Footnotes to Plato from the foothills of the Superstition Mountains
At a gathering of Boston academicians some years back, by way of a conversational opener, I said to Professor X, "I understand you teach at the University of L." The good professor replied, "I conduct classes at the University of L." I found that to be a very good distinction, one borne out by my own experience.
Related post: Can Philosophy be Taught?
60 Minutes last night did a segment on the Khan Academy, an online source of short tutorials in mathematics, science, and other subjects. A wonderful resource for homeschoolers and anyone interested in filling in the gaps in his education. I viewed a couple of algebra and a couple of probability lectures last night and found them to be of high quality. Recommended by Bill Gates.
I once had a graduate student with whom I became friends. Ned Flynn, to give him a name, one day told me that after he finished high school he wanted to follow in his father's footsteps and get a job with the railroad. His mother, however, wanted something 'better' for her son. She wanted him to go to college, which he did, in the desultory fashion of many. He ended up declaring a major in psychology and graduating. After spending some time in a monastery, perhaps also at the instigation of his Irish Catholic mother, and still not knowing quite what to do with himself, he was accepted into an M.A. program in philosophy, which is where I met him. After goofing around for several more years, he took a job as a social worker, a job which did not suit him. Last I saw him he was in his mid-thirties and pounding nails.
His complaint to me was that, had he followed his natural bent, he would have had fifteen or so years of job seniority with the railroad, a good paycheck, and a house half paid for. Instead, he wasted years on studies for which he had no real inclination, and no real talent. He had no discernible interest in the life of the mind, and like most working class types could not take it seriously. If you are from the working class, you will know what I mean: 'real' work must involve grunting and sweating and schlepping heavy loads. Those who work on oil rigs or in the building trades do real work. Reading, writing, and thinking are activities deemed effete and not quite real. When my mother saw me reading books, she would sometimes tell me to go outside and do something. That use of 'do' betrayed her working class values. What she didn't realize was that by reading all those fancy books I was putting myself in a position where I could live by my wits and avoid the schlepping and grunting. Of course, the purpose of the life of the mind is not to avoid grunt work, with which I have some acquaintance, but to live a truly human life, whether one fills one's belly from it or not.
Overeducation' is perhaps not the right word for cases like my former student Ned. Strictly speaking, one cannot be overeducated since there is and can be no end to true education. The word is from the Latin e-ducere, to draw out, and there can be no end to the process of actualizing the potential of a mind with an aptitude for learning. Perhaps the right word is 'over-credentialed.' It is clear that what most people in pursuit of 'higher education' want is not an education, strictly speaking, but a credential that will gain them admittance to a certain social and/or economic status. 'Education as most people use it nowadays is a euphemism for a ticket to success, where the latter is defined in terms of money and social position.
My library extends through each room of my house, except the bathrooms. (I suspect that in the average household, where the only purpose of reading could be to inspire excretion, it is the other way around.) If I weren’t pro-Israel I would say that my library commits territorial aggression against my wife’s ‘Palestinian’ books; her few shelves are either occupied territories or under threat of occupation. My bibliomaniacal blogger-buddies would turn green with envy if ever they laid eyes on my library. So I shall have to protect them from descent into this, arguably the deadliest, of the seven deadly sins.
Many of my books were acquired on the cheap from used bookstores in college towns such as Boston-Cambridge and Bloomington, Indiana. I used to really clean up when disgruntled graduate students packed it in, dumping costly libraries purchased with daddy’s money into the used book dens.
Among the used books I scored were plenty of copies of philosophical classics used in undergraduate courses. I always used to get a kick out of the marginalia, if you want to call them that. Mostly it was the absence of marginalia that caught my eye, an absence corresponding to the paucity of thought with which the reading was done. The rare marginalium was usually pathetic. Here is a passage from Thomas Paine, The Age of Reason (1794):
Revelation is a communication of something which the person to whom that thing is revealed did not know before. For if I have done a thing or seen it done, it needs no revelation to tell me I have done it or seen it, nor to enable me to tell it or to write it. (LLA, p. 13)
That’s not the best writing in the world, but the thought is clear enough. Our brilliant student’s comment? "Word Play!" ‘Word Play!’ is ever on the lips of boneheads who cannot or will not comprehend any piece of well-constructed prose. The litany of the blockhead: Word Play! Semantics! Hairsplitting!
One good thing about student marginalia was that it never extended very far since the reading never extended very far: the obscene magic marker underlining typically ceased three or four pages into the text.
One of the many drawbacks of teaching is that one could never get the little effers to do the reading especially if one used primary sources, refusing to dumb things down with comic books, audiovisual 'aids,' etc.: once they saw that genuine effort was demanded, they wimped out. All my preaching about being athletes of the mind availed nothing, falling on dead ears, like pearls before swine. Or am I being too harsh?
