What, can rigid, cold calculating mathematics possibly have in common with subtle, creative, lofty, imaginative art This question faithfully mirrors the state of mind of most people, even of most educated people, when they regard the numbers and symbols that populate the world of mathematics. But the great leaders of mathematics thought have frequently and repeatedly asserted that the object of their pursuit is just as much an art as it is a science, and perhaps even a fine art. Maxime Bocher, an eminent mathematician living at the beginning of this century, wrote: "I like to look at mathematics almost more as an art than as a science; for the activity of the mathematician, constantly creating as he is, guided although not controlled by the external world of the senses, bears a resemblance, not fanciful, I believe, but real, to the activities of the artist—of a painter, let us say." Rigorous deductive reasoning on the part of the mathematician may be likened here to the technical skill in drawing on the part of the painter. Just as one cannot become a painter without a certain amount of skill, so no one can become a mathematician without the power to reason accurately up to a certain point.
"Yet these qualities, fundamental though they are, do not make a painter or a mathematician worthy of the name, nor indeed are they the most important factors in the case. Other qualities of a far more subtle sort, chief among which in both cases is imagination, go into the making of a good artist or a good mathematician."
If mathematics wants to lay claim to being an art, however, it most shows that it possesses and makes use of at least some of the elements that go to make up the things of beauty. Is not imagination, creative imagination, the most essential element of an art Let us take a geometric object, such as the circle. To the ordinary man, this is the rim of a wheel, perhaps with spokes in it. Elementary geometry has crowded this simple figure with radii, chords, sectors, tangents, diameters, inscribed and circumscribed polygons, and so on.
Here you have already an entire geometrical world created from a very rudimentary beginning. These and other miracles are undeniable proof of the creative power of the mathematieian; and, as if this were not enough, the mathematician allows the whole circle to "vanish", declares it to be imaginary, then keeps on toying with his new creation in much the same way and with much the same gusto as he did with the innocent little thing you allowed him to start out with. And all this, remember please, is just elementary plane geometry. Truly, the creative imagination displayed by the mathematician has nowhere been exceeded, not even paralleled, and, I would make bold to say, now even closely approached anywhere else.
In many ways mathematics exhibits the same elements of beauty that are generally acknowledged to be the essence of poetry. First let us consider a minor point: the poet arranges his writings on the page in verses. His poem first appeals to the eye before it reaches the ear or the mind; and similarly, the mathematician lines up his "formulas and equations so that their form may make an aesthetic impression. Some mathematicians are given to this love of arranging and exhibiting their equations to a degree that borders on a fault. Trigonometry, a branch of elementary mathematics particularly rich in formulas, offers some curious groups of them, curious in their symmetry and their arrangement.
The superiority of poetry over other forms of verbal expression lies first in the symbolism used in poetry, and secondly in its extreme condensation and economy of words. Take a poem of universally acknowledged merit, say, Shelley"s poem "To Night". Here is the second stanza: Wrap thy form in a mantle gray, star-in wrought! Blind with thine hair the eyes of Day; Kiss her until she be wearied out; Then wander oer city, and sea, and land, Touching all with thine opiate wand—Come, long-sought !
Taken literally, all this is, of course, sheer nonsense and nothing else. Night has no hair, night does not wear any clothes, and night is not an illicit peddler of narcotics. But is there anybody balmy enough to take the words of the poet literally The words here are only comparisons, only symbols. For the sake of condensation the poet doesn"t bother stating that his symbols mean such and such, but goes on to treat them as if they were realities.
The mathematician does these things precisely as the poet does. Take numbers, for example, the very idea of which is an abstraction, or symbol. When you write the figure 3, you have created a symbol for a symbol, and when you say in algebra that is a number, you have condensed all the symbols for all the numbers into one all-embracing symbol. These, like other mathematical symbols, and like the poets symbols, are a condensed, concentrated way of stating a long and rather complicated chain of simple geometrical, algebraic, or numerical relations.In the writer"s opinion, what is the most fundamental element that makes a good artist or mathematician