Game Theory and Mathematical Economics as Art Forms
An alternative way of viewing game theory and mathematical economics is as art forms. The distinction between the common conceptions of science and art is in any case not sharp; perhaps our disciplines are somewhere in between. Much of art portrays the artist’s subjective view of the world; art is successful when the view expressed by the artist finds an echo in the minds of his audience, when the audience empathizes with what the artist is expressing. For this to happen, the artist’s statement must have some universality; it must express some insight of a general nature, must be related to the audience’s experience—in brief, there must be some objectivity in it.
The case for thinking of mathematics itself as an art form is clear. Mathematics at its best possesses great beauty and harmony. The great theorems of, for example, analytic number theory are reminiscent of Baroque architecture or Baroque music, both in their intricacy and in their underlying structure and drive. Other sides of mathematics are reminiscent of modern art in their simplicity, spareness and elegance; the most lasting and important mathematical ideas are often also the simplest.
A characterization of art that I find very apt is "expression through a difficult (or resistive) medium." (I heard this from my friend M. Brachfeld; he said he had read it somewhere, but could not remember where.) The medium may be stone or rhyme or meter or a musical instrument or canvas and paint, or the less well-defined but no less demanding medium of the novel. The resistiveness of the medium imposes a kind of discipline that enables—or perhaps forces—the artist to think carefully about what he wants to express, and then to make a clear, forthright statement.
In game theory and mathematical economics, the resistive medium is the mathematical model, with its definitions, axioms, theorems and proofs. Because we must define our terms, state our axioms and prove our theorems precisely, we are forced into a discipline of thought that is absent from, say, verbal economics.
If one thinks of mathematics as art, then one can think of pure mathematics as abstract art, like a Bach fugue or a Pollock canvas (though often even these express an emotion of some kind); whereas game theory and mathematical economics would be expressive art, like a cubist painting or Tolstoy’s War and Peace. We strive to make statements that, while perhaps not falsifiable, do have some universality, do express some insight of a general nature; we discipline our minds through the medium of the mathematical model; and at their best, our disciplines do have beauty, simplicity, force and relevance.
Robert Aumann (Nobel Laureate in Economics 2005)
"What is Game Theory Trying to Accomplish", Frontiers of Economics, Basil Blackwell, Oxford, 1985.
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
Betrand Russell (Nobel Laureate in Literature 1950)
"The Study of Mathematics", Mysticism and Logic: And Other Essays, Longman, London, 1918.
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. [...] The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.[...] It would be quite difficult now to find an educated man quite insensitive to the aesthetic appeal of mathematics. It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind—we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.
Godfrey Hardy (Fellow of the Royal Society 1910)
A Mathematician's Apology, Cambridge University Press, 1940.