TUESDAY, SEPTEMBER 25, 2018
Semantics and paraphrase: Once upon a time, more than twenty years ago, it began to occur to us that our society's guardians had perhaps abandoned their posts.
The great, dark trees of incoherence cast a deep shade all around. As far as a person could see in the course of a day, or even in the course of a week, there was nothing but incomprehension. It was incompetence and inanity pretty much all the way down.
There was no Fox News at this time. There was no MSNBC. Within the big woods of "cable news," there was only CNN—CNN, and the rational animals who made their livings reciting script upon its various programs.
What made us start to think that the guardians had abandoned their posts? We think first of the great Medicare non-discussion of 1995 and 1996.
Every night, pundits would gather on CNN to pretend to debate the Republican Party's Medicare proposal. This proposal was being advanced by the new House speaker, Newt Gingrich.
Night after night, month after month, the discussion which wasn't a discussion would break down along mandated line—on Crossfire, let's say:
On Crossfire, the two disputants "from the left" would say that Gingrich had proposed cutting $270 billion from the Medicare program.
The two disputants "from the right" would say that no one was cutting the Medicare program at all. According to these disputants, the GOP plan had merely proposed "slowing the rate at which the Medicare program would grow."
Eventually, Republicans began to claim it was demagogic to use the term "Medicare cuts" at all. They began insisting that journalistic use of this term was the latest example of "liberal bias."
In response to this extended attack, journalists began employing a string of euphemisms. These alternative terms were used instead of the allegedly demagogic term, "Medicare cuts."
No one ever quite explained what was wrong with this traditional term. Medicare "cuts?" This term had always been used, within both parties, to describe budget proposals of the type the GOP had made.
But so what? This non-discussion pseudo-discussion went on, night after night, for well over a year. As journalists scrambled to find softer terms, no one untangled the conceptual mess at the heart of this pseudo-discussion.
In policy terms, this badly bungled pseudo-discussion formed the heart and soul of the non-debate pseudo-debate which led up to November 1996 election, in which President Clinton won re-election over Candidate Dole.
Clinton won, Dole lost. But this non-discussion pseudo-discussion lay at the heart of that campaign. Night after night, for month after month, our journalists and cable news pundits performed a scripted non-conversation which spread confusion all over the land.
This non-conversation was, pure and simple, a semantic dispute. The two parties to this discussion agreed on all relevant facts.
That said, no one came forward to clarify this stultifying conceptual mess. Later, it occurred to us that this had been a good example of the guardians leaving their posts.
No logician ever stepped forward to straighten out that semantic conceptual mess. It fell to us to unpack this nonsense in a Baltimore Sun op-ed.
It fell to us, and to Al Franken, who was then still a comedian. Franken clarified this pitiful mess in a comical but instructive part of his 1996 best-seller, Rush Limbaugh is a Big Fat Idiot and Other Observations.
Two comedians had been able to clarify this braindead semantic dispute! As far as we know, no journalists ever did, except perhaps for Maraniss and Weisskopf—and no professor ever stepped forward to serve in a guardian role.
None of our nation's brilliant logicians stepped in to untangle this mess. No other professor came forward to serve in the guardian role.
A few years later, a similar situation obtained when a twenty-month presidential campaign foundered on the basic logic of paraphrase and quotation.
We refer to Campaign 2000, coverage of which which began in earnest in March 1999. As a few graybeards may still recall, that entire campaign turned on the claim that one of the candidates, Candidate Gore, "had a problem with the truth."
Allegedly, this meant that Candidate Gore was like his boss, President Clinton. Clinton had only recently escaped removal from office in his Senate impeachment trial.
For twenty months, that whole campaign turned on the basic logic of paraphrase and quotation. Starting in March 1999, mainstream journalists, again and again, paraphrased and "quoted" statements by Candidate Gore in ways which were designed to show his "problem with the truth."
Had Al Gore said he invented the Internet? Again and again, for twenty straight months, that's what our journalists said.
Had he said he inspired Love Story? Had he said he discovered Love Canal? Had he said he grew up on a farm, when he really grew up in a fancy hotel—even in the Ritz Carlton?
In September 2000, did the candidate lie when he told a joke about a union lullaby? Did he lie about the cost of his pet dog's arthritis pills?
These claims were widely bruited in September 2000, when new polling had made it seem that Candidate Gore was pulling away from Candidate Bush. Early in October, new claims of new lies appeared in the wake of the first Bush-Gore debate, undermining initial impressions that Gore had outperformed Bush.
Simply put, the acts of bogus paraphrase never stopped. Had the candidate "told Time magazine last year that he enacted the Earned Income Tax Credit, which of course went into law before he was ever in Congress?" Lawrence O'Donnell revived that groaner very late in the campaign, appearing on the high-profile syndicated program, The McLaughlin Group.
AL GORE, LIAR! From March 1999 through November 2000, it was the central "journalistic" narrative of Campaign 2000.
The press corps' crescendo of claims turned on highly tendentious acts of paraphrase and quotation. These presentations raised the most basic questions about the logic of these practices, but no logician rose to serve in the time-honored guardian role.
How silent were the professorial lambs? Back in 1978, Professor Bok had published a widely-praised book, Lying: Moral Choice in Public and Private Life.
As of Campaign 2000, this highly-regarded book was still in print. Indeed, a new paperback edition had appeared in 1989—and another new edition appeared in 1999!
That said, Professor Bok had nothing to say about the claims against Candidate Gore. Needless to say, no other logician or ethicist stepped forward to discuss the endless claims being lodged by our mainstream "press corps." Darlings, it isn't done!
These are just two examples—examples from long ago. That said, people are dead all over the world because these guardians walked off their posts during Campaign 2000. There's no limit on the disgust we should feel for these cosseted, useless figures.
Our logicians and ethicists had nothing to say about these long-running episodes. That said, these professors had long since been reassigned to posts outside the public square—to posts in mahoganied academic lounges, or perhaps to posts in the south of France.
The public was badly in need of their help, but the public's need wouldn't be served.
These guardians' refusal to serve continues to this day. The perpetual silence of these upper-class lambs helped give us our President Trump.
The polis will always need guardians! So Plato declared long ago in his famous though tedious book, The Republic. It represents one of the very few things Plato clearly got right.
The polis needs guardians, Plato declared. Tomorrow, we'll review his statements on this obvious point.
Our logicians and ethicists ought to be serving in a guardian role! But these people walked off their posts long ago. The silence of these useless people will be explored in our posts all week.
Tomorrow: Plato gets it right!
Short, medium and long: Long ago, when this site was still young, we posted three reports, of varying length, concerning the Medicare non-discussion discussion.
"The Speaker's new language" was our Medicare magnum opus. In it, we quoted Franken's book at some length.
Our shortest treatment of the matter bore the attractive title, "A tale of three numbers." For links to our three reports, you can just click here.
In these reports, we unpacked the basics of this semantic gong-show. That said, our nation's famous logicians offered no help at any point in this process.
Our logicians were locked in their aeries, as they have been for some time. Clownishly, they were discussing the set of all sets not members of themselves. Their refusal to serve helps explain how Donald Trump got where he is.
Years later, Paul Krugman linked to one of our Medicare reports to help clarify this matter. We can't remember when he did it, although you could find it on line.
THE INCOMPLETENESS FILE: Digest of reports!
MONDAY, SEPTEMBER 24, 2018
New chapter starts tomorrow: What did Godel demonstrate in his "incompleteness theorems?" Are these theorems useful, important, insightful, even coherent?
Can general readers hope to know the answers to such questions? Below, you see links to last week's reports from the incompleteness file:
New chapter starts tomorrow: What did Godel demonstrate in his "incompleteness theorems?" Are these theorems useful, important, insightful, even coherent?
Can general readers hope to know the answers to such questions? Below, you see links to last week's reports from the incompleteness file:
Tuesday, September 18: Incompleteness meets incoherence! A lucid writer intones.Tomorrow, we start our award-winning "guardians file." To review reports from all previous files, use links provided below:
Wednesday, September 19: What the Sam Hill is a "logical system?" No general reader will know!
Thursday, September 20: What the heck is a "formal system?" Once again, Joe Average won't know!
Friday, September 21: Do Godel's theorems even make snese? Flying spaghetti monsters!
Monday, September 10: Digest of reports: The Godel file.
Monday, September 17: Digest of reports: The Platonist file.
Monday, September 24: Digest of reports: The incompleteness file.
BREAKING: The Washington Post gets it right!
SATURDAY, SEPTEMBER 22, 2018
Plus, the Post and the Times get it wrong: All praise to the Washington Post's Joe Heim, who authored an outstanding report in Thursday's print editions.
In our view, it should have been an outstanding front-page report. Instead, it got pushed inside, to page A10, where it was an outstanding report about an appalling state of affairs.
Heim reported on the squalid culture which obtained at DC-area private schools in the 1980s, when Brett M. Kavanaugh was a student at Georgetown Prep. We think you should read every word.
Heim produced an outstanding report about an appalling state of affairs. This morning, the Post offers an outstanding front-page report about the drinking and debauchery, during that era, engaged in by Mark Judge, another victim of Georgetown Prep's appalling culture, and a likely victimizer to boot.
Parts of today's report are simply astounding. The report was written by Fisher and Stein. We think you should read every word, especially the part about former "Marriage and Sex teacher" Bernie Ward.
These were, and are, outstanding reports about a disgraceful state of affairs. That said, one day before Heim's report appeared, the Post offered a peculiar front-page report about Diane Feinstein—a puzzling report which the New York Times has finally matched, and has managed to top, on today's front page.
Let's focus on today's report in the Times. It was written by Nicholas Fandos, who's more than three years out of college. (Harvard, class of 2015.)
