Beyond the edge of imagination

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.

By the early 18th century, square roots of negative numbers had become a routine feature of mathematical practice. But their status remained shady. Rene Descartes had dismissively termed them “imaginary” and the name has stuck, lending them still an aura of unreality. Yet this bizarre artifact of mathematical play has turned out to have profound implications for our understanding of the material world. In the late 19th century, physicists were astounded to discover that electrical circuits could be understood by equations involving imaginary numbers. So too, they permeate the equations that underlie audio signal processing and holography. Most famously, Stephen Hawking has suggested the concept of “imaginary time,” a temporal analog of the square root of minus one.

Mathematicians have long since gotten over the trauma of the imaginaries and have moved on to even more arcane constructs. In “The Art of the Infinite,” Robert and Ellen Kaplan take us on a grand tour, leading us from the terra firma of the simple counting numbers (one, two, three, four and so on) through the discovery of the rationals, the irrationals, the negatives and the complex numbers that combine the ordinary, or real, numbers and the imaginaries to generate a two dimensional number-space known as the complex plane. It is here that the famous Mandelbrot set lives, that enigmatic emblem of chaos theory.

Once unleashed, the mathematical imagination reveals itself as a protean force, seemingly able to spin from nothing intricately structured webs. The Kaplans also introduce us to that other great pillar of the mathematical edifice, geometry and its own chimeric spectacle, “projective geometry.” Here “points” become “lines” and “lines” become “points” and all parallels meet at infinity, a point that is no longer special but just another node on a shimmering logical mesh. Just when the limits of sense seem almost upon us, the Kaplans push us one step further, into the domain of the transfinite numbers: the set of numbers that comes after infinity. No other artifact of mathematical imagining has so stretched the bounds of credulity. Are we, here -- in the realm of infinity plus one, infinity squared and infinity raised to the power of infinity -- beyond reason?

Anyone interested in a serious introduction to mathematics will delight in these volumes. The Kaplans’ background in languages and linguistics inclines them to a depth of literary allusion that few writers in any technical field can match. Robert Kaplan’s prior book, “The Nothing That Is: A Natural History of Zero,” remains, for my money, the best popular mathematics book ever written. As one of America’s leading mathematicians, Mazur reveals himself also as a delicate stylist who can not only evoke the history of algebra but also comment upon the common ground it shares with tulips in Arab cultural life. Because both books aim to impart not merely a vague understanding of mathematics but also a concrete comprehension of specific parts of the corpus, be prepared with a pencil in hand. Like meditation or yoga, mathematics is a practice: One must do it for oneself.

Unlike physics, which aims at a complete theory of physical reality, mathematics has no implied endpoint. For that reason, mathematicians are free in a way that natural scientists are not. Yet without constraints the mind is apt to veer into fantasy. Aware of this pitfall, the Kaplans note that mathematicians “can’t after all just say that anything we choose is a number.” When proposing new types of objects, they “must show that the franchise has been legitimately extended.” Like a trip to a wildlife park of the mind, these books enable us to see up close some of the more bizarre and bewildering extensions of human imagination. If, in the end, our feet appear to have left the ground, we may rest assured that for mathematicians, the stratosphere is higher still.