The Music of Pythagoras

by Kitty Ferguson

  Russell was born in 1872. In the years leading up to World War I, he tackled a question that would engage him for most of his life: whether mathematics can be, to a significant degree, reduced to logic, with one true statement implying the next. It is perhaps conventional wisdom that this is precisely the way mathematics works, but to assume so betrays a naive view. The issue is complex, and Russell knew it was. Though his place among academics was more as philosopher than mathematician, in Principles of Mathematics and a three-volume work that he co-authored with Alfred North Whitehead, Principia Mathematica, his goal was to re-found mathematics on logic alone.2 There is nothing anti-Pythagorean about faith in mathematical logic. It was on other issues that Russell took on both Pythagoras and Plato.

  Vehemently rejecting the idea that humans have any grounds for discussion of an ideal world beyond what can be extrapolated in a reasonable manner from what we experience with our five senses, Russell was convinced that “what appears as Platonism is, when analyzed, found to be in essence Pythagoreanism.” It was from Pythagoras that Plato got the “Orphic elements” in his philosophy, “the religious trend, the belief in immortality, the other-worldliness, the priestly tone, all that is involved in the simile of the cave, his respect for mathematics, and his intimate intermingling of intellect and mysticism.” Russell blamed Pythagoras for what he saw as Plato’s view that the realm of mathematics was a realm that was an ideal, of which everyday, sense-based, empirical experience would always fall short.

  Russell’s chapter on Pythagoras was part of a hefty tome of nearly nine hundred pages, his 1945 History of Western Philosophy. He wrote it to appeal to a wide, nonacademic readership, but it was no innocent survey without an agenda. His fascination with language, with analyzing it down to its minimum requirements, transforming sentences into equations to wring from them the most trimmed-down, unmistakable message possible, had made him a master at the manipulation of language, and—it must be said—the manipulation of readers. Careless reader he sometimes was, and sometimes careless thinker, but hardly ever careless writer. His chapter about Pythagoras is peppered with tongue-in-cheek understatements, making it easy to miss the fact that he intended this clever, seductive, amusing prose to undermine not only some of the prized tenets of the mathematical sciences but also belief in God.

  The book traced philosophy from Thales to himself, and Russell tried to show how this long history had culminated in, and finally found a corrective in, his own philosophy. In this context, he did not treat Pythagoras as just one more philosopher in the table of contents. The book’s final paragraph, long past the chapter devoted entirely to Pythagoras, states: “I do not know of any other man who has been as influential as he was in the sphere of thought.” The co-author of Principia Mathematica, Alfred North Whitehead, also believed Pythagoras’ influence had been tremendous, the very bedrock of European philosophy and mathematics.

  Russell agreed with those who thought that Pythagoras was the first to use mathematics as “demonstrative deductive argument,” rather than merely a practical tool of commerce and measurement. This, he thought, made Pythagoras a founding father of the line of mathematical thinking that would lead to all of modern mathematics including his own. “Pythagoras was intellectually one of the most important men that ever lived, both when he was wise and when he was unwise,” Russell wrote. “Unwise” referred to the fact that Pythagoras and Pythagoreanism seemed to Russell also to have had a mystical side, and when that encouraged Plato to introduce the Forms, the inheritance went sour.

  Just as other sciences had their roots in false beliefs—astronomy in astrology; chemistry in alchemy—mathematics, wrote Russell, had begun with “a more refined type of error,” the belief that although mathematics is certain, exact, and applicable to the real world, it nevertheless can be done by thought alone with no need to observe the real world. He had a point. Think of the ten-body cosmos. Even though the Pythagoreans discovered the ratios of musical harmony by listening (one of the senses) and observing where they were putting their fingers on the strings of the lyre (involving both sight and touch), they proceeded in an unfortunate way that involved trusting thought, not checked by observation. What Russell insisted had emerged as a result was a view of the realm of mathematics as an ideal from which sense-based, empirical knowledge would always fall short. Once that was in the air, lamented Russell, goodbye to the idea that observation of the real world was a useful guide to truth.

