Of Monopoles and One-Sided Coins

A much deplored marvel is the variety of the sciences: we have them as different as chemistry, psychology, and anthropology. But somebody isn’t just a chemist, but, say, an organic chemist, or a sensory psychologist, or a cultural anthropologist. Many problems require an interdisciplinary approach. At least the leader of such a team should be familiar with the various disciplines involved. What kind of education would such a scientific generalist have? According to [1] a difficulty is that one needs to go into different sciences deeply enough to go beyond the subject matter so as to acquire the habits of mind peculiar to each: “These habits, and not the subject matter, are what distinguish the sciences — for how else can we distinguish the chemical physicist from the physical chemist, the mathematical biologist from the biomathematician!”

And all this plays just within science. When we zoom out, we find not just different habits of mind: there is a culture, a temperament that is different. In an interview [2] Herbert Robbins, the mathematician, gave some impressions of the professors he encountered as an undergraduate. One of these was a famous literary critic. He would walk into the classroom with a briefcase full of books and lecture on the poets of the Romantic period. He’d take out a book, read a poem, and then comment on it. This kind of scholarship left Robbins cold. On the other hand, the mathematician Marston Morse deeply impressed him. Although Robbins hardly knew what Morse was talking about, it was clear that this prof “was on fire with creation”.

This illustrates the contrast between two cultures: the Culture of Appreciation and the Culture of Discovery. Another student, one who knows what Morse was talking about, might judge that his soul was more improved by the experience of “The Rime of the Ancient Mariner” than by being able to discover, say, the Morse Homology. The temperament of such a student would attract him to the Culture of Appreciation.

Mathematics lies firmly in the Culture of Discovery. But even within that subculture there are significant variations. For example, R.L. Moore was an extreme Discoverer. Mary Rudin reports [3] that when she completed her PhD in the 1940s under Moore, she had not read a single mathematics paper. The Moore Method was to give graduate students lists of statements. Some were true, some were false; the students were to find out which was which. Some were easy to prove or find a counterexample to. Some were hard; the students were to find out which was which. Of the latter problems there were some of which Moore would know whether they were true, of some he wouldn’t. The students would work on whatever problem of the list they wanted. This way Rudin settled a conjecture that was, without her knowing it of course, famous. The written report was accepted as her PhD thesis. This is the way Moore intended his method to work out. Moore did not condone any Standing On Shoulders, not even of Giants.

Meanwhile, at the University of Chicago, president Robert Maynard Hutchins had forced through the so-called Great Books program. When Hutchins became president he had brought with him Mortimer Adler who served as a sort of Chief Ideologue. This program, at the extreme end of the Culture of Appreciation, was a pop version of the Oxford Greats and Modern Greats programs. “Pop version” because in Greats it goes without saying that one masters Greek so as to be able to study the Great Masters (Homer, Aeschylos, Euripides, Plato, Aristotle, …) in the original and that one is fluent in Latin so that the more difficult texts such as the Aeneid of Vergilius can be tackled. The Great Books of Chicago include more or less the same authors, but were read in English translations.

Another Chicago innovation was to embrace the Scientific Revolution by including Copernicus, Gilbert, Kepler, Galileo, Harvey, Bacon, Descartes, Pascal, Newton, Huygens, Lavoisier, Fourier, Faraday, and Darwin. A constraint of the selection was that the number of Great Books added up to the magical number of one hundred; more than enough to fill up one’s undergraduate years.

I agree that the study of Political Science should include Plato (Great Books vol. 7), Aristotle (vol. 8 and 9), Machiavelli, Hobbes (vol. 23), Locke (vol. 35), de Montesquieu (vol. 38), and Volume 43 with the Federalist Papers and Mill. But I find it astonishing that Adler apparently also thought that, likewise, the study of mathematics should include (in vol. 34) “Mathematical Principles of Natural Philosophy” by Isaac Newton (commonly chummily referred to as “The Principia”) and, in vol. 31, “The Geometry” by René Descartes.

I was enticed by Adler’s slogan “Learn from the Masters, not from their students”. Indeed, I thought, why settle for less when I can get analytical geometry from the great Descartes himself? Accordingly, I took the unusual step of actually finding “La Géometrie” and checking it out. Before dismissing this book, let me acknowledge that I have now seen for myself that Descartes was a genius. In the first place the quality of the prose. When I get the impression that Plato was a good writer, I am really seeing that somehow Plato manages to make the translator write a beautiful text. In the case of Descartes the original is French, so that I have first-hand experience of the text. More importantly, at the time algebraic notation was in such a primitive state that Descartes was the first to write the formulas in the form that is familiar to us. But the great idea that Descartes is justly credited for is only implicitly present in the book.

