FRIDAY, NOVEMBER 12, 2021
What did Gödel prove? This morning, we got to thinking about the types of highfalutin, upper-end discussions to which we're all expected to bow.
Over the summer, we vowed to restrict ourselves to the analysis of such fare. Eventually, though, the nonsense from the lower slopes caused us to abandon our pledge.
(Michael Corleone said it best: "Every time I think I'm out, they pull me back in.")
At some point, we hope to return to those higher-level discussions. That said, the one question we'd like to see answered concerns—who else?—Kurt Gödel.
Our question would be this:
What did Gödel actually show in his "incompleteness theorems?" What did he actually prove?
Also, is it possible that he didn't prove anything at all? Is it possible that his alleged finding was really an example of the type of "conceptual confusion" the later Wittgenstein discussed?
Our interest in this question stems from several sources:
On the one hand, we've seen no one who is able to explain what Gödel proved in a way which would be comprehensible for the general reader. Then too, there are the nuggets which appear in the "Gödel-made-easy" books.
We refer to these two books, each of which was allegedly aimed at general readers:
Incompleteness: The Proof and Paradox of Kurt Gödel. Rebecca Goldstein, 2005.
Journey to the Edge of Reason: The Life of Kurt Gödel. Stephen Budiansky, 2021.
In our view, neither author was able to explain the incompleteness theorem(s) in a way which would be comprehensible to the general reader. At some point, inquiring minds want to know why that assignment seems to be so hard.
On the other hand, Goldstein included a remarkably puzzling paragraph about the so-called "liar's paradox" and the way "the mind crashes" in the face of its wicked complexity. Due to its sheer incoherence, it remains the most interesting paragraph we've read in the past twenty years.
Meanwhile, Budiansky offers a comical account of Bertrand Russell's tussles with the liar's paradox. Both writers draw a connection between the liar's paradox, Russell and Gödel. In the face of their material, we can't help flirting with the unthinkable:
Is it possible that this whole mishegoss is just a ball of confusion, as the later Wittgenstein seems to have semi-alleged?
Gödel is widely described as the greatest logician since Aristotle. That said, what the heck did he actually prove? And why can't anyone seem to describe his famous theorem(s) in a way which would be comprehensible to the general reader?
What, if anything, did Gödel prove? We're assigning Kevin Drum the task of handling the submissions!
Tomorrow: Kenosha, flat and round