The land of "abstract objects:" Again, let's start with our basic questions. For example:
In a world which is crawling with top-flight logicians, why is our public discourse bedeviled by hopeless illogic? From 1994 through 1996, why did no logician step in to critique this iconic claim by Newt Gingrich:
No one is cutting the Medicare program. We're just slowing the rate at which the program will grow.For roughly two years, the public discourse came to a halt as reporters and pundits struggled with that claim by the Gingrich-led Republican party. Our journalists were hopelessly overmatched by the claim. But where were our many elite logicians?
In some ways, our question is easily answered. Our elite logicians concern themselves with loftier topics than these. As we showed you just last week, W. D. Hart offered these thoughts in his 2010 book, The Evolution of Logic:
HART (page 59): Perhaps more generally a quantifier is a second-level function whose value at an (n + 1)-ary first-level concept is an n-ary concept, unless n is zero, in which case its value is a truth value, an object. In that case, quantifiers would be second-level functions sometimes having first-level concepts as values and sometimes objects as values. When the value of a first-level concept at an object is truth, Frege says the object falls under the concept. Perhaps the concept:falls-under is a binary second-level concept whose first argument is an object and whose second is a first-level concept. In that case, second-level concepts could also have arguments of different levels.When issues like these are being resolved, few will have time for the affairs of the world.
Then too, we have Professor's Goldstein's explanation of Godel's incompleteness theorems. According to the New York Times review of Goldstein's 2005 book, this was magically lucid work:
GOLDSTEIN (page 175): The Godel number that will correspond to the sequence of p1 followed by p2 is:According to the Times review, that was part of an exposition which was magically lucid for the general reader. That said, when logicians have matters like this to discuss, who has time for the piddling issues affecting the health of tens of millions of people?
GN(p1, p2) = 7387398776738467398827343980675846758
Through Godel's inspired contrivances, all of the logical relations that between propositions in the formal system become arithmetic relations expressible in the arithmetical language of the system itself. This is the essence of the heart-stopping beauty of the whole thing. So if, for example, wff1 logically entails wff2, then GN(wff1) will bear some purely arithmetical relation to GN(wff2). Suppose, say, that it can be shown that if wff1 logically entails wff2, then GN(wff2) is a factor of GN(wff1). We would then have two ways of showing that wff1 logically entails wff2; we could use the rules of the formal system to deduce wff2 from wff1; or we could show that GN(wff1) can be obtained from GN(wff2) by multiplying by an integer. Suppose that GN(wff1) = 195589 and GN(wff2) = 317. 317 is a factor of 195589, since 317 multiplied by 617 = 195589. So that wff1 logically entails wff2 could be demonstrated either by using the formal rules of proof to arrive at wff2 from wff1, or, alternatively, by using the rules of arithmetic to arrive at GN(wff1) = 195589 by multiplying 617 by 317 = GN(wff2). The metasyntactic and the arithmetic collapse into one another.
For better or worse, our elite logicians have developed elite concerns. They leave the affairs of the world to people like the Washington Post's Monica Hesse, who recently explored the metaphysics, hermeneutics and ontology of Stephen Miller's deeply troubling yet highly evocative hair.
Our journalists have been exploring such pitiful topics for decades. Our logicians live in the clouds, perhaps in the land where numbers and circles have the "perfect, timeless existence" alluded to in The New Yorker's unintentionally comic review of Goldstein's surprising book.
So it goes with us rational animals in this, the deeply dangerous era of Donald J. Trump. According to Professor Horwich, this lunacy persists, at least in part, for an all-too-human reason:
When the later Wittgenstein came along and offered advances in rational conduct, our philosophers turned him away, concerned because he'd said that their previous efforts had been built on definable types of illusions. According to Professor Horwich, the guild has chosen to turn him out, to reaffirm the old ways.
This brings us to the second major "paradox" explored in Professor Goldstein's 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel. On Wednesday, we explored "the liar's paradox," a form of chicken-scratching on the approximate level of the tree which famously falls in the forest when no one is around.
The notion that modern logic is built around the mysteries of this ancient riddle—well sir, Trump can sing the entire theme song from Green Acres before the laughter subsides.
