The ancient Cretan's tale: According to Rebecca Goldstein, Kurt Godel—he of Godel's astonishing incompleteness theorems—was "the greatest logician since Aristotle."
Goldstein is hardly alone in that assessment. And since Godel was born in 1906, that assessment covers a lengthy time span of well over two thousand years.
A tiny irony lurks here. According to Goldstein (and everyone else), Godel's greatness leads us back to the realm of "ancient paradox"—specifically, to an ancient saying in which an outspoken Cretan was said to have said that all Cretans are liars.
From this ancient formulation, Godel is said to have spun his logical gold. In the passage shown below, Goldstein compares the fame of Albert Einstein with the relative obscurity of his friend, the great logician Godel:
GOLDSTEIN (page 65): We know a great deal about the preoccupations that led Einstein to his special theory of relativity. It is all part of the public record of the scientist who performed the role of the professional genius in the collective imagination of the world...In Godel's hands, "ancient paradox" was the fuel which produced the greatest advances in logic in over two thousand years. This helps explain why Goldstein's favorably-reviewed 2005 book carried this title:
But Godel's genius was never put on public display the way Einstein's was. The sources of [Godel's] inspiration, the play of mind, revealing how ancient paradox could be transformed into a proof for conclusions shot through with meta-overtones, are unknown. He must somehow have glimpsed the metamathematical potential of logic, even when logic was, as it was then, far less mathematically respectable than his own work would render it.
Incompleteness: The Proof and Paradox of Kurt GodelGodel ran on paradox—ancient paradox at that. Paradox was the fuel which drove the greatest advance in logic in well over two thousand years.
So the story is told. But if Godel actually is that "greatest logician since Aristotle," his use of paradox may help us answer our own basic questions—our questions about the role of the elite logician in the affairs of the suffering world.
Our questions run like this:
Why do our elite logicians seem to have so little to say about the everyday logic, or lack of same, which defines our clown-like public discourse?
Also this, as incomprehensible as it may seem:
Is it possible that our greatest elite logicians have never been all that sharp? Is it possible that the world of elite logic resembles a house of cards?
So the later Wittgenstein may possibly have seemed to suggest. At any rate, we'll continue to consider that trio of questions, today and all through the week.
According to Goldstein (and everyone else), how astounding were Godel's achievements in logic? What made him the greatest since Aristotle? What made him a figure whose genius can sensibly be placed alongside Einstein's?
What made Godel so great? In the passage shown below, Goldstein starts to define the ancient paradox which lies at the heart of his astonishing work.
As we move through additional text from Goldstein, we'll see the esteem in which she holds the great Godel—but we'll continue see the outline of the "ancient paradox" which lay at the heart of his work:
GOLDSTEIN (page 49-50): Paradoxes, in the technical sense, are those catastrophes of reason whereby the mind is compelled by logic itself to draw contradictory conclusions. Many are of the self-referential variety; troubles arise because some linguistic term—a description, a sentence—potentially refers to itself. The most ancient of these paradoxes is known as the "liar's paradox," its lineage going back to the ancient Greeks. It is centered on the self-referential sentence: "This very sentence is false.""This very sentence is false!" Goldstein describes that as a "self-referential sentence." Dating its lineage back to the Greeks, she says it creates a paradox—a "catastrophe of reason."
As she continues, Goldstein starts fleshing out this idea. We were amazed the first time we read the short passage which follows:
GOLDSTEIN (continuing directly): ...It is centered on the self-referential sentence: "This very sentence is false." This sentence must be, like all sentences, either true or false. But if it is true, then it is false, since that is what it says; and if it false, well then, it is true, since, again, that is what it says. It must, therefore, be both true and false, and that is a severe problem. The mind crashes."The mind crashes," Goldstein says. In essence, that's the point we plan to make concerning the apparent lack of competence of the modern elite logician, who tends to break down in these ways.
We first read Goldstein's book somewhere around its publication in 2005. We were amazed to see a ranking philosophy professor making these presentations some 52 years after the publication of the later Wittgenstein's seminal work, Philosophical Investigations.
"This sentence must be, like all sentences, either true or false?" It was the essence of the later Wittgenstein's work to say that many sentences are neither true nor false. According to Wittgenstein, that's especially true of the kinds of sentences we generate "when doing philosophy."
Must every sentence be true or false? According to Wittgenstein, many (apparent) sentences are simply incoherent.
These sentences are neither true nor false. Instead, these sentences don't make any definable sense, no matter how they may appear.
This is a very basic idea; it's hardly unique to Wittgenstein. But in the year 2005, there was a major philosophy professor cruising along as if this elementary notion had never been unloosed on Earth!
On its dust jacket, Goldstein's book was favorably blurbed by a trio of ranking scholars. It was well reviewed in the New York Times, as is required within the guild for work done by the elect.
(We'll look at that amazing New York Times review in the weeks ahead.)
Goldstein's book was widely praised within the tents of the clan. (For the New Yorker review, click here.) Along the way, readers were asked to dance around the maypole of the silliest "paradox" of them all, a construct on the approximate level of the (chestnut) tree which falls in the forest when no one is around.
This part of Professor Goldstein's book seemed amazingly silly to us. But as she continued, she oohed and aahed about the brilliance of the logical work to which the silly "ancient paradox" was brilliantly put by Godel.
How brilliant was Godel's use of this paradox? According to Professor Goldstein, his work was as brilliant as this:
GOLDSTEIN (continuing directly): Paradoxes like the liar's play a technical role in the proof that Godel devised for his extraordinary first completeness theorem. Godel was able to take the structure of self-referential paradoxicality, the sort of structure that causes our minds to crash when considering "This very sentence is false"—and turn it into an extraordinary proof for one of the most surprising results in the history of mathematics. This itself seems almost paradoxical. Paradoxes have always seemed specifically designed to convince us that we are simply not smart enough to take up whatever topic brought us to them. Godel was able to twist the intelligence-mortifying material into a proof that leads us to deep insights into the nature of truth and knowledge and certainty.Deep insights into the nature of truth! It also purees and dices!
"This very sentence is false?" To Goldstein, this intelligence-mortifying construct causes our minds to crash. That said, leave it to Godel! In his hands, this silliest of all the paradoxes "leads us to deep insights into the nature of truth and knowledge and certainty," and what's not to like about that?
"This very sentence is false!" According to Goldstein, such paradoxes seem designed to convince us that we're "simply not smart enough" to deal with the problems they cause.
Goldstein said it first! Essentially, though, that's the possibility we plan to float about the elite logician and his or her puzzling work.
Is it possible that our elite logicians have never been all that sharp? Could this raise the ultimate question about Aristotle's original assessment of our kind, at least as it has been interpreted down through the ages?
We humans are "the rational animal," Aristotle is said to have said. Is it possible that this assessment has always been fundamentally wrong, perhaps even comically so? Is it possible that we've been "seeing ourselves from afar" when we praise ourselves in that way? Is it possible that Professor Harari's gloomier view may turn out to be more nearly right?
Are we really the rational animal? Or could it be that, in the end, we're more about "gossip" and "fiction?" Tomorrow, we'll consider the silliness of that tired old Cretan's tale, the ancient paradox which has long caused our greatest elite minds to crash.
Tomorrow: Sillily silly all the way down
Coming: Lord Russell's "set of all sets!"