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.
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.