WEDNESDAY, JUNE 22, 2022
Drums along the theorems: We should have known that Kevin Drum would stab us in the back!
We'd selected Kevin to serve as judge of those who took The Gödel Challenge. Under the rules, contestants would send him their attempts to explain Gödel's incompleteness theorems.
He would decide if anyone had been able to explain the theorems in a way which made actual sense.
Instead of playing by these rules, Kevin has gone ahead and explained Gödel's work by himself! Or at least he has tried to do so. Now, you'll have to be the judge—and that's hardly fair to you.
First question: Has Kevin explained Godel's theorems in a way you can understand? Beyond that, have you come away with any idea why anyone is supposed to care about Gödel's hugely significant work?
You'll have to answer those questions yourselves. For today, we'll note an intriguing part of what Kevin says—something we assume is completely correct.
Kevin's elucidation comes in four parts. At the start of Part 3, he alarmingly tells us this:
So far, what Gödel has done is inventive and easy to understand. What comes next is world historically insightful and more or less impossible to understand for non-mathematical laymen. But let's go ahead with a simplified version.
Kevin has made an interesting claim in that passage. He has said that Godel's work is "more or less impossible to understand for non-mathematical laymen."
This suggests the first point we'd be inclined to make about Gödel the logician. Even if his work turns out to make sense, it's going to play zero role in the daily lives of non-specialists.
Gödel's work is "more or less impossible to understand for non-mathematical laymen!" That doesn't mean that his work is "wrong" in some respect. Imaginably—you may have to imagine hard—his work could even be useful in some way, in some technical realm.
That said:
Kevin says that Godel's work is "more or less impossible [for you] to understand." Given the role of logic in our nation's daily discourse, we'd call that an interesting statement—and as he continues his explanation of Godel's work in Parts 3 and 4, he keeps repeating variants of this assessment:
"In day-to-day use, Gödel's theorem plays no role," he says at one point. A bit later, he offers this:
"Working mathematicians go through life never knowing anything about Gödel, who is of interest mostly to abstract logicians."
Even if we assume that Godel's work makes sense on its own terms, it seems to be disappearing into the ether. Then, at last, the coup de grace:
Long story short, Gödel's theorem is both enormously important but also of little use in real life. This is the way of things.
Whether it makes sense or not, Gödel's theorem is of little use. Of little use in real life!
If that is true, it isn't immediately clear why Gödel's theorems would be "enormously important," or even important at all. But this does suggest the initial point we've been aiming at in our first two presentations this week:
You'll recall the starting point for this week's exploration. We were working from this presentation by the leading authority on Gödel's ballyhooed work:
Kurt Friedrich Gödel (1906 – 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Frege.
According to the leading authority, Gödel was one of the three most significant logicians in all of western history. That said, we've now been told, we'll assume correctly, that his work "is of little use in real life."
Indeed, his work is of so little use that it's "of interest mostly to abstract logicians," whoever they may be. Forget about the average shlub! Even most working mathematicians are said, we assume correctly, to "go through life never knowing anything about Gödel" at all!
What does it mean when the most significant logician of the past century—one of the two most significant logicians of the past several thousand years—has produced work which is said to be of interest to almost no one? Among other things, it could mean this:
It could mean that no one pays any attention to the problems of what might be called "daily logic." This may help explain the three million logical errors which infest our failing national discourse on an hourly basis, and then on into the night.
We haven't answered the basic question—does Gödel's work even make sense? For reasons we'll eventually cite, we still wouldn't assume that it does. We wouldn't assume that it doesn't.
That said, no one has heard of the two most significant logicians of the past several thousand years! The professors withdraw while the journalists flail. Mister Trump rises to power.
"We haven't answered the basic question—does Gödel's work even make sense?"
ReplyDeleteOh dear. "Basic question"? Why is it suddenly "the basic question"? What do you care, dear Bob?
Does it help you forget about Dark Lord Putin remotely hypnotizing the owners of your neighborhood gas stations and supermarkets?
Somerby should have read Drum’s comments section.
ReplyDeleteHaving studied the Axiom of Choice, and being generally familiar with Godel's Incompleteness Theorem, I tend to agree with Bob. I'm just not convinced of their huge importance. The Axiom of Choice is only significant when one is dealing with infinite sets -- which have not real world counterpart.
ReplyDeleteMust Godel’s work be important to you, to be important to anyone?
