Teaching – Page 2 – Maverick Philosopher

Infinity and Mathematics Education

Time for a re-post. This first appeared in these pages on 18 August 2010.

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A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie); on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical. If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory. Cantor sought to achieve an exact mathematics of the actually infinite. But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's Elements, David Hilbert's Foundations of Geometry, Richard Dedekind's Continuity and Irrational Numbers, Cantor's Contributions to the Founding of the Theory of Transfinite Numbers, etc. Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc. Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of constructing the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to supplant a textbook-driven approach, but that the latter ought to be supplemented by the foregoing. I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No!

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite. The countably infinite has nothing to do with the potentially infinite. I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity. In so doing they took a lot of the excitement and wonder out of it.

So what did you learn? You learned how to solve problems and pass tests. But how much actual understanding did you come away with?

Higher Remediation

My friends in the teaching trenches tell stories that lead me to believe that so-called 'higher education' is now little more than 'higher remediation.'

On Teaching Philosophy

It is pointless to supply answers to unasked questions or to promote questioning among those who have all the answers.

How Little They Remember

A man hereabouts with a passion for chess got my number. We've become friends.

He told me he took a course in the philosophy of religion way back when. I pressed him on details. All he remembers is the old professor walking into the room, flipping a switch, and intoning "Let there be light!"

The chess player's forgetfulness reminds me of a story.

An eager young nun and a wise old nun were discussing teaching. The young nun was waxing enthusiastic over the privilege, but also the responsibility, of forming young minds. The old nun took a glass of water, inserted her forefinger, and agitated the water. Suddenly she removed her finger and the water immediately returned to its quiescent state.

"So much for the forming of young minds," said the older and wiser one.

A Protreptic Puzzler

A curious passage from Aristotle's Protrepticus:

. . . the fact that all men feel at ease in philosophy, wishing to dedicate their whole lives to the pursuit of it by leaving behind all other concerns, is in itself weighty evidence that it is a painless pleasure to dedicate oneself wholeheartedly to philosophy. For no one is willing to engage in exhausting work for a long time. (#53, p. 24)

To set the Stagirite straight, I should like to shunt his shade into some Philosophy 101 classroom for a spell.

Is Suggestibility Always Bad?

Belonging to a community of believers reinforces one in one's belief. If the belief is true and good, then so is the suggestibility that sustains and reinforces it.

If we weren't suggestible, we wouldn't be teachable by that highest form of teaching, indirect teaching by example. As the Danish Socrates wrote,

The essential sermon is one's own existence. (Søren Kierkegaard, Journals, #1056)

On Indoctrination

Is indoctrination ever a good thing?

Presumably, to indoctrinate is to teach one doctrine as if it is true, as opposed to presenting a variety of different doctrines on the same topic without endorsing any one of them. In general, indoctrination ought not be done at the college level: Competing positions should be presented fairly and objectively and students should be encouraged to think matters through themselves and form their own opinions. But this point demands careful qualification.

For surely indoctrination is legitimate in some subjects such as mathematics and the hard sciences. No one could fault a math or science teacher for failing to give equal time to the views of numerologists, alchemists, astrologists, flat earthers and geocentrists. And in political science classes short shrift should be given to 9-11 'truthers' and other conspiracy enthusiasts. Their views may be discussed in passing, but to present them as if such theories are serious contenders in the arena of ideas makes a mockery of the search for truth, which presumably is what universities ought to be about. Certain views are beyond the pale and ought not be dignified by being taken seriously, e.g., Holocaust denial, the allegations made in the protocols of the Elders of Zion, the views of NAMBLA members, and so on.

But even in philosophy some indoctrination could well be justified, in logic, for example. One is justified in teaching introductory standard logic dogmatically without bringing in Hegelian and Marxist and dialetheist critiques of the law of non-contradiction, say. But not only in logic. To borrow an epithet from Arthur Collins, eliminative materialism is a 'lunatic" philosophy of mind. I would cover it in a philosophy of mind course, but I would not present it as a possible view that one might justifiably hold; I would present it as not merely false but as incoherent. And I would take myself to be justified in doing so. Of course, I would present the doctrine and the arguments thought to support it accurately; but I would not present it as if it were one epistemically possible view among others. So in that sense I would be engaged in legitimate indoctrination: if not by the promotion of the true view, at least by the rejection of false or incoherent ones.

