EXPLANATION: What was Gödel talking about?

THURSDAY, SEPTEMBER 1, 2022

Top scholars, strange explanations: "Love is strange," Mickey & Sylvia said.

(To hear them say it, click here.)

At the highest academic levels, explanations can be very strange. We think of Rebecca Goldstein's 2005 book about Kurt Gödel, the Austrian "logician and mathematician" who is widely described as "the greatest logician since Aristotle."

Gödel composed his "incompleteness theorem" (or theorems) in 1931, when he was just 25. But what was Gödel talking about in those famous theorems?

Goldstein's book—Incompleteness: The Proof and Paradox of Kurt Gödel—was an attempt to answer that question for non-specialists.

Goldstein, of course, is no slouch. She's a high-ranking philosophy professor and a well-regarded novelist. Her book was festooned with blurbs from high-level public intellectuals, praising her ability to make Gödel's work accessible for general readers.

For ourselves, we don't have the slightest idea what Gödel was talking about. Beyond that, we lack confidence in the belief that Gödel's famous theorems even make elementary sense. 

You may think that any such thought is insane. Our response would go something like this:

Who's being naïve now, Kay?

Our concern begins with the following paragraph from Goldstein's book. It's part of a passage in which Professor Goldstein starts to describe or explain  Gödel's theorem.

Goldstein links Gödel's theorem to an ancient paradox. Her presentation starts like this 

GOLDSTEIN (pages 49-50): Paradoxes, in the technical sense, are those catastrophes of reason whereby the mind is compelled by logic itself to draw contradictory conclusions. Many are of the self-referential variety; troubles arise because some linguistic term—a description, a sentence—potentially refers to itself. The most ancient of these paradoxes is known as the "liar's paradox," its lineage going back to the ancient Greeks. It is centered on the self-referential sentence: "This very sentence is false." This sentence must be, like all sentences, either true or false. But if it is true, then it is false, since that is what it says; and if it false, well then, it is true, since, again, that is what it says. It must, therefore, be both true and false, and that is a severe problem. The mind crashes. 

We rank that as the most interesting paragraph of the current century. We say that because Professor Goldstein plainly isn't some shlub. She's a high-ranking philosophy professor whose work is praised, all over the dust jacket, for its sheer lucidity.

Her presentation comes from the highest ends of the academic world. Also, it makes no sense. Let's run through this nonsense again:

In that presentation, Goldstein refers to an "ancient paradox" widely known as The Liar's Paradox. In reality, this ancient "paradox" operates on the level of a silly carnival sideshow. But as we see, Goldstein will go on to link The Liar's Paradox to  Gödel's great theorem.

Why would we compare this "paradox" to a silly card trick? Let's run through it again:

In Goldstein's accurate presentation, The Liar's Paradox centers on this (allegedly) confounding collection of words:

"This very sentence is false."

If that sentence is true, then it's false, Goldstein says. But if it's false, that means that it's true!

In the face of this "catastrophe of reason," Goldstein says "the mind crashes." Sadly, attempts at explanation can be strange, even at the highest levels of academic authority.

What makes this card trick so silly? Let's get clear about the factor which makes the collection of words in question seem so confounding, so strange. In doing so, we'll employ the type of (rather simple-minded) analysis the later Wittgenstein would have employed, along with some of his language.

This very sentence is false! On the surface—in what we might call its "surface grammar"—that collection of words seems to resemble such familiar statements as these:

Rudy Giuliani's statement about the Dominion voting machines is false. 

The very first sentence in John Smith's new book is demonstrably false! 

What you just said is absurd, bogus, ludicrous—false. 

"Barack Obama was born in Kenya?" We're sorry, but that's plainly false.

Such statements are wholly recognizable parts of everyday public discourse. Please note:

In each instance, someone has already made a statement. Then, a second party comes along and says that this statement is false.

In the slightly twee language of Wittgenstein's Philosophical Investigations, this is a very familiar type of "language game." The "language game" goes like this: 

A familiar type of occurrence: 
Person A makes a statement (Statement A). Then, Person B comes along and, in a second statement (Statement B), he says Statement A is false. 

This sort of thing happens all the time. Everyone is familiar with this sort of exchange.

Continuing, please notice this:

On the surface, the collection of words which makes Goldstein's mind crash seems to resemble the statements we've posted. On the surface, it looks like someone is declaring some statement to be false.

But in the case which makes Goldstein's mind crash, there is no pre-existing statement! No one has actually made a statement—a statement which can then be challenged! And, until someone has actually made a statement, there is no way to declare a statement false.

In the circumstance which makes Goldstein's mind crash, there is no pre-existing statement to be declared false! It's a bit like conducting a boxing match in the absence of an opponent.

