To Infinity and Beyond

Infinity has many faces. The layman often perceives it as a kind of “number” larger than all numbers. For some primitive tribes infinity begins at three, for anything larger is “many” and therefore uncountable. The photographer’s infinity begins at thirty feet from the lens of his camera, while for the astronomer—or should I say the cosmologist—the entire universe may not be large enough to encompass infinity, for it is not at present known whether our universe is “open” or “closed,” bounded or unbounded. The artist has his own image of the infinite, sometimes conceiving it, as van Gogh did, as a vast, unending plane on which his imagination is given free rein, at other times as the endless repetition of a single basic motif, as in the abstract designs of the Moors. And then there is the philosopher, whose infinity is eternity, divinity, or the Almighty Himself. But above all, infinity is the mathematician’s realm, fir it is in mathematics that the concept has its deepest roots, where it has been shaped and reshaped innumerable times, and where it finally celebrated its greatest triumph.

Like most other sciences, European mathematics came to a virtual standstill during the long, dark Middle Ages. It was not until the sixteenth century that the notion of infinity—long since forgotten as a scientific issue and having become instead the subject of theological speculations—underwent its revival. And one of the first questions to be tackled was once again that of finding an approximation to the value of π.

Central to the development of the calculus were the concepts of convergence and limit, and with these concepts at hand it became at last possible to resolve the ancient paradoxes of infinity which had so much intrigued Zeno. For example, the runner’s paradox is explained by the following observation: By first covering one-half the distance between the runner’s starting and end points, then half the remaining distance, and so on, he will cover a total distance equal to the sum:

A series is obtained from a sequence by adding up its terms one by one. From a finite sequence a₁, a₂, a₃, …, a _n we obtain the finite series, or sum, a₁ + a₂ + a₃ + …, + a _n . But for a_n infinite sequence a₁, a₂, a₃, …, a _n , …, a problem arises: How should we compute its sum? We cannot, of course, actually add up all its infinitely many terms; but we can, instead, sum up a finite, but ever increasing, number of terms: a₁, a₁ + a₂, a₁ + a₂ + a₃, and so on. In this way we obtain a new sequence, the sequence of partial sums of the original sequence. For example, from the sequence 1, 1/2, 1/3, …, 1/n, … we get the sequence of partial sums 1, 1 + 1/2 = 1.5, 1 + 1/2 + 1/3 = 1.83333 …, and so on. If this sequence of partial sums converges to a limit S, then we say that the infinite series a₁ + a₂ + a₃ + … converges to the sum S. For the sake of brevity, we also use the phrase, “the series has the (infinite) sum S.”

If the harmonic series is the most celebrated of all divergent series, the same distinction for convergent series goes, without reservation, to the geometric series. We have already met this series in connection with the runner’s paradox. In a geometric sequence, or progression, we begin with an initial number a and obtain the subsequent terms by repeated multiplication by a constant number q: a, aq, aq², …, aq ⁿ , …. The constant q is the common ratio, or quotient, of the progression. Sometimes our progression is terminated after a certain number of terms, in which case, of course, we omit the final dots. Such finite geometric progressions appear quite frequently in various situations. Perhaps the most well known is compound interest: If one deposits, say, $100 in a savings account that pays 5% annual interest, then at the end of each year the amount of money will increase by a factor of 1.05, yielding the sequence $100.00, 105.00, 110.25, 115.76, 121.55, and so on (all figures are rounded to the nearest cent).¹ On paper, at least, the growth is impressive; alas, inflation will soon dampen whatever excitement one might have derived from this growth!

Infinite geometric sequences and series arise not only in pure mathematics but also in geometry, physics, and engineering, and at least one contemporary artist, Maurits C. Escher, has based on them many of his works. We shall examine some of these in the following chapters. Meanwhile, let us take a brief look at some other series, several of which mark important milestones in the history of mathematics. We have seen that the harmonic series 1 + 1/2 + 1/3 + 1/4 + … diverges. But the corresponding series with the squares of the natural numbers has baffled mathematicians for many years; among them were several of the Bernoulli brothers, who all failed to find its sum, although it had been known for some time that the series converges.¹ It was the great Swiss mathematician Leonhard Euler (1707–1783) who finally solved the mystery.

