But was this a crazy idea? Many people have heard of Einstein, who enjoyed his walks with Godel.
Aristotle's name is also well known. That said, we'll guess most people wouldn't know that Aristotle is said to have been our greatest logician, at least until Godel came along.
(For our previous Godel File report, click here.)
Godel is hailed as our second greatest logician because of his incompleteness theorems. We'll take a first peek at those theorems tomorrow—but for today, we tip our hat to Rebecca Goldstein.
Goldstein wrote the book on Godel; she did so in 2005. Along the way, she outlined Aristotle's work as a logician, in a footnote on page 55:
GOLDSTEIN (page 55): Aristotle is commonly acknowledged as the father of logic. His work in logic is laid out in the Prior Analytics, which is part of the posthumous consortium known as the Organon. The philosopher had the seminal insight that in a deductive logical argument, some words are logically relevant while others are not. The irrelevant words can be dispensed with by making them variables...Goldstein goes on to offer a "stock syllogism" as an example of what she's talking about. She then says, "The move toward denoting logically irrelevant words with variables was a move toward generality and thus toward the science of logic."
At this point, we start in surprise. Has someone actually made a science of logic? If you read American newspapers, or if your watch our "cable news" programs, you might be greatly surprised by any such allegation.
As everyone must understand by now, "the science of logic" plays almost no role in our modern American discourse. Within our failing culture's pseudo-discussions, basic information and basic facts are almost wholly verboten. But anything resembling logic is typically frowned upon too.
This destructive state of affairs can't be blamed on Aristotle. Because Godel lived in modern times, we'll be a bit less forgiving when it comes to him.
Sad! Modern logicians have created a system in which their highly abstruse ruminations have nothing to do with the bungled logic of our public discussions. Godel lived until 1978. At some point, he probably should have stepped in.
At any rate, our modern logicians walk to work in the clouds, where they engage in high-brow endeavors which even they may not be equipped to explain. This brings us to a fascinating part of Godel's horrifically tragic story, as told by Goldstein in 2005 and then by Jim Holt in the derivative title essay of his well-received new book.
We refer to a possibly crazy idea to which Godel was apparently devoted throughout the course of his life. We refer to Godel's alleged belief in "Platonism," an alleged doctrine with which Goldstein says Godel "fell in love" when he was in his late teens.
In our previous report, we learned an unfortunate fact. Godel, who died of self-starvation, exhibited signs of severe mental illness throughout the course of his highly unusual life.
"At the age of five, he seems to have suffered a mild anxiety neurosis," Holt writes in his new book, drawing on Goldsetin's work. "At eight, he had a terrifying bout of rheumatic fever, which left him with the lifelong conviction that his heart had been fatally damaged."
By normal reckoning, Godel's apparent paranoia eventually led to his death. But then, as noted yesterday, he had always struck the Princeton community as being extremely odd.
In her book, Goldstein cites an array of peculiar ideas Godel is said to have expressed. Our continuing question is this:
Should it seem strange to think that the western world's second greatest logician was apparently in the grip of serious mental illness? Asking a slightly different question, should it seem strange that "Aristotle's successor" was in the grip, throughout his life, of some extremely peculiar ideas?
In theory, the mental illness, however tragic, shouldn't matter at all. If a person achieves some great intellectual breakthrough—in the physical sciences, let's say—that achievement isn't negated by the tragedy of a severe mental illness.
That said, Godel's achievements didn't come in the realm of physical science. Indeed, Goldstein quotes him telling one Princeton-based scholar, "I don't believe in natural science." She quotes him telling Thomas Nagel that he doesn't believe in evolution, offering Stalin's similar disbelief as supporting evidence.
She quotes him saying this to Noam Chomsky: "I am trying to prove that the laws of nature are a priori." Should it seem strange to hear that history's second greatest logician was bruiting such notions around?
In fairness, these peculiar-sounding statements were made during Godel's adult life. He formulated his "incompleteness theorems" when he was just 23.
If those theorems make sense and are actually important in some way, later delusions and mental illness can't vitiate the achievement. That said, we might want to consider the lifelong belief in "Platonism" which, according to Goldstein's book, played the key role in Godel's thinking from his teenage years on.
What the heck is Platonism? According to Goldstein, Godel "fell in love" with the doctrine while he was still a teen.