These high-flying kids today! Fandos seems troubled by the thought that Feinstein honored a pledge of confidentiality to Christine Blasey Ford, a constituent who says she was the victim of a sexual assault when she was just 15. On this basis, Fandos suggests, at several points, that Feinstein, who is 85, may have been slipping a bit in the noggin when she behaved in this manner—when she honored her pledge to Blasey Ford.
Absent the suggestions of senility, the Washington Post's Sean Sullivan authored an equally puzzling, front-page report on Wednesday. Which part of "I made a pledge of confidentiality" don't these scribes understand?
Reading these scribes' reports, we'll admit that we have no idea. But all week long, we've seen cable news hosts who seemed to be similarly challenged concerning this bone-simple ethical point.
Let's be fair! The idea that Feinstein should have broken her pledge seemed to gain wide purchase in the past week or so—and this wasn't simply a cable news talking-point of the right.
Some Democrats seemed to suggest that Feinstein should have outed Blasey Ford too. In this part of Sullivan's report, he blew right past an obvious irony as he described this state of affairs:
Such highly principled people! Later in his report, Sullivan quoted a few of the three million Republicans who have pretended that they never heard about Feinstein's pledge to Blasey Ford. That said, a range of highly principled Democrats also seem to have wandered onto the list of those who favor breaking such a pledge concerning a report of a sexual assault:
On the brighter side, Sullivan (like his editor) seems to be ready for a cable news hosting spot. As he records all these criticisms, he challenges no one to reconcile their criticism with the ethics of Feinstein's pledge. But so it went, again and again, on cable news this week.
On cable news this week, somnolent hosts like Anderson Cooper repeatedly let hard-bitten pseudo-conservatives play dumb about the circumstances surrounding Feinstein's behavior. Why didn't she bring it up earlier, these con men kept asking, skipping right past the fact of her pledge with their host's acquiescence.
On Wednesday morning, Sullivan adopted this stance. Today, Fandos takes us one step farther, suggesting that Feinstein honored her pledge because she's getting soft in the head—because she seems to be senile.
This is how Aristotle's "rational animals" will behave when Plato's "guardians" abandon their posts. When our logicians devote their lives to pondering 2 + 2 = 4, children all over the newsosphere are going to reason this way.
Next week: The guardians file
One more example: Last weekend, on AM Joy, Joy Reid and Zerlina Maxwell criticized Feinstein's behavior. They failed to discuss the ethics of breaking a pledge of confidentiality concerning a sexual assault.
Can they explain why Feinstein should have broken her pledge? If so, they didn't bother.
You can see their exchange in the first two minutes of this tape. We no longer expect much better from Reid. We do expect better from Maxwell.
Plus, the Post and the Times get it wrong: All praise to the Washington Post's Joe Heim, who authored an outstanding report in Thursday's print editions.
In our view, it should have been an outstanding front-page report. Instead, it got pushed inside, to page A10, where it was an outstanding report about an appalling state of affairs.
Heim reported on the squalid culture which obtained at DC-area private schools in the 1980s, when Brett M. Kavanaugh was a student at Georgetown Prep. We think you should read every word.
Heim produced an outstanding report about an appalling state of affairs. This morning, the Post offers an outstanding front-page report about the drinking and debauchery, during that era, engaged in by Mark Judge, another victim of Georgetown Prep's appalling culture, and a likely victimizer to boot.
Parts of today's report are simply astounding. The report was written by Fisher and Stein. We think you should read every word, especially the part about former "Marriage and Sex teacher" Bernie Ward.
These were, and are, outstanding reports about a disgraceful state of affairs. That said, one day before Heim's report appeared, the Post offered a peculiar front-page report about Diane Feinstein—a puzzling report which the New York Times has finally matched, and has managed to top, on today's front page.
Let's focus on today's report in the Times. It was written by Nicholas Fandos, who's more than three years out of college. (Harvard, class of 2015.)
These high-flying kids today! Fandos seems troubled by the thought that Feinstein honored a pledge of confidentiality to Christine Blasey Ford, a constituent who says she was the victim of a sexual assault when she was just 15. On this basis, Fandos suggests, at several points, that Feinstein, who is 85, may have been slipping a bit in the noggin when she behaved in this manner—when she honored her pledge to Blasey Ford.
Absent the suggestions of senility, the Washington Post's Sean Sullivan authored an equally puzzling, front-page report on Wednesday. Which part of "I made a pledge of confidentiality" don't these scribes understand?
Reading these scribes' reports, we'll admit that we have no idea. But all week long, we've seen cable news hosts who seemed to be similarly challenged concerning this bone-simple ethical point.
Let's be fair! The idea that Feinstein should have broken her pledge seemed to gain wide purchase in the past week or so—and this wasn't simply a cable news talking-point of the right.
Some Democrats seemed to suggest that Feinstein should have outed Blasey Ford too. In this part of Sullivan's report, he blew right past an obvious irony as he described this state of affairs:
SULLIVAN (9/19/18): Democratic senators on the Judiciary Committee were tepid about Feinstein. "She did her best," said Sen. Mazie Hirono (Hawaii). "I can't fault her," said Sen. Kamala D. Harris (Calif.). "Extremely difficult circumstances," noted Sen. Richard J. Durbin (Ill.)Based on that passage, some Democrats apparently felt that Feinstein should have broken her pledge to Blasey Ford. In a wonderful bit of unintentional irony, Sullivan attributes that report to "a Democrat with direct knowledge of internal Senate dynamics"—a Democrat who spoke to Sullivan on the basis of a pledge of confidentiality!
Privately, some Democratic senators wished that Feinstein had come to them sooner with the allegation, according to a Democrat with direct knowledge of internal Senate dynamics. The Democrat spoke on the condition of anonymity to be candid.
It was late July when Feinstein received the letter from Ford detailing the allegations from decades ago against Kavanaugh. Ford is a constituent of Rep. Anna G. Eshoo (D-Calif.), who relayed the letter to Feinstein.
Ford was insistent on confidentiality. It was not until a private meeting last Wednesday, after a report by the Intercept, that Feinstein revealed the letter to her Democratic colleagues on the Judiciary Committee. In a Washington Post article published Sunday, Ford told her story publicly for the first time.
Such highly principled people! Later in his report, Sullivan quoted a few of the three million Republicans who have pretended that they never heard about Feinstein's pledge to Blasey Ford. That said, a range of highly principled Democrats also seem to have wandered onto the list of those who favor breaking such a pledge concerning a report of a sexual assault:
SULLIVAN: Feinstein's decision to keep the accusation away from her own party, at a moment when liberals were applying immense pressure to defeat Kavanaugh and moderate Democratic senators were debating whether to support him, has triggered second-guessing among Democrats. Now, some lawmakers simply want to turn the page.Running to Feinstein's left, de Leon thinks a senator should break a pledge of confidentiality to a person like Blasey Ford. In other words, Feinstein should have "outed" Blasey Ford. Just as "Father" once famously did, we lefties now know best!
"I'm not going to go back and revisit that," said Sen. Doug Jones (D-Ala.), who represents a Republican state and is up for reelection in 2020. "I just think we need to deal with where we are now, not where we might have been."
[...]
Feinstein's challenger this November is Kevin de León, a state lawmaker running to her left who finished a distant second in California's all-party primary. He slammed Feinstein last week for "failure of leadership" and questioned why she waited to give information about the accusation to the FBI.
On the brighter side, Sullivan (like his editor) seems to be ready for a cable news hosting spot. As he records all these criticisms, he challenges no one to reconcile their criticism with the ethics of Feinstein's pledge. But so it went, again and again, on cable news this week.
On cable news this week, somnolent hosts like Anderson Cooper repeatedly let hard-bitten pseudo-conservatives play dumb about the circumstances surrounding Feinstein's behavior. Why didn't she bring it up earlier, these con men kept asking, skipping right past the fact of her pledge with their host's acquiescence.
On Wednesday morning, Sullivan adopted this stance. Today, Fandos takes us one step farther, suggesting that Feinstein honored her pledge because she's getting soft in the head—because she seems to be senile.
This is how Aristotle's "rational animals" will behave when Plato's "guardians" abandon their posts. When our logicians devote their lives to pondering 2 + 2 = 4, children all over the newsosphere are going to reason this way.
Next week: The guardians file
One more example: Last weekend, on AM Joy, Joy Reid and Zerlina Maxwell criticized Feinstein's behavior. They failed to discuss the ethics of breaking a pledge of confidentiality concerning a sexual assault.
Can they explain why Feinstein should have broken her pledge? If so, they didn't bother.
You can see their exchange in the first two minutes of this tape. We no longer expect much better from Reid. We do expect better from Maxwell.
THE INCOMPLETENESS FILE: Do Godel's theorems even make sense?
FRIDAY, SEPTEMBER 21, 2018
Flying spaghetti monsters: Are Godel's "incompleteness theorems" actually "important?"
Do they carry any social significance? In the end, do they even make sense?
We'll admit to being doubters on the last of those points. Consider a part of Rebecca Goldstein's book which we'll explore in more detail at some later point.
Goldstein's book, designed for general readers, appeared in 2005. It bore this title: Incompleteness: The Proof and Paradox of Kurt Godel.
In our view, the general reader won't likely emerge from this book with the ability to discuss these supposedly transplendent theorems. For ourselves, we were surprised by the way Goldstein, a philosophy professor, leaned on the concept of "paradox" in her discussions, not excluding this rumintaion on a famous "abstract object:"
The paragraph continues from there. The reference to "Russell" is a reference to Lord Russell, eventual husband of Lady Ottoline—that is to say, to Bertrand Russell—who came up with this world-class groaner back in 1901.
