Beauty in the Equation

(cc) devin.berg,

“It is more important to have beauty in one’s equations
than to have them fit the experiment.”
–Paul Dirac.

There is a general notion that “beauty” is something that belongs to the arts and humanities and has more or less been banished from science.  Yet, in the last hundred years, many prominent artists and humanists* have banished beauty from their consideration (e.g. the Dadaist, Fluxus, and much of the Postmodern) and many prominent scientists and mathematicians have in their way proclaimed beauty as an inspiration and guide to scientific discovery.

The theory of beauty (aesthetics) proposed by twentieth century humanists tends to go from muddy to muddled.  On the other hand, a scientist like Werner Heisenberg, in his essay “The Meaning of Beauty in the Exact Sciences,” gives a straight forward definition of beauty as “the proper proportion of the parts to the whole, and to each other.”  Now the humanist might complain that this is overly simplistic, but from Aristotle to the present the notion that the integrity and proportionality of the parts to the whole has been a steady guide for artists, craftspeople, scientists and engineers.

In general, one will find a particular scientific theory more elegant than another by the greater diversity of phenomena that it can unify.  The progress of science is always in this direction.  Prior to Newton, one would not have suspected that the rising of a balloon and the falling of an apple were examples of the same phenomena: gravity.  Prior to Maxwell and Faraday the scientists exploring magnets and those exploring static electricity did not suspect that the phenomena they were exploring would turn out to be different aspects of the same thing.  Among the other great unifications are Darwin’s discovery of an underlying principle that unifies the diversity of living species and Einstein’s unifications of space and time, and matter and energy.  In our own era, the goal of theoretical physics is guided by the belief in a final unifying theory, a theory of everything that will harmonize quantum mechanics with general relativity and gravity with the other fundamental forces.

In addition to the idea of unity in variety, Heisenberg’s definition speaks of proper proportion.  Proportionality refers to ratios, and by extension to that which is rational.  Thus the fundamental faith of science, that the world is ultimately rational, is an extension of this principle.  The original of the idea that beauty has to do with proper proportion may have come from Pythagoras’ discovery of the mathematical relationships of musical tones.  An idea that many consider one of the most beautiful ever, and from which the musical scales we still used were ultimately derived.

The humanist again might be quick to complain that such examples as music and the golden mean are exceptions, that in the real world there is little basis for the idea that beauty refers to correct proportions.  As an example, most people would say that a butterfly is more beautiful than a human louse.  But on what basis could one say that a butterfly is more properly proportioned than a human louse?  We associate butterflies with summer days and flowers, we associate the louse with parasitism and an itchy head.  Does not the beauty we attribute to the butterfly really come from these associations, which in the end are quite subjective?  And, of course, there is also the butterfly’s colorfulness. Pleasant associations and colorfulness are attributes of our common understanding of beauty – and one could argue that they are more central to it than the idea of proportionality.  So can one justify the aesthetic ideal of proportionality?

As an undergraduate student, I took an entomology class from a professor who was a renowned authority on the human louse.  He would get rhapsodic when showing slides of the louse’s anatomy and would occasionally say how beautiful the louse was.  My fellow students would roll their eyes.   The louse’s internal organs, however, are wonderfully proportioned to enhance its ability to survive its particular lifestyle.  All living creatures are wonderfully proportioned in this sense – they are all beautiful in their own way.  It takes a certain kind of detachment, however, to find beauty in a louse; indeed, it takes the kind of detachment required of a good scientist.

One of our most common uses of the word beauty is in reference to attributes that we find in other people, attributes that make others loveable to us.  We also commonly use the word beauty to refer to what makes a person erotically attractive to us.  While there seems to be some connection between the use of the word beauty to refer to that which is lovable and also that which is sexually desirable, they don’t mean quite the same thing. And both uses of the word seem nearly opposite the kind of detachment that finds beauty in the form of a louse.  Perhaps rather than having only the one word “beauty” for these different understandings, we should have different words.  One of the reasons, I suspect, that the idea of beauty is so muddled is that we try to make one word carrying many different meanings.

