. In part for that reason, they have a standard traditional greeting:

"I saw you from afar."

Theoretically, this greeting was this group's attempt to compensate for smallness of stature. The greeted party was so large and imposing that he'd been seen from afar!

We never used that textbook, but the anecdote stuck in our heads. As it turns out, it had probably surfaced through the work of Marlin Perkins, later of TV's Wild Kingdom fame.

In 1965, Perkins and his wife, Carole Morse Perkins, had written a children's book about the Kalahari. The book was well received in a short review in the New York Times:

BERKVIST (5/9/65): This delightful little book, 50-odd pages of well-chosen words and photographs, opens a window on the world of the Bushmen, a remarkable handful of people who have somehow managed to strike a precarious bargain with nature in South Africa's vast, inhospitable Kalahari Desert.

"During their stay in the Kalahari," the reviewer went on to say, "the authors watched [the Bushmen] meet the harsh demands of desert life with good-humored dignity."

You can possibly guess the name of the Perkins book: "I Saw You From Afar!" The Times explained this traditional greeting as that textbook later did:

"The title echoes a highly complimentary form of greeting among Bushmen, most of whom are less than five feet tall and like to be told otherwise."

So the Times reviewer said—and so it went as the western world reached out to indigenous peoples. But the anecdote always stuck in our heads. At long last, we may know why:

Long ago and far away, Aristotle, the father of logic, is said to have made an basic assessment, one which became iconic:

"Man [sic] is the rational animal."

So Aristotle is commonly said to have said. Or words to that effect!

As the centuries passed, this flattering assessment came to define our species' basic self-assessment, at least within western culture. It strikes us as a near relation to "I saw you from afar."

Are we humans really "rational" in some definitive way? Aristotle, the father of logic, may have been wrong about that. And what better illustration, one might imagine, than the tragedy of Kurt Godel, who is commonly said to be our second greatest logician?

How odd! The western world's second greatest logician was severely mentally ill and given to crazy ideas! In the face of this slightly odd state of affairs, along come the popularizers, with their slavish, scripted enablers within the mainstream press corps.

The popularizers pretend to explain Godel's "incompleteness theorems." In turn, the journalists say that they've managed to make Godel's ideas and achievement just amazingly clear.

Along the way, we gullibles are told that the second greatest logician in history devoted his life to a key idea. The great logician was a Platonist, we're told. This "doctrine" is then made stunningly clear:

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...

Godel believed that numbers and circles have a perfect, timeless existence! At least, that's the way Jim Holt explains the "doctrine" in the title essay of his new, perhaps slightly plagiarized book, When Einstein Walked With Godel: Excursions to the Edges of Thought.

Holt derives his material, with lax attribution, from Rebecca Goldstein's 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel. In that general interest book, Goldstein describes the great logician''s key belief like this:

GOLDSTEIN (page 44): Platonism is the view that the truths of mathematics are independent of any human activities...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.

See there? According to Goldstein, our second greatest logician believed that 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 that make up that reality!

We humans! We've been seeing ourselves from afar for an extremely long time! At long last, along came the later Wittgenstein, making something resembling that claim about the various proclamations and assertions which constitute historical philosophy.

Wittgenstein said our allegedly greatest minds—not excluding himself, in his earlier efforts—had been seduced and led to ruin by conceptual confusion.

Back in 1969, we told Professor Cavell that we'd give him and Albritton fifty year to get this whole thing straightened out.
After that, we'd have to speak up, we said. Or words to that effect.

Stanley Cavell died in June, at age 91. Even as this loss occurs, the inanity of modern American journalism has finally overwhelmed even us.

This morning, we watched the clowning on Morning Joe as the children burned oodles of time trying to guess who wrote the op-ed column. We just can't go there any more. We're finally saying goodbye to all that, at least for the most part.

We've decided to move on to the larger story which has been lurking here. We've started with Godel this week because Holt's favorably-reviewed new book has brought the topic to mind.

"I saw you from afar," the Bushmen are said to say. We're forced to say that we hear an echo in Aristotle's famous remark.

Over here in the western world, we self-flattering human beings have hidden behind that remark. Now, we're going to bring you the never-before-told, dramatic true stories of human intellectual failure at its highest levels.

In truth, the extended tale we're going to tell is largely a comical tale. If you could ignore the endless wars and the endless killing, it would be humorous all the way down.

Along the way, we'll offer two paradigms—a pair of competing capsule accounts of our ballyhooed human race. One will come from Aristotle, one from Professor Harari:

The Aristotelian account: We humans are the rational animal.

The Harari heuristic: Our human species, Homo sapiens, is a slightly improved great ape. We drove all other human species into extinction because, through chance mutations, we developed 1) the ability to "gossip," and 2) the ability to promulgate, and believe and affirm, irrational, sweeping group "fictions."

