Professor takes Platonist challenge: Rebecca Goldstein is a ranking philosophy professor.
She's also a highly-regarded novelist. For our money, her inclination to tell the human story didn't serve her especially well when she wrote her 2005 general interest book, Incompleteness: The Proof and Paradox of Kurt Godel.
Goldstein goes into substantial internal detail about the way Godel "fell in love" when he was still a teen—about the "ecstatic transfiguration" produced by his love affair with Platonism.
She goes on and on, then on and on, about this "transfigurative intellectual love." At times, though, she also makes it sound like Godel had sworn allegiance, at this point in his life, to some revolutionary political group.
According to Goldstein, Godel "had become a Platonist in 1925," the year during which he turned 19. He "was already a committed Platonist in 1926," the year in which he began attending meetings of the super-elite discussion group called the Vienna Circle.
Godel was moving in lofty academic circles at a very young age. It's also true, according to Goldstein, that he didn't want the Circle to know about his commitment to Platonism. In effect, he was an Alger Hiss hiding within a circle of people all named Whittaker Chambers.
Back in 2005, Jim Holt reviewed Goldstein's book for The New Yorker. He told a shortened version of this slightly comical story in that review, which has now become the title essay of his own new book, When Einstein Walked with Godel:
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. In the philosophical world of nineteen-twenties Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science...Members of the Vienna Circle had no idea that a Platonist was lurking among them! For better or worse, Goldstein tells the story of the "clandestine Platonist" at much greater length, and with much more dramatic flair.
Gödel was introduced into the Vienna Circle by one of his professors, but he kept quiet about his Platonist views. Being both rigorous and averse to controversy, he did not like to argue his convictions unless he had an airtight way of demonstrating that they were valid...
At any rate, Godel was "a committed Platonist" by the time he was 20. He was lurking among the "logical positivists," unwilling to let them know about his deepest beliefs.
By the time he was 24, Godel had formulated and publicized his "incompleteness theorems," the theorems which are said to identify him as the greatest logician since Aristotle. Essentially, Godel saw these theorems as the proof of his beloved Platonism. Or so Goldstein says, we would assume correctly.
As of 1930, this silliness was consuming one of the western world's highest intellectual elites. Along the way, we may puzzle a bit at the stakes involved in this war of the worlds.
As we saw yesterday, Godel, the second greatest logician, was trying to figure out how we can know that 2 + 2 equals 4. He'd fallen in love with Platonism, and wanted to prove that the doctrine was true.
Can major battles of this type really revolve around matters like 2 + 2? As we proceed today and tomorrow, we'll see other apparently vapid puzzlements move to center stage in this remarkable tale.
Before proceeding, we ought to admit it—this story of the "committed Platonist" strikes us as amazingly silly. Today, though, we ask a more challenging question:
In the course of writing her book, was Goldstein able to explain the nature of this "doctrine?" To what beliefs had Godel committed when he committed to Platonism, if in clandestine fashion?
As we saw yesterday, Holt made virtually no attempt to describe this powerful doctrine in his review of Goldstein's book. Was Goldstein, a ranking philosophy professor, able to clarify matters further, writing at much greater length?
In her first attempt at taking the Platonist challenge, Goldstein had, rather unhelpfully, offered the formulation shown below. We've shown you this passage before. Try to ignore the technical language:
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...There it stands. In this, her first pass at the Platonist challenge, Goldstein tells us this:
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 Platonist believes in "the objective existence of mathematical reality," whatever that is supposed to mean. But wait—there's more.
According to the Platonist, the truths of mathematics are determined by the reality of mathematics! More specifically, the truths of mathematics are determined by the nature of the entities which make up that reality.
So this ranking professor has said. Presumably, everyone can see how unhelpful this first attempt at explication was. That leaves us asking an obvious question:
Does Goldstein go on, in her book-length text, to clarify, unpack, elucidate or explain the essence of this alleged doctrine? Does she ever do a better job helping us understand the nature of the doctrine to which the second greatest logician in history is said to have committed his life?
What the heck is Platonism? Does Goldstein ever provide an answer which is, quoting from the blurbs on her book, lucid, accessible, clear?
We're going to say that she doesn't. To give you a sense of what we mean, let's take ourselves to page 87 of her general interest book.
Friend, let's start with a basic admission. There's no way to present an excerpt from this book without exciting a possible objection. The reader may suspect that a tiny shard of Goldstein's brilliantly lucid exposition has been taken out of some larger context.