The experience of beauty always has two poles: the subject that experiences and the object that is experienced.  Different people take delight in vastly different objects.  This fact has often been used as evidence that beauty is purely a matter of taste. But is this not an oversimplification? Calling beauty purely a matter of taste does little to explain the complex phenomenon of the experience of beauty.  I suggested earlier that a simple equation that explains a great deal is more beautiful than a complex equation that doesn’t really explain much.  An equation’s explanatory power is not a matter of personal taste.  Not everyone experiences beauty in the contemplation of mathematical equations, but those who do can often provide solid reasoning for their judgment of the difference between the more and less beautiful equation.  If beauty were merely a matter of taste, any such reasoning would be fatuous.

Albert Einstein wrote “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.”  This would seem to be a very different idea of beauty than the one proposed by Heisenberg.  I would suggest though, that it is a necessary addition.  Not only every organism, but every workable machine fulfills the requirement of Heisenberg’s definition of beauty – every good machine and every viable organism requires a proportionality of its parts and an integration of all the parts into a functioning whole.

But is there not something more to the beauty of a great work of art or to the works of nature, than this simple beauty of the machine?  I would suggest that the difference is that a machine is not mysterious – there is always someone, hopefully the repairperson, who understands it fully.  A great work of art is never fully understood, not even by the person who creates it.  There have been thousands of articles written about Shakespeare’s Hamlet, and thousands more will undoubtedly be written in the future.  Hamlet the character and Hamlet the play are filled with ambiguities.

Ambiguity is the life blood of a work of art.  It is what keeps us coming back.  Compare a crossword puzzle to a good poem.  A crossword puzzle can be incredibly engaging and enjoyable to work on, but the moment the last box is completed, the value of the puzzle plummets to nothing.  When finished, a crossword puzzle is utterly without ambiguity.  A good work of art, through its ambiguities, makes sure that we can never fill in the last box.  One can gaze at the Mona Lisa’s smile for ages and still not be sure just what it conveys.

As ambiguity is to art, mystery is to Nature.  For a few hundred years now, science has penetrated the layers of Nature’s mystery, but as each layer is penetrated a new layer is discovered.  And this is not likely to end.  Even if physicists should complete the quest for a final theory of physics, it has become increasingly clear that the theory will not answer the question of why and how the universe came to have the properties described by that theory.  Like the Mona Lisa’s smile, Nature’s mystery refuses final resolution.

From the point of view of mystery, we can see that Heisenberg’s definition of beauty may be more a definition of value in general than a correct definition of beauty.  Integrity and proportionality, wholeness and rationality, applies equally to the good, the true and the beautiful; the role of mystery, however, is quite different in each.  A good machine or good legal system seeks to banish ambiguity; a true principle or theory acknowledges the presence of mystery, but seeks to penetrate it as far as possible.  The beautiful object or idea celebrates the mysterious and invites us to contemplate it.

If you enjoyed this article, please consider a donation.The truth of science plays a dual role in relation to other values.  One the one side, it is eminently practical and its discoveries are the stock-in-trade of the engineer, who turns these discoveries into a society’s economic goods.  But on the other side, its discoveries expand our sense of the world we live in, revealing its incredible diversity and the connections that hold this great diversity together — in short, revealing its beauty.  Using the classic definition of beauty, that it is “unity in variety,” it is not difficult to see how beauty can lead us to truth, and how truth can lead to beauty.

Connecting the world of ideas and the world of the senses, the physicist Murray Gell-Mann said at a Noble Prize banquet in his honor, “The beauty of the basic laws of natural science, as revealed in the study of particles and of the cosmos, is allied to the litheness of a merganser diving in a pure Swedish lake, or the grace of a dolphin leaving shining trails at night in the Gulf of California.”  To this, I’ll say Amen! and leave it at that.

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*Note: ‘humanist’ is used here to mean scholars in the humanities. It does not refer to the modern secular humanist movement.


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