When we first encountered Goldstein's book, it surprised us to think that a capable philosophy professor would still be trafficking in this antique hocus-pocus about these "abstract objects"—about "abstract objects" which may or may not be "members of themselves."
We were even more surprised to see her marveling about this pseudo-paradox, which is even more simple-minded:
We'll discuss these matters in the weeks ahead, possibly next week. For ourselves, if Godel's theorems turn on piddle like this, we'll float the shocking possibility that they may not make any real sense.
We know it's shocking to hear such claims about the genius theorems Goldstein gushes about. Then again, this "greatest logicians since Aristotle" seems to have been mentally ill his entire life; eventually died of self-starvation; and believed all sorts of crazy idea, perhaps including the crazy idea that numbers and circles live "a perfect, timeless existence" somewhere, apparently in an "abstract" realm we can access through something resembling ESP.
Do the theorems of this unfortunate man actually make any sense? For now, we'll vote with the doubters. Meanwhile, when Jordan Ellenberg discussed Goldstein's book for Slate, he offered these remarks, among others:
When our greatest logicians devote their lives to the antics of spaghetti monsters, should we be surprised by the sheer stupidity which obtains all over the national discourse engineered by corporate journalists? We'll be focusing on that question next week. For today, let's visit an early part of Goldstein's book, where she starts to get something right.
When Holt summarized Goldstein's book, he profiled the strangeness of Godel. Again, we ask you to marvel at the highlighted part of this pile:
We mention this for a reason. Near the start of her book, Goldstein gives a weirdly decent explanation of this potent conundrum. She speaks about a different fact—the fact that 5 + 7 = 12—but, as you can probably see, the basic logic of all such statements is pretty much the same.
Citizens, we encountered this same traditional groaner as college freshmen ourselves! How can we know that 7 + 5 = 12? Professor Nozick raised this "problem" in the introductory course, Phil 3: Problems in philosophy.
How do we know that 7 + 5 = 12? One wag in the back of the class dared to ask himself this:
What makes a proposition like “2 + 2 = 4” true? Using a slightly tougher example, Goldstein offers this:
How do we know that 1 + 1 = 2? Simple! Among other factors, we would be strongly disinclined to accept alleged counterexamples! Let's think in terms of marbles.
"Two" is simply the name we give to the number of marbles you'll typically have if you start with one marble, then receive one additional marble. If you counted your marbles at that point and found you had three marbles, we would assume that you hadn't noticed the addition of the third marble. Beyond that, we wouldn't accept such counterexamples as these:
How do we know that 1 + 1 = 2? We know it because we know what we mean when we make the familiar statement. All other addition facts follow from there. No flying spaghetti monsters, abstract or not, need apply!
Goldstein made a decent play on page 17. In our view, her book goes downhill from there, biographical writing excluded.
People, one plus one equals two! As our greatest thinkers argued this point, war came to Europe again.
Next week: The guardians file
Now for the rest of the story: After we freshmen took Phil 3, we all decided to abandon philosophy as a major. Nozick, who was only 26 at the time, went on to become a huge star. (He was always very nice to us pitiful freshmen.)
We switched back after sophomore year. Historical inevitability seemed to take over from there.
Flying spaghetti monsters: Are Godel's "incompleteness theorems" actually "important?"
Do they carry any social significance? In the end, do they even make sense?
We'll admit to being doubters on the last of those points. Consider a part of Rebecca Goldstein's book which we'll explore in more detail at some later point.
Goldstein's book, designed for general readers, appeared in 2005. It bore this title: Incompleteness: The Proof and Paradox of Kurt Godel.
In our view, the general reader won't likely emerge from this book with the ability to discuss these supposedly transplendent theorems. For ourselves, we were surprised by the way Goldstein, a philosophy professor, leaned on the concept of "paradox" in her discussions, not excluding this rumintaion on a famous "abstract object:"
GOLDSTEIN (page 91): Russell's paradox concerns the set of all sets that are not members of themselves. Sets are abstract objects that contain members, and some sets can be members of themselves. For example, the set of all abstract objects is a member of itself, since it is an abstract object. Some sets (most) are not members of themselves. For example, the set of all mathematicians is not itself a mathematician—it's an abstract object—and so is not a member of itself. Now we form the concept of the set of all sets that aren't members of themselves and we ask of ourselves: is it a member of itself?...Now we form the concept of the set of all sets that aren't members of themselves? But why in the world would we do that?
The paragraph continues from there. The reference to "Russell" is a reference to Lord Russell, eventual husband of Lady Ottoline—that is to say, to Bertrand Russell—who came up with this world-class groaner back in 1901.
When we first encountered Goldstein's book, it surprised us to think that a capable philosophy professor would still be trafficking in this antique hocus-pocus about these "abstract objects"—about "abstract objects" which may or may not be "members of themselves."
We were even more surprised to see her marveling about this pseudo-paradox, which is even more simple-minded:
"This very sentence is false."Good God! The later Wittgenstein returned to England hoping to remove these flying spaghetti monsters from the pseudo-discourse in which he himself had trafficked as the early Wittgenstein. We were surprised to see a ranking professor still shoveling these snowstorms around.
We'll discuss these matters in the weeks ahead, possibly next week. For ourselves, if Godel's theorems turn on piddle like this, we'll float the shocking possibility that they may not make any real sense.
We know it's shocking to hear such claims about the genius theorems Goldstein gushes about. Then again, this "greatest logicians since Aristotle" seems to have been mentally ill his entire life; eventually died of self-starvation; and believed all sorts of crazy idea, perhaps including the crazy idea that numbers and circles live "a perfect, timeless existence" somewhere, apparently in an "abstract" realm we can access through something resembling ESP.
Do the theorems of this unfortunate man actually make any sense? For now, we'll vote with the doubters. Meanwhile, when Jordan Ellenberg discussed Goldstein's book for Slate, he offered these remarks, among others:
ELLENBERG (3/10/05): In his recent New York Times review of Incompleteness, Edward Rothstein wrote that it’s “difficult to overstate the impact of Gödel’s theorem.” But actually, it’s easy to overstate it: Goldstein does it when she likens the impact of Gödel’s incompleteness theorem to that of relativity and quantum mechanics and calls him “the most famous mathematician that you have most likely never heard of.” But what’s most startling about Gödel’s theorem, given its conceptual importance, is not how much it’s changed mathematics, but how little. No theoretical physicist could start a career today without a thorough understanding of Einstein’s and Heisenberg’s contributions. But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel’s work. So far, I’ve done it myself.You can read the rest of what Ellenberg wrote. For now, we're just saying!
When our greatest logicians devote their lives to the antics of spaghetti monsters, should we be surprised by the sheer stupidity which obtains all over the national discourse engineered by corporate journalists? We'll be focusing on that question next week. For today, let's visit an early part of Goldstein's book, where she starts to get something right.
When Holt summarized Goldstein's book, he profiled the strangeness of Godel. Again, we ask you to marvel at the highlighted part of this pile:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind...[T]he members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.Sadly, strangely, possibly dumbly, the greatest minds in Europe were puzzling hard over this:
What makes a proposition like "2 + 2 = 4" true?Seriously though, folks! From 1901 right up through Godel's arrival at college, that's what our allegedly greatest minds were struggling to figure out!
We mention this for a reason. Near the start of her book, Goldstein gives a weirdly decent explanation of this potent conundrum. She speaks about a different fact—the fact that 5 + 7 = 12—but, as you can probably see, the basic logic of all such statements is pretty much the same.
Citizens, we encountered this same traditional groaner as college freshmen ourselves! How can we know that 7 + 5 = 12? Professor Nozick raised this "problem" in the introductory course, Phil 3: Problems in philosophy.
How do we know that 7 + 5 = 12? One wag in the back of the class dared to ask himself this:
Who is this "problem in philosophy" a problem for?Or words to that effect! On the world's most exalted comedy stages, we've occasionally recalled one subsequent discussion. We did so just a few years ago, with comedy-loving Clarence Page as an opening act:
PHILOSOPHICALLY TORTURED TEACHING ASSISTANT: Students, how can we know that 7 + 5 equals 12?Did that exchange really take place? Memory sometimes plays tricks. But we're fairly sure that we remember the paper we finally wrote on this topic, and it resembled the explanation Goldstein supplies early in her book.
INNOCENT FRESHMAN: Miss Cummings told us? In second grade?
FRUSTRATED TEACHING ASSISTANT (tearing his hair as he stares out the window, seeming to contemplate the abyss): No, no, students, you're missing my point! How do we know that 7 + 5 equals 12?
[Pregnant pause]
PUZZLED FRESHMAN: Same answer?
What makes a proposition like “2 + 2 = 4” true? Using a slightly tougher example, Goldstein offers this:
GOLDSTEIN (page 17): The rigor and certainty of the mathematician is arrived at a priori, meaning that the mathematician neither resorts to any observations in arriving at his or her mathematical insights nor do these mathematical insights, in and of themselves, entail observations, so that nothing we experience can undermine the grounds we have for knowing them. No experience would count as grounds for revising, for example, that 5 + 7 = 12. Were we to add up 5 things and 7 things, and get 13 things, we would recount. Should we still, after repeated recountings, get 13 things we would assume that one of the 12 things had split or that we were seeing double or dreaming or even going mad. The truth that 5 + 7 = 12 is used to evaluate counting experiences, not the other way around.Goldstein is on the right track. That said, and stating the obvious, it makes more sense to explore the logic of this conundrum through the simplest possible example: 1 + 1 = 2.
How do we know that 1 + 1 = 2? Simple! Among other factors, we would be strongly disinclined to accept alleged counterexamples! Let's think in terms of marbles.
