When mathematicians wander: Jim Holt got off easy.
Back in 2005, he wrote a slightly-disguised review of Rebecca Goldstein's new general interest book, Incompleteness: The Proof and Paradox of Kurt Godel. Holt's review appeared in The New Yorker, an upper-end general interest magazine.
In her general interest book, Goldstein had told the story of Godel's life. She'd also tried to explain his "incompletenesss theorems," on the basis of which he's often been called the greatest logician since Aristotle.
First, though, Goldstein tried to explain the doctrine which, she said, lay at the heart of Godel's intellectual life from the time of his first year in college. Because he was only writing a review, Holt described the doctrine very briefly, and then quickly moved on.
We'd have to say he got off easy! Here's his key passage again:
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...Platonism is built around the belief that "abstractions like numbers and circles have a perfect, timeless existence independent of the human mind." So said Holt, in The New Yorker, and then he quickly moved on.
Do numbers and circles have a perfect, timeless existence independent of the human mind? Friend, do you have even the slightest idea what that word sequence might possibly mean? We'll go ahead and answer for you:
No, you don't have the slightest idea. And neither does anyone else!
Because he was only writing a review, Holt got to leave things right there. Today, Holt's review, edited to remove most references to Goldstein, is the title essay of his own new book, When Einstein Walked with Godel: Excursions to the Edge of Thought.
Just this once, let's be honest! Holt's account of Godel's beloved "doctrine" is straight outta Jabberwocky. You don't know what the heck it means, and neither does anyone else.
But so what? When Holt's new book appeared, fronted by that title essay, major reviewers stood in line to exclaim, for the ten millionth, about how amazingly lucid and clear Holt's science/math writing is. Reviewers swore that Holt's writing was brutally lucid and clear.
In effect, reviewers swore that Holt had made Godel and Einstein easy. This has been standard journalistic practice dating at least to Einstein's own general interest book about relativity—to the brilliant physicist's failed attempt to make his own theories clear.
(More on that effort below.)
This is standard journalistic behavior—and Goldstein benefited from this practice when her book about Godel appeared. Her own account of Platonism is so murky that it seems to have come from the third planet beyond Jabberwocky. That said, the usual suspects stood in line to say how lucid her writing was.
Three major academic stars blurbed Goldstein's book on its jacket. You already know what they said:
In his dust jacket blurb, Alan Lightman praised Goldstein for her "penetrating, accessible, and beautifully written book."
Brian Greene went one step further. He said Goldstein's account of Godel was "remarkably accessible."
In a New York Times review, Polly Shulman said that Goldstein's writing was "surprisingly accessible." Meanwhile, back on the book's dust jacket, Stephen Pinker said this:
This book is a gem...Rebecca Goldstein, the gifted novelist and philosopher, offers us not just a lucid expression of Godel's brainchild but a satisfying and original narrative of the ideas and people it touched. Written with grace and passion, Incompletenesss is an unforgettable account of one of the great moments in the history of human thought."Lucid" was Pinker's word of choice. At Salon, Laura Miller stepped in to top him on this part of the color wheel, calling the book "eminently lucid."
This constitutes a familiar practice within several modern guilds. (Goldstein praises Holt's new book on that book's dust jacket!) Within the burgeoning publishing world of modern science-and-math-made-easy, the professors praise each other in these ways, as do the major reviewers.
Are these blurbs ever accurate? Back in 2005, Miller said that Goldstein's "masterful" book provided "an eminently lucid explanation of Gödel’s theorem and its implications.”
Does anyone think that Miller, a general interest reviewer, could string two coherent words together about Godel's highly abstruse theorems? We'd be very surprised if she could, but if she can, it's hard to believe that her ability stems from Goldstein's widely praised book.
At this point in our explorations, we haven't examined Goldstein and Holt's attempts to explain, elucidate, unpack or describe Godel's actual theorems. This week, we've been trying to see if either writer could explain, elucidate or describe the alleged "doctrine" called Platonism, which is said by Goldstein to lie at the heart of all Godel's ruminations.
