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

29 comments:

  1. "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."

    All students who take geometry in 7th or 8th grade, learn to do mathematical proofs. The point of that exercise is not to prove things about angles and sides of triangles, but to teach children how mathematical facts are established and how logic works, using triangles as examples.

    Because every reader will have had that experience, they will have some idea of what Holt is talking about when he says that 2+2=4 can be derived in a logical system.

    Somerby says that Holt "defines or pretends to define" Godel's theorems. Why cannot Holt be describing them instead of defining them? Somerby makes a claim for Holt that Holt himself does not assert. Holt is writing about Godel, not defining anything he said.

    Then Somerby says: "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.' ”

    Somerby is never careful about gender pronouns, but now all of a sudden he is talking about "she" won't understand this or that. A sneaky way of emphasizing Somerby's assertion that the reader won't grasp what Holt is saying -- because women presumably don't do math well, and if some man might understand Holt, surely no woman would. Then the pronoun "she" is returned to the closet, never to be used again (until Somerby wants to talk about Stormy once more).

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

    Who are Plato's guardians? The average reader will have no way of knowing who Somerby is talking about now. Couldn't Somerby behave in a manner more respectful of what his readers do and do not know? Or is this another reference to Bob Dylan?

    ReplyDelete
    Replies
    1. "All students who take geometry in 7th or 8th grade, learn to do mathematical proofs."

      Nigga please.

      US students don't take geometry until the 10th grade and even then they are too stoned on the base cultural distractions to learn to do anything. What kind of crack are you smoking fucking dumbass?

      Delete
    2. Better trolls please

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  2. PBS has released a statement saying that Bert & Ernies aren't gay. In what universe do puppets have sex lives?

    ReplyDelete
  3. The Cretan is lying.

    ReplyDelete
  4. One slightly well known person is still reading the Daily Howler.

    "The truth is that we humans aren’t really all that smart. We’re basically overclocked apes with a few extra cognitive tricks tossed in, and those tricks aren’t especially sophisticated."

    https://www.motherjones.com/kevin-drum/2018/09/robot-puppies-will-be-mans-best-friend-a-decade-from-now/

    ReplyDelete
    Replies
    1. Caesar, thanks for the link.

      What's just as interesting in that post is that Drum used an old, Somerby technique - put the offensive or stupid remarks in bold type. And boy, were those remarks by Fowler pathetic. Good thing he’s racking up all those bux, so he can afford such nonsense. In fact, I’ll go so far as to say that he himself received “comfort” from his new animatron.

      As far as Bob’s post goes, I agree with him. Holt is a dolt, at least in this example microscopized by Bob.

      I didn’t know what a “logical system” was either, until I looked it up. If Wikipedia is any guide, while it seems mainly to refer to mathematical constructs, I think it can refer to anything anyone wants to imagine, as long as it’s logically consistent with the system so imagined.

      In fact, I’ll go so far as to say that Creationists have their own logical system. The Bible is the Word of God, so it must be true. So for example, they have to explain why the story of Noah’s Ark requires two of every single species to be within walking distance of Noah’s house. How is that possible? It doesn’t matter. The Bible is the Word of God, so it can’t be false. I actually saw a pastor use that argument on YouTube.

      The closest Holt got to this was “But the truth of this proposition can be seen only from outside the logical system.”

      And as always, pay attention readers: Bob is criticizing the countless people who think Holt is tits when it comes to describing the beauty and accessibility of Holt’s prose. Perhaps Bob’s right. In the main, we’re better at gossip than reasoning.

      Leroy

      Delete
  5. "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. "

    Anyone who doesn't understand the term "logical system" is too stupid to be reading *anything*.

    What is difficult about this? Does Somerby actually understand the term, and is he just concerned for the poor uneducated dumbass who is subjected to the mental violence of having to ponder an erudite term like "logical system?"

    Did Somerby ever take algebra or calculus? Anyone with the remotest sense of these subjects understands, first, that mathematics is more than just arithmetic, and, second, that it involves deriving true statements from others, using LOGICAL rules used SYSTEMATICALLY. For example, it allows us to solve equations where a term is unknown; it allows us to prove a theorem. The manipulations of mathematics (using its SYSTEM OF LOGICAL RULES) can be used to describe and predict events in the real world.

    But Somerby pretends to speak for the "common man", the "general reader", who apparently just feels in his gut that all of them egghead perfesers are full of it. 2+2=4? Chortle.

    ReplyDelete
  6. "As the weeks and months proceed, we'll return to Harari's account, reviewing his claims in more detail"

    OMG.

    ReplyDelete
    Replies
    1. Ha!
      Thanks 2:57. I Needed a laugh.

      Delete
  7. There is a term fading into the foggy distance that presents itself to memory occasionally. What is it? Oh yes: "achievement spaces, or differences, or gaps" or something.

    Seems that TDH abandoned that topic, throwing all those schoolchildren under the bus, like so many others have done so many times before. What a shame. And there are 50 states and several TUDAs that would make for at least several weeks of listing NAEP scores showing gaps.

    ReplyDelete
  8. Using Aristotle's logic on Somerby's premise:

    Humans are not rational.
    Somerby is a human.
    Therefore, Somerby is not rational.

    Somerby thinks he can prove humans are not rational by citing examples of humans behaving irrationally. That is a logical fallacy. You must show it for all humans in order for it to be true. In mathematics, you only need a single counterexample to render a statement untrue.

    But Somerby disproves it by his own argument. He proves that someone is irrational by using rational analysis, namely his own. That proves that he at least is rational. Apparently, Harari is as well, since he has reasoned out the truth about Homo sapiens.

    And this club of sages to which Somerby and possibly Harari belong is quite elite.

    ReplyDelete
  9. Another statement that has hade me puzzled, lo, these many years, is "the exception that proves the rule." What the hell does that mean? It seems to me that if there is a "rule" it shouldn't have exceptions. If a "rule" has an exception, it could conceivably have more than one exception. Therefore, the "rule" is not a "rule."

    ReplyDelete
    Replies
    1. @4:01 wikipedia has a nice discussion https://en.wikipedia.org/wiki/Exception_that_proves_the_rule

      Delete
    2. This is a maxim of legal interpretation. It means that a stated exception implies the existence of an unstated general rule. For instance, if you see a sign that says, "No parking 5P to7P", you are justified in assuming that there is a general rule that parking is allowed, with the exception of the two hours. The sign doesn't actually say that you can park from 7:01P to 4:59P, but you may infer that you may.

      Delete
  10. Chaikovsky, Serenade for Strings
    Chamber Orchestra Basel
    Concertgebouw, Amsterdam

    https://www.youtube.com/watch?v=rKwb9P1JtC4

    ReplyDelete
  11. "An obedient line of upper-end journalists said this all made perfect sense."

    Hey Bob. This is a perfectly typical pretentious lib-zombie shit. Your own tribal shit. What did you expect?

    ReplyDelete
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  13. This comment has been removed by the author.

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