"Two" is simply the name we give to the number of marbles you'll typically have if you start with one marble, then receive one additional marble. If you counted your marbles at that point and found you had three marbles, we would assume that you hadn't noticed the addition of the third marble. Beyond that, we wouldn't accept such counterexamples as these:
Haystack Calhoun does the math:What would we say to such counterexamples? We would say they aren't what we mean! In each case, that simply isn't what we mean when we say 1 + 1 = 2!
A farmer has one haystack. He adds to it a second haystack. He sees that he still has one (larger) haystack. The farmer declares that, at least on the farm, 1 + 1 = 1.
Porky Pig adds to the wealth:
A farmer has one (male) pig. He adds one (female) pig. Months later, he finds that he has eight pigs. The farmer declares that, at least on the farm, 1 + 1 = 8.
The evaporation monologues:
A chemist has one beaker of a chemical. He adds a second beaker of a different chemical. The beaker's contents go "poof" and all the liquid disappears. When he added the second beaker, he ended up with no beakers. The chemist declares that, at least in the lab, 1 + 1 = 0.
How do we know that 1 + 1 = 2? We know it because we know what we mean when we make the familiar statement. All other addition facts follow from there. No flying spaghetti monsters, abstract or not, need apply!
Goldstein made a decent play on page 17. In our view, her book goes downhill from there, biographical writing excluded.
People, one plus one equals two! As our greatest thinkers argued this point, war came to Europe again.
Next week: The guardians file
Now for the rest of the story: After we freshmen took Phil 3, we all decided to abandon philosophy as a major. Nozick, who was only 26 at the time, went on to become a huge star. (He was always very nice to us pitiful freshmen.)
We switched back after sophomore year. Historical inevitability seemed to take over from there.
THE INCOMPLETENESS FILE: What the heck is a "formal system?"
THURSDAY, SEPTEMBER 20, 2018
Once again, Joe Average won't know: Friend, are you a general reader? That is to say, are you a non-specialist in the fields of mathematics, mathematical logic, theoretical physics and the like?
Friend, if you're a general reader, let's consider the title essay of Jim Holt's new book.
The new book is called When Einstein Walked with Godel: Excursions to the Edge of Thought. The title essay is called When Einstein Walked with Godel—and friend, we're telling you this:
Ignore the various things you read about how "readable" Holt's essays are! Friend, if you're a general reader, there is exactly zero chance that you'll emerge from that title essay with even the slightest idea what Kurt Godel's "incompleteness theorems" are alleged to be all about.
As we showed you yesterday, there's zero chance you'll have any idea! Ignore what reviewers have said!
Alas! Several layers of academia, journalism and the publishing world are involved in the creation of this strange state of affairs. Before we look at Rebecca Goldstein's first attempt at explaining Godel's theorems—it's Goldstein's "Godel made easy" book which Holt reviewed in his title essay—let's take a minute to consider, once again, who these high-ranking players are.
We'll start with Godel himself, the man "who has often been called the greatest logician since Aristotle."
Who the Sam Hill was Kurt Godel? As it turns out, he seems to have been mentally ill throughout the whole course of life. (At age 72, he died of self-starvation.)
During his adult years in Princeton, he was famous for believing all sorts of crazy ideas. Among them, perhaps, was his foundational belief in "Platonism"—his ardent belief, in Holt's formulation, that "numbers and circles have a perfect, timeless existence" somewhere. (We're able to access this perfect world through some form of ESP.)
Should it seem strange that our greatest logician can be described in this way? We'll examine that question in more detail next week.
For now, let's continue assembling our list of players. Let's consider the circle of thinkers among whom Godel was moving when he devised his iconic theorems, when he was just 23.
According to the profiles offered by Holt and Goldstein, Godel was moving among the Vienna Circle, a group which is said to have included some of the western world's greatest thinkers. As Europe suffered between two wars, these thinkers were puzzling over how we can know that 2 + 2 = 4. They were also puzzling over how we can know that 4 is an even number.
Later, one of their descendants was puzzling over the question of how we can know that 317 (or 17, or 7) is a prime. Godel, our second greatest logician, was apparently puzzling out these crucial topics too.
Friend, do you find it odd to think that our greatest thinkers were puzzling over such questions? We find that odd (and unimpressive) too, just the way you do!
We find that unimpressive, a point we'll discuss next week. But at this point, we must consider the role in this story which gets played by the publishing industry. We must also consider the work of our own modern-day professors and upper-end journalists.
Our publishing business is awash in "Einstein made easy" books (and the like). None of these books has ever managed to make Einstein easy, including the 1916 "Einstein made easy" book written by Einstein himself.
(Einstein, our greatest theoretical physicist, was not a skilled popular writer.)
No one can understand these books, but professors keep turning them out. They take turns blurbing each other's books, telling us rubes how "lucid" these "accessible" books really are. In response, major reviewers stand in line to say beautifully readable these amazingly easy books are!
As any lover of humor would, we've found this fandango fascinating for a great many years. Next week, we'll consider the real-world problems our savants ignore as they produce unreadable books about 2 + 2 equaling 4 and about how we can know such facts.
Quick question! When our ranking professors behave in these ways, should we really be surprised by the intellectual chaos which characterizes our journalism? When our greatest thinkers behaved (and behave) in these ways, should we really be surprised by the low-IQ mugging and clowning which gets presented on corporate cable each night, as our nation slides into the sea?
(And each morning, on Morning Joe, whose entire panel flipped today concerning the need for an FBI probe of what the accuser has said. The panel moved from yesterday's "no" to today's full-throated "yes." We'd use the accuser's name, except the Times is calling her "Blasey" and the Post is still calling her "Ford.")
When the title essay to his new book first appeared, Holt was reviewing Professor Goldstein's 2005 "Godel made easy" book. Because Goldstein is a highly regarded novelist as well as a ranking philosophy prof, it may have seemed like a great idea to have her write a book about the life and the work of this puzzling, disordered man.
As we noted yesterday, Holt's treatment of the "incompleteness theorems" will be totally incoherent for the general reader. For our money, the general reader won't likely be able to make hide nor hair of Goldstein's treatment either.
Holt wrote a book review for The New Yorker; by way of contrast, Goldstein had written a complete book. In our view, the general reader will have little chance of understanding Godel's theorems from reading that book, but for obvious reasons, we can't reproduce Goldstein's full presentation in the way we could do with Holt.
(We also think the professor went places which we found astounding. "This very sentence is false?" It's stunning to think that ranking professors can still find meaning in places like that. More on that starting tomorrow.)
Where Holt wrote an incoherent essay, Goldstein wrote a hard-to-read book. For our money, the general reader will almost surely emerge from that book with no idea what those "incompleteness theorems" are actually all about.
For today, we'll only show you the way Goldstein introduced the theorems. A person might claim that this is unfair, although we aren't sure it is.
On page 23 of Goldstein's book, she stops discussing Albert Einstein and turns to the young Kurt Godel. As she introduces Godel, she marvels at how young he was when he devised his iconic theorems. She almost seems to say that the theorems are easy to state:
The two conclusions Godel reached "are rendered in more or less plain English," Goldstein writes. "Delightfully," the overall strategy of his proof isn't formidably technical!
Goldstein makes it sound like Godel and his theorems won't be all that hard! Then, she quotes the Encyclopedia of Philosophy's "crisp statement of the two theorems." The passage she quotes goes like this:
According to Goldstein, this account of Godel's theorems has been "rendered in more or less plain English." We trust and believe that you, a general reader, can see that this just isn't so.
How does the Encyclopedia define or describe the first theorem? In plain English, it goes like this:
Citizens, can we talk? The general reader will have no idea what a "formal system" is! Beyond that, this general reader will have little idea what "number theory" is.
In part for these reasons, this general reader won't be able to imagine what a formal system "adequate for" number theory is. The general reader will have no idea what that passage is talking about.
However "crisp" this statement may be, this statement will be clear as mud to the general reader. It contains the kind of technical language which may not look like technical language. But this language is guaranteed to leave the general reader on the outside, haplessly looking in.
Briefly, let's be fair. This passage represents Goldstein's first attempt at describing these iconic theorems. This strikes us as a strange first attempt but, at least in theory, Goldstein could have continued on from there to unpack these theorems in a way the average Joe could actually understand.
For our money, that doesn't happen in Goldstein's book. Along came Holt, to offer the crazily incoherent summary we perused in full in yesterday's report.
On page 26, Goldstein reassures the general reader. She does so in this passage, in which she once again plays the "plain English" card:
In our view, things don't get a whole lot better for the general reader in what follows from there. Things seem technical all the way down. It seems to us that the general reader will likely be forced to quit.
Citizens, let's review:
Our greatest logician was mentally ill and possessed of crazy ideas. At the heart of his prolix theorems was his apparently crazy belief that numbers and circles live a perfect existence somewhere.
In turn, our philosophy professors seem to have no idea how to explain these prolix theorems (which "say an awful lot") to the general reader. But they produce books which claim to have done that anyway. When they do, journalists rush to say that they understood every word. And it all began with our greatest thinkers pondering 2 + 2.
When we see this cultural pattern unfold, are we surprised by the utter incoherence displayed by lesser thinkers on corporate cable? Are we surprised that our broken, pre-rational public discourse has now helped to place a Donald J. Trump in the White House?
Seeing ourselves from afar, we humans still tend to believe, say and suggest that we're the rational animal. In our view, this profoundly iconic notion qualifies as "Aristotle's [gigantic large howling] error."
Tomorrow, we'll debase Godel a tiny bit more, prepping a bit for next week. We'll also see Professor Goldstein do something amazingly rare.
