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
Gödel, schmodel.
ReplyDeleteForget Gödel, dear Bob. What do the anthropologists living in the cave inside your head say about Mrs Pelosi's Big War next year? Anything? Please tell us, dear Bob.
Nobody knows what the hell you're talking about
DeleteGood for you, dear Nobody.
DeleteMao, you're a brown noser for American conservatives, and when you have nothing left to say you smear bullshit on your mouth so you seem like you're keeping busy
DeleteTsk, tsk, tsk.
DeleteSay what you will about Mao, but he sure loves the Establishment.
DeleteHere we see our best and brightest, relentlessly getting to the heart of the most intricate aspects of politics.
DeleteOh wait, wrong thread.
(no offense intended just a Friday funny)
Bob won't read this, but it's not hard to explain WHAT Godel proved. It harder to explain HOW he proved it.
ReplyDeleteThe first step is to understand that the natural numbers can be generated from a set of axioms. This is comparable to how geometry is derived from a set of axioms. Eg. see Peano Postulates https://en.wikipedia.org/wiki/Peano_axioms
Starting with these axioms, one can prove, e.g., that 1+1=2. One can also prove many more complicated things. E.g., a few years ago, a mathematician finally proved Fermat's Last Theorem. https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
What Godel proved is that there are statements about natural numbers that are true but which cannot be proved within the system. This is puzzling concept. How can we know something is true if it can't be proved? You'll have to follow Godel's actual proof for the answer.
Is Bob asking whether this result have real-world consequences? That's a deep question. I don't know the answer off hand.
If memory serves, we believe wikipedia is lying to you, David. It's not that there are propositions "that are true but which cannot be proved within the system". It's that there are propositions that can be neither proved or disproved. They are neither true nor false. They are "undecidable".
DeleteDavid is correct. There are undecidable statements which are nonetheless true.
DeleteI wouldn't be so quick to bash wikipedia for accuracy when it comes to non-political content.
ReplyDelete"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem
How to apply to "real-world" though?
DeleteI'll take a stab.
Can a language describe the ways it fails to pass information accurately?
Can the human brain comprehend its own limitations?
These are perhaps futile attempts, but I tried.
Brains are given to animals for survival. Survival of the species. Whether the human brain fits the purpose, tsk, it's too early to tell.
DeleteWell, it fit the purpose well enough when humans were making rudimentary tools and fighting for daily survival.
DeleteWould Bob had survived as a caveman if he had taken the afternoon off to ponder about incompleteness? Obviously things are much different in the Information Age.
Dear Bob appears to feel that egghead logic and philosophy are indeed useless. We tend to concur.
DeleteBut hey, didn't Vonnegut predict that big brains will ruin this species? Sorry, we've forgotten the title of that novel.
Galápagos, google tells us...
DeleteI didn't read any Vonnegut until just a few years ago. Cat's Cradle was fantastic.
DeleteYeah, the sixties. And then it all stopped. Since the eighties it's been like medieval times, nothing remarkable. Obey, consume, conform. And no end in sight.
DeleteThanks Ronald Reagan.
DeleteThanks to Ronald Reagan.
Delete"Also, is it possible that he didn't prove anything at all?"
ReplyDeleteGodel's incompleteness theorem has applications in computer science. Somerby is not a computer scientist, so he doesn't know that. But it is fair to ask what leads him to conclude that there must not be any applications just because he doesn't know what they are -- disregarding the statements of many experts about the importance of Goedel and his work.
This is an example of when scoffing at the expertise of others makes Somerby look like a huge fool.
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