So says Professor Harari in his widely-praised best-seller, Sapiens: A Brief History of Humankind, or words to that effect.

Along the way, we'll describe the building blocks of the Harari heuristic in more detail. For today, we'll only say this:

The idea that numbers have a perfect timeless existence is a remarkably good example of what Harari means when he describes the group "fictions" which gave our particular human species a competitive advantage. In Harari's account, this advantage came from the new ability to believe and affirm sweeping claims which don't make observable sense.

So how about it? Does the number 3 have a perfect timeless existence independent of the human mind? Do circles share that perfect existence with their good friends, the numbers?

According to Holt, that's what the second greatest logician in history believed from the time of his youth. According to Goldstein, he didn't believe in evolution, in part because Stalin said.

Numbers and circles have a perfect existence! Holt makes little attempt to explain what such a "belief" could possibly consist in or mean. But then again, so what?

At the New York Times, at the Wall Street Journal, rational animals stood in line to say how clear it all is. This has been going on forever, and it's a comical tale.

Next week: A Platonist among them!

THE GODEL FILE: A Platonist was hiding among them!

THURSDAY, SEPTEMBER 6, 2018

But was this a crazy idea? Many people have heard of Einstein, who enjoyed his walks with Godel.

Aristotle's name is also well known. That said, we'll guess most people wouldn't know that Aristotle is said to have been our greatest logician, at least until Godel came along.

(For our previous Godel File report, click here.)

Godel is hailed as our second greatest logician because of his incompleteness theorems. We'll take a first peek at those theorems tomorrow—but for today, we tip our hat to Rebecca Goldstein.

Goldstein wrote the book on Godel; she did so in 2005. Along the way, she outlined Aristotle's work as a logician, in a footnote on page 55:

GOLDSTEIN (page 55): Aristotle is commonly acknowledged as the father of logic. His work in logic is laid out in the Prior Analytics, which is part of the posthumous consortium known as the Organon. The philosopher had the seminal insight that in a deductive logical argument, some words are logically relevant while others are not. The irrelevant words can be dispensed with by making them variables...

Goldstein goes on to offer a "stock syllogism" as an example of what she's talking about. She then says, "The move toward denoting logically irrelevant words with variables was a move toward generality and thus toward the science of logic."

At this point, we start in surprise. Has someone actually made a science of logic? If you read American newspapers, or if your watch our "cable news" programs, you might be greatly surprised by any such allegation.

As everyone must understand by now, "the science of logic" plays almost no role in our modern American discourse. Within our failing culture's pseudo-discussions, basic information and basic facts are almost wholly verboten. But anything resembling logic is typically frowned upon too.

This destructive state of affairs can't be blamed on Aristotle. Because Godel lived in modern times, we'll be a bit less forgiving when it comes to him.

Sad! Modern logicians have created a system in which their highly abstruse ruminations have nothing to do with the bungled logic of our public discussions. Godel lived until 1978. At some point, he probably should have stepped in.

At any rate, our modern logicians walk to work in the clouds, where they engage in high-brow endeavors which even they may not be equipped to explain. This brings us to a fascinating part of Godel's horrifically tragic story, as told by Goldstein in 2005 and then by Jim Holt in the derivative title essay of his well-received new book.

We refer to a possibly crazy idea to which Godel was apparently devoted throughout the course of his life. We refer to Godel's alleged belief in "Platonism," an alleged doctrine with which Goldstein says Godel "fell in love" when he was in his late teens.

In our previous report, we learned an unfortunate fact. Godel, who died of self-starvation, exhibited signs of severe mental illness throughout the course of his highly unusual life.

"At the age of five, he seems to have suffered a mild anxiety neurosis," Holt writes in his new book, drawing on Goldsetin's work. "At eight, he had a terrifying bout of rheumatic fever, which left him with the lifelong conviction that his heart had been fatally damaged."

By normal reckoning, Godel's apparent paranoia eventually led to his death. But then, as noted yesterday, he had always struck the Princeton community as being extremely odd.

In her book, Goldstein cites an array of peculiar ideas Godel is said to have expressed. Our continuing question is this:

Should it seem strange to think that the western world's second greatest logician was apparently in the grip of serious mental illness? Asking a slightly different question, should it seem strange that "Aristotle's successor" was in the grip, throughout his life, of some extremely peculiar ideas?

In theory, the mental illness, however tragic, shouldn't matter at all. If a person achieves some great intellectual breakthrough—in the physical sciences, let's say—that achievement isn't negated by the tragedy of a severe mental illness.

That said, Godel's achievements didn't come in the realm of physical science. Indeed, Goldstein quotes him telling one Princeton-based scholar, "I don't believe in natural science." She quotes him telling Thomas Nagel that he doesn't believe in evolution, offering Stalin's similar disbelief as supporting evidence.

