THE INCOMPLETENESS FILE: What the Sam Hill is a "logical system?"

WEDNESDAY, SEPTEMBER 19, 2018

No general reader will know: According to the headline on the New York Times review, Jim Holt's new book is a collection of essays which "make sense of the infinite and the infinitesimal."

It's Holt's "conviviality, and a crispness of style, that distinguish him as a popularizer of some very redoubtable mathematics and science,“ the gushing reviewer said, marching in upper-end lockstep.

Indeed, it wasn't just the New York Times making these mandated statements. According to the headline on the Christian Science Monitor review, "When Einstein Walked with Gödel"—that's the title of Holt's new book—"is science writing at its best."

The essays in Holt's new book "all wonderfully achieve [his] stated goal," which includes "enlighten[ing] the newcomer," the Monitor's reviewer said. "This is considerably more difficult than it sounds, and Holt does a beautifully readable job."

Holt's collection of essays wasn't reviewed by the Washington Post, but the reviewer for the Wall Street Journal completed the rule of three. Holt is "one of the very best modern science writers," this third reviewer opined. He specifically singled out Holt's "wonderful title essay."

That's the very essay we've been discussing—the essay in which Holt tries to explain Kurt Godel's "incompleteness theorems."

Reviewers seemed to agree. Holt's work is "beautifully readable," especially for "the newcomer"—for the general reader. But then we turn to that title essay, the one in which Holt attempts to explain Godel's theorems.

According to Holt, those theorems have established Godel, by widespread agreement, as "the greatest logician since Aristotle." An obvious question arises:

How "beautifully readable" is Holt's explanation of those iconic theorems? To what extent is Holt's account of those theorems "science writing at its best?"

As we noted yesterday, Holt explains those theorems in two extremely long paragraphs. As we showed you yesterday, the first of those paragraphs, by far the shorter of the two, reads as shown below in Holt's title essay, which first appeared in The New Yorker in 2005.

Below, you see the first of the two paragraphs in which Holt explains Godel's theorems. By the end of this paragraph, our greatest logician since Aristotle is, for reasons which don't quite get explained, pondering 2 + 2:

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 1920s 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. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.

As this first long paragraph ends, the greatest thinkers in Europe are puzzling over a knotty problem. According to Holt's own language, they're trying to explain "what makes a proposition like 2 + 2 = 4 true."

Without so much as chortling even once, Holt proceeds from there:

One group of Europe's most brilliant thinkers is said to have said that the truth of 2 + 2 = 4 lies in the fact that "it could be derived in a logical system according to certain rules." Without attempting to explain what that technical word salad means, Holt proceeds to say what the youthful Godel believed:

The youthful Godel is said to have thought that the truth of 2 + 2 = 4 lies in the fact that "it correctly describes some abstract world of numbers"—presumably, the world in which "abstractions like numbers and circles have a perfect, timeless existence independent of the human mind." So the greatest logician thought, as opposed to the other great thinkers.

Scotty, beam us down! Despite the gushing of those reviewers, no general reader will have any idea what that paragraph means.

What does it mean to say that 2 + 2 = 4 "can be derived in a logical system according to certain rules?" No general reader has the slightest idea, but Holt doesn't stop to decipher the claim. He merely compares it to what Godel is said to have thought—a belief which is said to involved the perfect existence of circles.

In these ways, our greatest thinkers puzzled out 2 + 2. Last May, major journalists stood in line to say how "beautifully readable" Holt's new book is, especially the "wonderful title essay" in which this hodgepodge appears.

In such ways, we see a modern, high-end display of "Aristotle's error." These reviewers aren't reflecting the ancient claim that "man [sic] is the rational animal." Rather, they're acting out the "Harari heuristic," which holds that our warlike species, Homo sapiens, gained control of the planet when, through a set of chance mutations, our ancestors developed the ability to "gossip" and the ability to invent and affirm sweeping group "fictions."

As the weeks and months proceed, we'll return to Harari's account, reviewing his claims in more detail. For today, we'll only note an obvious fact—even by the end of this first paragraph, Holt's opaque, highly technical writing will have left any general reader several light-years behind.

Alas! Whether they know it or not, general readers will already be at sea by the end of that first paragraph. Most specifically, such readers will have no idea what a "logical system" is.

Nor will such readers have any idea what it means to say that a nursery school fact like 2 + 2 "can be derived in a logical system according to certain rules." Already, Holt may as well be writing in some form of ancient Etruscan.

In the paragraph which follows, Holt starts explaining those "incompleteness theorems." When he does, a large pile of Sandstorm arrives.

At the end of this pig-pile of abstruse phrases, Godel's two theorems get defined. The general reader will have zero idea what Holt is talking about:

HOLT (continuing directly): 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. But how could one demonstrate that mathematics could not be reduced to the artifices of logic? Gödel’s strategy—one of preternatural cleverness and, in the words of philosopher Rebecca Goldstein, “heart-stopping beauty”—was to use logic against itself. Beginning with a logical system for mathematics, one presumed to be free of contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of doublespeak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another. In fact, as Gödel showed, a numerical formula could even be made to say something about itself. Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying when it asserts, "I am nor provable?" No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.

Ar the end of this, the world's longest paragraph, Holt defines, or pretends to define, Godel's two "incompleteness theorems." Despite the subsequent, mandated gushing of our journalistic elites, no general reader will have any idea what Holt is talking about.

Consider the various snares and traps that reader has encountered during this long forced march to the sea:

We're told that Godel wanted to demonstrate that "mathematics could not be reduced to the artifices of logic." The general reader will have no idea what such a reduction might look like.

In pursuit of this puzzling end, we're told that Godel "beg[an] with a logical system for mathematics, one presumed to be free of contradictions." The general reader won't know what "a logical system" is. He won't know what it means for such a creature to be "adequate for mathematics."

We're now told that Godel came up with "a formula that said something about numbers." On the pain of impending death, the general reader won't be able to imagine an example of any such formula making any such statement. Nor will she have any idea what it might mean to produce "a formula that, while ostensibly saying something about numbers, also says, 'I am not provable.' ”

A bit later on, the reader seems to be told that Godel produced "a self-referential formula" which generated a proposition whose truth "can be seen only from outside the logical system." No general reader has any idea what Holt is talking about.

At any rate, atthe end of this long harangue, Holt describes Godel's first incompleteness theorem. Excitement builds for the general reader. Then he's told that the theorem says this:

"No logical system can capture all the truths of mathematics."

That would be an exciting claim. Except, do you remember the problem with which the general reader started? He or she has no idea what a "logical system" is!

In these two paragraphs, Holt explains, or pretends or attempts to explain, Godel's two "incompleteness theorems." For unknown reasons, this Olympian hodgepodge was first offered to general readers in the pages of The New Yorker. Few subscribers could have had any idea what Holt was talking about.

Thirteen years later, Holt's piece was published as the title essay in a collection of his work. Mainstream reviewers stood in line to praise it for being readable, especially for newcomers to the subject matter.

What a long, strange journey it has been through those lengthy paragraphs! We started with Europe's greatest thinkers pondering the fact that 2 + 2 equals 4. We were told that the greatest logician since Aristotle believed that circles and numbers and other such critters have a perfect, timeless existence, an existence we can access through some version of ESP.

Eventually, an avalanche of technical language landed on our newcomer heads. But so what? An obedient line of upper-end journalists said this all made perfect sense.

We're going to say that all these groups are providing textbook illustrations of "Aristotle's error." Also this:

When our journalists behave in the manner described, they're helping us see how things fall apart when Plato's guardians fail.

Tomorrow: Goldstein's first attempt