THE SOUL OF THE LOGICIAN: The "paradoxicality" rules!


Children of paradox:
Did it make sense for Europe's great minds to imitate Wittgenstein's tics?

In her well-received 2005 book about Godel's incompleteness theorems, Rebecca Goldstein rolls her eyes, at considerable length, at the way this aping went down. (See yesterday's report.)

In fairness, Goldstein draws her material from published sources, and the conduct does seem absurd. That said, Professor Von Wright saw this phenomenon in a kinder, gentler light in his Biographical Sketch of Wittgenstein, a portrait which is part of Norman Malcolm's 1958 book, Ludwig Wittgenstein: A Memoir:
VON WRIGHT (page 19): Because of the depth and originality of his thinking, it is very difficult to understand Wittgenstein's ideas and even more difficult to incorporate them into one's own thinking. At the same time the magic of his personality and style was most inviting and persuasive. To learn from Wittgenstein without coming to adopt his forms of expression and catch-words and even to imitate his tone of voice, his mien and gestures, was almost impossible. The danger was that the thoughts should deteriorate into a jargon...

Wittgenstein's most characteristic features were his great and pure seriousness and powerful intelligence. I have never met a man who impressed me so strongly in either respect.
Von Wright was a friend and admirer of Wittgenstein. Writing roughly five decades later, Goldstein sounded a bit like a detractor, perhaps and possibly of the type Professor Horwich would later describe.

Whatever! Moving along to a central concern, why has Wittgenstein's later work tended to wither on the vine? Why has his later work proven hard to be quite hard to explain and apply?

We'll add one important point to Von Wright's explanation. It isn't just "the depth and originality" of Wittgenstein's later work which has made it hard to employ and apply. It's the fact that his seminal later work, Philosophical Investigations, is in fact extremely jumbled and extremely hard to follow.

Simply put, Wittgenstein didn't "produce a good book," an unfortunate fact to which he confessed in his preface to the Investigations. In truth, the basic ideas of the book were quite poorly expressed, while his mannerism and his "mien"—"the magic of his personality and style"—were apparently quite compelling and easy to imitate.

Question! Was Mr. NAME WITHHELD, our freshman year teaching assistant, aping Wittgenstein's famous intensity when he tore his hair at a third-story window, tortured by the maddening logic of 7 plus 5 making 12?

We can't answer your question! But should all that foolishness have transpired over the question of 7 + 5? And what might that foolishness have to say about the smarts of our greatest logicians extending all the way back through the annals of time?

Is it possible that our greatest logicians have never been all that sharp in any significant sense? To us, the aping of Wittgenstein's tics does seem relevant to this question. Then too, there's the work by Professor Goldstein herself on the power of paradox and "paradoxicality."

The full title of Professor Goldstein's book is this: Incompleteness: The Proof and Paradox of Kurt Godel. Perhaps because Goldstein is also a highly-regarded novelist, she pushes the role of "paradox" hard as she seems to struggle to make her book readable for the non-specialist.

In the following passage, she's discussing one of Godel's peculiar beliefs—the belief, in Goldstein's words, "that there was a vast conspiracy, apparently in place for centuries, to suppress the truth 'and make men stupid.' "

In fairness, anyone who has read a major newspaper or watched a "cable news" show will perhaps be tempted to join Godel in this gloomy belief. Conceivably, enrollment in an introductory philosophy course could trigger similar musings.

For today, ignore all that! As she continues, Goldstein is direct enough to attribute Godel's peculiar belief to "paranoia." But so believed our greatest logician. Goldstein further reacts:
GOLDSTEIN (page 48-49): That the greatest logician since Aristotle should have followed reason so unwaveringly to such illogical conclusions has struck many people as paradoxical. But, as I hope will become ever clearer in the chapters to come, the paradoxes in Godel's personality were at least partially provoked by the world's paradoxical responses to his famous work...

[T]he metamathematical import of his theorems, which to Godel was their most important aspect, was disregarded. Even more paradoxically, the racier currents in the culture, hawking postmodern uncertainty and the false mythology of all absolutes, scooped his theorems up...reinterpreting them so that they precisely negated the convictions that Godel...had so passionately wanted to demonstrate.
Paradox was everywhere as Goldstein surveyed the scene. But then she turned to "technical paradoxes," entities which lay at the heart of Godel's stupendous work:
GOLDSTEIN (page 49-50): Paradoxes, in the technical sense, are those catastrophes of reason whereby the mind is compelled by logic itself to draw contradictory conclusions. Many are of the self-referential variety; troubles arise because some linguistic term—a description, a sentence—potentially refers to itself. The most ancient of these paradoxes is known as the "liar's paradox," its lineage going back to the ancient Greeks. It is centered on the self-referential sentence: "This very sentence is false."
"Paradoxes like the liar's play a technical role in the proof that Godel devised for his extraordinary first completeness theorem," Goldstein goes on to say. "Godel was able to take the structure of self-referential paradoxicality...and turn it into an extraordinary proof for for one of the most surprising results in the history of mathematics."

Yes, paradoxicality! According to Nexis, the word last appeared in the New York Times in July 1990—in an "On Language" column in an "On Language" column criticizing the use of such terms.

That's a mere aside. A bit later in her book, Goldstein turns to "Russell's paradox," the formulation which lured Wittgenstein back into philosophy when he was still rather young:

"Russell's paradox concerns the set of all sets that are not members of themselves," Goldstein correctly if murkily says.

When we first encountered Goldstein's book, we were surprised, perhaps amazed, by some of what we found. Citing one key example, we were amazed to find that a ranking philosophy professor could still be treating these paradoxes in the reverential way Goldstein does some 52 years after the appearance of Philosophical Investigations.