Tomorrow: A (near) perfect statement by Goldstein
Once again, Joe Average won't know: Friend, are you a general reader? That is to say, are you a non-specialist in the fields of mathematics, mathematical logic, theoretical physics and the like?
Friend, if you're a general reader, let's consider the title essay of Jim Holt's new book.
The new book is called When Einstein Walked with Godel: Excursions to the Edge of Thought. The title essay is called When Einstein Walked with Godel—and friend, we're telling you this:
Ignore the various things you read about how "readable" Holt's essays are! Friend, if you're a general reader, there is exactly zero chance that you'll emerge from that title essay with even the slightest idea what Kurt Godel's "incompleteness theorems" are alleged to be all about.
As we showed you yesterday, there's zero chance you'll have any idea! Ignore what reviewers have said!
Alas! Several layers of academia, journalism and the publishing world are involved in the creation of this strange state of affairs. Before we look at Rebecca Goldstein's first attempt at explaining Godel's theorems—it's Goldstein's "Godel made easy" book which Holt reviewed in his title essay—let's take a minute to consider, once again, who these high-ranking players are.
We'll start with Godel himself, the man "who has often been called the greatest logician since Aristotle."
Who the Sam Hill was Kurt Godel? As it turns out, he seems to have been mentally ill throughout the whole course of life. (At age 72, he died of self-starvation.)
During his adult years in Princeton, he was famous for believing all sorts of crazy ideas. Among them, perhaps, was his foundational belief in "Platonism"—his ardent belief, in Holt's formulation, that "numbers and circles have a perfect, timeless existence" somewhere. (We're able to access this perfect world through some form of ESP.)
Should it seem strange that our greatest logician can be described in this way? We'll examine that question in more detail next week.
For now, let's continue assembling our list of players. Let's consider the circle of thinkers among whom Godel was moving when he devised his iconic theorems, when he was just 23.
According to the profiles offered by Holt and Goldstein, Godel was moving among the Vienna Circle, a group which is said to have included some of the western world's greatest thinkers. As Europe suffered between two wars, these thinkers were puzzling over how we can know that 2 + 2 = 4. They were also puzzling over how we can know that 4 is an even number.
Later, one of their descendants was puzzling over the question of how we can know that 317 (or 17, or 7) is a prime. Godel, our second greatest logician, was apparently puzzling out these crucial topics too.
Friend, do you find it odd to think that our greatest thinkers were puzzling over such questions? We find that odd (and unimpressive) too, just the way you do!
We find that unimpressive, a point we'll discuss next week. But at this point, we must consider the role in this story which gets played by the publishing industry. We must also consider the work of our own modern-day professors and upper-end journalists.
Our publishing business is awash in "Einstein made easy" books (and the like). None of these books has ever managed to make Einstein easy, including the 1916 "Einstein made easy" book written by Einstein himself.
(Einstein, our greatest theoretical physicist, was not a skilled popular writer.)
No one can understand these books, but professors keep turning them out. They take turns blurbing each other's books, telling us rubes how "lucid" these "accessible" books really are. In response, major reviewers stand in line to say beautifully readable these amazingly easy books are!
As any lover of humor would, we've found this fandango fascinating for a great many years. Next week, we'll consider the real-world problems our savants ignore as they produce unreadable books about 2 + 2 equaling 4 and about how we can know such facts.
Quick question! When our ranking professors behave in these ways, should we really be surprised by the intellectual chaos which characterizes our journalism? When our greatest thinkers behaved (and behave) in these ways, should we really be surprised by the low-IQ mugging and clowning which gets presented on corporate cable each night, as our nation slides into the sea?
(And each morning, on Morning Joe, whose entire panel flipped today concerning the need for an FBI probe of what the accuser has said. The panel moved from yesterday's "no" to today's full-throated "yes." We'd use the accuser's name, except the Times is calling her "Blasey" and the Post is still calling her "Ford.")
When the title essay to his new book first appeared, Holt was reviewing Professor Goldstein's 2005 "Godel made easy" book. Because Goldstein is a highly regarded novelist as well as a ranking philosophy prof, it may have seemed like a great idea to have her write a book about the life and the work of this puzzling, disordered man.
As we noted yesterday, Holt's treatment of the "incompleteness theorems" will be totally incoherent for the general reader. For our money, the general reader won't likely be able to make hide nor hair of Goldstein's treatment either.
Holt wrote a book review for The New Yorker; by way of contrast, Goldstein had written a complete book. In our view, the general reader will have little chance of understanding Godel's theorems from reading that book, but for obvious reasons, we can't reproduce Goldstein's full presentation in the way we could do with Holt.
(We also think the professor went places which we found astounding. "This very sentence is false?" It's stunning to think that ranking professors can still find meaning in places like that. More on that starting tomorrow.)
Where Holt wrote an incoherent essay, Goldstein wrote a hard-to-read book. For our money, the general reader will almost surely emerge from that book with no idea what those "incompleteness theorems" are actually all about.
For today, we'll only show you the way Goldstein introduced the theorems. A person might claim that this is unfair, although we aren't sure it is.
On page 23 of Goldstein's book, she stops discussing Albert Einstein and turns to the young Kurt Godel. As she introduces Godel, she marvels at how young he was when he devised his iconic theorems. She almost seems to say that the theorems are easy to state:
GOLDSTEIN (page 23): He is Kurt Godel, and in 1930, when he was 23, he had produced an extraordinary proof in mathematical logic for something called the incompleteness theorem—actually two logically related incompleteness theorems.Tell the truth! Reading that passage, it sounds like it won't be hard to make Godel easy!
Unlike most mathematical results, Godel’s incompleteness theorems are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof’s overall strategy, delightfully, is not. The two conclusions that emerge at the end of all the formal pyrotechnics are rendered in more or less plain English. The Encyclopedia of Philosophy’s article “Godel’s Theorem” opens with a crisp statement of the two theorems:
The two conclusions Godel reached "are rendered in more or less plain English," Goldstein writes. "Delightfully," the overall strategy of his proof isn't formidably technical!
Goldstein makes it sound like Godel and his theorems won't be all that hard! Then, she quotes the Encyclopedia of Philosophy's "crisp statement of the two theorems." The passage she quotes goes like this:
GOLDSTEIN (continuing directly): "By Godel's theorem, the following statement is generally meant:That quoted passage is attributed the Encyclopedia of Philosophy. We'll suggest you consider this:
"In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable. (This statement is occasionally referred to as Godel’s first theorem.)
"A corollary to the theorem is that the consistency of a formal system adequate for number theory cannot be proved within the system. (Sometimes it is this corollary that is referred to as Godel’s theorem; it is also referred to as Godel’s second theorem.)"
According to Goldstein, this account of Godel's theorems has been "rendered in more or less plain English." We trust and believe that you, a general reader, can see that this just isn't so.
How does the Encyclopedia define or describe the first theorem? In plain English, it goes like this:
In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable.Friend, that passage simply isn't written in plain English. We hope you could already see that.
Citizens, can we talk? The general reader will have no idea what a "formal system" is! Beyond that, this general reader will have little idea what "number theory" is.
In part for these reasons, this general reader won't be able to imagine what a formal system "adequate for" number theory is. The general reader will have no idea what that passage is talking about.
However "crisp" this statement may be, this statement will be clear as mud to the general reader. It contains the kind of technical language which may not look like technical language. But this language is guaranteed to leave the general reader on the outside, haplessly looking in.
Briefly, let's be fair. This passage represents Goldstein's first attempt at describing these iconic theorems. This strikes us as a strange first attempt but, at least in theory, Goldstein could have continued on from there to unpack these theorems in a way the average Joe could actually understand.
For our money, that doesn't happen in Goldstein's book. Along came Holt, to offer the crazily incoherent summary we perused in full in yesterday's report.
On page 26, Goldstein reassures the general reader. She does so in this passage, in which she once again plays the "plain English" card:
GOLDSTEIN (page 26): [Godel’s theorems] are the most prolix theorems in the history of mathematics. Though there is disagreement about precisely how much, and precisely what, they say, there is no doubt that they say an awful lot and that what they say extends beyond mathematics, certainly into metamathematics and perhaps even beyond. In fact, the mathematical nature of the theorems is intimately linked with the fact that the Encyclopedia of Philosophy stated them in (more or less) plain English. The concepts of “formal system,” “undecidable,” and “consistency” might be semi-technical and require explication (which is why the reader should not worry if the succinct statement of the theorems yielded little understanding); but they are metamathematical concepts whose explication (which will eventually come) is not rendered in the language of mathematics.Finally! Three pages later, Goldstein notes that the general reader has no idea what a "formal system" is. For the record, she offers her first definition of the term on page 129 [sic].
In our view, things don't get a whole lot better for the general reader in what follows from there. Things seem technical all the way down. It seems to us that the general reader will likely be forced to quit.
Citizens, let's review:
Our greatest logician was mentally ill and possessed of crazy ideas. At the heart of his prolix theorems was his apparently crazy belief that numbers and circles live a perfect existence somewhere.
In turn, our philosophy professors seem to have no idea how to explain these prolix theorems (which "say an awful lot") to the general reader. But they produce books which claim to have done that anyway. When they do, journalists rush to say that they understood every word. And it all began with our greatest thinkers pondering 2 + 2.
When we see this cultural pattern unfold, are we surprised by the utter incoherence displayed by lesser thinkers on corporate cable? Are we surprised that our broken, pre-rational public discourse has now helped to place a Donald J. Trump in the White House?
Seeing ourselves from afar, we humans still tend to believe, say and suggest that we're the rational animal. In our view, this profoundly iconic notion qualifies as "Aristotle's [gigantic large howling] error."
