FRIDAY, SEPTEMBER 2, 2022
Where does the number 2 live? Professor Goldstein encountered a collection of words—a collection which made "the mind crash."
What collection of words could possess such power? As we noted yesterday, the collection of words was this:
"This very sentence is false."
This very sentence is false! We feel safe in telling you that no one has said that to you, not even once, in the normal course of daily activity. No one has ever said any such thing, no matter how long you've lived!
This very sentence is false! To borrow from Wittgenstein, it's the sort of thing a person will say "only when doing philosophy."
That doesn't necessarily mean that something is "wrong" with this ancient collection of words. But just to refresh your recollection, here's the way Professor Goldstein discussed that collection of words in her ecstatically-blurbed 2005 book, Incompleteness: The Proof and Paradox of Kurt Gödel:
GOLDSTEIN (pages 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 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.
Paradoxes like the liar's play a technical role in the proof that Godel devised for his extraordinary first completeness theorem...
According to Professor Goldstein, that collection of words—This very sentence is false—produces "a catastrophe of reason."
"The mind crashes" in the face of that catastrophic collection of words. "A severe problem" has arisen—and not only that:
According to Professor Goldstein (and many others), "the greatest logician since Aristotle" built his famous incompleteness theorem(s) out of catastrophic material like The Liar's Paradox. Such puzzlements "play a technical role in the proof that Gödel devised for his extraordinary first completeness theorem."
Given the silliness involved in the power she (and others) attribute to that silly collection of words, this leads us to wonder about the ultimate cogency of Gödel's famous work itself.
That said, we don't have to investigate Gödel's (broadly inexplicable) work to see the foolishness involved in Goldstein's claim about that collection of words.
As we noted yesterday, The Liar's Paradox functions at the approximate level of an amusing but silly card trick. It's astounding to think that the mind of a major intellectual would crash in the face of the silliness involved in this ancient carnival act.
That said, Professor Goldstein is a major public intellectual, and the book in which her mind proceeded to crash was published in 2005. Nor is there anything new about this:
At the highest levels, our public intellectuals—in this case, our ranking philosophy professors—have been engaged in this sort of foolishness since the dawn of time.
Alfred North Whitehead issued the famous statement all the way back in 1929. “The European philosophical tradition...consists of a series of footnotes to Plato," he said.
Arguably, Whitehead was describing a problem. Plato lived at the very dawn of the west. In the end, very few things he ever said ended up making actual sense.
Thousands of years have passed since Socrates, Plato and Aristotle—denizens of a tiny Athenian state—began the task of trying to figure the universe out.
A handful of people lived in the Athens of their time; these were three of the brightest. Inevitably, they got a million things wrong, but Professor Goldstein correctly described the state of play which persists to this very day.
Gödel was a Platonist, Goldstein correctly (though fuzzily) wrote:
GOLDSTEIN (page 44): Gödel'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...
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.
Strange but true! Major intellectual figures—people like Gödel—advocate something called "mathematical Platonism" to this very day.
It isn't Professor Goldstein's fault that she has a hard time explaining or describing this alleged school of thought. It is her fault that she doesn't come out and call attention to the general incoherence surrounding this alleged "view."
Note the giant fuzziness which prevails even as Goldstein sets out. According to Goldstein, Platonism is the view that "the truths of mathematics" are determined by "the reality of mathematics."
Beyond that, "the truths of mathematics" are determined by "the nature of numbers." The truths of mathematics are determined by the nature of numbers, which are "abstract entities" but are also "real."
According to the Platonists, the truths of mathematics are determined by the reality of mathematics! A skeptical figure might venture this thought:
Intellectual giants can hardly go wrong while espousing such foofaw as that!
Through no direct fault of her own, Goldstein is already knee-deep in nonsense as she starts attempting to describe this alleged school of thought. As she continues, things may get even worse.
Rather quickly, she's quoting a famous passage from an eminent British mathematician, G. H. Hardy (1877-1947).
As Goldstein notes, Hardy "expressed his own Platonist convictions" in a classic 1940 essay, A Mathematician's Apology. On page 46 of her book, Goldstein quotes Hardy offering such thoughts as these:
I believe that mathematical reality lies outside of us, that our function is to discover or observe it...317 is a prime, not because we think so, or because our minds are shaped in one way or another, but because it is so, because mathematical reality is built that way.
"Mathematical reality lies outside of us?" Mathematical reality is "built a certain way?"
Statements like these may conjure images of some sort, but do we feel sure that we know what such statements actually mean?
Meanwhile, why is 317 a prime? Stating the blindingly obvious, the number 317 is a prime for two obvious reasons, pretty much full stop:
The number 317 is a prime because it can't be divided evenly by some other number, and because that's the definition of a prime!
Because there's nothing mysterious about any of this, it's hard to know why mathematicians and physicists are often inclined to wander further afield, introducing peculiar imagery about where this "mathematical reality" lies.
That said, our greatest mathematicians and physicists will often wander into such lands when they venture across the boundaries which define the academic fields in which they plainly excel. Here's science writer John Holt, trying to describe what Gödel , as a very young man, would have said he believed about this:
HOLT: 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...
In Holt's rendering, Gödel believed that numbers and circles enjoy "a perfect, timeless existence independent of the human mind."
It's hard to know what that might mean, and we can always imagine that Holt is simply mischaracterizing what Gödel really believed. But mathematicians have long been drawn to such peculiar formulations, and it's a very short walk from there to mocking questions like these:
Where does the number 2 live? Do the circles live next door?
This morning's presentation started with a simple question. Given the fact that The Liar's Paradox is little more than a carnival trick, how is it possible that similar "paradoxes" could have played a key role in Gödel's famous theorems?
Given the fact that The Liar's Paradox is little more than a carnival trick, how can it be that a major public intellectual was describing it, in 2005, as "a catastrophe of reason" which makes "the mind crash?"
By any normal measure, Rebecca Goldstein is very bright. How can it be that she was describing The Liar's Paradox that way in a book which was praised for its lucidity by such major figures as Stephen Pinker, Alan Lightman and Brian Greene? So our question goes.
Alas! The woods are lovely, dark and deep, but we humans have always been drawn to remarkably fuzzy thinking, even at the very highest academic levels. When even our most brilliant mathematicians put their slide rules down, it often turns out that they're inclined to walk this way.
Some such thinking will turn out to be "mystical;" some might be thought of as "metaphysical." But this fuzzy thinking has never left us, dating to the years when the breezes off the Aegean cooled the originators of the European philosophical tradition.
The later Wittgenstein explored the fuzzy thinking which has infested high-end academic work since the dawn of time. Eventually, we'll try to summarize his account of the way "linguistic illusions" lead on to "muddled thinking."
It may seem strange to think that major public intellectuals could engage in the kind of work found in Goldstein's account of The Liar's Paradox, but Wittgenstein explored the ways such jumbled thinking persists.
As Wittgenstein sensed (but struggled to explain), bungled thinking is deeply bred in the bone of our species. Pulling one comment by Wittgenstein out of its immediate context, bad explanation at the highest levels is "as much a part of our natural history as walking, eating, drinking, playing."
Indeed, obvious errors made at the top can go unnoticed for hundreds of years! Next week, we'll illustrate this surprising point with a chapter from Einstein himself—with a chapter in which one of history's greatest intellectual tried attempts to explain a basic part of his own work.
His explanation didn't make sense. More than a hundred years have passed. To this day, no one has noticed.
Next week: Einstein explains