If one were to oppose all indoctrination, then one would have to present every extant view on every issue as if it had a legitimate claim on our attention. But this would encourage the view in students that all views are equally good, which is obviously not the case. For example, in the philosophy of mind, eliminative materialism, behaviorism, and type-type identity theory are all very bad theories with eliminativism being the worst and the identity theory being the best of the three. But nothing hinges on this example. I could give many from different areas of philosophy. The point is that a pedagogic posture of studied neutrality with respect to every view is as bad as an extreme doctrinalism in which contentious positions are tendentiously promoted.

One can see from these sketchy remarks that the issue is not easily sorted out. Teaching that promotes relativism and skepticism, that leaves the student with the notion that all views are equally good or that nothing can be known is bad teaching. Equally bad is teaching that merely foists opinions on students without inculcating habits of critical thought or without fairly presenting the debates surrounding reasonably debatable issues. (Not all issues, however, are reasonably debatable.) Navigating between the Scylla of of the one and the Charybdis of the other is no easy task.

Docendo Discimus

Teaching, we learn.

As it stands, a maxim, and true as far as it goes. But in need of qualification which, when added, makes it a maxim no longer. Brevity is essential to the maxim as it is to the aphorism and the epigram.

Closer to the truth is the following. Teaching, we learn; but only up to a point beyond which studying without having to teach is much to be preferred if the goal is an advance in understanding and erudition.

I never knew logic so well as after having taught it for a couple of years. But then the maxim lost its truth.

On Relevance in Education

From the mail bag:

I have taught high school and college-aged kids for many years, and am very often lobbed the relevance question. The logical coherence of the concept of God. Theories of space and time. Classic questions in epistemology and metaphysics. "How is this relevant," they ask. It annoys me. I make an impotent gesture toward the intrinsic value of knowledge, but am always left frustrated by having to defend what is so obvious to me –and to everyone else prior to the mid twentieth century–the indelible importance of these topics. Maybe you can help me out?

I don't know how much help I can be, but here are some thoughts.

1. The philosophy teacher has a problem the calculus instructor, say, does not. The latter does not have to show the relevance of his subject or motivate an interest in it. Perhaps two thirds of the students before him are engineering majors who need no convincing of the relevance of higher mathematics to their career goals. They are interested in mathematics, if not for its own sake, then for the sake of its use. The philosophy teacher, however, has not only to teach his subject but also, unlike the mathematics professor, to argue its relevance and motivate interest.

2. At this point lame justifications of philosophy come thick and fast. It teaches critical thinking; it is good preparation for law school, etc. I knock the crutches out from under these lame justifications in Should One Stoop to a Defense of Philosophy and the Humanities? As I say there:

Philosophy is an end in itself. This is why it is foolish to try to convince philistines that it is good for something. It is not primarily good for something. It is a good in itself. Otherwise you are acquiescing in the philistinism you ought to be combating. [. . .]

To the philistine's "Philosophy bakes no bread" you should not respond "Yes it does," for such responses are patently lame. You should say, "Man does not live by bread alone," or "Not everything is pursued as a means to something else," or "A university is not a trade school." You should not acquiesce in the philistine's values and assumptions, but go on the attack and question his values and assumptions. Put him on the spot. Play the Socratic gadfly. If a philistine wants to know how much you got paid for writing an article for a professional journal, say, "Do you really think that only what one is paid to do is worth doing?"

3. "I make an impotent gesture toward the intrinsic value of knowledge, but am always left frustrated by having to defend what is so obvious to me . . ." Most of the people who need to have this explained to them are not equipped to appreciate any explanation. So we humanists are in a tough spot. One of the conclusions I came too early on was that philosophy simply cannot be a mass consumption item at the college level. Although I didn't mind, and actually enjoyed, teaching logic courses, which can be of some use to the masses, I loathed teaching Intro to Philosophy and other philosophy courses designed to satisfy breadth requirements.

Part of the problem is that college level is so low nowadays that it has become a joke to speak of 'higher education.' People are not there to become educated human beings but to garner credentials that they believe will help them get ahead economically and socially. Nothing wrong with that, of course, but then why waste time on the pursuit of truth for its own sake? The average person has no intellectual eros; what he wants and needs is job training.

4. There is an irony here. People like you and me and thousands of others would never have had the opportunity to make a living from teaching philosophy if the level had not sunk so low, not so much because our level is low, but because there would simply have been no jobs for us if 'higher' education had not metastazised in the 1960s and beyond. So while we complain about the low level of our students, we ought to bear in mind that we have students in the first place and are not selling insurance or writing code because of the democratization of 'higher' ed.

