Magisterial Voices of the PastSeptember 26, 2005
Philip J. Davis
Musings of the Masters: An Anthology of Mathematical Reflections. Raymond G. Ayoub, editor, Mathematical Association of America, Washington, DC, 2004, 274 pages, $47.95.
We have here a collection of 17 articles by famous mathematicians of the recent past. Raymond Ayoub, a noted number theoretician and emeritus professor at Pennsylvania State University, assembled the articles, in addition to nicely translating the seven not originally written in English. Ayoub, who is a knowledgeable, cultured, and widely read anthologizer, has added a general introduction, portraits, and brief biographies of his 17 "masters," as well as summary commentaries on each article in which he often puts in his own takes on these historic texts.
The most recent of the articles is that of André Weil (1978), the oldest that of J.J. Sylvester (circa 1868). The dates of the original talks or publications average out to around 1940, but of course, the thoughts set down were already in the minds of the writers somewhat earlier. Only a few of the articles were written after the full-blown emergence of the electronic digital computer as a significant factor in the development of mathematics. Of the 17 articles, I was previously acquainted with only three, so I thank Professor Ayoub for this introduction and his glosses on these introspections of the past. Unfortunately, with the time and space allowed me by SIAM News, I am able to comment on only a few of them.
In writing about his selections, Ayoub states that with some notable exceptions mathematicians have not risen "to the challenge to write more extensively about their subject." He locates the reason for this partially in the culture of the mathematical community, many of whose members view "the writing of textbooks or popular works on mathematics with a slight degree of scorn." While he expresses the reasonable hope that his selections will be accessible to "the general reader who need not have a technical knowledge of mathematics," I doubt that this is the case. I had serious trouble getting a grip on one article and considered several others to be fuzzy beatings-around-the-bush. Creative mathematicians are not always the best spokespeople for the ideas they espouse.
In his introduction, Ayoub calls pure mathematics "sacred" and applied mathematics "secular." I like this theological distinction; it hits the elitists' nail right on the head. In future writings, I am going to appropriate Ayoub's "sacred," but plan to intensify the word "secular" to "profane."
I am appalled by the exclusive focus of past (and present) writers who discuss the descriptive and predictive aspects of applied mathematics while overlooking its prescriptive potentialities--the power to put in by fiat seemingly arbitrary mathematical arrangements like those that, since the days when the Babylonians were recording the price of onions in clay, have often come to characterize social structures and individual actions.
The past is a different land, a different culture, which we attempt to grasp by reading the written record and by considering the physical and social structures that we have inherited. In reading the various musings of Ayoub's masters--and there is no doubt that they were masters of their trade?I feel that I am already partially alienated from this culture. How so? Here are some examples.
I am bored by the endless speculations on why mathematics is true or on what heaven the square root of minus one resides in. I am much more interested in why mathematics is useful to humans and why some of its applications feed into actions that are inhuman.
I am disappointed that writers have harped on the existence of just four schools of the philosophy of mathematics--logicism, intuitionism, formalism, platonism--when in fact during the period from 1868 to 1978, and earlier, there were many philosophies, to which people often adhered informally or unconsciously in various degrees and combinations. Some of these philosophies have been laughed out of court; some (such as the phenomenology of Husserl) are difficult for me to understand; and none is adequate to explain the richness of mathematics and of the mathematical mind at work.
What is the locus of mathematical creativity? What can an individual do to enhance it? I yearn for something more concrete than Poincaré's idea that a "subliminal self," a self that is not inferior to the conscious self, constantly grinds away at a problem and then outputs involuntarily. I am dubious about his assertion that "mathe-matical invention is a process in which the human mind appears to borrow least from the external world."
I read in Francesco Severi's piece (1953) that Leonardo da Vinci loved mathematics, while from Wilhelm Maak's piece (1949) I learned that Goethe hated mathematics. Apparently, you can be a great genius without having to be the prototypical mad mathematician.
And yet, past thought, thought of the self-examining type, contains much that I can agree with and that I would like to see stressed. Hermann Weyl (1968) quotes the philosopher Ernst Cassirer (1874–1945) with approval: "Man no longer lives in a merely physical universe, he lives in a symbolic universe." To which we might add that this universe is increasingly symbolic, mathematized, chipified, and robotized; it is a universe in which some of us, when attempting a telephone transaction, end up shrieking in frustration, "I want to talk to a real person and not to a machine!" And this shriek immediately calls to mind the Turing test for determining whether a computer thinks, and a key question for our age: What does it mean to be human? Are we only, as AI guru Marvin Minsky once asserted, only "meat machines," and is it mathematics that is driving us to that condition?
