Silliness all the way down: Should the silly old "liar's paradox" make the elite mind crash?
So says Rebecca Goldstein and pretty much everyone else. As we noted yesterday, Goldstein described the problem in her 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel.
It's a puzzling book which was favorably reviewed within the guild.
In her book, Goldstein described the ancient "liar's paradox," which "causes our minds to crash." She hailed the historical brilliance through which Godel was able to turn this "intelligence-mortifying material" into "a proof that leads us to deep insights into the nature of truth."
It took a giant to handle this paradox. Once again, this is the passage in question:
GOLDSTEIN (page 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.In a footnote, Goldstein explained the origins of the liar's paradox. For the record, the paradox tracks back to this:
Paradoxes like the liar's play a technical role in the proof that Godel devised for his extraordinary first completeness theorem. Godel 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. This itself seems almost paradoxical. Paradoxes have always seemed specifically designed to convince us that we are simply not smart enough to take up whatever topic brought us to them. Godel 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.
GOLDSTEIN (page 49): Here is the textual reference the paradox is derived from: "One of themselves, even a prophet of their own, said, 'The Cretans are always liars.'...This witness is true." (Titus 1:12-13).That's where the so-called liar's paradox got its start. In its modern form, Goldstein, and everyone else, presents it as shown below:
"This very sentence is false."This very sentence is false! According to Goldstein, that troubling sentence is false if it's true, and if it's false, then it's true! That means the sentence is both true and false—and, according to Goldstein, "that is a severe problem."
Indeed, it's a severe "philosophical" problem! In our first semester as a college freshman, we studied six such "philosophical problems," including the eternal head-scratcher, "How can we possibly know that 7 plus 5 equals 12?"
(Miss Cummings told us in second grade, we mused. But this wasn't considered a proof.)
This very sentence is false! To Goldstein, this is an example of "self-referential paradoxicality, the sort of structure that causes our minds to crash."
More specifically, Goldstein says it's the sort of structure which causes the minds of our elite logicians to crash. According to Goldstein, it took "the greatest logician since Aristotle" to wrestle this beast to the ground.
When we first read Goldstein's book, we were amazed to see a ranking philosophy professor promulgating such manifest foolishness. In our view, it was astounding that a ranking philosophy professor could have been presenting such manifest nonsense in the year of our lord 2005, with her work being lauded in blurbs by three high-ranking academic authority figures.
Beyond that, it was amazing to think that such manifest nonsense was being promulgated in the year of the later Wittgenstein 52—that is, some 52 years after the publication of the later Wittgenstein's seminal work, Philosophical Investigations.
The later Wittgenstein had been very hot when were philosophy majors, back in the late 1960s. We were amazed to see a ranking professor who seemed to have gained nothing at all from that famous figure's murky but highly instructive later work.
Our view? Goldstein's presentation is manifest nonsense, bordering on something resembling insanity. The problem in the professor's work starts with this odd claim:
"This sentence must be, like all sentences, either true or false."Every sentence must be either true or false? On what planet has this elite yet crashing mind been dwelling all these years?
Is it true? Must every sentence (every statement) be either true or false? That sounds like an obvious statement of fact, but it should be scored as grossly misleading or as just plain wrong.
The essence of the later Wittgenstein's work takes us to an important point. Some sentences—especially those we create "when doing philosophy"—are in fact neither right nor wrong!
Instead, such sentences are incoherent. They may seem to display the apparent form of familiar, coherent claims, but, just as a matter of fact, they're nonsense all the way down.
The silly "This very sentence is false" is one such (apparent) sentence. Let's compare it to other, perfectly sensible sentences which display a roughly similar form:
Some (apparent) sentences don't make definable sense:Those five sentences may seem to share a common form. But the bottom four senteces, when offered in a sensible context, will in fact make perfect sense.
"This very sentence is false."
"The first sentence in this morning's news report is false."
"The second sentence on page 32 is false."
"Every statement you've made today has been false."
"Trump said Obama was born in Kenya. That statement was utterly false."
The first sentence doesn't make sense at all.
What makes the bottom four sentences different from the first? In each of the bottom four sentences, a two-step process has occurred. Some external statement was made, and this statement was then declared false.
In each case, you can repeat, or point to, the pre-existing sentence which is said to be false. You can look at this morning's news report and see what the first sentence said. You can turn to page 32 and read the second sentence.
In each of these cases, a two-step process occurs. First, someone mas made a statement, expressing it in the form of a sentence. Then, someone says that the statement in question is false.
That doesn't happen in the formulation which makes the elite mind crash. The familiar phrase "is false" appears in the first sentence listed above. But no external statement has been made. There is no pre-existing statement which is being described as false.
Let's make this as simple as possible. You can't sensibly say that a statement is false until a statement has been made.
That said, what statement is being declared false in Goldstein's intelligence-mortifying example? No such statement has been made! No statement is available to be rejected as false.
It's amazing to think that a ranking professor was still piddling around with such nonsense in 2005. It's especially amazing when we see major intellectuals on the dust jacket of her book, hailing the professor for her brilliantly lucid work.
Beyond that, we reach the most amazing idea of all—the idea that the greatest logician since Aristotle built "his extraordinary first completeness theorem" out of the blatant silliness being put on display here. We're left with the basic questions we've been raising all along:
Is it possible that our elite logicians have never been all that sharp? More generally, is it possible that we self-impressed human beings are barely "rational" at all?
Goldstein's dance with the liar's paradox is silly all the way down. The foolishness comes into stark relief at a later point in her book.
On page 166, Goldstein is finally trying to explain the essence of the reasoning which lies behind Godel's theorems. We'll look at some of that material tomorrow. For today, consider the highlighted material at the end of this passage, in which Goldstein returns to the liar's paradox:
GOLDSTEIN (page 166): By tradition, the liar's paradox is attributed to the Cretan Epimenides, who reputedly said something implying: All Cretans are liars. This sentence, in itself, isn't paradoxical, except insofar as it suggests that what Epimenides was saying was something like this:This very sentence is true! Goldstein is able to see that this (apparent) sentence is extremely strange, in that it lacks any content.
This very sentence is false.
Now that sentence, as we've already seen, is true if and only if it's false—not a good situation, logically speaking. Godel's strategy involves considering an analogue to that paradoxical sentence, viz. the proposition:
This very sentence is not provable within this system.
Let's call this sentence G. G, unlike its analogue, isn't paradoxical, though it is, like all self-referential sentences, somewhat strange. (Even the non-paradoxical self-referential This very sentence is true is mystifyingly strange. What's it saying? What's its content?)
She's right! But the same is true of her original sentence, the one which makes the mind crash. That (apparent) sentence lacks any content too. Putting it another way, it's neither true nor false.
When we see our ranking logicians fumble and flounder in these ways—when we see their minds continue to crash decades after Wittgenstein's instructive if jumbled later work—might we sensibly wonder about the basic mental ability of our human race?
Presumably, we no longer have to wonder why these elite logicians don't step forward to help us with the broken everyday logic which leads our nations onward toward wars. Frankly, these people don't seem especially sharp, and they seem to live in a bubble.
This very sentence is false? We're sorry, but the ancient "liar's paradox" is just a silly old parlor game; it's piddle all the way down. And yet, according to Goldstein, the greatest logician since Aristotle built his astounding life's work around its mortifying structure!
He then proceeded to starve himself to death. We'd have to say that tragic event was another bad sign.
Tomorrow: It's so easy to understand Goldstein (or so the New York Times said)
Friday: Onward and downward to "Russell's paradox," concerning the set of all sets...