The Liar's Paradox has been around forever. It does indeed date to antiquity. It has confounded college freshmen since the dawn of time.

Also, it operates on the level of a silly card trick. It's astonishing to think that a major philosophy professor in the post-Wittgenstein 21st century could think that it represents a "catastrophe of reason"—could suggest that such silly-bill nonsense can and does make "the mind crash."

Left on its own, the paragraph we've posted above represents an indictment of Professor Goldstein on the occasional bad day. That said, the paragraph can't be left on its own. In the book which scholars praise for its lucidity, Goldstein continues as shown:

GOLDSTEIN (pages 49-50): Paradoxes, in the technical sense, are those catastrophes of reason whereby the mind is compelled by logic itself to draw contradictory conclusions. Many are of the self-referential variety; troubles arise because some linguistic term—a description, a sentence—potentially refers to itself. The most ancient of these paradoxes is known as the "liar's paradox," its lineage going back to the ancient Greeks. It is centered on the self-referential sentence: "This very sentence is false." This sentence must be, like all sentences, either true or false. But if it is true, then it is false, since that is what it says; and if it false, well then, it is true, since, again, that is what it says. It must, therefore, be both true and false, and that is a severe problem. The mind crashes. 

Paradoxes like the liar's play a technical role in the proof that Gödel  devised for his extraordinary first completeness theorem. Gödel was able to take the structure of self-referential paradoxicality, the sort of structure that causes our minds to crash when considering "This very sentence is false"—and turn it into an extraordinary proof for one of the most surprising results in the history of mathematics... Gödel  was able to twist the intelligence-mortifying material into a proof that leads us to deep insights into the nature of truth and knowledge and certainty.

Really? Gödel was able to take the foolishness of The Liar's Paradox and "twist [it] into a proof that leads us to deep insights into the nature of truth and knowledge and certainty?"

On its face, this notion seems absurd. That said, scholars who try to explain Gödel's theorem routinely cite The Liar's Paradox as they stumble about, attempting to make his work accessible to the general reader.

Here, for example, is Professor Douglas Hofstadter. This passage appears early in his endless, Pulitzer Prize-winning book, Gödel, Escher, Bach: An Elegant Golden Braid (1979):

HOFSTADTER (page 17): In its absolutely barest form, Gödel's discovery involves the translation of an ancient paradox in philosophy into mathematical terms. That paradox is the so-called Epimenides paradox, or liar paradox. Epimenides was a Cretan who made one immortal statement: "All Cretans are liars." A sharper version of the statement is simply "I am lying" or, "This statement is false."...It is a statement which rudely violates the usually assumed dichotomy of statements into true and false, because if you tentatively think it is true, then it immediately backfires on you and makes you think it is false. But once you've decided it is false, a similar backfiring returns you to the idea that it must be true. Try it! 

"Try it!" Hofstadter says. This is precisely the sort of manifest dumbness which animates that interesting paragraph from Goldstein's later book.

The dust jacket of Goldstein's book is festooned with blurbs in which major public intellectuals praise her brilliance and her lucidity. (Brian Greene is one such advocate.) Within the book, the reader will find manifest foolishness of the type we've described.

Along the way, the reader is told that the greatest logician since Aristotle built his great pair of theorems out of the kind of foolishness involved in The Liar's Paradox. At this point, the mind should crash—or should at least consider doing same.

What did Gödel actually say, discover, demonstrate, find? Like you, we have no idea. We've never seen an account of his work, including Goldstein's, which make his theorems accessible to general readers. Based on decades of brutal experience, this leaves us wondering about a possible "catastrophe of reason" in which the greatest logician since Aristotle was offering incoherent work.

As a sensible person, the reader will be inclined to think that no such thing could be possible. The reader will assume that our highest intellectual authorities couldn't possibly be singing the praises of work which, in the end, simply doesn't make sense.

The later Wittgenstein had a standard retort to any such sense of assurance. His retort would have gone something like this: 

Who's being naïve now, Kay?

Wittgenstein said that much of our highest-order academic work is built upon the type of "bewitchments of language" (Professor Horwich: the "linguistic illusions") found in that ridiculous passage from Professor Goldstein's book. 

If you find it hard to believe that our highest academic authorities could have functioned that way down through the ages, we'll invite you to come back tomorrow. We'll be probing this embarrassing question:

Where does the number 2 live? 

Love is strange, Mickey & Sylvia said. Wittgenstein noticed how strange it can get when our top academics start offering explanations of the way the world works.

Tomorrow: Where does the number 2 live?

Next week: Einstein himself explains