And Now For Something Completely Different


What can one say about a book on infinity that hasn’t been said before? An infinite number of things, presumably, but I’ll make this brief.

The book, Approaching Infinity, is by philosopher Michael Huemer. Perhaps you’ve heard of him — but why? If you’re a libertarian, but not a philosopher or “into philosophy,” it’s likely because of his well-received book, The Problem of Political Authority (2013).

If you’re a libertarian and, though not a philosopher, are into philosophy, you may also be aware of Huemer’s excellent online-available essays on the right to own a gun and the right to immigrate. (I imagine readers on both the Left and Right are now gnashing their teeth.)

Huemer, like Robert Nozick before him, is clearly better described as a philosopher who is a libertarian than as a “libertarian philosopher.”

But Huemer is nothing if not prolific. Libertarians who are really into philosophy may even be aware of his criticism of Ayn Rand, his argument that we sometimes have a duty to disregard the law, his argument that attorneys have a moral obligation not to defend unjust causes, his criticism of the US government’s War on Drugs, and his essay on why people are irrational about politics (also a TED talk!).

But — and this is the point I want to stress — even though he’s published much of interest to libertarians, Huemer, like Robert Nozick before him, is clearly a person better described as a philosopher who is a libertarian than as a “libertarian philosopher.” His first book, Skepticism and the Veil of Perception (2001), dealt with epistemology (the field of study that led to his hiring at University of Colorado, Boulder); his second, Ethical Intuitionism (2005), focused on ethics. Now, having covered epistemology, ethics, and politics, Huemer, in Approaching Infinity, turns to the philosophy of mathematics (with an occasional nod to some issues in the philosophy of science). Clearly a well-rounded guy, philosophically speaking.

Also an iconoclast:

  • Although most philosophers since Descartes have opposed direct realism (the view that we are directly aware of real, physical objects), Huemer argues for just that point of view.
  • Although most modern philosophers oppose ethical intuitionism, the view that we can have direct knowledge of objective moral truths, Huemer again argues for exactly that.
  • Although most people readily accept political authority, and most philosophers are not anarchists, Huemer argues both against political authority and for a capitalist version of anarchy.

So it should surprise no one that Huemer, in analyzing some foundational issues in mathematics in order to solve various paradoxes of infinity, is willing to advance bold claims.

Almost everyone is familiar with at least some infinity paradoxes. We’ve all heard about Zeno and why that ball coming at you will never reach you, or why the hare can never catch the tortoise. Any you’re probably aware that strangeness results when even simple arithmetic is applied to infinity. E.g., ∞ = ∞ +1. Subtract ∞ from both sides: 0 = 1.

But I had no idea there were at least 17 different paradoxes associated with infinity. From Hilbert’s hotel to Gabriel’s horn . . . from Thomson’s lamp to Benardete’s paradox . . . from ancient Greek problems to dilemmas developed only in the past century . . . Huemer describes them all and then starts to evolve some background needed to solve the infinity paradoxes. There are discussions of actual and potential infinities, of Georg Cantor’s set theory, of the theory of numbers, of time and space, of both infinity and infinitesimals. Of the metaphysically impossible and the logically impossible. Of the principle of phenomenal conservatism (which Huemer introduced in his epistemology text), and even of the synthetic a priori.

Huemer, in analyzing some foundational issues in mathematics in order to solve various paradoxes of infinity, is willing to advance bold claims.

In building the background to handle the infinity paradoxes, Huemer argues that extensive infinities (including the cardinal numbers) can exist but not as specific magnitudes. Thus, the positive integers are infinite, in the sense that for any such number you can find higher positive integers, but not in the sense that there is a number “infinity” that is higher than all the positive integers. You cannot add and subtract “infinity” as I did in the previous paragraph. And he argues that while extensive magnitudes (time, space, volume) can sensibly approach infinity in this understanding, infinite intensive magnitudes (such as temperature, electrical resistance, attenuation coefficient, etc.) are metaphysically impossible. This distinction allows several paradoxes to be solved, or avoided.

