What the heck is an abstract object?


How the professors have failed: Late in life, we’ll finally admit it:

Since perhaps the late 1960s, we’ve conducted a love affair with hopelessly bad explanation.

We got our start in college, reading the later Wittgenstein. But the love affair continued. Example:

Recently, we decided to reread Professor Goldstein’s 2005 Godel-made-easy book, Incompleteness: The Proof and Paradox of Kurt Godel.

Rebecca Goldstein is a philosophy professor and a novelist; she has taught at a variety of institutions, including Barnard and Columbia. Incompleteness is part of the Atlas Books “Great Discoveries” series; it’s designed as an explanation of Godel for general readers.

We decided to revisit Incompleteness after looking at Goldstein’s new book, Socrates at the Googleplex, a sequel to our own abandoned 1998 project, Socrates Reads. In the course of our revisit, we’ve come to this conclusion:

We’ve seen a lot of bad explanation. Incompleteness plainly contains some of the very worst.

We don’t mean to single Goldstein out. In Incompleteness, she offers a string of standard presentations straight from the playbook of the philosophy guild. In our view, these are the types of presentations the later Wittgenstein tried to bring to an end.

Tried and failed! As with the journalists, so too here—the academic guild marches on!

Let’s skip through some of the wonderful nonsense parents pay a lot of money to have dumped on their kids' heads. About a third of the way through her book, Goldstein starts describing Russell’s Paradox.

This is the way she starts:
GOLDSTEIN (page 90): Russell’s famous paradox is of the self-referential variety. The liar’s paradox—this very sentence is false—is of the same variety...

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. ...
We’d describe that as wonderful nonsense—and no, we haven’t omitted some key preceding material. Let’s focus on these statements:
“Sets are abstract objects that contain members.”
“The set of all mathematicians is...an abstract object.”

By obvious inference:
“A set is an abstract object that contains members.”
Do you have the slightest idea what any of that verbiage means? More to the point, does Goldstein herself? Do her colleagues?

According to Goldstein (and many others), a set is “an abstract object that contains members.” On its face, that’s a somewhat peculiar statement.

If a set is “an object,” why would you call it “abstract?” Similarly, if a set is “abstract,” why would you call it an object?

At this point, are we already heading off the rails? Just asking! But on its face, “abstract object” is a somewhat strange locution. But then, so is the phrase “contains members.”

Say what? “Contains members” looks like a phrase we all understand. Surely, we all know what it means if we’re told that a social club contains 35 members, to offer one clumsy-sounding example.

In fact, that formulation is rarely used in normal English. It’s a rather peculiar locution, despite its familiar parts.

Is the phrase “contains members” actually rare? It sounds like everyday English!

Using Nexis, we checked. In the last ten years, these are the only times some variant of that phrase has appeared in the hard-copy New York Times:
June 16, 2009: Flocks of ducks or geese can contain members of several distinct species, for example, and if they are confronted by a predatory hawk or a hunter, they all take off together.

October 7, 2008: The public urge for punishment...is based in instincts that have had a protective and often stabilizing effect on communities throughout human history. Small, integrated groups in particular often contain members who will stand up and, often at significant risk to themselves, punish cheaters, liars and freeloaders.

August 1, 2004: Martin's first Air Force team, containing members of the academy's first senior class, went 9-0-1 in the regular season, then played a 0-0 tie with Texas Christian in the Cotton Bowl.
According to Nexis, those are the only uses of that phrase in the New York Times in the past ten years. One of the three occurred on Bloomsday, helping establish our point!

That doesn’t mean there’s something wrong with what Goldstein wrote. But “abstract object” isn’t the only unusual formulation in her statement, which is only seven words long.

For people who want to avoid incoherence, that could be a warning sign. That said, let’s return to the basics:

“A set is an abstract object.” Do you have any idea what that means? Luckily, Goldstein provides an example. Go ahead—puzzle on this:

“The set of all mathematicians is an abstract object.” Do you have any idea what that statement means?

Is Goldstein picturing a list of all living mathematicians? In theory, you could compile such a list. But would such a list be an “abstract object?” And would it be what Goldstein means?

“Sets are abstract objects that contain members.” If you aren’t inclined to challenge failed authority, you might let that first (apparent) statement slide by.

If you do, you’ll be farther off in the weeds with each successive move. You’ll have no idea what’s being said, no idea how to paraphrase.

Goldstein’s glorious incoherence doesn’t start on page 90. Much earlier, she starts trying to define a key term:
GOLDSTEIN (page 44): [Godel’s] commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical, realism. It is also known as mathematical Platonism...

Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference, and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract, entities (numbers, sets, etc.) that make up that reality. The structure of, say, the natural numbers (which are the regular old counting numbers, 1, 2, 3, etc.) exists independently of us, according to the mathematical realist; and the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares— are as objective as are, according to the physical realist, the physical properties of light and gravity.
That is perfect gibberish. But according to Goldstein (and many others), this perfect nonsense is very important. This perfect nonsense defines Platonism, an august-sounding “view.”

What the heck is Platonism? Hang on for the ride! According to Goldstein, Platonism consists in these beliefs:

According to Platonism, the fact that 4 is an even number is an “objective” truth. Evenness is an “objective property” of the number 4.

If you feel like you may not know what that means, you deserve a pat on the back. All through her book, Goldstein acts as if the word “objective” somehow explains itself.

(It doesn’t. We’d call that her foundational act of incoherence.)

That said, what exactly does it mean when Goldstein says that evenness is an “objective property” of the number 4?

According to Goldstein, that jumble of verbiage seems to mean something like this:

According to Platonism, the fact that 4 is an even number is a “truth of mathematics” which is “independent of any human activities.” That truth is “determined by the reality of mathematics.” It’s determined by the nature of the number 4, which is “a real though abstract entity.”

(At least she didn’t call it an “abstract object!”)

The truths of mathematics are determined by the reality of mathematics? How could someone dispute that?

We would be inclined to call that passage perfect nonsense. Our prediction: Subjected to coherent questioning, Goldstein couldn’t explain that passage in a million years.

She’s moving words on a ouija board, as her colleagues have done for centuries. The later Wittgenstein tried to address this ridiculous process.

Slowly but surely, his rather inarticulate efforts have been consigned to the dustbin. Professors just wanna have fun!

A few pages later, Goldstein says what is shown below. This is absolutely perfect nonsense, of the type Wittgenstein tried to discuss.

We highlight the world-class howler:
GOLDSTEIN (page 49): Paradoxes, in the technical sense, are those catastrophes of reason whereby the mind is compelled by logic to draw contradictory conclusions. Many are of the self-referential variety; troubles arise because some linguistic item—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.” That sentence must be, like all sentences, either true or false. But if it true, then it is false, since that is what it says; and if it is false, well then, it is true, since, again, that is what it says It must then be both true and false, and that is a severe problem. The mind crashes.
“That sentence must be, like all sentences, either true or false?”

Who ever said that every sentence has to be either true or false? Playing with dolls, you can imagine such a world. But we don’t live in a world like that, as Goldstein plainly knows.

Indeed, she herself had just presented a sentence which was neither true nor false. Here it is:

“This very sentence is false.”

Goldstein calls that a sentence; there’s no particular reason not to. But rather transparently, that sentence is neither true nor false; that sentence is incoherent. But then, our world is full of incoherent sentences, of a wide array of types. You can find them in our newspapers every day of the week.

If you’ll forgive the gendered sound of the following statement, Goldstein is playing with dolls. She’s playing with dolls in the way her colleagues, male and female, do every day of the week.

During the past several decades, we have often asked a question. As every manner of illogic has come to define our public discourse, we have asked where the professors and the logicians are.

Answer: They’ve been playing with dolls in the south of France, as Goldstein does all through her astonishing book.

We recommend this particular book as a sample of perfect nonsense. What kinds of people spend their lives presenting such incompetent work? The kinds of people whose biographies, by the world’s leading authority, contain passages of this type:
THE WORLD’S LEADING AUTHORITY: She married Harvard cognitive psychologist Steven Pinker in December 2007. They met after Pinker mentioned her in his book Words and Rules, where his example of an irregular verb form "familiar enough to block a regular version, but not quite familiar enough to sound natural” is the participle "stridden" in Goldstein's novel The Late-Summer Passion of a Woman of the Mind. Pinker recounts that Goldstein saw the book, contacted him, and they "had tea together.” Goldstein says, "When I first met Steve in the flesh, I said that the way he thinks had so completely changed the way I think—particularly what I had learned from him about cognitive psychology and evolutionary psychology—that I said, 'I don’t think I’ve had my mind so shaken up by any thinker since David Hume.' And he very modestly said, 'That can’t be the case.' But it was the case. So I can certainly say that Steve has profoundly influenced the way I think.”
Pinker, of course, is considered a giant—although, since we’re being honest today, we have to say that we’ve sometimes found his writing less than coherent.

But good God! Have you ever seen a worse highlighted passage?

(On what basis can we say that Goldstein made those statements to Pinker? Good God! She gave that account to Salon. To convince yourselves, click here.)

In recent decades, you have been desperately failed by these hapless, incompetent people. Parents’ bank accounts have been drained to expose the nation’s teen-agers to their wandering drivel.

Our “logicians” have failed us, both in their technical incompetence and in their relentless failure to address the real world.