Go ahead—keep that possible objection in mind! Then proceed to read this passage, in which Goldstein tells us what a possible fool believes:
GOLDSTEIN (page 87): For a Platonist, mathematical truth is the same sort of truth as that prevailing in lesser realms. A proposition p is true if and only if p. "Santa Claus exists" is true if and only if Santa Claus exists. "Every even number greater than 2 is the sum of two primes" is is true if and only if there is no even number greater than 2 that is the sum pf two primes (even if we can never prove it).What does a Platonist believe? With what sorts of beliefs did our second greatest logician fall in transfigurative love?
According to Goldstein, a Platonist believes the following. A Platonist believes that the proposition "Santa Claus exists" is true if and only if Santa Claus does exist!
No, really. That's what it says!
Friend, you must be a Platonist if that's what the doctrine is! Every person on your block is a committed Platonist too.
"Santa Claus exists" is true if and only if Santa Claus exists? Everyone believes that statement—and everyone believes the other two examples provided in that peculiar passage.
To marvel further at the type of work which routinely emerges from our academic elites, we turn to the footnote on that same page. In that footnote, Goldstein further discusses the third example from the passage we've already posted, the example involving the sum of two primes.
According to that footnote, "the Prussian mathematician Christian Goldbach (1690-1764) had conjectured that every even number greater than 2 is the sum of two prime numbers."
So far, so perfectly clear. But in her footnote, Goldstein further discusses Goldbach's conjecture. As she does, she tries again to let us know what a Platonist believes and asserts:
GOLDSTEIN (page 87): Goldbach's conjecture has been confirmed for every even number that has ever been checked; however, no proof has of yet been discovered for the universal conclusion that every even number greater than 2 is the sum of two primes. The fact that Goldbach's conjecture remains unproved means (at least according to the Platonist) that lurking out there beyond the point where mathematicians have checked there might be a counterexample: an even number that isn't the sum of two primes. Then again (according to the mathematical Platonist), there may not be a counterexample: every even number may be the sum of two primes, without there being a formal way of proving that this is so. A Platonist asserts that there either is or isn't a counterexample, irrespective of our having a proof one way or the other."A Platonist asserts that there either is or isn't a counterexample, irrespective of our having a proof one way or the other?"
Tell the truth! At least on its face, does that make any sense?
"A Platonist asserts that there either is or isn't a counterexample?" Who wouldn't make that "assertion?" And why would it take a Platonist to make the other assertions Goldstein lists in that passage about the conjecture?
Goldbach's conjecture remains unproved (and unrefuted)? Why would it take a Platonist to say that there might be a counterexample which hasn't yet been checked? Wouldn't anyone say the same thing? Ecstatic transfiguration to the side, what's love of Platonism got to do with it?
Goldstein's examples in this footnote bring us back to Santa Claus, who can only accurately be said to exist if he does exist. We flash on our second greatest logician trying to figure how we can know that 2 + 2 equals 4.
Misguided respect for academic authority will induce the trusting soul to assume that there must be some lofty explanation for these peculiar presentations. We'll strongly suggest that, in matters like these, such trust will be leading us wrong.
At substantial length, Goldstein tells a novelist's tale in her book—a tale of the powerful intellectual love which produced a committed Platonist. Respect for intellectual authority may incline us to believe that some actual great beliefs lie at the heart of this tale.
On the other hand, yesterday, in Holt's text, we saw our second greatest logician puzzling over 2 + 2 = 4. Today, we see a high-ranking professor offering what seems like perfect twaddle concerning the existence of Santa Claus.
In Goldstein's book, we also encounter the passage shown below. In this passage, Goldstein gives another example of what is, or was, at stake in this battle. Today, we'll highlight that passage:
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...According to the Platonist, the claim that the number 4 is even is an "objective" claim. So is the claim that 25 is a perfect square. (That is, that it's the product of 5 x 5.)
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.
These claims are "objective," the committed Platonist cries. But so does everyone you've ever known, along with all those you've never met. This account of what a Platonist believes doesn't seem to make any sense—and yet, this is the way a ranking academic authority explain this ancient "doctrine," with three academic stars blurbing how lucid she is.
According to Goldstein and Holt, Platonism is the doctrine which defined the world of our second greatest logician, a man who seemed to be mentally ill throughout his life and who was famous for believing a long list of crazy ideas. According to Goldstein, it was on this peculiar plain outside Troy that a giant battle was waged involving our second greatest logician and a lofty academic Circle.
Does any of this make any sense? Respect for authority tells us it must.
Experience tells us something different. Then too, there's the passage from G. H. Hardy, another alleged Platonist, which Goldstein quotes early on.
Tomorrow: When mathematicians stray, or how can we humans possibly know that 317 is a prime?