DeleteTDH is making numerous logical errors himself in his whole schtick about whether anyone can understand Godel, and whether this has anything to do with all the 'logical errors' that are made all the time by everyone. One reason is that logic doesn't apply all that much to politics, because other factors come into play, like different people having different self-interests, and because there are no answers to what must be done to anywhere near mathematical certainty. Also, politics is about power, and there is a logic to gaining and keeping power, but that logic is not consistent with doing what is best for all persons. TDH is on a path to nowhere here. And is it true that just like the countless lists of the best movies, the best movies on Netflix, the best songs of the 80s, ad infinitum, there are also lists of the best logicians of the last thousand years?
DeleteYeah, well said, dear AC/MA.
DeleteBlack Square by Malevich was sold (google tells us) for $85 million. And yet it's just a black square. Dear Bob must be flabbergasted. A great reason to denounce Aristotle; nicht wahr, dear Bob?
Why don't you, dear Bob, worry about things you understand. Like media hacks and their dumb talking points.
Mao,
DeletePlease remember to forget to mention the Supreme Court your fascist hero with the orange hairpiece put together, sides with your Establishment bosses on the regular.
Thanks in advance.
So it turns out Mao’s hero is the best thing to happen to the Establishment since Reagan. If that’s a coincidence, I don’t know what isn’t.
DeleteThe old saw is that democracy depends on an educated population. This is true but we don't have one. Research comparing wealthy and to non wealthy influence in government gives a completely flattened curve to American democracy At the most powerful levels, the wealthy pick policy. So much for the next election solving everything. You have to confront the wealthy.
ReplyDeleteAlso, considering that elections are run by Facebook ads, tv, radio spots, wouldn't it make more sense to be aware of marketing and misdirection techniques than simply "logic"? Yes it's condescending to say that I know philosophy and you don't. But it's a little condescending also to insist that voters simply start with "logic" before attempting bigger ideas.
I should add, to some degree you can organize without confrontation. You can just find people to join your organization.
DeleteLarge confrontations will happen if you are doing it right you just have to be ready and pick your battles.
Maybe even some sociology...
DeleteBob keeping Gumby Theatre alive.
ReplyDeleteHow can Somerby say that Godel is "of little use" when he himself doesn't understand how Godel's work is applied in fields such as computer science and modeling? This point was raised several times in Drum's comments. Drum at least reads his own comments, so perhaps he can set Somerby straight about that, but it is frustrating that the communication on this website is so one-way. When Somerby makes an error, he just goes on making it and making it, because he won't listen to feedback.
ReplyDeleteBack in the early days of the space race, people questioned why money was being wasted on space exploration when there were serious problems on Earth. After the technology developed specifically for space was applied to other problems, people had a better sense that there was more benefit to space travel than simply putting a flag on the moon. The same is true of the Mars program, even though we have not set foot there yet.
But it takes someone with a technical background to point out how incomprehensible, perhaps obscure knowledge makes our lives better in pragmatic ways. Somerby doesn't know enough to make those connections, but others have been, and they DO exist, as they do for even the most obscure academic pursuits.
Back when Senator Proxmire used to give awards for the biggest waste of government funding, some of the things he considered useless turned out to be important, even if he didn't recognize it, limited as he was by a non-technical education. One of the papers he singled out was by George Sperling, a member of the National Academy of Science. It was mathematical. This was, of course, a political stunt. But so is Somerby's approach here. And most likely for the same political purposes. Proxmire won Joe McCarthy's seat in WI, and perhaps needed to appeal to conservative rural voters, not just Democrats. So he adopted an anti-intellectual, pragmatic stance that would appeal to populist voters in both parties. That doesn't make him right.
Somerby, on the other hand, has no excuse for his continual attacks on expertise, knowledge, higher education.
Here are the comments of Janna Levin on Godel. She is a professor of Physics and Astronomy at Barnard College and has also written a book about Godel:
ReplyDelete"SC: To what other areas, outside of mathematics and logic, such as physics, theology, or philosophy, did Gödel contribute?
JL: He most directly, himself, personally contributed to physics. Obviously, his ideas had big implications that inspired other people to carry on, in probably all fields like theology, philosophy, physics, metaphysics, artificial intelligence, computer science—to everything he really was very influential. But he, himself, contributed to physics in a very interesting way, and it was kind of a similar idea to throwing a wrench in the works a bit.