A fascinating section of the text discusses various forms of impossibility. Sometimes philosophers note that X is physically impossible, given the laws of the universe as we now understand them, but nonetheless that it could be possible in a similar but slightly different possible world — say, with a slightly different Coulomb constant. But at other times X is deeply physically impossible. Consider these two alternatives described by Huemer:

Compare this pair of questions:

A. If I were to add a teaspoon of salt to this recipe, how would it taste?

B. If I were to add a teaspoon of salt to this recipe in an alternative possible world in which salt is a compound of plutonium and mercury and we are sea creatures who evolved living on kelp and plankton, how would it taste?

Huemer notes that it’s not merely that we have no idea about how to answer B but that, more importantly, even if we could answer B, answering it gives us no intuitions, is of no help in trying to figure out the answer to A. Though Huemer makes this point in the context of determining what counts as a solution to an infinity paradox, it also has direct application to various thought experiments in other areas of philosophy and to what counts as a helpful or unhelpful thought experiment. (On this see my own work, “Experiment THIS!: Libertarianism and Thought Experiments.”)

Related to the paradoxes of infinity are the problems of infinite regress. You may have heard of the problem of the regress of causes: asked what caused A, you explain that it was caused by B. But what caused B? C caused B. But … here is an infinite regress. Does this imply that we never really understand what caused A?

There are other interesting infinite regresses: of reasons, of truths, of resemblances, etc. Huemer offers helpful insights here as well, elaborating various factors that determine whether such infinite regresses are vicious or benign.

Did I mention that Huemer can be iconoclastic? Consider these passages from Approaching Infinity:

  • There are certain philosophical assumptions that tend to generate strong resistance to my views, and these assumptions are commonly accepted by those interested in issues connected with science and mathematics . . . I have in mind especially the assumptions of modern (twentieth-century) empiricism . . . the doctrine that it is impossible to attain any substantive knowledge of the world except on the basis of observation.”
  • “In the original, core sense of the term ‘number,’ zero is not a number. . . . Why is zero not a number in the original sense? Because a number, in the primary sense, is a property that objects can have, whereas zero is not a property that objects can have.” Huemer extends the concept of number to include zero but explains why such an “extension” does not work for “infinity” as a number.
  • There are reasons to doubt that sets exist. No one seems to be able to explain what they are, they do not correspond to the ordinary notion of a collection, and core intuitions about sets, particularly the naive comprehension axiom, lead to contradictions.”

In his final chapter, Huemer, taking to heart Nozick’s concerns about coercive philosophy, offers readers his own thoughts about problems that remain: which of his answers leave him concerned or unsatisfied, arguments that are incomplete, areas for further exploration.

As in his earlier books on ethics, epistemology, and politics, Huemer’s style is as easy and enjoyable as his logic is rigorous. Intelligent laypeople who are interested in philosophy can follow his thoughts without difficulty. No Hegel here.

Because I have little background in the philosophy of mathematics, I approached Huemer’s latest effort with trepidation, despite having very much enjoyed his three earlier books. But now that I’ve read it, I highly recommend it. The best news: before finishing Approaching Infinity, you’ll have to read halfway through it, and before that one-quarter of the way, and before that one-eighth, and before that. . . . Yet despite this you can read it through to the very end, and be enthralled on every page.

Editor's Note: Review of "Approaching Infinity," by Michael Huemer. Palgrave Macmillan, 2016, 275 pages.

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The Pains of Proflish


A student taking an advanced degree at a world-renowned institution sent me a news item about a math professor at Michigan State University who (allegedly, always allegedly) took off his clothes in the middle of class and ran around naked, shouting things like, “There is no f*cking God!”

No, I’m not going to claim those words as an invitation to comment on the linguistic habits of scientific atheists. To paraphrase Richard Nixon, I could do that, but it would be wrong. But I’m not sure how wrong it would be to take it as a commentary on the linguistic habits of college professors (of the which I am one). It seems to me that during the past 30 years we’ve done a lot of running around naked, intellectually speaking, and what has been revealed has not been impressive.

