Is God a Mathematician? (4 page)

From the perspective of the questions that are at the focus of the present book, once we strip the Pythagorean philosophy of its mystical clothing, the skeleton that remains is still a powerful statement about mathematics, its nature, and its relation to both the physical world and the human mind. Pythagoras and the Pythagoreans were the forefathers of the search for cosmic order. They can be regarded as the founders of pure mathematics in that unlike their predecessors—the Babylonians and the Egyptians—they engaged in mathematics as an abstract field, divorced from all practical purposes. The question of whether the Pythagoreans also established mathematics as a tool for science is a trickier one. While the Pythagoreans certainly associated all phenomena with numbers, the numbers themselves—not the phenomena or their causes—became the focus of study. This was not a particularly fruitful direction for scientific research to take. Still, fundamental to the Pythagorean doctrine was the implicit belief in the existence of general, natural laws. This belief, which has become the central pillar of modern science, may have had its roots in the concept of Fate in Greek tragedy. As late as the Renaissance, this bold faith in the reality of a body of laws that can explain all phenomena was still progressing far in advance of any concrete evidence, and only Galileo, Descartes, and Newton turned it into a proposition defendable on inductive grounds.

Another major contribution attributed to the Pythagoreans was the sobering discovery that their own “numerical religion” was, in fact, pitifully unworkable. The whole numbers 1, 2, 3,…are insufficient even for the construction of mathematics, let alone for a description of the universe. Examine the square in figure 6, in which the length of the side is one unit, and where we denote the length of the diagonal by
d.
We can easily find the length of the diagonal, using the Pythagorean theorem in any of the two right triangles into which the square is divided. According to the theorem, the square
of the diagonal (the hypotenuse) is equal to the sum of the squares of the two shorter sides:
d
²
= 1
²
+ 1
²
, or
d
²
= 2. Once you know the square of a positive number, you find the number itself by taking the square root (e.g., if
x
²
= 9, then the positive
x
= v 9 = 3). Therefore,
d2
= 2 implies
d
= v2 units. So the ratio of the length of the diagonal to the length of the square’s side is the number v2. Here, however, came the real shock—a discovery that demolished the meticulously constructed Pythagorean discrete-number philosophy. One of the Pythagoreans (possibly Hippasus of Metapontum, who lived in the first half of the fifth century BC) managed to prove that the square root of two cannot be expressed as a ratio of any two whole numbers. In other words, even though we have an infinity of whole numbers to choose from, the search for two of them that give a ratio of v2 is doomed from the start. Numbers that can be expressed as a ratio of two whole numbers (e.g., 3/17; 2/5; 1/10; 6/1) are called
rational numbers
. The Pythagoreans proved that v2 is not a rational number. In fact, soon after the original discovery it was realized that neither are v3, v17, or the square root of any number that is not a perfect square (such as 16 or 25). The consequences were dramatic—the Pythagoreans showed that to the infinity of rational numbers we are forced to add an infinity of new kinds of numbers—ones that today we call
irrational numbers.
The importance of this discovery for the subsequent development of mathematical analysis cannot be overemphasized. Among other things, it led to the recognition of the existence of “countable” and “uncountable” infinities in the nineteenth century. The Pythagoreans, however, were so overwhelmed by this philosophical crisis that the philosopher Iamblichus reports that the
man who discovered irrational numbers and disclosed their nature to “those unworthy to share in the theory” was “so hated that not only was he banned from [the Pythagoreans’] common association and way of life, but even his tomb was built, as if [their] former comrade was departed from life among mankind.”

Perhaps even more important than the discovery of irrational numbers was the pioneering Pythagorean insistence on mathematical proof—a procedure based entirely on logical reasoning, by which starting from some postulates, the validity of any mathematical proposition could be unambiguously established. Prior to the Greeks, even mathematicians did not expect anyone to be interested in the least in the mental struggles that had led them to a particular discovery. If a mathematical recipe worked in practice—say for divvying up parcels of land—that was proof enough. The Greeks, on the other hand, wanted to explain why it worked. While the notion of proof may have first been introduced by the philosopher Thales of Miletus (ca. 625–547 BC), the Pythagoreans were the ones who turned this practice into an impeccable tool for ascertaining mathematical truths. The significance of this breakthrough in logic was enormous. Proofs stemming from postulates immediately put mathematics on a much firmer foundation than that of any other discipline discussed by the philosophers of the time. Once a rigorous proof, based on steps in reasoning that left no loopholes, had been presented, the validity of the associated mathematical statement was essentially unassailable. Even Arthur Conan Doyle, the creator of the world’s most famous detective, recognized the special status of mathematical proof. In
A Study in Scarlet,
Sherlock Holmes declares that his conclusions are “as infallible as so many propositions of Euclid.”

On the question of whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt—mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the feeble human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician—
mathematics was God!

The importance of the Pythagorean philosophy lies not only in its actual, intrinsic value. By setting the stage, and to some extent the
agenda, for the next generation of philosophers—Plato in particular—the Pythagoreans established a commanding position in Western thought.

