THE INCOMPLETENESS FILE: Incompleteness meets incoherence!

TUESDAY, SEPTEMBER 18, 2018

Lucid writer intones:
Subscribers to The New Yorker had a major treat in store.

Or at least, so it seemed.

Their February 28, 2005 issue had arrived in the mail. It featured a lengthy essay in which a writer named Jim Holt discussed a pair of new books.

One of the books concerned Albert Einstein, an extremely famous theoretical physicist. The other new book concerned Kurt Godel, a "logician" who isn't well-known by the average shlub at all.

This May, Holt's New Yorker essay, lightly edited, reappeared as the title essay in his own new book, When Einstein Walked with Godel: Excursions to the Edge of Thought.

As seems to be required by law, Holt's new book was praised by major reviewers—was praised for its lucidity. In the thirteen years since that essay appeared, Holt had become a "made man" in New York publishing circles.

Within those circles, Holt is now reflexively praised for the clarity of his writing about difficult science and math. As you can see, Wikipedia even headlines him as a "philosopher!" That's how silly and mandatory this sort of thing has become.

Holt's book of essays was praised this year for its brilliant lucidity. As we noted last Friday, the same was true of the new book about Kurt Godel which he discussed in The New Yorker back in 2005.

Hurrah! That new book, by Rebecca Goldstein, had been described as "accessible"—but also as "surprisingly accessible," even as "remarkably accessible."

It had been praised as a "lucid expression" of Godel's ideas—but it had also been hailed as "eminently lucid." So it goes within our tightly scripted academic journalistical complex.

Goldstein's treatment of Godel's ideas had been widely praised. Now, a writer at The New Yorker was going to boil matters down even further! Subscribers would finally get a chance to understand Godel's "incompleteness theorems," on the basis of which, Holt now said, Godel has often been called "the greatest logician since Aristotle."

Truth to tell, nothing dimly resembling that occurred in Holt's piece. In fairness, Goldstein hadn't been especially lucid when it came to explaining Godel's theorems either.

For the general reader, Goldstein's treatment of Godel's theorems would almost surely have been extremely hard to follow. When Holt took his turn in The New Yorker, his attempt to describe those "incompleteness theorems" was almost comically incoherent—incoherent all the way down.

Today, the book by Holt which features that essay is being praised by major journalists for its brilliant clarity. In this way, a comical aspect of our journalism—indeed, of our upper-end culture's most basic attempt at rationality—has once again been put on display, for perhaps the ten millionth time.

Holt's essay appeared in early 2005. Its author discussed Einstein's theory of relativity, then turned to Godel's theorems. In this week's reports, we'll speak of Godel alone.

Back in May, readers of the New York Times and the Wall Street Journal were told that Holt's rather obvious incoherence is an example of brilliant lucidity.

In this way, we've been given another look at the classic misassessment we've now christened as "Aristotle's error." We've been given another look at the way we humans, at least in the west, keep "seeing ourselves from afar."

Thanks to his incompleteness theorems, Godel has often been described as the greatest logician since Aristotle. But what did Godel actually say in his theorems? What was he trying to show?

In his essay for The New Yorker, Holt addressed those basic questions in two enormously long paragraphs. Today, we'll examine the first of those paragraphs, transcribing it as it appears in Holt's current book.

We'll start with an apology. In the past two weeks, we've already posted the start of the first paragraph in question. Before we show you Holt's full paragraph, we'll revisit that part, for review:
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...
According to Holt, Godel had come to believe that "abstractions like numbers and circles had a perfect, timeless existence independent of the human mind." As it turns out, Godel was "seduced" by this "doctrine" as a mere freshman in college.

Already, a perceptive reader might suspect that she's being directed by "a guide...who only has at heart [her] getting lost." What in the world does a person believe when he believes that "numbers and circles have a perfect, timeless existence independent of the human mind?" What does it mean to "believe," to be seduced by, such a peculiar notion?

Already, a perceptive reader should be asking such questions. But Holt just kept plowing ahead.

