In 1980, Kenneth Boulding (former president of both the American Association for the Advancement of Science and the American Economic Association), wrote about the conflict between academic specialization on the on hand, and broad liberal arts inquiry on the other. He noted, "It may well be that the only answer to this problem is redundancy, inefficiency, extravagance, and waste. One could argue indeed that the main reason for getting rich, that is economic development, is that it permits the human race to indulge in these last four delights." [B]
Mathematics is richer — and if Boulding. is correct, mathematicians are richer — because of our long heritage of broad inquiry. When we study or teach great mathematics, we gladly share the lore of the broad intellectual interests of great mathematicians. Who among us doesn't know that Cantor's theology consumed as much — if not more — of his life than mathematics? That Pythagoras advanced a musical theory as well as an eponymous theorem? that Newton dabbled in alchemy? That Galois' political interests were the reason for cutting short his brief, mathematical life?
At the same time, we don't (except in exceedingly rare cases) study these interests directly. Few math students or professors have read Cantor's theology or Galois' political writings. For this reason, I welcomed the chance to get my hands on Ayoub's Musings of the Masters.
On the one hand, I shouldn't have judged the book by its cover (or its title). The subtitle of the book ("an Anthology of Mathematical Reflections") might be better phrased "an Anthology of Philosophical Reflections on Mathematics". And in spite of the Renaissance-evoking word "Masters" and a jacket that nods to Roman, Greek, and Gothic traditions, each of the seventeen authors included is a twentieth century mathematician. I was sorry to miss out on the likes of Cantor, Galois, and Euler (who after all Dunham names the "Master of us all" [D]). I also missed authors from outside the Western European/U.S. tradition. Where are the Russian and Polish mathematicians? Since reading this collection, for example, I have often been pointed to Hugo Steinhaus' "Between spirit and matter mediates mathematics" [S], which has not yet been translated from its original Polish.
On the other hand, this volume contains a great deal of food for thought, and — to continue the metaphor — I gobbled it up. What is mathematics? How do we do mathematics? What does it mean to be a scholar? These introspective questions are the bread and butter of the collected essays.
There is much in this collection that deserves a wide audience: an audience who would read and chew on the material together. Let me give you some of my favorite examples.
Mathematics as a sacred undertaking
Many of the authors in Musings, regardless of their theological beliefs, see their discipline as supernatural (in the sense of being outside of natural laws) and any endeavors in this discipline as almost sacred. So we read Birkoff's words about faith in scientific thought:
If [my comments about scientific thought] are worthy of serious attention it is not because of their novelty, but rather because in their aggregate they rise above the details of the numerous specialized fields of knowledge and sustain the scientist in his unceasing and ardent search after truth. [p.114]
Hilbert weighs in:
The tool which governs the mediation between theory and practice, between thought and observation is mathematics; it builds the bridge and carries more and more of the load. It thereby happens that the basis of our entire present day culture, in so far as it is based on investigations dealing with nature, can be found in mathematics. [p. 125]
Lichnerowicz speaks about scholarship in any scientific discipline:
What then distinguishes a scholar from a technician who is a grand chief engineer…? Broadly speaking, if you will permit me to speak frankly, the one can find the pinnacle of his career by becoming the CEO of his firm without being a traitor to his vocation while the other not. … A scholar is a person who participates actively in the scientific adventure, who is a crusader in the scientific enterprise… The scholar is by definition one who does not place a monetary value on the results of his work but delivers them freely to all. [pp. 188-194].
Late in the book, Levy considers (and refutes) the widespread belief in God as a way of understanding the world, and then turns to mathematics. These quotations are taken from the course of several pages:
This belief [in God] does not help at all in explaining the mystery of the world. We do not explain a mystery by imagining a greater one. . . . And yet, little by little, I discovered that science never does anything but explain one mystery by another. … I often asked myself if the necessity of the laws of nature is not comparable to the laws of mathematics. We do not refer to God in order to explain that the sum and product of two integers are independent of their order. It could not be otherwise. [pp. 224-227]
Do other scholars view their disciplines with the same reverence and awe? These essays would make wonderful fodder for a cross-disciplinary reading group.