What the heck is Platonism? We've already seen what Holt said. According to Goldstein's first bite at this apple, Platonism involves the belief that "the truths of mathematics are determined by the reality of mathematics"—and as we showed you yesterday, the project goes downhill from there, all the way to an impossibly strange rumination about the way the Platonist would evaluate the claim that Santa Claus exists.
Please understand—Goldstein isn't some second-rate shlub who got hauled in from the cold. As we explained in an earlier post, she lives a perfect timeless existence at or near the very top of modern academic elites.
She's a ranking philosophy professor, and a highly-regarded novelist. This helps explain why her book was blurbed so favorably by other elites—unless you think that writing like this really does deserve to be praised as transplendently lucid:
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."Santa Claus exists" is true if and only if Santa Claus exists? If that explains the doctrine with which Godel "fell in love" as a teen, then the whole world is a Platonist, including you and yours.
Casey Stengel is said to have said it when he managed the 1968 Mets: "Can't anybody here play this game?" the gent is said to have asked.
The later Wittgenstein said the same thing about a wide range of major "philosophers," not excluding himself in his own earlier phase. In time, we'll be perusing this major jailbreak which, according to Professor Horwich, is being strategically ignored.
That will come at a later date! For today, we're going to see what can happen when brilliant people decide to play out of position.
The greatest shortstop would probably make an extremely poor tight end. In 1968, Rod Laver was the world's top-rated male tennis player. There's little reason to think he could have helped Stengel's hapless Mets.
So too in the worlds of mathematics and physics! Consider what happened when Goldstein quoted G. H. Hardy, who was, by all accounts, a brilliant mathematician.
Who the heck was G. H. Hardy? The leading authority on his life answers your question here.
By all accounts, Hardy was a brilliant mathematician. For better or worse, he also crossed over to do some "philosophizing" in his iconic 1940 essay, A Mathematician's Apology.
According to Goldstein, Hardy, "an English mathematician of great distinction, expressed his own Platonist convictions" in this "classic" text. She seems to think that the passage she quotes in her book will help us understand this alleged doctrine.
Below, you see the passage Goldstein quotes, on page 46 of her book. We'll especially focus on the way Hardy puzzles over how we humans can know that 317 is a prime:
HARDY (1940): I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it...In our view, that passage is the work of a brilliant mathematician who's playing way out of position. In effect, Hardy—a brilliant mathematician—becomes your Uncle Charlie at Thanksgiving dinner, going on and on.
[T]his realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more than what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but "2" or "317" has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy—I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
Step by step, a brilliant mathematician leads us away from clarity in that jumpy passage. This is the sort of thing which can happen when mathematicians wander far afield.
Hardy muddles his thinking in that passage at an array of points. At one point, he refers to the number 2 as a "mathematical object."
Do you have any idea why a person would want to do that?
That passage starts with Hardy saying that "mathematical reality" (whatever that is) "lies outside us." Consider:
You surely know what Homer meant when he said the battle between Achilles and Hector took place "outside the walls of Troy." But are you sure you understand what Hardy means when he says that "mathematical reality" (whatever that is) is somehow found "outside us?"
That's a rather unusual formulation. Are you sure you know what it means? Can you think of any conceivable way to disagree with that peculiar statement?
In this passage, Hardy plays with dueling "isms"—with "realism" and "idealism." This will almost surely work to further confuse the general reader. Indeed, we'd advise you to check your wallet even when full-fledged "logicians" start burying you in such jargon.
Eventually, the rubber meets the road. Hardy, a brilliant mathematician, chooses to tell us this:
"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."
What makes 317 a prime? According to Hardy, 317 is a prime because it is so! (Hardy's emphasis), not because our minds are shaped in some way.
Friend, would you have any idea how to disagree with that? Has it ever occurred to you to think that 317 (or, more simply, 7 or 17) is a prime because "your mind is shaped in some way," whatever that might mean?
Do you understand what you're being told when you're told that 317 (or, more simply, 17) is a prime because it is so? Are you really completely sure that you're being told anything at all? Does that sound a bit like Uncle Charlie arguing some favorite political point?