5. I am an elitist, but not in a social or economic or racial sense. Everyone who has what it takes to profit from it ought to have the opportunity to pursue real education — which is not to be confused with indoctrination in leftist seminaries — in institutions of higher — no 'sneer' quotes — education. Equality of opportunity! But of course there will never be equality of outcome or result because people are not equal.

Philosophy — the real thing, not some dumbed-down ersatz — cannot be a mass consumption item. It is for the few. But who those few are cannot be decided by criteria of race or sex or age or religion or national origin. High culture is universal and belongs to all of us, even though we individually and as members of groups are not equal in our ability to contribute to it.

Nice but Dumb

I can't believe that this old 16 September 2004 post from my first weblog languished there so long before being brought over, today, to my newer digs.

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My cat Caissa – named after the goddess of Chess – was feeling under the weather recently, so I took her to the vet for some blood work. The twenty-something receptionist at Caring Critters was nice enough but she stumbled over my name. But I was in a good mood, so I didn’t mind it too much. She didn’t even try to pronounce it which I suppose is better than mangling it. I don’t cotton to being called Valenzuela, Valencia, Vermicelli, Varicella, Valparaiso or Vladivostok. Don’t make me into an Hispanic. In these parts, if your are not Hispanic you are an ‘Anglo.’ That doesn’t sit well with me either.

Perhaps I should be happy that I do not rejoice under the name of Znosko-Borovsky or Bonch-Osmolovsky. Nor do I stagger under such burdens as Witkiewicz, Brzozowski, or Rynasiewicz. The latter is the name of a philosopher I knew when he taught at Case Western Reserve University. Alvin Plantinga once mentioned to me, sometime in the late '80s, that he had been interviewed at Notre Dame, except that ‘rhinoceros’ was all Plantinga could remember of his name.

Actually, none of these names is all that difficult if you sound them out. But apparently no one is taught phonics anymore. Damn those liberals! They’ve never met a standard they didn’t want to erode. I am grateful to my long-dead mother for sending me to Catholic schools where I actually learned something. I learned things that no one seems to know any more, for example, grammar, Latin, geography, mathematics. The next time you are in a bar, ask the twenty-something ‘tender whether that Sam Adams you just ordered is a 12 oz or a pint. Now observe the blank expression on her face: she has no idea what a pint is, or that a pint is 16 oz, or that there are four quarts in a gallon, or 5,280 feet in a mile, or 39.37 inches in a meter, or that light travels at 186, 282 miles/sec, or that a light-year is a measure of distance, not of time.

Even Joan Baez got this last one wrong in her otherwise excellent song, Diamonds and Rust, a tribute to her quondam lover, Bob Dylan. The irony is that Joanie’s pappy was a somewhat distinguished professor of physics! In a high school physics class we watched a movie in which he gives a physics lecture.

I was up in 'Flag' (Flagstaff) a few years back to climb Mt. Humphreys, the highest point in Arizona at 12,643 ft. elevation, (an easy class 1 walk-up except for the thin air) and to take a gander at the moon through the Lowell Observatory telescope. While standing in line for my peek, I overheard a woman say something to her husband that betrayed her misconception that the moon glows by its own light. She was astonished to learn from her husband that moonlight is reflected sunlight. I was astonished at her astonishment. One wonders how she would account for the phases of the moon. What ‘epicycles’ she would have to add to her ‘theory’!

The Professor-Student ‘Non-Aggression Pact’

William J. Bennett and David Wilezol, Is College Worth It? (Thomas Nelson 2013), p. 134:

Knowing that students prefer to spend more time having fun than studying, professors are more comfortable awarding good grades while requiring a minimum amount of work. In return, students give favorable personal evaluations to professors who desire to be well received by students as a condition of preserving their employment status. Indeed, the popularity of the student evaluation, which began in the 1970s, has had a pernicious effect.

I would say so. Here is an anecdote to illustrate the Bennett thesis. In early 1984 I was 'up for tenure.' And so in the '83 fall semester I was more than usually concerned about the quality of my student evaluations. One of my classes that semester was an upper-level seminar conducted in the library over a beautiful oak table. One day one of the students began carving into the beautiful table with his pen.

In an abdication of authority that part of me regrets and a part excuses, I said nothing. The student liked me and I knew it. I expected a glowing recommendation from him and feared losing it. So I held my tongue while the kid defaced university property.

Jeff H. and I had entered into a tacit 'non-aggression pact.' (And I got tenure.)