Al Gore said he invented the Internet! As you may be prepared to admit by now, the logicians let that howler go. They were pondering abstract objects, and entities which are and aren't real!

People are dead all over the world because the logicians have been writing these books, because they’ve perhaps been living the lives defined by such horrible quotes.

Concerning the term “abstract object:” Within the art world, there is one use of the term “abstract object” which is fairly common and quite easy to explain. According to Nexis, this would be the most recent usage in the New York Times:
November 30, 2008: Equally thought-provoking is Charles Simonds's monumental abstract object hanging from the ceiling in the front gallery. Made of clay, polyurethane, metal and wood, it looks like an asteroid floating in space, or possibly a giant gnarly tree root. It is beautifully finished, if you care to get up close to the menacing object and take a look.
In this passage, the term refers to a physical object in a piece of sculpture, an object which isn’t representational.

That isn’t what Goldstein was talking about. What does her usage mean?


  1. To help unconfuse Bob, why doesn't he click on the following link to the main repository of human knowledge:


    1. hardindr, that must have taken you 10 seconds to google. Bob's too busy for that. Easier to just write a long diatribe..

    2. I realize that my comment didn't directly deal with Bob's topic, i.e. popularizers aren't doing a good job with math, logic, or physics. However, Bob never points to other sources of information that might correct his lack of understanding on these topics. I find that very frustrating.

    3. I doubt Somerby really doesn't understand what is meant by an abstract object. He is pointing out that this is a technical definition (philosophers know and agree on what is meant) lodged in a book purportedly for non-experts. The author doesn't bother about whether the reader will understand what is meant and neither do most readers.

      Hardindr assumes Somerby must be confused if he complains that a nontechnical reader should be confused. David in CA assumes Somerby doesn't understand sets. Neither of you are understanding that his point is about how readers brush right on past sentences that are unclear without trying to understand them, and that because of people's tendency to ignore their own incomprehension, a great deal of nonsense is accepted in many areas of life.

      Somerby is not complaining that a philosophy book for neophytes is confusing him. He is complaining that it will confuse most people and yet people don't object to it -- they even call such authors brilliant.

      He is making an analogy between these sorts of books popularizing technical subjects and the kind of garbage that is handed down by Maddow and others in our media, where people accept similarly unproven, illogical, even factually incorrect statements routinely without examination and no one objects -- our pundits keep collecting their huge salaries.

      Most of all, he is saying that someone (besides him, obviously) should be objecting to the level of nonsense served up every day by our so-called experts, whether in politics or philosophy or physics.

      If you cannot understand what he is saying here, I don't think you are ready to volunteer for that job. If you (like our trolls) point out that Somerby is unclear, I think you need to go back to school and get another degree. He really isn't that opaque, except to our wilfully obtuse trolls.

    4. In all the years I have read Somerby, whenever he has brought his issues with science/mathematics/philosophical popularizers, he has never made it clear that he understands the topics he has complained about. Many times he has said that the supposed "expert" professors in such subjects during his undergrad time at Harvard also confused him. Maybe Bob really does understand these concepts, maybe he doesn't; I honestly can't tell, though what you have written has made me consider an alternative way to read Somerby's frequent complaints. I am not disagreeing with his complaints, I just find it annoying that he will not point to some kind of alternative to these bad popularizations, if he knows of them.

      As an aside, I have no degrees, and I ain't a troll, bub.

    5. And I don't think you are the one calling Somerby a hypocrite for being unclear either.

    6. Hardindr, aren't you making his point? If we have to look up abstract object in wikipedia in order to understand the author's explanation for abstract object, isn't she failing?

  2. All communication is a consensual act. Words mean what people agree they mean. See Hilary Putnam for philosophical exposition (extending Quine).

    The point of this post seems to be that people who can untangle the confusion in today's discourse, because of their training in logic, critical thinking, even cognitive psychology, should be applying their minds to that practical use. Unfortunately, no one pays them to do that and it can ruin a career. Some of us who have tried have been hounded at our place of employment by stalkers who take their animosity into real life -- meaner than the fairly tame trolls we have here, but just as dedicated. It just isn't worth it to take that risk. Elizabeth Loftus got death threats and had to have security guarding her when she spoke to the public.

    As far as school tuition, I've found that no matter how carefully I try to communicate with students, what they take away from a class (or a textbook) is different than what I said, and different than what I intended. I console myself by believing that whatever bit of knowledge, inspiration or improved thinking they do attain is more than what they arrived with. Whether you think it is worth the tuition depends on how much you value the things higher education correlates with, such as higher salary, less likelihood of being conned into a high risk mortgage, better health practices, and so on -- due not to content of college courses but to acquired habits of mind and behavior that are self-protective in real life.