I can’t say I was surprised by the news my fellow Watcher sent me. What did surprise me was the reported reactions of the professor’s class. (No, I didn’t mean “were the reported reactions”; I meant was; the number of the verb follows that of the subject, which is what, and which is singular.) “We were literally scared for our lives,” one student said. “The police took about 15 minutes to get here, and during this time he continued walking around screaming.” The complaint was echoed by another student: "It took them more than 15 minutes to arrive. It could have turned into something very bad if he had a weapon on him. It was pretty infuriating to have to wait that long." And that second student wasn’t even in the professor’s presence; the professor was out in the hall, by that time, and the student was in a classroom.

The fact that the troubled pedagogue was naked didn’t seem to have allayed these young people’s fears. And as for the 15 minutes: I’m no fan of the police, but look at your watch and picture yourself getting a call, leaving your office, traveling across one of the nation’s largest college campuses, locating the place where an incident is taking place, clambering upstairs, and confronting some nut who’s running around naked . . . Now look at your watch again. Think you could make it in 15 minutes? Think that somebody has a right to complain bitterly at this complete abdication of police responsibility? Think that you and I and a bunch of fit young college kids concerned with a naked, middle-aged man possess a right to have cops show up in less than 15 minutes?

I think I’d rather take off my clothes and run around like a maniac than to utter the complaints of those college students.

But if you’re thinking just about words, and not about guts, the worst part of this report is the eight words that say, “The professor’s name has not yet been released.” Not released by whom? And why not? Everybody on the scene knew who he was. Their reactions were reported at length. A blurry picture of his apprehension was included in the news report. So why not his name?

During the past 30 years we professors have done a lot of running around naked, intellectually speaking, and what has been revealed has not been impressive.

Pity? Perhaps. But this pity, this verbal delicacy and restraint, is by no means evenly distributed. If Joe Blow from Kokomo has a fight with his girlfriend, gets a little drunk, drives down the street, and gets nailed by a passing cop, no one will withhold his name from publicity — or his mugshot either, in some jurisdictions.

The day after the scary incident, anonymous students identified the professor as a certain John McCarthy. The day after that, the really loony thing happened. An article about the affair appeared in the MSU student newspaper. You can tell MSU standards of journalism by contemplating the following sentence, which is about the weekly meeting of the “steering committee” of the university’s president: “At the Steering Committee meeting Tuesday, the conversation turned to mathematics professor John McCarthy, which students said he had a mental breakdown during a class Monday.”

“Which students said he had a mental breakdown . . .” OMG — now we know what kind of grammar MSU is teaching.

Well, let’s see what intellectual level MSU’s president is operating on. For other people, the serious issue introduced by the professor’s actions might be, “Did MSU know that at least one of its senior professors might be crazy? Does MSU have any way of discovering how many of its senior professors actually are crazy?” But that was not the issue that President Anna K. Simon wished to discuss. For her, we learn, “an incident Monday brings in to [sic] question the impact and role of social media.”

Huh? As far as I can make out from Simon’s murky remarks, murkily reported, the problem is information control: “’The complication of social media, with everyone with a camera and a cell phone, is one that we continue to struggle with in terms of information because the event would not, under (normal) circumstances, trigger one set of alerts,’ Simon said. ‘There’s also the need for more crisp communication about what the outcome was. Whether that would have controlled some of the rumors, tweets and other things, I’m not quite sure.’”

Did Michigan State know that at least one of its senior professors might be crazy? Does Michigan State have any way of discovering how many of its senior professors actually are crazy?

Let’s look at this in another way. Suppose you’re concerned about the quality of some public institution. You want to find out whether there’s any quality control. You learn that a teacher, policeman, bureaucrat, or other publicly employed personality, may have done something egregiously stupid and wrong, and perhaps illegal, while exercising his or her official duties. She’s said to have told her students to vote for Obama. He’s said to have beaten a homeless person for “resisting” some “order.” She’s accused of making a “questionable” transfer of city funds. He allegedly takes off his clothes in front of his students and runs around screaming.