The problem is not that students are given an opportunity to comment upon and complain about their teachers. The problem is the use to which student evaluations are put for tenure, promotion, and salary 'merit-increase' decisions. My chairman at the time was an officious organization man, who would calculate student evaluation averages to one or two decimal places, and then rank department members as to their teaching effectiveness. Without getting into this too deeply for a blog post, there is something highly dubious about equating teaching effectiveness with whatever the student evaluations measure, and something absurd about the false precision of calculating averages out to one or two decimal places.

Jones is a better teacher than Smith because her average is 3.2 while his is only 3.1? Well, no, but if the chairman is asked to justify his decision, he can point to the numbers. There is mindless quantification, but it takes someone more thoughtful than an administrator to see it.

I strongly recommend the Bennett-Wilezol book to anyone thinking of attending college or thinking of bankrolling someone's attendance. Here is a review.

How Willfully Stupid Can a Willfully Stupid Liberal Be?

There is no limit.

German Home Schoolers Seek Asylum in USA

Home-schooling is illegal in Germany. So, "In 2008, Uwe and Hannelore Romeike left Germany with their five children and came to the United States asking for refugee status as an oppressed minority."

So they left Germany to seek asylum in Left-Fascist Amerika. There is a touch of irony here. Well, we are not as far gone as the "land of poets and thinkers." (Heinrich Heine) Not yet, leastways.

The reason for the disallowance of home schooling is that the powers that be don't want the formation of "parallel societies" (Parallelgesellschaften). That's a real knee-slapper given the green light to Muslim immigration and the Islamization of Germany. No "parallel societies" unless they are politically correct parallel societies.

The Pee Cee, you see, are 'inclusive.' Even unto their own extermination. The Germans seem especially PC-whipped.

It is perhaps not irrelevant that the Romeikes are Christians. Nor that ". . . one of the oldest universities in Germany inaugurated the country's first taxpayer-funded department of Islamic theology. The Center for Islamic Theology at the University of Tübingen is the first of four planned Islamic university centers in Germany." (Ibid.)

Read about the Romeikes here. It turns out that their request for asylum was denied.

L.A. Schools: No Suspension/Expulsion for Willful Defiance

I said a few entries back that liberals lack common sense. Here is further proof, as if further proof is needed:

This week, the Los Angeles Unified School District—the second-largest in the
nation—decided to end the practice of suspending or expelling students for
"willful defiance," starting this fall. District officials said the practice
disproportionately affects minority students' education and leads to more
disciplinary problems for students down the line.

Both the policy and the justification for it are insane. That the policy is crazy is self-evident to anyone of sound mind. The justification too is completely crack-brained. It assumes that the only reason minority students are disproportionately affected by the old expulsion rule is because they are unjustly discriminated against on the basis of their skin color. But that is obviously false: the minorities are disproportionately affected and 'overrepresented' among the ones expelled because they are disproportionately trouble-causing. It is not their skin color, but their bad behavior that explains why they get expelled and suspended more often.

Liberals cannot see this because they are blinded by their politically correct notion that all groups are equal in every respect and so differential outcomes have to be chalked up to racism. Too many liberals are willfully stupid people in willful defiance of common sense and we ought to expel them from the precincts of the reasonable before they do any more damage to educational institutions.

Contemporary liberals have something like the opposite of the Midas Touch. Everything King Midas touched turned to gold. Everything a liberal touches turns to dreck.

Or can you think of a counterexample?

Observations on the Joys of Teaching

Teaching is the feeding of people who aren't hungry.

Teaching philosophy is the feeding of people who are neither hungry nor know what food is.

Teaching is like agitating water in a glass with one's forefinger. As long as the finger is in motion, the water is agitated; but as soon as the finger is removed, the water returns to its quiescent state.

Philosophy, like a virgin, is wasted on the young.

The classroom is a scene of unreality. No one takes it quite seriously. Not the students, from whom little is expected and less demanded. Not the teachers, who waste their time in discipline and remediation.

According to an apocryphal story about George Santayana, one day, while lecturing at Harvard, he suddenly intuited the absurdity of teaching. Stopping in mid-sentence, he walked out of the classroom never to return. The truth is less dramatic: he dutifully finished the semester, turned in his grades, resigned his professorship, and embarked for Rome where he spent the rest of his life in cultured retirement.

"I would rather eat dry bread than teach." Franz Schubert, quoted in Maurice J.E. Brown, Schubert: A Critical Biography (New York: Da Capo, 1988), p. 233.

"I would rather sweep the streets than teach children!" Ralph E. Hone, Dorothy L. Sayers: A Literary Biography (Kent: Kent State University Press, 1979), p. 24. Hone is quoting Sayers.

The quotations borrowed from Dr. Gilleland, antediluvian, bibliomaniac, and curmudgeon.