In the passage we've posted above, Holt said that's what Godel believed. He didn't try to explain what that peculiar formulation might possibly mean. Instead, he moved on to describe a major dispute within the intellectual world of Godel's Vienna.

What follows is the first of the two lengthy paragraphs in which Holt explains, or attempts or pretends to explain, Godel's "incompleteness theorems." Warning! The second graf, which we'll review tomorrow, is almost twice as long as the first.

According to recent reviews in the Times and the Journal, this paragraph appears within the title essay of a book in which the writing is brilliantly incisive and clear. Additional warning! By the end of this paragraph, the greatest logician since Aristotle is asking himself how we can know that 2 + 2 equals 4!

That's what Godel is asking himself! People, we're just saying:
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.
What makes a proposition like “2 + 2 = 4” true?

As the people of Europe struggled and groaned between two deeply destructive world wars, that's the type of question the western world's most brilliant thinkers were laboring to resolve!

You've now seen the first of the two paragraphs in which Holt explains, or attempts to explain, Godel's "incompleteness theorems." In this first paragraph, Holt basically sets the stage for his ultimate explanation. That will come in the second paragraph, which is roughly twice as long.

What makes a proposition like “2 + 2 = 4” true? In this, the first of his two paragraphs, Holt—the brilliant, incisive writer—sets the stage for spelling it out. This is what he has said:

According to one group of thinkers in Godel's Vienna, "2 + 2 = 4" is true because this rather familiar arithmetical proposition "can be derived in a logical system according to certain rules."

Tell the truth, dear general reader: Do you have the slightest idea what that statement means?

Tell the truth, New Yorker subscriber: Do you understand what it means, even in a general sense, to "derive [an arithmetical proposition] in a logical system according to certain rules?"

Friend, of course you don't! But if that's what one group of thinkers were thinking, at least one other brilliant thinker was brilliantly thinking this:

According to the greatest logician since Aristotle, "2 + 2 = 4" is true because it "correctly describes [an] abstract world of numbers." Perhaps more precisely, it correctly describes one aspect of the perfect, timeless existence enjoyed by numbers and circles outside the human mind!

Europe was struggling between two wars. Tearing their hair in the loftiest circles, the western world's most brilliant "thinkers" were laboring over this.

That being said, did Holt go on to make Godel's approach to this matter brilliantly clear? More specifically, was he able to explain Godel's "incompleteness theorems" in a way the general reader might find wonderfully clear?

That's what major reviewers have suggested. But at this point, does any of this seem especially clear?

Tomorrow, we're going to ask you to strap yourselves into your seats. We'll quickly revisit this first paragraph, into which a substantial amount of incoherence has already been poured.

Then, we'll look at Godel's endless succeeding paragraph, in which he attempts to describe the working of Godel's theorems in a way the general reader will be able to comprehend.

We humans! Seeing ourselves from afar once again, major reviewers have seemed to say that Holt did a wonderful job!

Tomorrow: "Beginning with a logical system for mathematics..."

28 comments:

  1. "As the people of Europe struggled and groaned between two deeply destructive world wars, that's the type of question the western world's most brilliant thinkers were laboring to resolve!"

    If you cannot pause and consider such things BETWEEN two world wars (especially when you don't know the second war is going to happen and you've just fought the war to end all wars), when can you consider such things?

    Somerby implies that theoretical subjects are a waste of time because they have no practical use. This view is so profoundly anti-scientific that it is usually found among conservatives. One of the premises of science (and all pursuit of knowledge) is that it be pursued for its own sake with trust that what is discovered will be useful in some way that is yet unknown. This has proven to be true many times in the past and is the reason why science today is pursued without reference to expected practical benefits of discovery. This is so basic to science that students are taught it in their introductory science classes. But, once again, Somerby seems not to have benefitted much from his Harvard coursework.

    But I suppose one cannot deride humans for daring to believe they can think and also respect the nature of inquiry for its own sake.

    Somerby is an ass today and for all time. That makes anything else he has to say on this subject moot. He is the last person who should be complaining that advertising on dustjackets is not always true, or whatever he is whining about today.