You’d like more facts. But how long do you have to spend just trying to confirm this person’s name? A week? A month? Three months? Forever? Unless there’s a miracle, the information control artists will keep you from knowing what it is until virtually everyone has forgotten the episode — and then the data will be stored in a closed file, no longer accessible to the public. In the meantime, you will be informed that personnel regulations do not allow release of that information, or, pending possible legal action, the city cannot comment on this case, or some other nonsense that never applies to a normal person in a normal job (or didn’t, until the “standards” of “public service” bureaucracies spread into big private companies). And, to top it off, some CEO will entertain the media by looking at her navel and meditating about how tough the times are, what with all these cameras and phones and computers around, ready to convey the truth to anyone online . . .

So what do you think? What are we supposed to say about that? What are we able to say, since if we do comment we can always be told that we do not have all the facts?

The chair of John McCarthy’s department presumably has all the facts. These facts lead him to be concerned “about the way some people made jokes about the incident. An incident like this often teaches us who we are and what we represent. I hope we can all use what transpired after this incident to reflect on our values and our role as members of an institution that strives to be among the best of the world.”

Gosh, don’t you feel guilty? Your making jokes about a figure of authority at an institution that strives to be among the best of the world has hurt the feelings of an institution that strives to be among the best of the world. Or something.

But to continue with college professors, which I can easily do, considering that I am one, have you been following the curious case of Professor Amy Bishop? She’s the one who was recently convicted of killing three of her colleagues and wounding three others at a meeting of the Biology Department at the University of Alabama, Huntsville. That happened in 2010, and there were plenty of witnesses, because she didn’t manage to kill them all, but it took two and a half years to convict her. I don’t know why, except that it may have something to do with the cultural and verbal universe in which she lived.

Perhaps the EEOC is still trying to find out whether the woman who wasted her brother and killed or did her best to kill six of her colleagues is in “unstable mental health.”

In 1986, in Massachusetts, where’s she’s from, she killed her brother Seth with a shotgun, then went to a local auto dealership and tried to commandeer a car so she could escape. Apparently because of her family’s ties to the local power structure, she wasn’t even questioned about the shooting for 11 days. Then it was called an “accident.” Eight years later, she was implicated in an attempt to pipe bomb an academic supervisor in Boston. He had suggested she was “mentally unstable.” Four months after the attempted bombing, investigators finally showed up at her house. She was uncooperative, and the investigation was inconclusive. It went away. Seven years later, she was arrested after assaulting a woman in a fight over a high chair at an International House of Pancakes in Peabody, MA. She was sentenced to probation and an anger management class (which she probably didn’t take). In the restaurant, she had yelled, “Don’t you know who I am? I’m Amy Bishop!”

Now she gets to the University of Alabama, Huntsville, where she is known as “difficult” by “some.” A good piece of reporting tells the story. Bishop didn’t publish very much; she listed her children as first and second authors on one of her publications; a student filed a grievance against her; she was detested by almost everyone.

Then, as our reporter says — and this is the cream of the jest:

In September 2009 Bishop filed a complaint with the Federal Equal Employment Opportunity Commission. Someone on her departmental tenure committee had called her "crazy" in her tenure review, and would not retract the statement when an administrator gave him a chance to back down. The anonymous professor maintained that Bishop's unstable mental health was apparent on their first meeting.

The EEOC is still looking into that complaint.

I have been unable to learn whether the federal agency is still looking into it. Perhaps it is still trying to find out whether the woman who wasted her brother and killed or did her best to kill six of her colleagues is in “unstable mental health,” or, in plain terms, insane, bonkers, off her rocker, completely gone, in the zone, out of her skull, a desperate lunatic, and otherwise, well, crazy, or if she is, whether anyone should have said it.

A Martian appears in your kitchen and tells you that the folks back on the slopes of Olympus Mons have been following the Amy Bishop story on their nightly news. He wants to know what is so weird and touchy about that word crazy. He wants to know how somebody who uses it in its clearest and most self-evident application could possibly be investigated by a government of 300 million people (which presumably ought to have other things on its mind), because the word might have been discriminatory against the woman who killed four people. What words would you use to explain this?

Maybe you wouldn’t be able to find them, but we professors would — or at least keep anyone else from doing so.