Tomorrow, we'll debase Godel a tiny bit more, prepping a bit for next week. We'll also see Professor Goldstein do something amazingly rare.
Tomorrow: A (near) perfect statement by Goldstein
THE INCOMPLETENESS FILE: What the Sam Hill is a "logical system?"
WEDNESDAY, SEPTEMBER 19, 2018
No general reader will know: According to the headline on the New York Times review, Jim Holt's new book is a collection of essays which "make sense of the infinite and the infinitesimal."
It's Holt's "conviviality, and a crispness of style, that distinguish him as a popularizer of some very redoubtable mathematics and science,“ the gushing reviewer said, marching in upper-end lockstep.
Indeed, it wasn't just the New York Times making these mandated statements. According to the headline on the Christian Science Monitor review, "When Einstein Walked with Gödel"—that's the title of Holt's new book—"is science writing at its best."
The essays in Holt's new book "all wonderfully achieve [his] stated goal," which includes "enlighten[ing] the newcomer," the Monitor's reviewer said. "This is considerably more difficult than it sounds, and Holt does a beautifully readable job."
Holt's collection of essays wasn't reviewed by the Washington Post, but the reviewer for the Wall Street Journal completed the rule of three. Holt is "one of the very best modern science writers," this third reviewer opined. He specifically singled out Holt's "wonderful title essay."
That's the very essay we've been discussing—the essay in which Holt tries to explain Kurt Godel's "incompleteness theorems."
Reviewers seemed to agree. Holt's work is "beautifully readable," especially for "the newcomer"—for the general reader. But then we turn to that title essay, the one in which Holt attempts to explain Godel's theorems.
According to Holt, those theorems have established Godel, by widespread agreement, as "the greatest logician since Aristotle." An obvious question arises:
How "beautifully readable" is Holt's explanation of those iconic theorems? To what extent is Holt's account of those theorems "science writing at its best?"
As we noted yesterday, Holt explains those theorems in two extremely long paragraphs. As we showed you yesterday, the first of those paragraphs, by far the shorter of the two, reads as shown below in Holt's title essay, which first appeared in The New Yorker in 2005.
Below, you see the first of the two paragraphs in which Holt explains Godel's theorems. By the end of this paragraph, our greatest logician since Aristotle is, for reasons which don't quite get explained, pondering 2 + 2:
Without so much as chortling even once, Holt proceeds from there:
One group of Europe's most brilliant thinkers is said to have said that the truth of 2 + 2 = 4 lies in the fact that "it could be derived in a logical system according to certain rules." Without attempting to explain what that technical word salad means, Holt proceeds to say what the youthful Godel believed:
The youthful Godel is said to have thought that the truth of 2 + 2 = 4 lies in the fact that "it correctly describes some abstract world of numbers"—presumably, the world in which "abstractions like numbers and circles have a perfect, timeless existence independent of the human mind." So the greatest logician thought, as opposed to the other great thinkers.
Scotty, beam us down! Despite the gushing of those reviewers, no general reader will have any idea what that paragraph means.
What does it mean to say that 2 + 2 = 4 "can be derived in a logical system according to certain rules?" No general reader has the slightest idea, but Holt doesn't stop to decipher the claim. He merely compares it to what Godel is said to have thought—a belief which is said to involved the perfect existence of circles.
In these ways, our greatest thinkers puzzled out 2 + 2. Last May, major journalists stood in line to say how "beautifully readable" Holt's new book is, especially the "wonderful title essay" in which this hodgepodge appears.
In such ways, we see a modern, high-end display of "Aristotle's error." These reviewers aren't reflecting the ancient claim that "man [sic] is the rational animal." Rather, they're acting out the "Harari heuristic," which holds that our warlike species, Homo sapiens, gained control of the planet when, through a set of chance mutations, our ancestors developed the ability to "gossip" and the ability to invent and affirm sweeping group "fictions."
As the weeks and months proceed, we'll return to Harari's account, reviewing his claims in more detail. For today, we'll only note an obvious fact—even by the end of this first paragraph, Holt's opaque, highly technical writing will have left any general reader several light-years behind.
Alas! Whether they know it or not, general readers will already be at sea by the end of that first paragraph. Most specifically, such readers will have no idea what a "logical system" is.
Nor will such readers have any idea what it means to say that a nursery school fact like 2 + 2 "can be derived in a logical system according to certain rules." Already, Holt may as well be writing in some form of ancient Etruscan.
In the paragraph which follows, Holt starts explaining those "incompleteness theorems." When he does, a large pile of Sandstorm arrives.
At the end of this pig-pile of abstruse phrases, Godel's two theorems get defined. The general reader will have zero idea what Holt is talking about:
Consider the various snares and traps that reader has encountered during this long forced march to the sea:
We're told that Godel wanted to demonstrate that "mathematics could not be reduced to the artifices of logic." The general reader will have no idea what such a reduction might look like.
In pursuit of this puzzling end, we're told that Godel "beg[an] with a logical system for mathematics, one presumed to be free of contradictions." The general reader won't know what "a logical system" is. He won't know what it means for such a creature to be "adequate for mathematics."
We're now told that Godel came up with "a formula that said something about numbers." On the pain of impending death, the general reader won't be able to imagine an example of any such formula making any such statement. Nor will she have any idea what it might mean to produce "a formula that, while ostensibly saying something about numbers, also says, 'I am not provable.' ”
A bit later on, the reader seems to be told that Godel produced "a self-referential formula" which generated a proposition whose truth "can be seen only from outside the logical system." No general reader has any idea what Holt is talking about.
At any rate, atthe end of this long harangue, Holt describes Godel's first incompleteness theorem. Excitement builds for the general reader. Then he's told that the theorem says this:
"No logical system can capture all the truths of mathematics."
That would be an exciting claim. Except, do you remember the problem with which the general reader started? He or she has no idea what a "logical system" is!
In these two paragraphs, Holt explains, or pretends or attempts to explain, Godel's two "incompleteness theorems." For unknown reasons, this Olympian hodgepodge was first offered to general readers in the pages of The New Yorker. Few subscribers could have had any idea what Holt was talking about.
Thirteen years later, Holt's piece was published as the title essay in a collection of his work. Mainstream reviewers stood in line to praise it for being readable, especially for newcomers to the subject matter.
What a long, strange journey it has been through those lengthy paragraphs! We started with Europe's greatest thinkers pondering the fact that 2 + 2 equals 4. We were told that the greatest logician since Aristotle believed that circles and numbers and other such critters have a perfect, timeless existence, an existence we can access through some version of ESP.
Eventually, an avalanche of technical language landed on our newcomer heads. But so what? An obedient line of upper-end journalists said this all made perfect sense.
We're going to say that all these groups are providing textbook illustrations of "Aristotle's error." Also this:
When our journalists behave in the manner described, they're helping us see how things fall apart when Plato's guardians fail.
Tomorrow: Goldstein's first attempt
No general reader will know: According to the headline on the New York Times review, Jim Holt's new book is a collection of essays which "make sense of the infinite and the infinitesimal."
It's Holt's "conviviality, and a crispness of style, that distinguish him as a popularizer of some very redoubtable mathematics and science,“ the gushing reviewer said, marching in upper-end lockstep.
Indeed, it wasn't just the New York Times making these mandated statements. According to the headline on the Christian Science Monitor review, "When Einstein Walked with Gödel"—that's the title of Holt's new book—"is science writing at its best."
The essays in Holt's new book "all wonderfully achieve [his] stated goal," which includes "enlighten[ing] the newcomer," the Monitor's reviewer said. "This is considerably more difficult than it sounds, and Holt does a beautifully readable job."
Holt's collection of essays wasn't reviewed by the Washington Post, but the reviewer for the Wall Street Journal completed the rule of three. Holt is "one of the very best modern science writers," this third reviewer opined. He specifically singled out Holt's "wonderful title essay."
That's the very essay we've been discussing—the essay in which Holt tries to explain Kurt Godel's "incompleteness theorems."
Reviewers seemed to agree. Holt's work is "beautifully readable," especially for "the newcomer"—for the general reader. But then we turn to that title essay, the one in which Holt attempts to explain Godel's theorems.
According to Holt, those theorems have established Godel, by widespread agreement, as "the greatest logician since Aristotle." An obvious question arises:
How "beautifully readable" is Holt's explanation of those iconic theorems? To what extent is Holt's account of those theorems "science writing at its best?"
As we noted yesterday, Holt explains those theorems in two extremely long paragraphs. As we showed you yesterday, the first of those paragraphs, by far the shorter of the two, reads as shown below in Holt's title essay, which first appeared in The New Yorker in 2005.
Below, you see the first of the two paragraphs in which Holt explains Godel's theorems. By the end of this paragraph, our greatest logician since Aristotle is, for reasons which don't quite get explained, pondering 2 + 2:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of 1920s Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.As this first long paragraph ends, the greatest thinkers in Europe are puzzling over a knotty problem. According to Holt's own language, they're trying to explain "what makes a proposition like 2 + 2 = 4 true."
Without so much as chortling even once, Holt proceeds from there:
One group of Europe's most brilliant thinkers is said to have said that the truth of 2 + 2 = 4 lies in the fact that "it could be derived in a logical system according to certain rules." Without attempting to explain what that technical word salad means, Holt proceeds to say what the youthful Godel believed:
The youthful Godel is said to have thought that the truth of 2 + 2 = 4 lies in the fact that "it correctly describes some abstract world of numbers"—presumably, the world in which "abstractions like numbers and circles have a perfect, timeless existence independent of the human mind." So the greatest logician thought, as opposed to the other great thinkers.
Scotty, beam us down! Despite the gushing of those reviewers, no general reader will have any idea what that paragraph means.