She quotes him saying this to Noam Chomsky: "I am trying to prove that the laws of nature are a priori." Should it seem strange to hear that history's second greatest logician was bruiting such notions around?

In fairness, these peculiar-sounding statements were made during Godel's adult life. He formulated his "incompleteness theorems" when he was just 23.

If those theorems make sense and are actually important in some way, later delusions and mental illness can't vitiate the achievement. That said, we might want to consider the lifelong belief in "Platonism" which, according to Goldstein's book, played the key role in Godel's thinking from his teenage years on.

What the heck is Platonism? According to Goldstein, Godel "fell in love" with the doctrine while he was still a teen.

"First exposure to Plato can be an extremely heady experience for those with a passion for abstraction," Goldstein writes. "It can amount to a sort of ecstasy."

Parenthetically, Goldstein says that she can remember her own first exposure to Plato. But what is the doctrine which may result? What is this "Platonism?"

In the opening essay of Holt's new book, he defines this doctrine, or he at least pretends or attempts to do so. As critics robotically praise the clarity of Holt's writing, he sketches the doctrine like this:

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...

Say what? According to Holt, Godel, who was still in his teens, was "seduced by" a certain notion. More specifically (or possibly less so), he was seduced by "the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind."

According to Holt, this "doctrine," which is called Platonism, "has always been popular among mathematicians." In fairness, this suggests that it isn't something the teen-aged Godel somehow dreamed up by himself.

At this point, our review of the Godel file reached a critical turn. We're told that the western world's second greatest logician was swept away by a seductive idea, an idea which lay at the heart of his subsequent work.

What did this greatest logician believe? According to Holt, he believed that numbers have a perfect, timeless existence independent of the human mind. So Kurt Godel believed!

Readers, how about it? Do you believe that the number 3 "has a perfect existence independent of the human mind?" Do you believe that the number 3 has a perfect existence at all?

Indeed, let's ask the more relevant question: Do you have even the slightest idea what Holt is talking about? Granting the fact that you can memorize Teacher's words and reproduce those words on the test, would you have even the slightest idea what you were talking about?

Readers may be inclined to suppose that Holt proceeds from there to explain this murky bit of word salad. In our view, such a supposition would be badly mistaken.

(To peruse his original New Yorker essay, you can just click here.)

Does Holt clarify this description? We'll set that question aside for another day. Instead, we'll turn to Goldstein, to the start of her attempt to explicate "Platonism."

Godel fell in love with the doctrine, she says. She lays out the doctrine as follows:

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.

Goldstein muddies her account with a fair amount of technical language. (How many general interest readers are likely to know what a "formal system" is?)

Beyond that, she seems to think, all through her book, that the terms "objective" and "subjective" explain themselves in contexts like this. Sadly, they very much don't.

Whatever! In this passage, Goldstein starts describing the doctrine with which Godel "fell in love" as a teen. According to Goldstein, the ardor Godel felt for this doctrine impelled him to devise the "incompleteness theorems" which made him famous among certain academics, while leaving him completely unknown to everyone else on the face of the earth.

Godel's ardor for this alleged doctrine forms the core of Goldstein's work. And how does she describe the doctrine? She describes it like this:

According to Goldstein, Godel fell in love with the view that "the truths of mathematics are determined...by the reality of mathematics."

So far, so completely no good! Explicating further, she says Godel adopted the view that "the truths of mathematics are determined by the nature of the entities that make up" the reality of mathematics.

As an overview, she starts by saying that Godel developed a "commitment to the objective existence of mathematical reality!"

Readers, tell the truth. Would you say that the essence of this key doctrine has begun to come clear? We're going to say that we're munching on salad—that it's word salad all the way down!

At this point, we'll raise two questions, after which we'll adjourn for the day. One of our questions will concerns Kurt Godel. The other question will concern writers like Goldstein and Holt, who are robotically praised, by scripted journalists, for the clarity of their work.

Regarding Godel, we'll ask this question: Is it possible that the western world's second greatest logician fell in love at an early age with a crackpot idea?

He believed many other crazy things in the course of his tragic life. Could this idea, which drove the work for which he's lauded, possibly have been the first of his crackpot ideas?

That's a question about our second greatest logician. Our second question concerns Holt and Goldstein—and it's built on a basic concept:

Here's the basic concept. Some claims or ideas are right or wrong. But some claims, and some ideas, don't rise to that level.

Such claims and ideas aren't right or wrong—they're simply incoherent!

This idea is stunningly basic. To what extent are general interest writers like Goldstein and Holt conversant with this idea?

Tomorrow: A Platonist in their midst (we didn't get there today)