The Aesthetic Impulse in Science and Mathematics---Merely a Matter of Words?October 1, 1999
Philip J. Davis
In memory of Gian-Carlo Rota
Beauty and Revolution in Science. By James W. McAllister, Cornell University Press, 1996, 248 pages, $29.95.
"The Phenomenology of Mathematical Beauty." By Gian-Carlo Rota, Synthèse, Vol. III, No. 2, May 1997.
"I cannot tell by what logick we call a toad ugly."---Sir Thomas Browne, Religio Medici.
What is this wonderful thing, this process called the scientific method? Philosophers of science, scientists themselves, have for centuries tried to come to grips with it. Is there such a thing? If so, what is it? Is it unique? Can it be taught? Is it a guide to further research?
Francis Bacon, Descartes, Thomas Huxley, Ernst Mach, Niels Bohr, the Logical Positivists, Dirac, Percy W. Bridgman, Michael Polanyi, Thomas Kuhn, Imre Lakatos are just a few of the hundreds of now departed who have all put in their two cents. Even I, in my naiveté, once formulated a two-word condensation of Bridgman's position (more about which later).
To do science, one needs a guide, a set of precepts to follow; otherwise, we're walking around in the total dark. So said Francis Bacon. And who can disagree?
Observe, analyze things down to their components, said Descartes; then synthesize, build up. Good thinking, René, but how does this help the researcher? Well, it tells the researcher not to rely too much on the holy texts, the received wisdom of the ages. To assert this much was, in Descartes' day, heretical. How could the results of millions of experiences, experiments, ideas and cross-influences, published or transmitted orally, be reduced to a few bullets displayed via an overhead projector, or even to a book-length treatise?
If there were a set of rules that scientists could follow, let us even say rules that could be expressed in a formal language, then we could close up shop, allow the housecat to put her paws on the button, and sit back to watch what emerges. I think that Descartes' Dream was to think hard in the present so that all further thought could be eliminated, and some of the TOE (theory of everything) physicists---or at least their popularizers---seem to be asserting the same.
Yet the search, or rather less a search than a dreamy yearning for condensation by unification, goes on and on, often carried on mostly by those outside science, but also by not a few on the inside. It would seem that those who desperately want to dispel and trivialize the ultimate mystery are slated to create mysteries of their own.
Beauty and Revolution in Science, by James McAllister, a lecturer in the Faculty of Philosophy at the University of Leiden, is one of the most recent discussions of the scientific method. It's a book that tackles a very contentious subject, and consequently I both agree with and question a lot of what the author says.
McAllister's book focuses fairly narrowly on the role of aesthetics in the development of science. It is an elaboration of two major theses. The first
is that aesthetics, or more precisely "aesthetic induction," plays an important and rational role in the creation and acceptance of physical theories---and this despite what is commonly admitted to be an uncertain correlation with experience. The second thesis is that revolutions in science are essentially "aesthetic ruptures" and that these ruptures can be given a rational basis.
In order to understand McAllister's carefully worked out line of argumentation, the reader will have to come to grips with some key words. I list a few: beauty, simplicity, symmetry, empirical, objective, analogy, aesthetic sensibility, rational, revolution. These words are basic terms we use in describing how we perceive the world. On the other hand, there is little agreement as to what they mean or how they are to be employed; they are terms that, on deeper elaboration, can lead rapidly to vicious definitional circles. Consider "analogy," that important component of conceptualizations; it is both useful and problematic. An electric current is like the flow of water: useful. Copernicus bolstered his presentation of heliocentrism by analogy with the contemporary social structure: The sun was analogous to the king in occupying a central position. Was this merely a useful nod to the status quo to avoid condemnation?
Later on, the world was said to be analogous to a clock that God remembered to wind up once in a while. Today, the mind is often a computer and the body is a "meat machine," from which analogies we are invited to draw physical conclusions and social policies.
In his elaboration of his two theses (the rationality of the aesthetic component and of scientific revolutions), McAllister discusses how aesthetic evaluation in science works. What is beauty? What is simplicity? What is the relation between beauty, simplicity, and empirical corroboration? He discusses the evolution of aesthetic perception and acceptance, and states which of the great scientific advances of the past constitute revolutions and which do not.
Consider also "rationality." To be rational commonly means to be logical or to base conclusions on reason, with the deeper meanings of reason or logic themselves up for grabs. To McAllister, the rationalist image of science asserts that
"there is a basis for forming and judging decisions and policies in scientific practice that does not depend on convention, fashion, or other local or historical phenomena."
What constitutes the individual planks in this---to my mind dubious---basis, I found hard to dig out of the book. To establish the rationality of acting on aesthetic impulse, the author argues simply that such actions have occasionally worked and therefore should not be excluded as a possibility.
Similarly, for revolutions:
"As long as there endures correlation between a theory's showing particular aesthetic properties and its demonstrating empirical success, it may be rational to continue choosing on those aesthetic criteria."
McAllister seems to admit as revolutions only changes that exhibit an "aesthetic rupture." On this basis he concludes (as other commentators have concluded) that Copernicus created a disturbance but not a revolution---for he maintained the ancient aesthetic---driven obsession with circularity. Kepler was the revolutionary. Relativity was not a revolution but only a completion of Newtonianism, since it maintained a deterministic stance, which would be abandoned by the revolutionary quantum theory.