To appreciate Cantor’s revolutionary ideas about the infinite, we must first make a brief excursion into the history of the number concept. The simplest type of numbers are, of course, the counting numbers 1, 2, 3,…. Mathematicians prefer to call them the natural numbers, or again the positive integers. Simple though they are, these numbers have been the subject of research and speculation since the dawn of recorded history, and many civilizations have assigned to them various mystical properties. An important branch of modern mathematics, number theory, deals exclusively with the natural numbers, and some of the most fundamental questions about them—for example, questions relating to the prime numbers—are without answer to this day. But without reservation, the single most important property of the natural numbers is this—there are infinitely many of them. The fact that there is no last counting number seems so obvious to us that we hardly bother to reflect upon its consequences. The entire system of calculations with numbers—our familiar rules of arithmetic—would have colapsed like a house of cards if there were a last number beyond which nothing else existed. Suppose for a moment that such a number did exist, say 1,000.

The discovery of these “holes” is attributed to Pythagoras, founder of the celebrated Greek school of mathematics and philosophy in the sixth century B.C. The life of Pythagoras is shrouded in mystery, and the little we know about him is more legend than fact. This is partially due to an absence of documents from his time, but also because the Pythagoreans formed a secret society, an order devoted to mysticism, whose members agreed upon strict codes of communal life. There is some doubt whether many of the contributions attributed to Pythagoras were indeed his own, but there is no question that his teaching has had an enormous influence on the subsequent history of mathematics, an influence which lasted for more than two thousand years. His name, of course, is immortally associated with the theorem relating the hypotenuse of a right triangle to its two sides, even though there is strong evidence that the theorem had already been known to the Babylonians and the Chinese at least a thousand years before him. The theorem says that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two sides: c² = a² + b² (Fig. 8.1). The Pythagorean Theorem is probably the most well know, and certainly the most widely used theorem in all of mathematics, and it appears, directly or in disguise, in almost every branch of it.

Georg Cantor was born in St. Petersburg (now Leningrad) on March 3, 1845. His parents had emigrated from Denmark; his father was a converted Protestant and his mother a born Catholic, but according to some evidence both were of Jewish origin. It may be that this multicultural background played a role in Cantor’s early interest in medieval theological arguments, particularly those concerning continuity and the infinite. From St. Petersburg the family moved to Germany, and it was there, at the University of Halle, that in 1874 he published his first significant work on the concept of infinity. This was only the first of a series of works to be published between 1874 and 1884, and it at once changed the entire foundation on which the concept had been based thus far.

A function may be thought of as a transformation, or “mapping,” from the x-axis to the y-axis, both of which are one-dimensional sets of points. In higher mathematics we also deal with transformations from a two-dimensional set of points to another two-dimensional set, that is, from one plane to another. One of the most interesting transformations of this kind is the transformation of inversion, or more precisely, inversion in the unit circle. Given a circle with center O and radius 1, a point P whose distance from O is OP = r is “mapped” to a point Q, lying on the same ray from O as P, whose distance from O is OQ = 1/r (Fig. 12.1). In this way, a one-to-one correspondence is established between the points of the original plane and those of the new plane: every point of the one plane is mapped onto a point of the other.¹ There is only one exception to this rule: the point O itself.

It is, of course, not just for the sake of beauty that mathematicians study inversion, for the subject turns up in many different branches of science, sometimes quite unexpectedly. We will discuss here one such case—the role inversion plays in cartography, the science of map making.

But let us return to ordinary geometry. Among the host of geometric figures around us, the regular polygons have always played a special role. A polygon (from the Greek words polys = many and gonon = angle) is a closed planar figure made up of straight line segments. A regular polygon is a polygon whose sides and angles are all equal. The simplest regular polygon is the equilateral triangle; next comes the square, followed by the pentagon, the hexagon, and so on. As we saw in Chapter 1, the Greeks were particulary interested in these regular polygons and used them to find an approximation for the number π. They knew, of course, that there exist infinitely many of these polygons; that is, for any given integer n ≥ 3, there exists a regular polygon having n sides—an “n-gon”, as mathematicians say.

We close Part II with an examination of two of the most revolutionary developments in modern mathematics—both directly related to infinity. The first of these, the creation of projective geometry, takes us back to the Renaissance, and it has its roots not in science but in art. During the Middle Ages, both science and art were subordinated to the religious and mythological beliefs of the time. Nature was depicted not as she really was, but as the observer’s fantasies and religious beliefs wanted her to be. Thus, the world believed in a sun that moved around the earth, not because the available evidence, based on an objective observation of the heavens, made such a conclusion inevitable, but because the Roman Catholic church decreed that it must be so. The earth itself was flat—despite mounting evidence to the contrary—because to believe in a round earth meant to let the poor creatures on the “other side” plunge into the abyss of infinite space. And a painter depicted his saints and heroes not in their natural perspective—that is, faraway figures appearing smaller than nearby ones—but according to their status in the Church hierarchy.