    ReplyDelete
    Replies
    1. And yet you keep coming back here to deposit your shit. Why don't you shove off if it bothers you so much.

      Delete
    2. Any pile of shit on the sidewalk needs a sign that says "Watch out for the shit" or else others may step in it.

      No one forced you to read my comment either. Happy to use shit analogies if that helps you think about things.

      Delete
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  2. "Then, we'll look at Godel's endless succeeding paragraph, in which he attempts to describe the working of Godel's theorems in a way the general reader will be able to comprehend."

    In the first sentence, he means "Holt's endless..." not Godel's. Things are generally clearer if you use the right names.

    ReplyDelete
  3. There is an irony to Somerby's repeated mention of tearing of hair between two wars and what the best thinkers were doing. Einstein was one of the people tearing his hair and he is credited with opening the door to the knowledge that created the atom bomb which ended WWII. Is that practical enough for Somerby?

    ReplyDelete
  4. Having studied far too much mathematics, I partially agree with Somerby. Although many fields of mathematics have turned out to have practical uses, they never found any practical use for Godel's Incompleteness Theorem. IMHO they never will.

    Here's another example of a useless result, the Banach-Tarski paradox:

    Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

    The mathematics behind this result are elegant. But, the disjoint pieces are not measurable sets. Unmeasurable sets do not correspond to anything in the real world. So, the result has no practical value.

    However, the lack of practical use doesn't mean that these two results lack value. They are beautiful human constructions that add to our culture.

    ReplyDelete
    Replies
    1. When you don't know the practical value of some discovery, it may be because it hasn't been found yet. Today's useless findings may be crucial to progress tomorrow, to address some problem that doesn't exist today.

      Delete
    2. There's an interesting debate ongoing about Gödel's theorem and how it relates to artificial intelligence.

      Delete
  5. "Tell the truth, New Yorker subscriber: Do you understand what it means, even in a general sense, to "derive [an arithmetical proposition] in a logical system according to certain rules?"

    Friend, of course you don't!"

    What is unclear about deriving a proposition from rules that exist in a logical system?

    Given that A=B and B=C. One derives the proposition that A=C. This is an illustration of what's called the transitive property. It is a proposition that can be derived from a set of rules in a logical system. That system, in this case, is called mathematics.

    Given A<B, B<C, and D<B. What true statements can be made? I can derive that A<C, D<C, but I cannot say whether A<D. Based upon the rules in this logical system, I cannot assert that A<D.

    Now, did I just manipulate a bunch of symbols, or did I uncover a fact about the real world? Well, that's partly what the debate is about, one that Somerby wants to ridicule, rather than engage in. And his projection of his own ignorance onto his readers is insulting, and just makes him look stupid.

    ReplyDelete
  6. Isaac Newton, in order to make progress with his physical theories about motion and gravitation, invented calculus. He found it more productive to represent his discoveries using mathematical notation. The interesting thing is that once the theorems are represented symbolically, further insights *about the physical world* can be derived by using the rules of mathematics. Thus, again, mathematics seems to be predictive of truths about the physical world that would otherwise remain unknown.

    ReplyDelete
    Replies
    1. Another example, of course, is Einstein. His work was principally formulated mathematically. The truth of his predictions was only later verified experimentally.

      Delete
  7. one of my favorite quotes goes something like:

    The discovery of E = mc^2 led to the atomic bomb as ineluctably as the invention of television led to the assassination of President Kennedy.

    ReplyDelete
    Replies
    1. There's a much more direct line from the discovery of the massive power that results from conversion of matter to energy and the development of the bomb than the television analogy, which is doubtful at best.

      Delete
  8. Hey Bob. Your creepy excitement about this particular theorem reminds me of maistream news media, their creepy excitement about all kinds of shit.

    ReplyDelete
  9. "As the people of Europe struggled and groaned between two deeply destructive world wars, that's the type of question the western world's most brilliant thinkers were laboring to resolve!"