How hard has it been to integrate Wittgenstein's later work into the realms of elite logic? Apparently, it's been very hard, and this leaves us asking our basic questions:

How sharp have our ranking logicians been? How sharp are we great apes at all?

Next week: Goldstein on the liar's paradox and on "the set of all sets"


  1. What's with your fascination with retarded geeks/idiot-savants, Bob? C'mon, leave 'em be, man. It takes all kinds.

  2. Wittgenstein’s book, “Philosophical Investigations”, wasn’t a good book, according to Wittgenstein himself (or so says Somerby), and its basic ideas were “quite poorly expressed”, acoording to what seems to be Somerby’s own judgment.

    And yet, Somerby claims that it has been very hard for logicians “to integrate Wittgenstein's later work into the realms of elite logic”, which leads him to wonder if elite logicians just aren’t really all that sharp.

    Another possibility suggests itself though, present in Somerby’s own post. If Wittgenstein wrote a bad, jumbled, and poorly expressed book, then that might explain why his ideas were never “integrated into the realms of elite logic.”

    Wittgenstein should have written a better book. Maybe *he* wasn’t all that sharp.

  3. There seems to be some disagreement about the premise that philosophers mostly rejected the later Wittgenstein. From Wikipedia:

    “Philosophical Investigations appeared as a book in 1953, and has since come to be recognised as one of the most important works of philosophy in the 20th century.”

    “In 1999 a survey among American university and college teachers ranked the Investigations as the most important book of 20th-century philosophy, standing out as "the one crossover masterpiece in twentieth-century philosophy, appealing across diverse specializations and philosophical orientations."

    “The Investigations also ranked 54th on a list of most influential twentieth-century works in cognitive science prepared by the University of Minnesota's Center for Cognitive Sciences.”

    From comments to Horwich’s piece:

    “I cannot state this strongly enough: the author of this article is mistaken that Wittgenstein isn't well-respected or that his ideas are "dismissed." Wrong, wrong, wrong! I urge anyone to look up WVO Quine, Hilary Putnam, D. Z. Philips, Peter Winch, Kai Nielsen, etc., etc. Basically, anyone who is within the Analytic tradition -- and most British and American professional philosophers are -- are strongly influenced by Wittgenstein, whether the early (i.e., Tractatus) or the later (i.e., Blue/Brown Book, Philosophical Investigations) Wittgenstein, and he does receive proper respect.”

  4. I wrote a serious comment about how Somerby is not seriously engaging any of this, needs to stop calling people apes and either critique Godel using Wittgenstein's actual comments or stop being an idiot. That disappeared when I clicked publish. I don't have the time or interest to write it again.

    1. Beautiful. Thanks for making me smile.

  5. How quaint the ways of paradox, at common sense she gaily mocks...(WSGilbert)

    Professor of Logic: "The most ancient of these paradoxes is known as the "liar's paradox," its lineage going back to the ancient Greeks. It is centered on the self-referential sentence: "This very sentence is false."

    Student: I don't understand that sentence because of the phrase 'this very sentence.' That phrase is here used pronominally, not adjectivally, right?

    Professor: Yes, it is a pronominal.

    Student: any pronoun (pro[for]noun) stands for some grammatical or real object, yes?

    Professor: Of course.

    Student: In your paradoxical statement, then, what does the pronominal phrase "this very sentence" stand for?

    Professor: It stands for the sentence "this very sentence is false," that's why we call it "self-referential."

    Student: Haven't you taught us as the starting point of logic that if A equals B and B equals C then A equals C?

    Professor: Indeed

    Student: So then in any logical system involving A, B , and C, B can always be substituted for A, right?

    Professor: Right

    Student: Now, we've agreed already that "This very sentence" (A) in your paragraph stands for, in your own words, " the sentence 'this very sentence is false' "(B)

    Professor: We have.

    Student: So then, substituting our B for our A, we get the logically equivalent sentence "the sentence 'this very sentence is false' is false."

    Professor: We do.

    Student: Now where's the paradox? Is there even any sense of the word "false" that could make it paradoxical?

    Professor: You understand perfectly. Self-referential statements have no logical basis.

    Student: But they do just fine in poetry:
    "The pay phone it was ringing,
    and it just about blew my mind.
    When I took the receiver of the hook
    this foot came through the line."--BDylan

    Student: Then, substituting B for A according to th basic rules of logic, we get this statement: "

  6. "The Set of all Sets."
    What is a "set?"
    A set is a grouping on an entirely subjective basis of various individual items. For example "the set of all words referring to vegetables starting with b, all animals staring with r, and all continents starting with a." The works of Rabelais, Joyce, Borges, and many lesser authors are crammed with such sets.
    What then is a "set of sets?"
    Obviously, a grouping on an arbitrary subjective basis of a number of sets.
    So you have an obviously nonsensical concept: the first "set" refers to something absolutely lacking in individual components (it being made up of groupings, not individuals) while the second group of "sets" is entirely made up of individual components. Thus the word "set" is given two distinct and contradictory meaning in the expression "set of all sets." Making the expression as logically meaningless as "The East (direction) is Red (color)."
    The Russel paradox (set of all sets not members of themselves) is so nonsensical as not even to rise to the level of a paradox.

  7. 4:33 - actually a set of sets is perfectly plausible. As long as one can define unambiguously what's in a set, it's kosher. E.g., the set of all letters in the alphabet has 26 elements. The set of all subsets of this set would be the set of all collections of letters. That's a valid set. It has 2^26 elements. Each element is a set of letters.

    1. And each set of letters is a group of elements, so all those "sets" together are still just a group of elements. Now go define unambiguously (specify the elements) indicated by the predicate (all sets not members of themselves). Is the number of such "sets" (like those abounding in. Rabelais) not infinite, therefore undefinable?

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