The great idea of analytic geometry is that geometric figures are characterized as sets of solutions of algebraic equations. But Descartes is concerned with contributing to mathematics as his contemporaries saw it. At the time mathematicians were still absorbed with mastering the achievements of antiquity. In geometry Apollonius of Perga (third century B.C.) was the state of the art. Descartes was concerned with advancing the state of the art. The way to do that in the early 17th century was to solve an unsolved problem of Apollonius. Descartes succeeded by the innovative method of translating the problem to algebra. At the time a work of genius, and it greatly helped his contemporaries. But as far as I can tell “La Géometrie” is only of interest to historians.

That was one disappointment. For my next visit I followed a suggestion of Adler’s [4], page 257:

Thus we have no hesitation recommending that you try to read at least some of the great scientific classics of our tradition. In fact there is no excuse for not trying to read them. None of them is impossibly difficult, not even a book like Newton’s Mathematical Principles of Natural Philosophy, if you are willing to make the effort [underlining mine].

My understanding had always been that Newton used his invention of differential calculus (“method of fluxions”) to show that Kepler’s laws followed from the laws of dynamics as discovered by Newton himself. That is indeed highly probable: in “The Method of Fluxions” we find the calculus and the necessary preliminary algebra in modern algebraic notation. Newton had completed the book in 1671, but it remained unpublished until 1736, after his death. The shock I experienced in leafing through the famous book of Newton [5] was that the mathematics was translated back from the algebraic form to geometrical diagrams à la Euclid. Truly a feat of genius, but what perverse genius! It seems that Newton wrote in a letter that he “designedly made Principia abstruse … to avoid being baited by Mathematickal Smatterers” ([6], p. 459).

Newton was indeed a genius: he absorbed the latest algebra, extended it, and enriched mathematics with his discoveries in the calculus. However, he managed to hide it from his contemporaries and from those who mistakenly turn to Principia for Newton’s mathematics. Instead, Newton showed himself as the all-time great virtuoso in Euclid, a style of mathematics that Newton’s Renaissance predecessors had already superseded.

So there, professor Adler: do you really want me to “make the effort” to decrypt the designedly abstruse Principia? With that amount of effort a first-year student can master volume 1 of The Feynman Lectures on Physics. With that amount of effort I could upgrade my ill-remembered wave mechanics and learn quantum field theory. I will probably do neither, but for me there is no doubt where the effort would go. But then I was not born in the Culture of Appreciation.

In arguing against Adler I am of course kicking in an open door: the Great Books idea has been discarded a long time ago. But this program is but one example of the universal and baleful phenomenon that education has been hijacked by bookish types. This has been so through the ages: educational reformers, from Comenius in the 17th century to Dewey in the 20th, have protested against this to no avail under the slogan “Things, not Words”. If you want to be a nurse, an engineer, or a doctor you have to spend most of your education in school, with books. Too bad, if you’re not a bookish type and inclined to think with your hands rather than with words.

Within the bookish world things can be improved as well. My PhD is from a university faculty where the allowed credits were from mathematics, physics, and astronomy. There were a half a dozen programs, including “Mathematics, with Physics and Astronomy”, “Physics and Mathematics, with Astronomy”, …; you can easily guess the other ones. My current university is different in requiring credits for courses outside the selected program. This doesn’t help: a typical choice is “The Contemporary Japanese Cinema”. Stronger measures are needed; perhaps a compulsory course (one course, but a big and demanding one) of the kind described by David Denby [7]. Compulsory, because we seem to be condemned by temperament to one of the two cultures. We need some help in avoiding to become the oxymoronic entity exemplified by the one-sided coin and the magnetic monopole.

[1]: “The Education of a Scientific Generalist” by Hendrik Bode, Frederick Mosteller, John W. Tukey, and Charles Winsor. Science vol. 109 (1949), pp. 553-558.

[2]: Mathematical People D. Albers and G. Alexanderson (eds.), Birkhäuser, 1985; p.287.

[3]: More Mathematical People D. Albers, G. Alexanderson, and C. Reid (eds.), Harcourt Brace Jovanovich, 1990; p. 291.

[4]: How to Read a Book by Mortimer J. Adler and Charles Van Doren, Simon and Schuster, rev. ed. 1967.

[5]: Mathematical Principles of Natural Philosophy by Isaac Newton, 3rd edition, 1726, translated by Andrew Motte, 1729. The first edition is Principia Mathematica Philosophiae Naturalis published in Latin in 1687.

[6]: Never at Rest by Richard S. Westfall, Cambridge University Press, 1983. A biography of Newton. Quote found in Coming of Age in the Milky Way by Timothy Ferris, Anchor, 1989, page 118.

[7]: Great Books by David Denby, Simon and Schuster, 1996.

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