What does it mean to say that 2 + 2 = 4 "can be derived in a logical system according to certain rules?" No general reader has the slightest idea, but Holt doesn't stop to decipher the claim. He merely compares it to what Godel is said to have thought—a belief which is said to involved the perfect existence of circles.
In these ways, our greatest thinkers puzzled out 2 + 2. Last May, major journalists stood in line to say how "beautifully readable" Holt's new book is, especially the "wonderful title essay" in which this hodgepodge appears.
In such ways, we see a modern, high-end display of "Aristotle's error." These reviewers aren't reflecting the ancient claim that "man [sic] is the rational animal." Rather, they're acting out the "Harari heuristic," which holds that our warlike species, Homo sapiens, gained control of the planet when, through a set of chance mutations, our ancestors developed the ability to "gossip" and the ability to invent and affirm sweeping group "fictions."
As the weeks and months proceed, we'll return to Harari's account, reviewing his claims in more detail. For today, we'll only note an obvious fact—even by the end of this first paragraph, Holt's opaque, highly technical writing will have left any general reader several light-years behind.
Alas! Whether they know it or not, general readers will already be at sea by the end of that first paragraph. Most specifically, such readers will have no idea what a "logical system" is.
Nor will such readers have any idea what it means to say that a nursery school fact like 2 + 2 "can be derived in a logical system according to certain rules." Already, Holt may as well be writing in some form of ancient Etruscan.
In the paragraph which follows, Holt starts explaining those "incompleteness theorems." When he does, a large pile of Sandstorm arrives.
At the end of this pig-pile of abstruse phrases, Godel's two theorems get defined. The general reader will have zero idea what Holt is talking about:
HOLT (continuing directly): Gödel was introduced into the Vienna Circle by one of his professors, but he kept quiet about his Platonist views. Being both rigorous and averse to controversy, he did not like to argue his convictions unless he had an airtight way of demonstrating that they were valid. But how could one demonstrate that mathematics could not be reduced to the artifices of logic? Gödel’s strategy—one of preternatural cleverness and, in the words of philosopher Rebecca Goldstein, “heart-stopping beauty”—was to use logic against itself. Beginning with a logical system for mathematics, one presumed to be free of contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of doublespeak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another. In fact, as Gödel showed, a numerical formula could even be made to say something about itself. Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying when it asserts, "I am nor provable?" No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.Ar the end of this, the world's longest paragraph, Holt defines, or pretends to define, Godel's two "incompleteness theorems." Despite the subsequent, mandated gushing of our journalistic elites, no general reader will have any idea what Holt is talking about.
Consider the various snares and traps that reader has encountered during this long forced march to the sea:
We're told that Godel wanted to demonstrate that "mathematics could not be reduced to the artifices of logic." The general reader will have no idea what such a reduction might look like.
In pursuit of this puzzling end, we're told that Godel "beg[an] with a logical system for mathematics, one presumed to be free of contradictions." The general reader won't know what "a logical system" is. He won't know what it means for such a creature to be "adequate for mathematics."
We're now told that Godel came up with "a formula that said something about numbers." On the pain of impending death, the general reader won't be able to imagine an example of any such formula making any such statement. Nor will she have any idea what it might mean to produce "a formula that, while ostensibly saying something about numbers, also says, 'I am not provable.' ”
A bit later on, the reader seems to be told that Godel produced "a self-referential formula" which generated a proposition whose truth "can be seen only from outside the logical system." No general reader has any idea what Holt is talking about.
At any rate, atthe end of this long harangue, Holt describes Godel's first incompleteness theorem. Excitement builds for the general reader. Then he's told that the theorem says this:
"No logical system can capture all the truths of mathematics."
That would be an exciting claim. Except, do you remember the problem with which the general reader started? He or she has no idea what a "logical system" is!
In these two paragraphs, Holt explains, or pretends or attempts to explain, Godel's two "incompleteness theorems." For unknown reasons, this Olympian hodgepodge was first offered to general readers in the pages of The New Yorker. Few subscribers could have had any idea what Holt was talking about.
Thirteen years later, Holt's piece was published as the title essay in a collection of his work. Mainstream reviewers stood in line to praise it for being readable, especially for newcomers to the subject matter.
What a long, strange journey it has been through those lengthy paragraphs! We started with Europe's greatest thinkers pondering the fact that 2 + 2 equals 4. We were told that the greatest logician since Aristotle believed that circles and numbers and other such critters have a perfect, timeless existence, an existence we can access through some version of ESP.
Eventually, an avalanche of technical language landed on our newcomer heads. But so what? An obedient line of upper-end journalists said this all made perfect sense.
We're going to say that all these groups are providing textbook illustrations of "Aristotle's error." Also this:
When our journalists behave in the manner described, they're helping us see how things fall apart when Plato's guardians fail.
Tomorrow: Goldstein's first attempt
THE INCOMPLETENESS FILE: Incompleteness meets incoherence!
TUESDAY, SEPTEMBER 18, 2018
Lucid writer intones: Subscribers to The New Yorker had a major treat in store.
Or at least, so it seemed.
Their February 28, 2005 issue had arrived in the mail. It featured a lengthy essay in which a writer named Jim Holt discussed a pair of new books.
One of the books concerned Albert Einstein, an extremely famous theoretical physicist. The other new book concerned Kurt Godel, a "logician" who isn't well-known by the average shlub at all.
This May, Holt's New Yorker essay, lightly edited, reappeared as the title essay in his own new book, When Einstein Walked with Godel: Excursions to the Edge of Thought.
As seems to be required by law, Holt's new book was praised by major reviewers—was praised for its lucidity. In the thirteen years since that essay appeared, Holt had become a "made man" in New York publishing circles.
Within those circles, Holt is now reflexively praised for the clarity of his writing about difficult science and math. As you can see, Wikipedia even headlines him as a "philosopher!" That's how silly and mandatory this sort of thing has become.
Holt's book of essays was praised this year for its brilliant lucidity. As we noted last Friday, the same was true of the new book about Kurt Godel which he discussed in The New Yorker back in 2005.
Hurrah! That new book, by Rebecca Goldstein, had been described as "accessible"—but also as "surprisingly accessible," even as "remarkably accessible."
It had been praised as a "lucid expression" of Godel's ideas—but it had also been hailed as "eminently lucid." So it goes within our tightly scripted academic journalistical complex.
Goldstein's treatment of Godel's ideas had been widely praised. Now, a writer at The New Yorker was going to boil matters down even further! Subscribers would finally get a chance to understand Godel's "incompleteness theorems," on the basis of which, Holt now said, Godel has often been called "the greatest logician since Aristotle."
Truth to tell, nothing dimly resembling that occurred in Holt's piece. In fairness, Goldstein hadn't been especially lucid when it came to explaining Godel's theorems either.
For the general reader, Goldstein's treatment of Godel's theorems would almost surely have been extremely hard to follow. When Holt took his turn in The New Yorker, his attempt to describe those "incompleteness theorems" was almost comically incoherent—incoherent all the way down.
Today, the book by Holt which features that essay is being praised by major journalists for its brilliant clarity. In this way, a comical aspect of our journalism—indeed, of our upper-end culture's most basic attempt at rationality—has once again been put on display, for perhaps the ten millionth time.
Holt's essay appeared in early 2005. Its author discussed Einstein's theory of relativity, then turned to Godel's theorems. In this week's reports, we'll speak of Godel alone.
Back in May, readers of the New York Times and the Wall Street Journal were told that Holt's rather obvious incoherence is an example of brilliant lucidity.
In this way, we've been given another look at the classic misassessment we've now christened as "Aristotle's error." We've been given another look at the way we humans, at least in the west, keep "seeing ourselves from afar."
Thanks to his incompleteness theorems, Godel has often been described as the greatest logician since Aristotle. But what did Godel actually say in his theorems? What was he trying to show?
In his essay for The New Yorker, Holt addressed those basic questions in two enormously long paragraphs. Today, we'll examine the first of those paragraphs, transcribing it as it appears in Holt's current book.
We'll start with an apology. In the past two weeks, we've already posted the start of the first paragraph in question. Before we show you Holt's full paragraph, we'll revisit that part, for review:
Already, a perceptive reader might suspect that she's being directed by "a guide...who only has at heart [her] getting lost." What in the world does a person believe when he believes that "numbers and circles have a perfect, timeless existence independent of the human mind?" What does it mean to "believe," to be seduced by, such a peculiar notion?
Already, a perceptive reader should be asking such questions. But Holt just kept plowing ahead.
In the passage we've posted above, Holt said that's what Godel believed. He didn't try to explain what that peculiar formulation might possibly mean. Instead, he moved on to describe a major dispute within the intellectual world of Godel's Vienna.
What follows is the first of the two lengthy paragraphs in which Holt explains, or attempts or pretends to explain, Godel's "incompleteness theorems." Warning! The second graf, which we'll review tomorrow, is almost twice as long as the first.
According to recent reviews in the Times and the Journal, this paragraph appears within the title essay of a book in which the writing is brilliantly incisive and clear. Additional warning! By the end of this paragraph, the greatest logician since Aristotle is asking himself how we can know that 2 + 2 equals 4!
That's what Godel is asking himself! People, we're just saying:
As the people of Europe struggled and groaned between two deeply destructive world wars, that's the type of question the western world's most brilliant thinkers were laboring to resolve!
You've now seen the first of the two paragraphs in which Holt explains, or attempts to explain, Godel's "incompleteness theorems." In this first paragraph, Holt basically sets the stage for his ultimate explanation. That will come in the second paragraph, which is roughly twice as long.