If projective geometry, through its principle of duality, has enormously enriched mathematics from an aesthetic point of view, the creation of non-Euclidean geometry has had an unparalleled intellectual impact on our entire scientific and philosophical thought. It marked the first serious doubt since Euclid’s time as to the validity of our fundamental mathematical premises; it shattered the age-old belief in the power of mathematics to show us the road to the ultimate and absolute truth; and it brought about a whole reexamination of our ability to understand the physical world in which we live. This reexamination has had consequences which far transcended mathematics; ultimately, it led to the theory of relativity and helped in shaping our modern views of the universe. The spark that ignited this intellectual revolution was once again the infinite; more specifically, the question: What happens to parallel lines very far away?

From times immemorial, man has aspired to the infinite. The first known attempt to reach infinity occurred in Babel and is told in Genesis: “And they said: ‘Come, let us build a city, and a tower, with its top in heaven.’” Their attempt was doomed to fail, for God, fearing that their aspirations may be too high, “confounded their language, that they may not understand one another’s speech” (Genesis 11:4). Ever since, the Tower of Babel has become an allegory of the infinite—or of man’s futile efforts to reach it.

At one time a famous mathematical curiosity, later a source of inspiration for artists, the strange properties of the Möbius strip have fascinated professionals and laymen alike ever since its discovery in 1865. Named after its creator, the German mathematician and astronomer August Ferdinand Möbius (1790–1868), it was the embryo of an entirely new branch of mathematics known as topology, the study of those properties of a surface which remain invariant when the surface undergoes a continuous deformation.

Everyone has, at one time or another, been fascinated by mirrors. Even animals. I remember once watching a cat staring with puzzlement at a mirror, no doubt wondering who that other fellow cat behind the shiny surface was. The laws of optics play here a subtle trick: a ray of light falling on a mirror is reflected at exactly the same angle at which it hit the mirror¹, creating the illusion that a hidden object appears from behind the mirror (Fig. 19.1).

We now arrive at one of the most beautiful applications of mathematics to art—a study of the various possibilities of filling the plane with infinite repetitions of a single artistic design. Such infinite patterns have captured the imagination of artists and craftsmen since the earliest recorded time and have provided the framework for the exquisite abstract art of the Moslems. It is in these designs that the interplay between geometry and art reaches its supreme level.

You will not find his name in many art books, for he was largely ignored by the art community. His pictures do not adorn the walls of the world’s great museums, for he loathed publicity. If you wish to see his art, look for it in books on mathematics or physics, for he felt a closer kinship to the world of science than to his fellow artists. Unknown but to a few throughout most of his lifetime, he suddenly rose to fame during the last fifteen years of his life, but his genius was not universally recognized until after his death. For if there has ever been an artist who depicted the mathematical curiosities of the world around us, it was Maurits Cornelis Escher.

During the twelfth century there evolved in central Europe a mystic movement of Jewish devotees, the kabbalists, whose belief in the transcendence of God led them to the Ein Sofi the infinite. According to the Kabbalah (in Hebrew: “tradition”), God is revealed to man only through His virtues and deeds, never directly as Himself; the many references to God in the scriptures are only allusions to His manifestations. In their search for spiritual fulfillment, the kabbalists were seeking a path to the divine spirit, if not to God Himself. This they achieved through a system of ten sephirot (literally: “spheres,” and also “enumerations”), emanating from the Ein Sof (“endless”), which became the symbol for the hidden God (Fig. 22.1). The upper sphere, the one closest to the Ein Sof, was called the “crown”; next came the spheres of “wisdom,” “intelligence,” “mercy,” and so on down to the lowest sphere, the “kingdom.” It is only through the ten sephirot, according to the kabbalists, that one can approach the divine spirit; perhaps we can find here a subtle reference to the mathematical idea of limit, of an infinite series whose sum we can only approach, never reach. The kabbalists depicted their sephirot in various geometric forms, as in Fig. 22.1.

From the dawn of recorded history, man has watched the skies above him, marveling at their mysteries and wondering about the myriads of stars that seem to be embedded, like tiny gems, in the celestial dome. What are those stars made of? How far are they? What message do they have for us? Questions such as these, inspired by man’s awe at the grandeur of the creation, were the first step in the creation of a science of the heavens, astronomy. It is a paradoxical fact that astronomy, which studies the farthest objects we can think of, was the first discipline of knowledge to become a full-fledged science in the modern sense of the word. Compare this with geology or biology—the disciplines concerned with our planet and its inhabitants—which emerged as true sciences only in the past few centuries. It seems that the farther a mystery lies, the greater the urge to solve it!