In this passage, you see a brilliant mathematician making little clear sense. Is the number 17 a prime just because we think it is? Do you have any idea why anyone would ever make such a claim? In the absence of any such idea, do you understand why Hardy seems to be aggressively "refuting" this claim?
Why is 317 a prime? Now that you've asked, we can explain it amazingly simply. The number 317 is a prime because it can't be divided evenly by the number 2, or by any other "natural number," as you will quickly be able to see if you just give it a try.
317 can't be divided evenly by any other number! Go ahead—you can try them all, though if you're arithmetically slick, you'll know that you only have to try these numbers: 2, 3, 5, 7, 11, 13, 17, 19.
After you try 19 and fail, you don't have to try any more. (Reason: 19 x 19 is larger than 317.) But go ahead—try them all! No other number will divide evenly into 317. That's the most straightforward, simple-minded answer to the (rather imprecise) question Hardy semi-poses in that peculiar passage.
Playing out of position, Hardy almost seems to fashion a tautology: 317 is a prime because it is so! Writing a general interest book, Goldstein presents this passage as if it will help us understand Hardy's "Platonist convictions" and the doctrine of Platonism as a whole.
We humans! If we weren't so inclined to defer to authority, we'd respond to this in the manner of the child who saw that the emperor forgot to put on his clothes. We'd marvel at the lunacy involved in Holt's lucid but ludicrous statement:
"[Gödel was] seduced 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...has always been popular among mathematicians."If we weren't so strongly inclined to defer to academic (and journalistic) authority, we'd react as a sensible person might. We'd marvel at the peculiar claim that mathematicians are inclined to think that numbers and circles "have a perfect, timeless existence" of some undisclosed kind. We'd wonder why a man like Hardy was throwing various "isms" around as he heatedly seemed to explain how we can know that 317 is a prime.
What was Hardy trying to say in that passage? We have no idea.
That said, the burden of clarity falls on the person who's making the lofty claim. It isn't your task, as Hardy's reader, to pretend to make sense of something he's said. Nor should you ever simply assume that something makes sense just because it's being said by a ranking academic.
It was Hardy's job to make his statement lucid! If Goldstein is going to quote him, it's Goldstein's job to let us know why "317 is a prime because it is so" isn't simply the holiday raving of a type of Uncle Charlie.
This brings us back to Einstein's general interest book, even as it points us toward the work of the later Wittgenstein.
Einstein is widely viewed as the most brilliant physicist at least since Newton. After he fashioned his theories of relativity, a publisher asked him to write a general interest book to explain what he had done to the general reader.
The book appeared in German in 1916, in English in 1920. Manifestly, it didn't "make Einstein easy." In his recent biography of Einstein, Walter Isaacson told the comical story which explains how this happened.
Einstein, the world's most brilliant physicist, wasn't a general interest writer! In effect, he was the greatest athlete of all time, but not real good at cooking.
According to Isaacson, as Einstein tried to make Einstein easy, he selected his cousin Elsa's teen-aged daughter as his focus group. "He read every page" to her, Isaacson writes, "pausing frequently to ask whether she indeed got it."
She kept saying she understood, "even though (as she confided to others), she found the whole thing totally baffling." So it went when the planet's most brilliant physicist briefly played out of position.
Friend, do you have the slightest idea what it means to believe in the doctrine of Platonism? Was this really some deep philosophical view? Or was it possibly one of the first of Godel's crazy ideas?
We ask because Godel is described as the greatest logician since Aristotle. What might it mean if our greatest logicians was in thrall to crazy ideas? What does it our highest ranking professors can't make out this fact?
Could it mean that we in the west, like the Bushmen of the Kalahari, have been "seeing ourselves from afar?" Could it light the way toward the work of the later Wittgenstein, which we plan to discuss in coming weeks, if we get there before Mr. Trump decides to start his war.
Are Goldstein and Holt amazingly lucid? It certainly isn't a moral failing, but no, we don't think they are.
That said, Goldstein and her partners in blurbing are among our highest academic elites. When you see the way our top professors perform, are you surprised that our journalists perform even worse? Are you surprised that our cultural breakdown has reached the point where Donald J. Trump holds such power?
Next week: The incompleteness file