"First exposure to Plato can be an extremely heady experience for those with a passion for abstraction," Goldstein writes. "It can amount to a sort of ecstasy."
Parenthetically, Goldstein says that she can remember her own first exposure to Plato. But what is the doctrine which may result? What is this "Platonism?"
In the opening essay of Holt's new book, he defines this doctrine, or he at least pretends or attempts to do so. As critics robotically praise the clarity of Holt's writing, he sketches the doctrine like this:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians...Say what? According to Holt, Godel, who was still in his teens, was "seduced by" a certain notion. More specifically (or possibly less so), he was seduced by "the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind."
According to Holt, this "doctrine," which is called Platonism, "has always been popular among mathematicians." In fairness, this suggests that it isn't something the teen-aged Godel somehow dreamed up by himself.
At this point, our review of the Godel file reached a critical turn. We're told that the western world's second greatest logician was swept away by a seductive idea, an idea which lay at the heart of his subsequent work.
What did this greatest logician believe? According to Holt, he believed that numbers have a perfect, timeless existence independent of the human mind. So Kurt Godel believed!
Readers, how about it? Do you believe that the number 3 "has a perfect existence independent of the human mind?" Do you believe that the number 3 has a perfect existence at all?
Indeed, let's ask the more relevant question: Do you have even the slightest idea what Holt is talking about? Granting the fact that you can memorize Teacher's words and reproduce those words on the test, would you have even the slightest idea what you were talking about?
Readers may be inclined to suppose that Holt proceeds from there to explain this murky bit of word salad. In our view, such a supposition would be badly mistaken.
(To peruse his original New Yorker essay, you can just click here.)
Does Holt clarify this description? We'll set that question aside for another day. Instead, we'll turn to Goldstein, to the start of her attempt to explicate "Platonism."
Godel fell in love with the doctrine, she says. She lays out the doctrine as follows:
GOLDSTEIN (page 44): Godel's commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher...Goldstein muddies her account with a fair amount of technical language. (How many general interest readers are likely to know what a "formal system" is?)
Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up that reality. The structure of, say, the natural numbers (which are the regular old counting numbers: 1, 2, 3, etc.) exists independent of us, according to the mathematical realist...and the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical properties of light and gravity.
Beyond that, she seems to think, all through her book, that the terms "objective" and "subjective" explain themselves in contexts like this. Sadly, they very much don't.
Whatever! In this passage, Goldstein starts describing the doctrine with which Godel "fell in love" as a teen. According to Goldstein, the ardor Godel felt for this doctrine impelled him to devise the "incompleteness theorems" which made him famous among certain academics, while leaving him completely unknown to everyone else on the face of the earth.
Godel's ardor for this alleged doctrine forms the core of Goldstein's work. And how does she describe the doctrine? She describes it like this:
According to Goldstein, Godel fell in love with the view that "the truths of mathematics are determined...by the reality of mathematics."
So far, so completely no good! Explicating further, she says Godel adopted the view that "the truths of mathematics are determined by the nature of the entities that make up" the reality of mathematics.
As an overview, she starts by saying that Godel developed a "commitment to the objective existence of mathematical reality!"
Readers, tell the truth. Would you say that the essence of this key doctrine has begun to come clear? We're going to say that we're munching on salad—that it's word salad all the way down!
At this point, we'll raise two questions, after which we'll adjourn for the day. One of our questions will concerns Kurt Godel. The other question will concern writers like Goldstein and Holt, who are robotically praised, by scripted journalists, for the clarity of their work.
Regarding Godel, we'll ask this question: Is it possible that the western world's second greatest logician fell in love at an early age with a crackpot idea?
He believed many other crazy things in the course of his tragic life. Could this idea, which drove the work for which he's lauded, possibly have been the first of his crackpot ideas?
That's a question about our second greatest logician. Our second question concerns Holt and Goldstein—and it's built on a basic concept:
Here's the basic concept. Some claims or ideas are right or wrong. But some claims, and some ideas, don't rise to that level.
Such claims and ideas aren't right or wrong—they're simply incoherent!
This idea is stunningly basic. To what extent are general interest writers like Goldstein and Holt conversant with this idea?
Tomorrow: A Platonist in their midst (we didn't get there today)