    Is this supposed to be a swipe at Holt, or at Gödel and the Vienna Circle? This circle may have been brilliant thinkers, maybe not. But if they were brilliant, it was in a particular area, mathematics or philosophy perhaps, and perhaps not in other areas. There were many other Europeans, some brilliant, some not, devoted to the public sphere trying to help the struggling masses. Not everyone has an aptitude for political or economic matters.

    Would you ask an artist to quit devoting his life to his art and become some sort of political activist, or would you demand that his art treat only political matters?

    How about a physicist who labors his whole life long, to the exclusion of other concerns like politics, to discover something?

    A historian who spends his life researching the past?

    A musician whose principal concern is the mastery of his or her art?

    The (implied) notion that intellectuals share some outsized portion of the blame for bad things, like WW1 or the rise of Hitler, or that they could have singlehandedly prevented it is whimsical. And "intellectuals" disagree about politics about as much as "non-intellectuals." Einstein and Gödel are a case in point.

    ReplyDelete
  10. Bob,
    After reading your bullshit, DinC checked with Wikipedia and seems convinced that he is a mathematical wizard.
    You are both full of shit.
    2 apples plus 2 apples = four apples.
    Not 4.1 apples.
    QED.

    ReplyDelete
    Replies
    1. Ah, 4:37. Yes, I copied the statement of Banach Tarski from wikipedia. In 1968 I knew enough mathematics, so that I was able to understand the derivation of the Banach Tarski paradox. As I recall, it required knowledge and understanding of several fields of mathematics, including set theory, the Axiom of Choice, and Group Theory. The version I learned had the pieces reconstituted to form a ball twice as big, rather than two balls of the same size. I assume both results are based on fairly equivalent demonstrations.

      Looking back, I regret the time and energy spent learning this stuff, since it was never of any practical use to me.

      P.S. The Axiom of Choice and its many equivalent statements are fascinating. But, any work that involves the Axiom of Choice is likely to be useless, since there's no real-world analogue.

      Delete
    2. Nothing that you learn is ever wasted. First, the mental exercise involved in learning keeps your brain fit and increases your cognitive ability. Second, learning opens doors so that when an opportunity comes along, you will be prepared to take advantage of it. No one can do everything they are qualified or capable of doing because our time is limited, but you cannot choose to do something if you are unqualified and do not have the ability to grasp that chance.

      I worked for several years in a dream job that existed at the intersection of two fields. I happened to be qualified for it because of a mid-life career change that left me with two different, unusual competences. As a result, I was recruited specifically for that job, at a high salary, putting all of my prior experience to good use. I would not have been eligible or on anyone's radar without the "respecialization" learning I invested in seeking that second career choice.

      You never know in advance what opportunities will come your way. It is better to study what interests you strongly and trust that when you are ready, there will be a chance to use that knowledge. You DO have to put yourself in a position to be hired for what you can do, but you cannot even do that if you don't have the training.

      So, regretting the knowledge that was never used strikes me as silly because it is probably only unused because you made a choice to pursue something else that you liked better, and were also trained to do.

      If an orchestra needs an English horn but you only play clarinet, you are going to miss out. But if you can play both, you have an option. The more you learn, the more options you will have.

      Delete
    3. Or, you could use a trumpet with a straight-mute.

      Delete
    4. Anonymous 8:01 PM: Oddly enough, I worked for several years in a dream job that existed at the intersection of t/w/o/ three fields. ... [and I could quote back the rest of that paragraph, with only the same numeric changes, in support of your comment.]

      And while my hobby of calligraphy begun in high school and further developed in college was never professionally employed (always a labor of love), it did teach me a lot about layout/design, and even font elements, that later stood me in good stead when PCs and WP software came along. Besides, even my co-workers enjoyed hand-callig'ed birthday/holiday cards.

      Delete
  11. Belle nuit, ô nuit d’amour
    Souris à nos ivresses ...

    https://www.youtube.com/watch?v=U2vI6Jjeu8Q

    ReplyDelete
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