Is mathematics discovered or invented? Are numbers and geometric forms "out there" in some transcendent space, above and beyond the material world, eternally awaiting our apprehension? Or are they the products of human imagination, the fruits of a creativity that expresses itself not in shades of paint but in structures of logical relations? This vast and perhaps unanswerable question has divided mathematicians for centuries, though most fall on the side of discovery. When dealing with numbers, it is difficult not to feel that here is a realm of inexorable being: Four flowers will wither, four people will die, four civilizations will inevitably fade, even four mountains will erode away, but four itself seems eternal and immutable, immune to contingency and decay.
Mathematical objects have always seemed in a class of their own. Immaterial and insubstantial, they are not subject to the laws that govern the physical world. So what exactly is their status? Is the number 15 a Platonic Ideal, a convenient fiction or merely a useful tool that enables us to enumerate, say, the number of apples in a basket? What about 15's reciprocal, the fraction 1/15; or its square root, which is known to be an irrational number, one that cannot be expressed as any kind of fraction? (To understand the square root of a number, N, imagine a square field of area, N; the square root of N is the length of each side of the field.)
What about the concept of negative numbers? Is minus 15 an actual number or something mathematicians have invented for their own perverse pleasure? How can one have minus 15 apples? The idea seems to make no sense at all. The best way to understand the negatives is to look at them in bookkeeping terms: If my debts exceed my income, then my balance will be negative. Historically speaking, it was the practicalities of bookkeeping that ushered the negatives into the mathematical mainstream, but once you've accepted them as a legitimate kind of number, they too become subject to the usual mathematical operations, including the square root. Which brings us to the enigmatic notion of the square root of minus 15, an entity that is neither whole nor fractional, neither rational nor irrational.
At different points in history, mathematicians have worried over all these types of numbers. Indeed, the history of mathematics may be seen as a process of territorial expansion in which mathematicians continuously push out the boundaries of what may be considered a legitimate object for investigation. Two marvelous new books, "Imagining Numbers" by Barry Mazur and "The Art of the Infinite" by Robert and Ellen Kaplan, wrestle with the philosophical status of mathematical entities. Neither resolves the age-old debate. Whether numbers are discovered or invented, these authors assert that mathematics is an enterprise that advances not so much by the power of reason as by the sheer vitality of human creativity.
Consider the case of the so-called "imaginary" numbers, whose story is enchantingly told by Mazur. By the early 16th century, mathematicians had realized there existed certain equations that lent themselves to a disturbing species of solution. Take the equation X2 + 1 = 0. At first glance a perfectly reasonable formula, but a slight recasting quickly reveals its aberrant nature, for this is equivalent to the equation X2 = -1. The solution, X, must therefore be the square root of minus one.
But what is the square root of a minus number? It certainly isn't a minus number itself, because, as French writer Stendhal fretted, minus times minus yields a plus (-3 x -3 = 9). In mathematics, as in natural language, a double negative becomes a positive. In the 16th century, such mathematicians as Girolamo Cardano had begun to realize that the solutions to a vast number of equations involved the square roots of negative quantities. Considering these mysterious roots, Cardano wrote that they must be neither positive nor negative but "some recondite third sort of thing." At one point in his great work, "Ars Magna," Cardano found himself forced to invoke the square root of minus 15 and told his readers that they would simply "have to imagine" this absurd proposition and dismiss the "mental tortures" it would no doubt provoke.