What makes a proposition like “2 + 2 = 4” true? In this, the first of his two paragraphs, Holt—the brilliant, incisive writer—sets the stage for spelling it out. This is what he has said:
According to one group of thinkers in Godel's Vienna, "2 + 2 = 4" is true because this rather familiar arithmetical proposition "can be derived in a logical system according to certain rules."
Tell the truth, dear general reader: Do you have the slightest idea what that statement means?
Tell the truth, New Yorker subscriber: Do you understand what it means, even in a general sense, to "derive [an arithmetical proposition] in a logical system according to certain rules?"
Friend, of course you don't! But if that's what one group of thinkers were thinking, at least one other brilliant thinker was brilliantly thinking this:
According to the greatest logician since Aristotle, "2 + 2 = 4" is true because it "correctly describes [an] abstract world of numbers." Perhaps more precisely, it correctly describes one aspect of the perfect, timeless existence enjoyed by numbers and circles outside the human mind!
Europe was struggling between two wars. Tearing their hair in the loftiest circles, the western world's most brilliant "thinkers" were laboring over this.
That being said, did Holt go on to make Godel's approach to this matter brilliantly clear? More specifically, was he able to explain Godel's "incompleteness theorems" in a way the general reader might find wonderfully clear?
That's what major reviewers have suggested. But at this point, does any of this seem especially clear?
Tomorrow, we're going to ask you to strap yourselves into your seats. We'll quickly revisit this first paragraph, into which a substantial amount of incoherence has already been poured.
Then, we'll look at Godel's endless succeeding paragraph, in which he attempts to describe the working of Godel's theorems in a way the general reader will be able to comprehend.
We humans! Seeing ourselves from afar once again, major reviewers have seemed to say that Holt did a wonderful job!
Tomorrow: "Beginning with a logical system for mathematics..."
Lucid writer intones: Subscribers to The New Yorker had a major treat in store.
Or at least, so it seemed.
Their February 28, 2005 issue had arrived in the mail. It featured a lengthy essay in which a writer named Jim Holt discussed a pair of new books.
One of the books concerned Albert Einstein, an extremely famous theoretical physicist. The other new book concerned Kurt Godel, a "logician" who isn't well-known by the average shlub at all.
This May, Holt's New Yorker essay, lightly edited, reappeared as the title essay in his own new book, When Einstein Walked with Godel: Excursions to the Edge of Thought.
As seems to be required by law, Holt's new book was praised by major reviewers—was praised for its lucidity. In the thirteen years since that essay appeared, Holt had become a "made man" in New York publishing circles.
Within those circles, Holt is now reflexively praised for the clarity of his writing about difficult science and math. As you can see, Wikipedia even headlines him as a "philosopher!" That's how silly and mandatory this sort of thing has become.
Holt's book of essays was praised this year for its brilliant lucidity. As we noted last Friday, the same was true of the new book about Kurt Godel which he discussed in The New Yorker back in 2005.
Hurrah! That new book, by Rebecca Goldstein, had been described as "accessible"—but also as "surprisingly accessible," even as "remarkably accessible."
It had been praised as a "lucid expression" of Godel's ideas—but it had also been hailed as "eminently lucid." So it goes within our tightly scripted academic journalistical complex.
Goldstein's treatment of Godel's ideas had been widely praised. Now, a writer at The New Yorker was going to boil matters down even further! Subscribers would finally get a chance to understand Godel's "incompleteness theorems," on the basis of which, Holt now said, Godel has often been called "the greatest logician since Aristotle."
Truth to tell, nothing dimly resembling that occurred in Holt's piece. In fairness, Goldstein hadn't been especially lucid when it came to explaining Godel's theorems either.
For the general reader, Goldstein's treatment of Godel's theorems would almost surely have been extremely hard to follow. When Holt took his turn in The New Yorker, his attempt to describe those "incompleteness theorems" was almost comically incoherent—incoherent all the way down.
Today, the book by Holt which features that essay is being praised by major journalists for its brilliant clarity. In this way, a comical aspect of our journalism—indeed, of our upper-end culture's most basic attempt at rationality—has once again been put on display, for perhaps the ten millionth time.
Holt's essay appeared in early 2005. Its author discussed Einstein's theory of relativity, then turned to Godel's theorems. In this week's reports, we'll speak of Godel alone.
Back in May, readers of the New York Times and the Wall Street Journal were told that Holt's rather obvious incoherence is an example of brilliant lucidity.
In this way, we've been given another look at the classic misassessment we've now christened as "Aristotle's error." We've been given another look at the way we humans, at least in the west, keep "seeing ourselves from afar."
Thanks to his incompleteness theorems, Godel has often been described as the greatest logician since Aristotle. But what did Godel actually say in his theorems? What was he trying to show?
In his essay for The New Yorker, Holt addressed those basic questions in two enormously long paragraphs. Today, we'll examine the first of those paragraphs, transcribing it as it appears in Holt's current book.
We'll start with an apology. In the past two weeks, we've already posted the start of the first paragraph in question. Before we show you Holt's full paragraph, we'll revisit that part, for review:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians...According to Holt, Godel had come to believe that "abstractions like numbers and circles had a perfect, timeless existence independent of the human mind." As it turns out, Godel was "seduced" by this "doctrine" as a mere freshman in college.
Already, a perceptive reader might suspect that she's being directed by "a guide...who only has at heart [her] getting lost." What in the world does a person believe when he believes that "numbers and circles have a perfect, timeless existence independent of the human mind?" What does it mean to "believe," to be seduced by, such a peculiar notion?
Already, a perceptive reader should be asking such questions. But Holt just kept plowing ahead.
In the passage we've posted above, Holt said that's what Godel believed. He didn't try to explain what that peculiar formulation might possibly mean. Instead, he moved on to describe a major dispute within the intellectual world of Godel's Vienna.
What follows is the first of the two lengthy paragraphs in which Holt explains, or attempts or pretends to explain, Godel's "incompleteness theorems." Warning! The second graf, which we'll review tomorrow, is almost twice as long as the first.
According to recent reviews in the Times and the Journal, this paragraph appears within the title essay of a book in which the writing is brilliantly incisive and clear. Additional warning! By the end of this paragraph, the greatest logician since Aristotle is asking himself how we can know that 2 + 2 equals 4!
That's what Godel is asking himself! People, we're just saying:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of 1920s Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.What makes a proposition like “2 + 2 = 4” true?
As the people of Europe struggled and groaned between two deeply destructive world wars, that's the type of question the western world's most brilliant thinkers were laboring to resolve!
You've now seen the first of the two paragraphs in which Holt explains, or attempts to explain, Godel's "incompleteness theorems." In this first paragraph, Holt basically sets the stage for his ultimate explanation. That will come in the second paragraph, which is roughly twice as long.
What makes a proposition like “2 + 2 = 4” true? In this, the first of his two paragraphs, Holt—the brilliant, incisive writer—sets the stage for spelling it out. This is what he has said:
According to one group of thinkers in Godel's Vienna, "2 + 2 = 4" is true because this rather familiar arithmetical proposition "can be derived in a logical system according to certain rules."
Tell the truth, dear general reader: Do you have the slightest idea what that statement means?
Tell the truth, New Yorker subscriber: Do you understand what it means, even in a general sense, to "derive [an arithmetical proposition] in a logical system according to certain rules?"
Friend, of course you don't! But if that's what one group of thinkers were thinking, at least one other brilliant thinker was brilliantly thinking this:
According to the greatest logician since Aristotle, "2 + 2 = 4" is true because it "correctly describes [an] abstract world of numbers." Perhaps more precisely, it correctly describes one aspect of the perfect, timeless existence enjoyed by numbers and circles outside the human mind!
Europe was struggling between two wars. Tearing their hair in the loftiest circles, the western world's most brilliant "thinkers" were laboring over this.
That being said, did Holt go on to make Godel's approach to this matter brilliantly clear? More specifically, was he able to explain Godel's "incompleteness theorems" in a way the general reader might find wonderfully clear?
That's what major reviewers have suggested. But at this point, does any of this seem especially clear?
Tomorrow, we're going to ask you to strap yourselves into your seats. We'll quickly revisit this first paragraph, into which a substantial amount of incoherence has already been poured.
Then, we'll look at Godel's endless succeeding paragraph, in which he attempts to describe the working of Godel's theorems in a way the general reader will be able to comprehend.
We humans! Seeing ourselves from afar once again, major reviewers have seemed to say that Holt did a wonderful job!
Tomorrow: "Beginning with a logical system for mathematics..."
THE PLATONIST FILE: Digest of reports!
MONDAY, SEPTEMBER 17, 2018
New chapter starts tomorrow: Friend, do you know what a Platonist is? How well were Rebecca Goldstein and Jim Holt able to explain the puzzling "doctrine" of Platonism?
Tomorrow, we start our "incompleteness file"—our reports on the efforts by Goldstein and Holt to explain Kurt Godel's "incompleteness theorems."
Those reports start tomorrow. For today, we offer these links to last week's reports from the Platonist file:
Tomorrow, we start the incompleteness file. With what degree of clarity did Holt explain Godel's theorems?
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.For links to our earlier reports from the Godel file, you can just click here.
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.
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:
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:
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:
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:
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:
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:
That said, many members of our tribe have followed King down that road. Inevitably, the Times felt the need to print this ridiculous letter:
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.
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 finalThe 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?This is the way we pseudo-liberals now play the game. But as we kept reading, that basic question recurred:
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.
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.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!
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.
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."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!
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.
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.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.
[...]
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.
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 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!
''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.
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...McEnroe played long ago. He was routinely called a jerk, which is what he routinely was.
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
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:
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 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:
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:
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:
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
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...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.
[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.
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
Subscribe to:
Posts (Atom)