In an article dated 1950, Raymond L. Wilder (whose subsequent writings are close to my own heart) pursues the theme of symbols:
"Much of our mathematical behavior that was originally of the symbolic initiative type drops to the symbolic reflex level. This is apparently a kind of labor saving device set up by our neural systems. It is largely due to this that a considerable amount of what passes for ‘good' teaching in mathematics is of the reflex type involving no use of symbolic initiative."
True, but let us notice that one of the goals of mathematics is to make itself so automatic that its processes can be carried out without thought. Witness the abacus or theorem proving via a computer. Witness the automatic, minimal-thought aspect that is built into such high-level packages as MATLAB.
Discussions of teaching methodology, like Ben Franklin's death and taxation, will never disappear. Jacques Hadamard, in an article written in 1905, talks about the pluses and the minuses of the socratic or the heuristic method of teaching and decries the "Here's what it is, boys and girls. Now copy it and learn it" approach. We used to argue about the "R.L. Moore method" of teaching math. Now we can argue till the cows come home about the pluses and the minuses of computer-assisted calculus and linear algebra, presented with or without PowerPoint, and the amount of math the "average" person should know for the country to stay ahead of its competitors.
Questions of whom to teach, what to teach, how to teach, where to teach, how to test, whom to pass, how much money and time to invest in teaching are still on the table. To underline ideas that have arisen since Hadamard wrote, just think of the new math, the new new math, and the current corruption of textbooks created by the spirit of political-correctism. (See "The Language Police Knock on Math's Door," SIAM News, July/August 2004.)
Hilbert's 1930 article "Logic and the Understanding of Nature" contains the well-known war cry:
"The true basis on which Comte could not succeed in finding an unsolvable problem, in my opinion, consists in the fact that there is no unsolvable problem at all. In place of the foolish ignorabimus is, in contrast, our slogan, We must know/We shall know."
Just one year later, as we have all read, Gödel appears to have shattered Hilbert's sanguine valedictory. And just where Gödel's Incompleteness Theorems leave the math enterprise today is grist for the mill of contemporary mathematical essayists, some of whom have said that it's of no consequence at all.
A precocious, philosophic Norbert Wiener at the age of twenty, a pre-Aberdeen Proving Ground Wiener, a pre-The Human Use of Human Beings Wiener, takes up the question of conflicts in moral standards, setting in italics his statement that when two parties have moral standards that irreconcilably antagonize one another, "then such a conflict can be settled only by the brute force suppression of the disputant on one side." He also poses the question, mooted for millennia by philosophers, theologians, economists, and moralists, "What is the highest good?" and concludes that while there is good and there is better, there is no best. To a point set topologist, this sounds like what goes on in an open set on the real line.
In a 1970 article, Paul Lévy, speculating on mathematics and theology and echoing Pascal, concludes that we cannot elucidate the mystery of the world. But after looking around him, he reports with some regret that science and mathematics are unlikely to replace religion as a source of consolation.
André Weil comes out in favor of the study of the history of mathematics and gives numerous reasons for its pursuit. I not only agree but find the study of history critical. In the present benighted period, when humanistic studies are being replaced everywhere by the pragmatics of the market, could the historians of our subject mobilize their base to the point that they would wave Weil's cogent reasons in front of a congressional committee and receive financial support for historical studies? I am reminded of the old quip that has Euclid, asked by a student about the use of all this stuff, saying to his TA, "Give the kid two bits, because he needs to profit from what he learns." In a period of social stress and turmoil--and what period is not?--it might be Euclid who appears as the anti-humanistic snob.
André Lichnerowicz opines (1955):
"[The scientific community] should not only teach science but inform society of the implications of its results, communicate also its hopes and fears, freed from the spirit of its work."
Amen, as I implied earlier regarding prescriptive applications.
John von Neumann, in an article (1947) that all mathematicians should read, ob-serves:
"As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideals coming from ‘reality', it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l'art pour l'art."
Again, amen, and in spades.
Well, these are the masters whose words resonated most in me, and I have taken the liberty of giving them my own spin. I encourage my readers to dip into Ayoub's book and to do the same.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at email@example.com.