Given that I have had no hands-on experience in the creation of physical theories, what is my reaction to these two theses? My opinion relies mostly on what I have been taught or have read, or on what physicist friends have told me over coffee.
I agree that aesthetics plays a role in the creation and acceptance of theory. Many physicists have asserted as much. Dirac was perhaps the most ardent and eloquent advocate for beautiful equations. I agree that the great discontinuities or ruptures that are often called revolutions in science have an aesthetic component. The switch from determinism to probabilism, and from the total objectivity of the physical world to the role of the observer, both of which are required by quantum theory, are such ruptures. But are they examples of an aesthetic impulse? They might be perceived as aesthetic or as anaesthetic, given human tendency to sugar-coat with sweet names experiences that are at first difficult to swallow.
I disagree that such a component (or indeed almost any component of theory formation) is rational. In fact, it appears to me that the whole development of science in the past four centuries is irrational. I use this word not in the psychiatric sense, but to mean "depending on inexplicable inspirations or epiphanies, and not entirely on prescribed agendas of thought." As an authority, I cite the words of Percy W. Bridgman, Nobelist in physics (1946), in his Reflections of a Physicist (1955):
"What appears to him [i.e., the scientist] to be the essence of the situation is that he is not consciously following any prescribed course of action, but feels complete freedom to utilize any method or device whatever which in the particular situation before him, seems likely to yield the correct answer. In his attack on his specific problem, he suffers no inhibitions of precedent or authority, but is completely free to adopt any course that his ingenuity is capable of suggesting to him. No one standing on the outside can predict what the individual scientist will do or what method he will follow. In short, science is what scientists do, and there are as many scientific methods as there are scientists."
Years ago, and for my own purposes, I opined that the "scientific method" was simply to "try everything." To which quantum physicist David Park appended "but don't waste time being stupid."
When the fundamental structures of the universe appear to be modeled in beautiful mathematics, therein is a profound mystery that some can merely accept but that others will want to explain.
After reading this provocative book, I wondered whether my disagreements boiled down simply to a matter of personal definitions of "aesthetic," "rational," and "revolution."
Now let's talk about the aesthetic component of (pure) mathematics, where I have had hands-on experience.
I've noticed that in tracing the historical development of specific mathematical ideas, I often come to a crucial point, a gap, where I ask myself, How did this mathematician happen on this beautiful step? I cannot answer, although students have pressed me; perhaps I am right not to answer, for to do so might be to deny the genius of the step.
The aesthetic component in mathematical research is great. It has been discussed by many authors, but I find Gian-Carlo Rota's article on the
subject particularly thoughtful. In limiting mathematics in his article to the sequence definition, theorem, proof, he finds less need to bring in empirical corroboration and seems also to evade the question of "importance."
Beauty, Rota asserts, can be found independently in each of the three items just mentioned. There are beautiful definitions, which, once made, can immediately inspirit a whole industry of investigations. There are beautiful theorems with ugly proofs, e.g., the prime number theorem. Picard's theorem asserting that an entire analytic function takes on all values with two possible exceptions has a most beautiful proof: five lines.
What are some of the hallmarks of mathematical beauty? Rota mentions: simplicity, a brilliant step, fruitfulness, revelation.
"Proof is beautiful," he writes, "when it gives away the secret of the theorem, when it leads us to perceive the actual and not the logical inevitability of the statement that is proved."
What are some further features? Beauty does not admit degrees of comparison: To state, for example, that the prime number theorem is more beautiful than Picard's theorem would be nonsensical.
Beauty should not be confused with elegance, which often resides in mere presentation.
The perception of beauty does not occur in a flash. To think that it does is what Rota calls the "light bulb error":
"The appreciation of mathematical beauty requires thorough familiarity with mathematics, and such familiarity is arrived at the cost of time, effort, exercise and Sitzfleisch."
As a consequence, the beauty that the educated public finds in mathematics is different from what the professional finds. And I would add---close to the position of the aesthetes of the late 19th century---that the perception of the aesthetic component in mathematics may be confined to an elite.
Rota also notes some features that I find contradictory: Beauty is objective even as truth is objective, he says. At the same time, he asserts that the perception of beauty depends upon particular schools and periods of history.
Returning to the necessary long experience that limits the general perception of mathematical beauty, Rota believed that teachers' attempts to arouse interest in mathematics on the basis of beauty are bound to fail. This position can be compared with the statement of James McNeill Whistler in his "Ten O'Clock Lecture" that art is "selfishly occupied with her own perfection only---having no desire to teach."
And, for altogether different reasons, Rota's stance, in the interests of a balanced presentation, can be compared with a statement cited by McAllister:
"I once heard Dirac say in a lecture, which largely consisted of students, that students of physics shouldn't worry too much about what the equations of physics mean, but only about the beauty of the equations. The faculty members present groaned at the prospect of all our students setting out to imitate Dirac."
---Steven Weinberg, Towards the Final Laws of Physics.
So much for aesthetics and education.
The book and the article under review bring up issues that deserve lengthier treatment. I hope that some readers will be tempted to give them further consideration.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.