Into Plato’s Cave

The famous British mathematician and philosopher Alfred North Whitehead (1861–1947) remarked once that “the safest generalization that can be made about the history of western philosophy is that it is all a series of footnotes to Plato.”

Indeed, Plato (ca. 428–347 BC) was the first to have brought together topics ranging from mathematics, science, and language to religion, ethics, and art and to have treated them in a unified manner that essentially defined philosophy as a discipline. To Plato, philosophy was not some abstract subject, divorced from everyday activities, but rather the chief guide to how humans should live their lives, recognize truths, and conduct their politics. In particular, he maintained that philosophy can gain us access into a realm of truths that lies far beyond what we can either perceive directly with our senses or even deduce by simple common sense. Who was this relentless seeker of pure knowledge, absolute good, and eternal truths?

Plato, the son of Ariston and Perictione, was born in Athens or Aegina. Figure 7 shows a Roman herm of Plato that was most likely copied from an older, fourth century BC Greek original. His family had a long line of distinction on both sides, including such figures as Solon, the celebrated lawmaker, and Codrus, the last king of Athens. Plato’s uncle Charmides and his mother’s cousin Critias were old friends of the famous philosopher Socrates (ca. 470–399 BC)—a relation that in many ways defined the formative influence to which the young Plato’s mind was exposed. Originally, Plato intended to enter into politics, but a series of violent actions by the political faction that courted him at the time convinced him otherwise. Later in life, this initial repulsion by politics may have encouraged Plato to outline what he regarded as the essential education for future guardians of the state. In one case, he even attempted (unsuccessfully) to tutor the ruler of Syracuse, Dionysius II.

Figure 7

Following the execution of Socrates in 399 BC, Plato embarked on extensive travel that ended only when he founded his renowned school of philosophy and science—the Academy—around 387 BC. Plato was the director (or
scholarch
) of the Academy until his death, and his nephew Speusippus succeeded him in that position. Unlike academic institutions today, the Academy was a rather informal gathering of intellectuals who, under Plato’s guidance, pursued a wide variety of interests. There were no tuition fees, no prescribed curricula, and not even real faculty members. Still, there was apparently one rather unusual “entrance requirement.” According to an oration by the fourth century (AD) emperor Julian the Apostate, a burdensome inscription hung over the door to Plato’s Academy. While the text of the inscription does not appear in the oration, it can be found in another fourth century marginal note. The inscription read: “Let no one destitute of geometry enter.” Since no fewer than eight centuries separate the establishment of the Academy and the first description of the inscription, we cannot be absolutely certain that such an inscription indeed existed. There is no doubt, however, that the sentiment expressed by this demanding requirement reflected Plato’s per
sonal opinion. In one of his famous dialogues,
Gorgias
, Plato writes: “Geometric equality is of great importance among gods and men.”

The “students” in the Academy were generally self-supporting, and some of them—the great Aristotle for one—stayed there for as long as twenty years. Plato considered this long-term contact of creative minds to be the best vehicle for the production of new ideas, in topics ranging from abstract metaphysics and mathematics to ethics and politics. The purity and almost divine attributes of Plato’s disciples were captured beautifully in a painting entitled
The School of Plato
by the Belgian symbolist painter Jean Delville (1867–1953). To emphasize the spiritual qualities of the students, Delville painted them in the nude, and they appear to be androgynous, because that was supposed to be the state of primordial humans.

I was disappointed to discover that archaeologists were never able to find the remains of Plato’s Academy. On a trip to Greece in the summer of 2007, I looked for the next best thing. Plato mentions the Stoa of Zeus (a covered walkway built in the fifth century BC) as a favorite place to talk to friends. I found the ruins of this stoa in the northwest part of the ancient agora in Athens (which was the civic center in Plato’s time; figure 8). I must say that even though the temperature reached 115 °F that day, I felt something like a shiver as I
walked along the same path that must have been traversed hundreds, if not thousands of times by the great man.

Figure 8

The legendary inscription above the Academy’s door speaks loudly about Plato’s attitude toward mathematics. In fact, most of the significant mathematical research of the fourth century BC was carried out by people associated in one way or another with the Academy. Yet Plato himself was not a mathematician of great technical dexterity, and his direct contributions to mathematical knowledge were probably minimal. Rather, he was an enthusiastic spectator, a motivating source of challenge, an intelligent critic, and an inspiring guide. The first century philosopher and historian Philodemus paints a clear picture: “At that time great progress was seen in mathematics, with Plato serving as the general architect setting out problems, and the mathematicians investigating them earnestly.” To which the Neoplatonic philosopher and mathematician Proclus adds: “Plato…greatly advanced mathematics in general and geometry in particular because of his zeal for these studies. It is well known that his writings are thickly sprinkled with mathematical terms and that he everywhere tries to arouse admiration for mathematics among students of philosophy.” In other words, Plato, whose mathematical knowledge was broadly up to date, could converse with the mathematicians as an equal and as a problem presenter, even though his personal mathematical achievements were not significant.