He became very close to Einstein when he was at the Institute. And Einstein once said, “I only go to the Institute for the pleasure of Gödel’s company.” I think people have probably heard that quote before. In fact, he was such a recluse, Einstein might have been one of the only people he was speaking to, for months at a time, on these walks. And he became interested in Einstein’s ideas about the relativity of space and time.
Einstein’s theory of a curved space-time is a theory that you and I, traveling at different speeds or in a curved space-time, or somebody near a black hole, or somebody else, might measure, literally, the passage of time differently. We might age differently from each other. I won’t notice a difference. But you might look at me, near a black hole, and think, I’m aging very slowly, like thirty years have elapsed for you and only one year has elapsed for me. So there’s this very strange but very real relativity of space and time in Einstein’s theory.
So what Gödel did, he was able to construct a very unusual space-time, one which was rotating, which is not what our universe is doing, but just a hypothetical—imagine a universe, which is rotating in some specific way—and he was able to show that he could find specific observers in that world who could travel back in time."
.....
The universe probably is rotating. I mean it is flying apart after the big bang but probably also rotating. I mean, every other part of it is. The earth rotates and also revolves around the sun, the sun revolves around the galaxy and the galaxy revolves around some galactic cluster (at least according to the goog).
DeleteBut the ability to prove things about an imaginary universe does what? And Einstein's theory of a curved space-time does what?
And then there is this, continuing in the interview with Levin:
ReplyDelete"SC: Did he not also come up with some ontological arguments on the existence of God?
JL: I think many times in Gödel’s life, he was interested in spiritual things. One was the transmigration of the soul, one was this kind of platonic reality, the existence of this pure, mathematical reality. And another was proving the existence of God. And he did struggle with trying to write these proofs of the existence of God.
I don’t think he ever got to a stage where he was particularly impressed with the outcome, and it seems that he had often kind of abandoned it. But later in his life, he did come back to it. And one of the things that bothered him so much—not just about the existence of God, but also about free will and the spirit and the soul—was this idea that he inspired that thought could kind of be mechanized, in a certain way. This is something that Alan Turing, the great British codebreaker who cracked the German Enigma code in World War II—it’s something that Alan Turing really brought forward.
He said, “Actually, I can mechanize this whole process of thinking about mathematics,” and he was totally inspired by Gödel. He was using Gödel coding and was very influenced by his results. And he basically invents the idea of the computer in the process. I build a machine. The machine mechanistically goes through certain operations, given certain input. And it performs mathematical operations. It thinks.
And Turing really believed he was going to build a machine that could think as well as a human being. And a lot of Gödel’s ideas get wrapped up in this. Now, Gödel could not argue with the correctness of Turing’s formalism and his mathematical proofs, but he was very disturbed by the idea that the human mind could be reduced to this mechanistic approach, even though he, himself, had sort of influenced this idea. And I think a lot of the trying to prove the existence of God was wrapped up in trying to reject Turing’s claim that we were reducible to machines."
https://www.simplycharly.com/interviews/janna-levin-kurt-godel-incompleteness-theorem-numbers-game/#.YrSLA3bMLSI
This, of course, would not impress Somerby, given that he is not religious, but the connection between Godel and Turing should. Somerby might be inclined to argue that Turing's work also made no contribution to anything (despite its importance in computer science), except that Turning managed to crack the Enigma code. He is credited with saving millions of lives by enabling the British to decipher German messages, especially about U-boat traffic in the Atlantic (a supply route for the Allies). Turing's anti-enigma machine was installed in 1940 and was breaking 84,000 messages per month (2 per minutes) by 1943. Broken messages provided information that changed the course of the war.
But math isn't important to Somerby, whether it is Godel, Einstein, or Turing.
Drum says it is "more or less impossible to understand for non-mathematical laymen".
ReplyDeleteThat makes me wonder what Drum's own background is. Does he have a PhD in math? or engineering? or even a BA? If not how can he explain that which is impossible for a layman to understand?
Yeah, I suppose there is a lack of ivory-tower “logicians” making political stands, though Chomsy, famously as always, expressed disappointment with his comrades when he was protesting in the various enterprises afoot at the time. But it’s not like the MSM gives flying fuck anymore about “public intellectuals.” They just don’t fit the narrative.
ReplyDeleteI wonder if Wittgenstein was an anarchist in the Chomskian sense – any institution which impedes human progress is allowed to be dismantled. Philosophers don’t seem to be saying much to the hoi-polloi, for reasons stated above, but what do our high-salaried, “professional” philosophers, actually produce for their work? The meaningful stuff has already been done.
Leroy