TUESDAY, JUNE 21, 2022
This is what he did: As we noted yesterday, he's one of the three most significant logicians in the history of the western world. Or at least, that's pretty much what the leading authority on the widely-ignored topic says.
We're discussing a time span of roughly 2500 years! Let's refresh ourselves:
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.
Aristotle, Frege and Gödel! Many people have heard of the first of the three. The other two, not so much!
Yesterday, we asked you what that somewhat peculiar fact might possibly mean. What does it mean when no one has heard of Frege and Gödel—when no one has heard of the two top logicians of the past (more than) two thousand years?
Today, we'll postpone an attempt at an answer. Instead, let's continue along with the leading authority as we start to learn what Gödel actually did.
According to that overview, Gödel was the most significant logician of the 20th century. Continuing along from the text shown above, this is what he did:
Gödel published his first incompleteness theorem in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any ω-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the natural numbers that can be neither proved nor disproved from the axioms. To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency.
Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming that its axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
No one has ever heard of Frege, or of Gödel either. Can you start to see why that might be? Does this start to suggest some thoughts about possible flaws, or even shortcomings, in our failed intellectual culture?
"...there are true propositions about the natural numbers that can be neither proved nor disproved..."
ReplyDeleteHmmm. There are statements that are neither provable nor disprovable. It seems to us, dear Bob, that there is no reason to call them "true", in this context. That's the whole point.
They can't be proven or disproven within the system. However, they are constructed in a way that, from outside the system, we can see that the statement is true. The statement is "This statement cannot be proven"
ReplyDelete"No one has ever heard of Frege, or of Gödel either. "
ReplyDeleteThis is untrue. Godel was taught in the first class in modeling in my graduate program. His theorem is important to computer science. Others have said the same thing here. Somerby doesn't listen and he imagines that his own ignorance encompasses "everyone," which is very far from the truth.
It is a waste of time to continue to discuss Somerby's obsessions, each time they roll around. Instead, lets ask why Somerby is so invested in taking obscure but important scholars and telling us that no one cares about them or their work. What does Somerby get out of doing this?
Yes, it is anti-intellectual and it does undermine the idea that there is such a thing as expertise -- even though the technology that surrounds us must convince at least some people that knowledge has contributed to our lives in various ways, some incredibly important. But why does Somerby seem to pick men who were considered insane in their later years? Is this some sort of defense mechanism now that he hiself is reaching old age without much to show for his life? If so, that is something to work out with a therapist. Maligning people like Godel, who made important contributions, just makes Somerby seem small, and kind of sad. It does nothing to diminish Godel, except among people as ignorant as Somerby is.
This cuts both ways. When Bob says "nobody"reads an author, and you say that you study him in academia, you are defending academics as people.
DeleteHowever, Bob is something of an egalitarian, or at least stresses even handedness. To him, the exclusion of the general public from reading Gödel (with maybe some exceptions due to the 1999 book) is the priority. In essence, he is saying that people who are not academics are also potential philosophers.
Being a little cynical one might say Bob isn't that impressed with Gödel. Maybe closer to the point, Bob isn't impressed with other writers who are impressed with him.
DeleteYes, as noted previously, Somerby thinks that no effort should have to be expended to understand something complex and technical. Being unimpressed by something that should be impressive to you, is a sign of stupidity and ignorance.
DeletePeople who are not academics may be potential philosophers, but they would have to expend the effort to understand. Expending such effort would make them no longer members of the general public but experts in what they are studying. Then Somerby would be unimpressed with them too.
So, I guess Bob didn’t watch the hearings.
ReplyDeleteWhat a surprise.
This is choice:
ReplyDelete“No one has ever heard of Frege, or of Gödel either. "
This is untrue. Godel was taught in the first class in modeling in my graduate program.”
Are you seriously having difficulty with the difference between “no one” and “a number of people greater than zero?” I became aware of Gödel via a popular book called “Gödel Escher and Bach.” You are really something, Cecilia.
DeleteI think you’re seriously having difficulties with being too literal as to “nobody” and that the reference point was Aristotle.
DeleteOnly people who took a humanities course in college know who Aristotle is. I’d bet there are more computer science majors these days. Is greatness among thinkers really a popularity contest, as Somerby suggests?
DeleteI learned about Aristotle in high school Western Civ, but that’s only because A is more popular than G.
ReplyDeleteOtherwise, psychology majors wouldn’t have to wait till graduate school to pretend to understand him.
There is a difference between cognitive science (especially computational modeling of cognitive processes) and psychology (which is 90% clinicians). Psych majors don't study Godel, even in grad school.
DeleteIt is odd that Somerby keeps mentioning Aristotle, when Aristotle has no practical relevance to modern life and got as much wrong as he did right in his natural philosophy (corrected by Isaac Newton, among others). Not long ago, Somerby was complaining that philosophy has no relevance, which is also wrong, given that philosophy majors go on to become lawyers and ethicists (now a career path in medicine and other fields making life and death decisions).
Over at Kevin Drum's blog there are a bunch of people in the comments who are demonstrating that they understand Godel and find value in his work. Better books for understanding Godel's theorem are mentioned. This is what a more intelligent discussion of Godel, one that takes his work seriously (unlike Somerby) looks like.
ReplyDeleteYes we all know Somerby can do no right! When he posted a short message that he was taking the day off, it was picked apart and criticized. Any and all references are scrutinized in ways that aren't even in context. Tongue-in-cheek comments are taken literally and extrapolated upon wildly. But I guess people need to pass the time somehow.
DeleteMaybe you have to be educated to find his anti-intellectualism offensive, or his constant nitpicking of female and black writers, or his unattributed borrowing from authors or his misogyny or his constant apologies for right wing atrocities (Roy Moore) or his ongoing bigotry. I assume he is amusing if none of that bothers you, and you din’t mind sophistry and mangled stats. Why don’t you just skip the comments so you can enjoy him untarnished?
DeleteHow would you know what is tongue in cheek?