On October 2, I was watching a CNN segment about why more security wasn’t provided to our diplomatic installation in Benghazi, when it was obvious that the place might be in danger from fanatic Muslims. The interviewer asked a professor — or someone who talked so much like a professor that he should immediately be given tenure — what he thought about all the warnings that came in, and apparently were not adequately heeded. Well, he said, “you have to parse the different kinds of violence that were taking place.”

That was his response.

What would you have to do to interpret that for your Martian friend?

I suppose you would start by noting that the key word was “parse.” In normal English, “parse” means to identify the grammatical functions of the words in a sentence. But in Proflish, the professor tongue, which is the status language of planet earth, the language to which all other languages aspire, “parse” means anything you want it to mean. In this case, it appears to mean something like “look at.”

Well, says the Martian, why can’t he just say “look at”?

That’s sort of a puzzler, but I can think of two, related reasons. One, he would be understood immediately, and that is not the goal of anyone speaking Proflish. Two, he would reveal the fact that he is saying nothing. Suppose I do look at or inspect various kinds of violence. Suppose I go further, and distinguish one kind of violence from another. So what? That isn’t enough. I haven’t really said anything. But a word like parse will keep everyone, or at least the interviewer, impressed with me. And that’s the point of talking, see? Ya see?

Yes, says the Martian. I’m parsing it all.

Editor's Note: Word Watch will comment on the presidential and vice presidential debates after the disease has run its course.

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All-American Johnny and the Educators


The debate over Johnny and the American educators goes on. Who’s more to blame for the problems in our public schools? The star maverick educator Michelle Rhee said the other night on a talk show she could see students getting paid for a good performance. The conservative education professionals simply say that “Johnny,” meaning most of the public school students, has to do better, to make our schools better. Parents say that educators will have to teach him better to make our schools better. Educators are beating up Johnny in the press to make their accusation strong. He needs to get real (which is as strong as their language gets). Ordinary lay folk know things aren’t right in America’s public schools. Some powerful politicians are demanding our schools be the best in the world.

We don’t need to be in an international contest. We have a lot to do here at home. “High schools are the downfall of American school reform,” said Jack Jennings, President of the D.C. Center on Education Policy. This disclosure pointed the finger. Educators knew the problem was this deep. They kept the scandal this buried. The public high schools in America number 27,000, and they have on their rolls millions of students; and 7,000 drop out every day. Administrators don’t know what to do about the high schools.

Powerful politicians are also making us accept over 100,000-plus foreign students from Latin America and Asia in our schools, who depress the test scores. They have trouble with the tests in English, and sometimes with discipline. Johnny is in a classroom of noisy students from many cultures and can’t get serious about paying attention when the teacher is busy keeping order in the class. Johnny doesn’t get rigorous tutoring at home and study discipline. He doesn’t score well on tests, which is not totally his fault.

Our high schools can’t do better when they’re like this, forced to be politically correct. Is this why Johnny doesn’t like being taught with “them”? Donal O’Shea, Dean of Faculty, Mount Holyoke College, and author of The Poincare Conjecture (referring to the formidable topological theorem the western world has been trying to prove for 100 years) dropped another bomb on American education in his March 2007 Forbes essay, saying that in 1985, of the million students who received bachelor’s degrees that year, only 16,000 majored in math or statistics. More disturbing, of the 1.4 million who received bachelor’s degrees in 2004, only 13,300 majored in math or statistics. More and more of our high schools are letting their students graduate with little or no science and math and serious humanities education. Graduates who go to college anyway mostly are woefully weak.

The public high schools in have on their rolls millions of students; and 7,000 drop out every day.

