Chapter 1: First Steps to Infinity
There is no smallest among the small and no largest among the large; But always something still smaller and something still larger.--Anaxagoras (ca. 500-428 B.C.)
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 sad 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, for 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.
Mathematical infinity begins with the Greeks. To be sure, mathematics as a science had already reached quite an advanced stage long before the Greek era, as is clear from such works as the Rhind papyrus, a collection of 84 mathematical problems written in hieratic script and dating back to 1650 B.C.1 But the ancient mathematics of the Hindus, the Chinese, the Babylonians, and the Egyptians confined itself solely to practical problems of daily life, such as the measurement of area, volume, weight, and time. In such a system there was no place for as lofty a concept as infinity, for nothing in our daily lives has to deal directly with the infinite. Infinity had to wait until mathematics would make the transition from a strictly practical discipline to an intellectual one, where knowledge for its own sake became the main goal. This transition took place in Greece around the sixth century B.C., and it thus befell the Greeks to be the first to acknowledge the existence of infinity as a central issue in mathematics.
Acknowledge--yes, but not confront! The Greeks came very close to accepting the infinite into their mathematical system, and they just might have preceded the invention of the calculus by some two thousand years, were it not for their lack of a proper system of notation. The Greeks were masters of geometry, and virtually all of classical geometry--the one we learn in school--was formulated by them. Moreover, it was the Greeks who introduced into mathematics the high standards of rigor that have since become the trademark of the profession. They insisted that nothing should be accepted into the body of mathematical knowledge that could nor be logically deduced from previously established facts. It is this insistence on proof that is unique to mathematics and distinguishes it from all other sciences. But while the Greeks excelled in geometry and brought it to perfection, their contribution to algebra was very meager. Algebra is essentially a language, a collection of symbols and a set of rules by which to operate with these symbols (just as a spoken language consists of words and of rules by which to combine these words into meaningful sentences). The Greeks did not possess the algebraic language, and consequently were deprived of its main advantages--the generality it offers and its ability to express in an abstract way relations between variable quantities. It is this fact, more than anything else, which brought about their horror infiniti, their deeply rooted suspicion of the infinite. "The infinite was taboo," said Tobias Dantzig in his classic work, Number--the Language of Science, "it had to be kept out, at any cost; or, failing this, camouflaged by arguments ad absurdum and the like."
Figure 1.1. The runner's paradox.
Nowhere was this fear of the infinite better manifested than in the famous paradoxes of Zeno, a philosopher who lived in Elea in the fourth century B.C. His paradoxes, or "arguments," as they were called, deal with motion and continuity, and in one of them he proposed to show that motion is impossible. His argument seems quite convincing: in order for a runner to move from one point to another, he must first cover half the distance between the two points, then half of the remaining distance, then half of what remains next, and so on ad infinitum (Fig. 1.1) Since this requires an infinite number of steps, Zeno argued, the runner would never reach his destination. Of course, Zeno knew full well that the runner would reach the end point after a finite lapse of time. Yet he did not resolve the paradox; rather, he left it for future generations. In this at least he was humble, admitting that the infinite was beyond his and his generation's intellectual reach. Zeno's paradoxes had to wait another twenty centuries before they would be resolved.
But while the Greeks were unable to grasp the infinite intellectually, they still made some good use of it. They were the first to devise a mathematical method to find the value of that celebrated number which we denote today by pi, the ratio of the circumference of a circle to its diameter. This number has intrigued laymen and scholars alike since the dawn of recorded history. In the Rhind Papyrus (ca. 1650 B.C.), we find its value to be (4/3)4, or very nearly 3.16049, which is within 0.6% of the exact value. It is indeed remarkable that the ancient Egyptians already possessed such a degree of accuracy. The Biblical value of pi, by comparison, is exactly 3, as is clear from a verse in 1 Kings vii 23: "and he made a molten sea, ten cubits from brim to brim, and his height was five cubits; and a line of thirty cubits did encompass him round about." Thus the error in the Biblical value is more than 4.5% of the true value of pi!
Figure 1.2. Regular inscribed
and circumscribing polygons.
Now all the ancient estimations of pi were essentially empirical--they were based on an actual measurement of the circumference and diameter of a circle. The Greeks were the first to propose a method which would give the value of pi to any degree of accuracy by a mathematical process, rather than by measurement. The inventor of this method was Archimedes of Syracuse (ca. 287-212 B.C.), the great scientist who achieved immortal fame as the discoverer of the laws of floating bodies and of the mechanical lever. His method was based on a simple observation: take a circle and circumscribe it by a series of regular polygons of more and more sides. (In a regular polygon, all the sides and angles are equal.) Each polygon has a perimeter slightly in excess of the circumference of the circle; but as we increase the number of sides, the corresponding polygons will encompass the circle more and more tightly (Fig. 1.2). Thus if we can find the perimeters of these polygons and divide them by the diameter of the circle, we will get a fairly close approximation to pi. Archimedes followed this procedure for polygons having 6, 12, 24, 48, and 96 sides, for which he was able to calculate the perimeters by using known methods. For the 96-sided polygon he found the value 3.14271 (this is very close to the 22/7 approximation often used in school). He then repeated the same procedure with polygons touching the circle from within--inscribed polygons--giving values that are all short of the true value. Again by using 96 sides, Archimedes found the value 3.14103, or very nearly 3-10/71. Since the actual circle is "squeezed" between the inscribed and circumscribed polygons, the true value of pi must be somewhere between these values.
Let there be no misunderstanding: Archimedes' method gave an estimation of pi far better than anything before him. But its real innovation was not in this improved value, but in the fact that it enabled one to approximate pi to any desired accuracy, simply by taking polygons of more and more sides. In principle, there is no limit to the degree of accuracy this method could yield--even though for practical purposes (such as in engineering), the above values are more than adequate. In modern language we say that pi is the limit of the values derived from these polygons as the number of sides tends to infinity. Archimedes, of course, did not mention the limit concept explicitly--to do so would have required him to use the language of algebra--but two thousand years later this concept would be the cornerstone around which the calculus would be erected.
1The papyrus is named after the Scottish Egyptologist A. Henry Rhind, who purchased it in 1858. It is now in the British Museum. See The Rhind Mathematical Papyrus by Arnold Buffum Chace, The National Council of Teachers of Mathematics, Reston, Virginia, 1979.
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