THE INCOMPLETENESS FILE: Incompleteness meets incoherence!

TUESDAY, SEPTEMBER 18, 2018

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

Or at least, so it seemed.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We'll start with an apology. In the past two weeks, we've already posted the start of the first paragraph in question. Before we show you Holt's full paragraph, we'll revisit that part, for review:

HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians...

According to Holt, Godel had come to believe that "abstractions like numbers and circles had a perfect, timeless existence independent of the human mind." As it turns out, Godel was "seduced" by this "doctrine" as a mere freshman in college.

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

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

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

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

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

That's what Godel is asking himself! People, we're just saying:

HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of 1920s Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.

What makes a proposition like “2 + 2 = 4” true?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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