It was not that astronomy came to a complete standstill during the Middle Ages. Many Arab and Jewish astronomers, working largely in Spain under the Islamic conquest but also in Persia and Turkey, made extensive observations of the stars and planets and used these observations to refine astronomical tables and almanacs. Even more significantly, these scholars rediscovered many of the Greek works in mathematics and astronomy and translated them into Arabic and thence into Latin. It is mainly through these translations that our knowledge of Greek science became possible. But important as these contributions were, they did not change man’s fundamental picture of the universe. This picture was essentially the Aristotelian-Ptolemaic one, according to which the immovable earth is at the center of a finite universe, made up of spherical shells in which the planets and stars are embedded.

Not quite ten years had passed since Bruno’s tragic death when an event took place that would completely vindicate him and his master, Copernicus. On January 7, 1610, Galileo Galilei (1564–1642), by then already a renowned scientist, aimed his new telescope at the planet Jupiter. To his amazement, he found the planet surrounded by four small objects, which he correctly identified as satellites circling their parent body. He named them the Medicean Stars, in honor of the Medici family in whose service Galileo hoped to be employed. Here, then, was an entire solar system in miniature—a retinue of small bodies circling a large one—and it gave strong, though indirect, support to the theory of a heliocentric system (which even at that time was far from being universally accepted). Even stronger evidence came when Galileo discovered that Venus, a planet closer to the sun than the earth, exhibits phases like the moon—a convincing proof that it must be circling the sun and not the earth.¹ He then directed his telescope (he called it a “spyglass”) at the moon and saw sights never seen before by the human eye—a heavenly body crisscrossed by valleys and mountains, by flat plains, and by “seas”—in short, an imperfect world not unlike our own and a far cry from the perfect crystal spheres of the Greeks.

The scientific community was not the only one to rejoice in the expanding horizons. Philosophers and authors, naturalists and poets—their imagination fired by the new vistas that the telescope has opened before them—now set to work to describe the new cosmology. Indeed, their imagination carried them to realms which even the most powerful telescope could not reach. Echoing Bruno, the English philosopher and poet Henry More (1614–1687) wrote his version of the plurality of worlds:

The Centre of each sever all world’s a Sunne…

About whose radiant crown the Planets runne,

Like reeling moths around a candle light;

These all together, one world, I conceit,

And that even infinite such worlds there be,

That inexhausted Good that God is hight,

A full sufficient reason is to me.

If you go out on a clear, moonless autumn night and look up to the constellation Andromeda, your eyes may catch a glimpse of a faint, diffuse smudge of light. In appearance it cannot compete with some of the more spectacular sights of the sky, such as the surface of the moon or the rings of Saturn, and even a telescope will not reveal much detail: you can make out the outline of an elliptical structure with a central condensed core, but that is about all. Yet before you dismiss this object as insignificant, stop and think: you are looking at the Great Nebula in Andromeda, a sister galaxy to our own Milky Way which, at a distance of 2,000,000 light-years, is the farthest object the unaided human eye can see. To all purpose and extent, you are looking at infinity.¹

In our story of the infinite we have looked mainly at the infinitely large, perhaps because it has received so much attention since Cantor’s pioneering work in the 1880s, perhaps also because there seems to be something about the infinitely large that captures the imagination in a way that the infinitely small cannot. This bias, however, is hardly justified. In the history of mathematics the infinitely small has played a role at least as important as its counterpart on the other extreme of the scale. If nothing else, it lies at the root of the notion of continuity, an idea that goes back all the way to the Greeks, whose philosophers heatedly debated the possibility of endless division. And much later, disguised as the infinitesimal, it would become the cornerstone around which the calculus was developed.¹ In any event, from a purely mathematical point of view the distinction between “large” and “small” is not really as fundamental as it may seem, since we can always use the function y = 1/x (or its two-dimensional equivalent, the transformation of inversion) to change the one into the other.

We have come to the end of our journey to the infinite. It has carried us from the “horror infiniti” of the Greeks, through the exultation in an infinite universe during the Renaissance, up to the mathematical breakthroughs of the nineteenth and early twentieth centuries, which finally demystified infinity and put it on a firm basis. We have also followed man's attempts to reach the infinite physically—from the Tower of Babel to Pioneer 10. And we have seen how artists and poets have depicted the infinite, each in his own way. It is this diversity, I believe, that makes the infinite—or any intellectual venture for that matter—so stimulating. Each of us is entitled to our own infinity.