Goldstein, a ranking philosophy professor, takes the liar's paradox very, very seriously. Later in her favorably reviewed and blurbed book, she describes "Russell's paradox," the conundrum which convinced the young Ludwig Wittgenstein to give up aeronautics and study logic instead:
GOLDSTEIN (page 90-91): Wittgenstein came from one of the wealthiest and most culturally elite families of Vienna, "the Austrian version of the Krupps, the Carnegies, the Rothschilds, whose lavish palace on Alleegasse had hosted concerts by Brahms and Mahler..." He was, in his intensity, preoccupations, ambitions and conflicts, indelibly stamped by the sensibilities of that intense, preoccupied, ambitious and conflicted city. While studying aeronautical engineering at the Technische Hochschule in Berlin, he had learned of Russell's paradox, and became interested in the foundations of mathematics.So what the heck was Russell's paradox? In truth, it was basically a tricked-out version of that ancient Greek/Cretan tale:
GOLDSTEIN (continuing directly): Russell's famous paradox is of the self-referential variety. The liar's paradox—this very sentence is false—is of the same variety. We get into trouble because some linguistic item talks about itself, at least potentially, and by reason of this self-referentiality we end up both asserting that some statement is true and that it is also false, which is logically impossible if anything is.This very sentence is false! Should we regard that as a "statement," as Goldstein does in that passage, or as a silly parlor trick? To see us vote for the latter approach, click here for Wednesday's report.
Regarding Russell's "famous paradox," Goldstein proceeds to the place where the rubber meets the road. As she does, it seems to us that we can hear the laughter of the gods:
GOLDSTEIN (continuing directly): Russell's paradox concerns the set of all sets that are not members of themselves. Sets are abstract objects that contain members, and some sets can be members of themselves. For example, the set of all abstract objects is a member of itself, since it is an abstract object. Some sets (most) are not members of themselves. For example, the set of all mathematicians is not itself a mathematician—it's an abstract object—and so is not a member of itself., Now we form the concept of the set of all sets that aren't members of themselves and we ask of this set: is it a member of itself? It either is or it isn't, just as the problematic sentence of the liar's paradox either is or isn't true. But if the set of all sets that aren't members of themselves is a member of itself, then it's not a member of itself, since it contains only sets that aren't members of themselves. And, if it's not a member of itself, then it is a member of itself, since it contains all the sets that aren't members of themselves. So it's a member of itself if and only if it's not a member of itself. Not good."Russell's paradox concerns the set of all sets that are not members of themselves." As Goldstein correctly notes, this is the famous conundrum which lured the young Wittgenstein to Cambridge.
Let's turn to Goldstein's account. For our money, it's amazing that a ranking philosophy professor could have produced such work in 2005, some 52 years after the publication of the later Wittgenstein's definitive text, Philosophical Investigations.
It's even more amazing that three academic stars—Pinker, Greene and Lightman—would have rushed to the back of her book to blurb her lucidity and praise her insight and skill.
To our ear, almost every part of that paragraph evokes the laughter of the gods. "Sets are abstract objects that contain members?" On the steepest slopes of Olympus, how the gods must have roared!
In our view, it's a remarkable fact that elite logicians still traffic in piddle like this. For today, we'll only enjoy the mirth which comes to us when we read this one small part of Goldstein's presentation:
"Sets are abstract objects..."Sets are abstract objects! Before we comment on that peculiar choice of words, let's get clear on one point. Goldstein is speaking here of a particular type of "set," though she fails to say so.
Your grandmother's special china set isn't, in any normal sense, some sort of "abstract object." It's a collection of physical objects—tea cups, saucers and plates—which are taken out of her cabinet for use on special occasions.
John Doe's full set of 1958 baseball cards is also, in the most straightforward sense, a collection of physical objects. And sure enough! Even as we sit here typing, Macy's is offering low, low prices on "Calvin Klein Modern Comforter Sets."
There too, the sets which you can buy at Macy's are collections of physical objects. No "abstract objects" are listed as part of the deal!
In fairness, Professor Goldstein isn't talking about that type of "set." She's discussing a different type of "set"—a type of set she chooses to describe as an "abstract object."
Tell the truth! Do you have any idea what an abstract object might be? Do you know why a person would choose to employ such an odd choice of words?
In our view, the type of "sets" to which Goldstein refers are more accurately described as "imaginary collections." You can imagine "the set of all mathematicians," though it would be hard to agree on, and list, all their names.
Once someone has explained what an "abstract object" is, you can even imagine "the set of all abstract objects," though you'd never be able to list every such set.