Secretary of Education Arne Duncan said recently that Johnny should read more. Although no one of his stature should have to say that reading is serious, it's clear that Johnny has to do much more reading to improve his articulation and language crafts — the kind of reading that doesn't always register on standardized tests. Harvard educated actor Matt Damon recently told a reporter, “We’re tying teacher salaries to how well kids are performing on tests; that kind of mechanized thinking has nothing to do with higher order [thinking].” He may be right, not only about tests in English but also about tests in math-based subjects. President Obama and his advisers put education on their plate the first thing in 2008, recognizing that the high schools were shockingly unhealthy, especially in the math and science departments. His team selected STEM subjects — science, technology, engineering, mathematics — to do the job. The team promised high school students who go for these subjects that they would be richly rewarded upon graduation from college as a STEM major. They’ll enjoy the good life, a professional career, prestige, and security all their lives. The team was romantic. Yet mathematics is the most important subject — and not just for STEM subjects, but for all the other disciplines too. Mathematics is the yeast in all of them. No subject can grow, get strong, become precise without it. Every subject has to establish its foundation on sturdy logic to survive. Mathematics is logic in its supreme form.

Johnny doesn’t know this about math. He is taught math in school as a set of mechanical exercises, found in a “manual,” a textbook, filled with them. The textbook is dully and poorly written. The author isn’t well-read, and doesn’t have to be. The book is written to sell to the state’s textbook adoption committee. Publishers fight hard to win the contract, which is quite big if the state is big and buys one math textbook to be used by all its high school algebra students. Johnny can’t comprehend this “adopted” textbook; the “whizzes” in class don’t read it either, but understand what the math formulas and the “mechanical writings” are saying. The mechanical math geeks are educators’ darlings. They test brilliantly and are “inventive” and become gadgeteers.

Johnny is taught math in school as a set of mechanical exercises, found in a “manual,” a textbook that is dully and poorly written.

Test-driven educators need to see students less than as machines, but more, particularly teens, as fragile souls always in need of constant anchoring. The horrors they can commit! Think of the 1999 Columbine High (Colorado) massacre of 12 students and a teacher by two male seniors (a front page story), and even the 2005 brouhaha at Monta Vista High (Culpertino, California) of Asian and white students over who’s best in math. Some whites at the school moved out of town. (Front page story, “The New White Flight”!) Teens easily crack under pressure. If only they were disciplined to channel their energy to better use, they would make high school a healthier world and also ensure that our pride and joy — the 18 of the 20 best universities in the world that are American — retain their strength. Joseph Nye of the Harvard Kennedy School said there are 750,000 foreign college students in American colleges. But then he said, “We have to do something about our secondary education.”

Remember that the 750,000 foreign students don’t have the cultural wherewithal to create brilliant American writing. That task belongs to Johnny, and he shouldn’t be thrown away. If he's not a math geek, Johnny still may learn how to contribute to American letters, which aren't brilliant enough.

The annual State Regents Exams for New York high school seniors reveal why educators should get real. The exams demand that to be college-ready each senior score at least 80 in math (last year many failed to solve the simplest of quadratic equations) and 75 in English Language Arts (two essays have to be written). The high school graduating rate for 2009 was 77%, but only 41% of the class was prepared for college. The two-year college was the only hope of many of the graduates not prepared for four-year college. Poor inner-city Johnny has it the worst — nothing, nobody to help him hope for a good score on the impossible tests; and no hope that the education system will take an interest in him.

Miracles do happen. A New York inner-city Johnny was picked to star in a Walmart ad that takes place in a school library. Johnny glows as he’s helped with his reading by a retired, lawyerly, grandfatherly looking gentleman smiling like one with the patience of Job. The ad runs twice nightly on the PBS “Tavis Smiley Show.” Thanks to Walmart for telling inner-city Johnny across America that “a mind is a terrible thing to waste.” The ’hood does have rich soil worth cultivating. Will other big businesses come in and help other overwhelmed students? Remember that the great English novelist Charles Dickens was born dirt-poor.

By the way, no one at our 18 most hallowed universities proved the Poincare Conjecture. Last year, a reclusive Russian mathematician, Gregori Perelman, proved it but refused the Clay Institute’s $1,000,000 prize. This caliber of confident mathematician tends to be shy, and to have other baggage, such as being incomprehensible at points in his lectures. One can’t expect the high school geek math teacher to be less handicapped. He mumbles at the blackboard. So Johnny’s best shot is to do a lot of good reading with a dictionary to get verbal competence and confidence in writing. That would be quite an achievement — more of an achievement than a brilliant score on a math test.

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