Mathematics and Infinity
Because of the self-imposed chronological limitations of this book, the combined effect of the essays is to give a snapshot into a moment (albeit a hundred-year moment) of mathematical history. There are several issues that the authors raise again and again. One of these is wonderment and delight at the coming together of what had long seemed different disciplines: arithmetic, algebra, and geometry. (Sylvester is particularly eloquent on this subject). Another, much more unsettling issue to the authors was the question of infinity. To be sure, I "grew up" (mathematically speaking), hearing rumors of the shock waves that Russell, Cantor, and Gödel sent through the mathematical community. To read the words of those present brings the whole issue to life. I was surprised and frankly shocked myself to read that Hilbert was an "intuitionist", asserting that there is no infinity. In his own words,
What the concept of "infinite" entails is something we must make clear; that "infinite" has no clear significance and without closer investigation has no meaning, since in general there exist only finite objects. There is no infinite oscillation, no infinitely rapid propagation of force or energy. Moreover, energy itself has a discrete nature and exists only in quanta. There exists in general no continuum that can be divided infinitely often.… [The infinite is] a gigantic abstraction, attainable only through the conscious or unconscious application of the axiomatic method.
So what does Hilbert make of his own work, which abounds with notions of infinity? It is merely a convenient device:
This perception of the infinite, which I have established in exhaustive detail, solves a series of basic questions. [p. 120]
I would no more have imagined such a sentence coming out of Hilbert's mouth than out of the Pope's! Other authors of the time make a point of disagreeing with Hilbert's (at that time) well-known stance, but admit that the question of the infinite is a problematic one for mathematicians. In an essay written 11 years before his celebrated A Mathematician's Apology [H], Hardy writes
I am sure that the vast majority of mathematicians will rebel against the doctrine — even if it is supported by some of themselves including mathematicians so celebrated as Hilbert and Weyl — that it is only the so-called 'finite' theorems of mathematics which possess a real significance. That 'the finite cannot understand the infinite' should surely be a theological and not a mathematical war-cry. [p. 52]
Mathematics: Pure and Applied
If the notion of infinity has died down — at least as a topic of conversation and controversy — the main questions of this book are still fresh and relevant. What is mathematics, and how do we do it? Just this January, Ingrid Daubechies began her 2005 Gibbs Lecture by proclaiming that among the sciences, mathematics is "type O: the universal donor". Some three dozen years earlier, Dame Mary Lucy Cartwright said much the same:
One of the distinctive characteristics of modern mathematics is its way of taking old mathematical ideas apart like watches, studying the parts separately, and putting these parts together again in new and interesting combinations and studying these complications in turn. I believe that this process has contributed enormously to this simplification in mathematics itself, and so made it more readily available for applications. [p. 13]
Much later in the book, von Neumann turns this view on its head when he questions whether mathematics is an empirical science — that is, whether it is based on observation. He concludes,
I think that… mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else, and in particular, to an empirical science. [p. 183]
Other authors address the nature of mathematics more indirectly. Severi uses Leonardo da Vinci as a case study for the question, "Was he a mathematician?" [His answer: no.] Maak addresses the nature of mathematics by arguing against Goethe's descriptions of the subject. Poincaré describes what it is like to do mathematics; the picture he draws of himself spending hours of fruitless thinking and work, followed by sudden revelation, will be familiar to many of his readers, whether mathematicians or no. His essay is both personal and touching, right down to the admission, "I am obliged to confess that I am absolutely incapable of doing addition without mistakes." [p. 22]
The editor of this collection, Raymond Ayoub, gives two introductions to each essay: the first gives a brief biography of the author and describes the historical setting for the essay, and the second summarizes the essay. These introductions are generally very helpful to the reader, and are likely to be helpful to an instructor's using the book in a classroom setting. The book itself is partitioned into four chapters, although I often found myself second-guessing the editor on which chapters were the best fit for which essays. The last chapter, "Miscellaneous", is indeed very random, including (in this order) Levy's refutation of the existence of God, Maak's one-sided argument with Goethe on the nature of mathematics; Severi's mathematical biography of Leonardo da Vinci, and Wiener's essay on the Highest Good (there isn't one; everything is relative).
Nonetheless, the essays included in this book are as thoughtful and deep as you would expect from authors of this stature. I would recommend this book to anyone interested in the philosophical thoughts of great mathematicians; I would also recommend it (and indeed, have already recommended it) to non-mathematicians who are curious about how mathematicians think about themselves and their discipline.
[B] K. Boulding, Graduate Education as Ritual and Substance, in The Philosophy and Future of Graduate Education, [William Frankena, editor] University of Michigan Press, Ann Arbor (1980) pp. 143-154.
[D] William Dunham, Euler: The Master Of Us All, by William Dunham. Mathematical Association of America, Dolciani Mathematical Expositions No. 22, 1999.
[H] G. H. Hardy, A Mathematician's Apology, Cambridge University Press, London, 1967.
[S] H. Steinhaus, "Between spirit and matter mediates mathematics," PWN, Warszawa-Wrocław 2000, 236-244 (Polish).
Annalisa Crannell teaches at Franklin and Marshall College.