Russell's paradox concerns the set of all sets that are not members of themselves! In our view, it also concerns grown men and women, typically of a certain class, amusing themselves by playing with dolls while ignoring the affairs of the world.
These woods are silly, and not very deep. Again and again, then again and again, Goldstein insists on the opposite view of the world which can be created when we pretend to address the imagined perils of "self-referential paradox," which causes the mind to crash.
"This problem that had stumped the great Lord Russell was obviously something worth thinking about." So Goldstein writes on page 93, explaining why the youthful Wittgenstein was drawn in by this puzzle.
Having said that, alas! By the time he wrote his first, early book, the still-youthful Wittgenstein had kicked Russell's paradox to the curb. Goldstein keeps rolling her eyes at such heretical work. According to Professor Horwich, our elite logicians are like that.
Our journalists deal with clothing and hair; with questions of who may or may not have had sex on one occasion in 2006; and of course with invented quotations which help advance preferred story lines. Our logicians deal with the set of all sets not members of themselves!
In this way, the dumbing proceeds. Eventually, after the lovin' is done, Donald J. Trump gets elected.
"Eventually, after the lovin' is done, Donald J. Trump gets elected."ReplyDelete
...and thank god for that. Amen.
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I’m sorry, but JFC, this is some of the most ignorant bullshit Somerby has ever come up with, and that’s saying a lot.ReplyDelete
Somerby refuses to even attempt to understand mathematics, because he would rather mock it apparently. His real or feigned inability to understand terms like “abstract objects” shows his utter ignorance of even rudimentary concepts of mathematics. Why is he doing this? To anyone with any knowledge of mathematics, Somerby looks like the biggest fool on the planet. He doesn’t have the decency or the intellectual honesty to study mathematics, even a little bit, to dispel his ignorance of it.
Should we expect the next book to appear from Somerby titled “debunking Lord Russell in three easy steps”, and those steps are
1. Quote something Russell said
2. Ask “friend, does this make sense?”
3. Of course it doesn’t!
QED, Russell is a fraud.
Somerby is the fraud here.
So, Bob, is a teacup an object that exists in the real world?ReplyDelete
How about the number “1?”
“No, only crackpot mathematical Platonists like Gödel believe that numbers have a timeless existence in the real world.”
Ah. So the number “1” is not a physical object?
“Of course not, silly.”
Thus it is a concept, it is abstract, not concrete. It is an item used in mathematics, and appears in many arithmetic problems and mathematical theorems. It is called an “object”, because it functions as an object for purposes of mathematics. It isn’t a physical object though. It must be “abstract.”
Ok. Never mind.
Someone pointed out a few days ago that, for a "set" or anything else to exist, it has to be capable of definition. An infinite series of "members" is incapable of definition and therefore cannot exist as a set. In the alleged paradoxes the unqualified (ie., lacking a limit to the number of its possible members) word "all" indicates that the number of members of such a "set" is infinite and thus that such a set is itself both an abstract object and a member of the "set of all nonexistent sets."ReplyDelete
The crucial nature of *limit* is the central concept in Plato's *Philebus*, Hegel, in the *Science of Logic* closely examines the concept of "infinity" and makes the crucial distinction between the "bad" infinity of unlimited self-extension and the "good" infinity of self-completing process (symbolized, if you will, by the contrast between a one-dimensional surface (line) extending without limit and a one-dimensional surface (oval) bending to eventually return into itself).
“An infinite series of "members" is incapable of definition and therefore cannot exist as a set.”Delete
@12:23 - the key to defining a set is that one can determine whether any object is or isn't an element of that set. E.g., the set of natural numbers is a well-defined set, even though it's infinite. "37" is an element of that set. My computer is not a element of that set.Delete
What is a “concrete object?” Why would anyone want to talk about things made of concrete???ReplyDelete
Our logicians just ain’t that sharp.
Bob mocks thge idea that, "Sets are abstract objects, giving as proof Your grandmother's special china set isn't, in any normal sense, some sort of "abstract object." It's a collection of physical objectsReplyDelete
Yes, the pieces of china are physical objects, but the set of these physical objects is an abstract object.
It's the terminology, not the definition of "abstract" that the blogger is mocking.ReplyDelete
It's the preoccupation with an extreme esotericism that he bemoans as well.
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