Harsh or not, it is blissful to repose in my Bradleyan reclusivity, far from the unreality of the classroom.
I am enjoying teaching quite a bit now that I no longer do it. With some things it is not the doing of it that we like so much as the having done it.
One day in class I carefully explained the abbreviation ‘iff’ often employed by philosophers and mathematicians to avoid writing ‘if and only if.’ I explained the logical differences among ‘if,’ ‘only if,’ and ‘if and only if.’ I gave examples. I brought in necessary and sufficient conditions. The whole shot. But I wasn’t all that surprised when I later read a student comment to the effect that Dr. V. can’t spell ‘if.’
An eager young nun and a wise old nun were discussing teaching over lunch. 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.
That, said the old nun, is what teaching is like.
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 construction 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?
Alfred North Whitehead's The Aims of Education and Other Essays (Macmillan, 1929) begins with this paragraph:
Culture is activity of thought, and receptiveness to beauty and humane feeling. Scraps of information have nothing to do with it. A merely well-informed man is the most useless bore on God's earth. What we should aim at producing is men who possess both culture and expert knowledge in some special direction. Their expert knowledge will give them the ground to start from, and their culture will lead them as deep as philosophy and as high as art. We have to remember that the valuable intellectual development is self-development, and that it mostly takes place between the ages of sixteen and thirty. As to training, the most important part is given by mothers before the age of twelve. A saying due to Archbishop Temple illustrates my meaning. Surprise was expressed at the success in after-life of a man, who as a boy at Rugby had been somewhat undistinguished. He answered, "It is not what they are at eighteen, it is what they become afterwards that matters."
That few today understand what education is is betrayed by the readiness of all too many to use 'educate' in place of 'inform.' Suppose you tell me about some petty fact. You have not 'educated' me, you have given me a scrap of information. The educated person is not the one whose head is stuffed with information, but the one whose experientially-honed judgment is capable of making sense of information. To become well-informed is not difficult; to become well-educated is a task of self-development for a lifetime.
The following is from Theodor Haecker's Tag-und Nachtbücher 1939-1945, translated into English by Alexander Dru as Journal in the Night (Pantheon Books, 1950), pp. 114-115.) I have made a couple of corrections in the translation. The following entry was written in 1940 in Hitler's Germany. The National Socialists seized power in 1933 and their 'one thousand year Reich' collapsed under the Allied assault in 1945. Haecker, a Christian, was bitterly opposed to the Nazi regime. Haecker's Journal provides keen insight into a dark time when an entire society went off the rails.
Continue reading “Theodor Haecker on the Teaching of Latin and Greek”
In one sense a philosophy is a set of conclusions, systematically set forth, on ultimate matters. To appreciate the conclusions, however, one must appreciate the arguments and counterarguments the sifting of which first led the philosopher to the conclusions. But to understand the arguments and counterarguments one must understand the issues and problems that they revolve around. Appreciation of the issues and problems, in turn, is rooted in wonder the presupposition of which is a contemplative detachment from the taken-for-granted.
And so we must distinguish: doctrines, arguments, problems, wonder. Philosophy as the study of the doctrines of the philosophers is philosophy in its most superficial sense. Studying that, one is not studying philosophy, but philosophies, and them in their most external form. Philosophy as the grappling with the arguments whose conclusions are the doctrines is closer to the real thing. Philosophy as the exfoliation and penetration of the problems themselves, under suspension of the need to solve them at all costs, is closer still to philosophy's throbbing heart. This is philosophy as aporetics. But without wonder there can be no appreciation of problems, let alone solutions. Thus we have it on the excellent authority of both Plato and Aristotle that philosophy begins in wonder.
Upshot? Teaching philosophy is well-nigh impossible. One can of course teach the lore of the philosophers, but that is not what philosophy is in its vital essence. And although argumentative and logical skills are impartable to the moderately intelligent, the aporetic sense, the feel for a philosophical problem, is not readily imparted regardless of the intelligence of the student. A fortiori, the wonder at the source of the aporetic sense is a gift of the gods, and nothing a mere mortal teacher can dispense.
So I propose to go Kant one better. Somewhere deep in the bowels of The Critique of Pure Reason, he remarks that "Philosophy cannot be taught, we can at most learn to philosophize." I say that neither philosophy as doctrinal system nor the art of philosophizing can be taught. For there is no one extant doctrinal system called philosophy, and neither the aporetic sense nor the wonder at its root can be taught. As I used to say in my teaching days, "Philosophy cannot be a mass consumption item." Logic perhaps, philosophy no.
Or to paraphrase a remark I once heard Hans-Georg Gadamer make, "Just as there are the musical and the unmusical, there are the philosophical and the unphilosophical." One cannot teach music to the unmusical or philosophy to the unphilosophical. The muse of philosophy must have visited you; otherwise you are out of luck.