    Pinker is not as highly regarded by cognitive psychologists as he is by the general public. He is a highly effective self-promoter (as are most Harvard faculty). There was a great deal of criticism of his popular books because they go way beyond the data and include a lot of speculation presented as fact. His books on language (his actual field) are better than his books on brain and behavior. In real life he is not a vaguely formal Britishy kind of guy who would take tea and be humble about changing someone's world view.

    1. You forgot to mention Susan Clancy, who was hounded out of Harvard...


  3. Poor Bob doesn't really understand the concept of a set. so he doesn't understand Russell's paradox and he certainly doesn't understand what Gödel did. Instead of starting with a book about Gödel, Bob should start with a very basic set theory book. One that I studied from many years ago was Naïve Set Theory by Paul Halmos.

    1. As a philosophy major, of course he understands the idea of a set.

  4. How about "abstraction" instead of object and "defined elements" instead of members

    1. That IS helpful!

      Now an "abstract object that contains members" is an "abstract abstraction that contains defined elements."

      I want to thank you, but where do I send the bouquet?

    2. not an abstract abstraction. Just an abstraction. Donate it

  5. I find Bob's typical postings far more incomprehensible than the philosophers, physicists and mathematicians he tends to ramble on about.

    1. You must know that there is an obvious solution to your problem.

  6. Is Bob really complaining about writers discussing complex issues are "incoherent?" Dear god Bob may not get satire but he does irony

    1. No, this is not what "Bob" is complaining about.

      He is complaining because people read right past incomprehensible statements in a variety of areas of life, without worrying about whether they understand them or not, accepting them at face value as if they were clear.

      For example, your remark is in the guise of a comment on the post but is actually just another humdrum expression of hostility toward the author of this blog. Tiresome, ugly, a waste of everyone's time, with nothing whatsoever to contribute to discussion here.

    2. Ah, but the problem with your presumption is that what is incomprehensible to Bob might not be incomprehensible to the entire planet.

      It could be that Bob is NOT the most intelligent person walking the planet, the sharpest tool in the shed, the brightest bulb in the box.

      We don't know. It's possible. It's never been disproven.

    3. He is criticizing popularizations aimed at everyday people. They are not geniuses either but they have the right to expect clear info in a book addressed to them. Just as we have the right to expect accuracy from Maddow.

    4. Ah, so Bob IS intelligent enough to understand. He just pretends it's all so incomprehensible in order to protect those unwashed rubes with lizard brains from "popularizations aimed at everyday people."

      And geez, Bobfan, take the advice of your tribe. You don't have to watch Rachel Maddow. But then how would you understand what Bob has devoted the remainder of his underachieving life to?

  7. And here in a nutshell is Bobism, in its full glory:

    "In recent decades, you have been desperately failed by these hapless, incompetent people. Parents’ bank accounts have been drained to expose the nation’s teen-agers to their wandering drivel."

    It is never enough for Somerby to take issue and raise alternative points in what would be an adult "discourse."

    Nope, gotta reach for the insults.

    And listen to the BOBfans whine when he gets it handed back to him.

    1. If you don't like his columns then don't read them. No one is forcing you to be here.

    2. Hi 5:53. Wondering what is the insult you saw? Maybe they really are "hapless, incompetent" and delivering "drivel". Are those the words you considered an insult? Are those words off limits in some way in adult discourse? I'm not meaning to get into a pissing match and not making a value judgement about what you said or criticizing you or saying you are wrong. Just wondering what the insult is because it doesn't seem insulting if it's true.

      Best regards,

  8. Numbers and such are abstract objects. I have no trouble understanding this. Maybe you should study some math before making a fool of yourself in public.

    1. Superficial response typifying what Somerby is complaining about.

    2. Anon 6:50 is too stupid to be reading this blog.

    3. This blog is designed to help general readers see the flaws in books like the one designed as an explanation of Godel for general readers.

  9. BTW a fine book on this topic was written some time ago shortly after Gödel published his famous result is "Gödel's Proof" by Ernest Nagel and James Newman. It's not trivial to read, because the proof itself is pretty subtle, but it really does give a pretty good idea of the concept underlying Gödel's work. Apparently the book was reissued recently, which speaks to its quality.

  10. Bob has "sometimes found his [Pinker's] writing less than coherent." I hope Bob will give us some examples.

  11. What do readers think is the latest or most important technological tool at Bob's disposal to better inform us? The internet, created in part by the initiative of his college acquaintance, or Nexis, launched in 1981?

  12. Bob, I will check your blog everyday now that you are writing on weekends which I love. As for this post, wonderful analysis, just wonderful.

  13. If you're interested in the role abstracta play within the ontology of art (enquiry into what sorts of things artworks are or could be), then check out my recent anthology Art & Abstract Objects (Oxford University Press 2013).