  Plato, as interpreted by Russell, had believed that anyone on a quest for truth had to reject all empirical knowledge and regard the five senses as untrustworthy, even false witnesses. Absolute justice, absolute beauty, absolute good, absolute greatness, absolute health, “the essence and true nature of everything”—the only way to reach that level of knowledge was, Plato had Socrates say, by means of “the mind gathered into itself.”3 Actually, there is no record of Pythagoras, or pre-Platonic Pythagoreans, insisting that truth about the universe must be discovered by thought alone, but, to Russell’s mind—although it was Plato who articulated the idea—its source was the Pythagoreans; it was implicit in the way they thought and the conclusions they reached. Russell was convinced that the idea of the superiority of thought and intellect over direct sense observation of the world would not have emerged at all had it not been for the combination of the Pythagorean view of numbers and Plato’s idea of Forms, which together created an unfortunate legacy that endures to the present and that has motivated people to look for ways of coming closer to what they saw as the mathematician’s ideal. “The resulting suggestions were the source of much that was mistaken in metaphysics and theory of knowledge. This form of philosophy begins with Pythagoras.”

  Bertrand Russell

  Having read Plato, one must take issue. He did not think of numbers and mathematics as Forms or “ideals” at all—not even as a sure path to discovering them. In his creation of the world-soul in his Timaeus, for example, and when Socrates taught about “recollection” in the Meno by drawing the square and the isosceles triangle for the untutored slave boy, mathematics for Plato was a way of reaching out toward the ultimate level of knowledge, toward the Forms, of trying to get there. It does not appear, in these passages, that Plato thought he was there or that numbers and mathematics were going to get him there. His pupils later thought of numbers as on the level of Forms, but even they did not necessarily believe human thinkers could reach that level of mathematics.

  Russell had another objection to Pythagoras. The Pythagorean insight that numbers and number relationships underlie all of nature—not created or invented by humans but discovered by them—was, he believed, a false vision and an enormous and tragic misstep in the history of human thought. Following that Pythagorean fantasy, mathematics was doomed always to have in it “an element of ecstatic revelation.” “Revelation” was, for Russell, an impossible concept. He wrote that those mathematicians who have “experienced the intoxicating delight of sudden understanding that mathematics gives, from time to time,” find the Pythagorean view “completely natural even if untrue.” In this he was ignoring the fact that neither the Pythagoreans nor any major mathematician from the late sixteenth century on, not even the ecstatically religious Kepler, ever claimed to have received a mathematical “revelation.” But Russell equated “discovery” of truth with “revelation,” and “revelation” with “illusion.” With that equation in mind, what seemed to be the discovery of the underlying level of mathematical reality equaled a leap of faith to a false “ideal world.” And, according to Russell, that idea had been foisted off on a gullible future.

  Russell nailed all this down by attributing to the “delighted mathematicians” a different idea (though many mathematicians would disagree with it): that mathematics is something created by mathematicians in the same way that music is something created by composers. This could have been an insightful parallel, had Russell followed up on it: From a background having to do with which tones and meters are possible, which sounds are
pleasant and which not—and much else that one might discover about hearing, sounds, and their effect on human emotions—a composer is still left with a vast number of choices. The result depends on the composer’s creativity and inventiveness in using basic, unchangeable material. Perhaps from a background of true mathematical possibilities, a mathematician likewise has a vast number of choices. Even if the uncharted territory one is exploring is not subject to choice or invention, the trails leading into it and across it are a matter of choice and creativity.

  Russell had something else in mind. He was opting for a different philosophy of mathematics, that mathematics is a human construction to impose logical order on the universe or draw a map through territory that is not inherently mathematical at all. He laid twofold blame on Pythagoras: first, for the Platonic idea that there is a realm not perceptible to human senses but perhaps to human intelligence, and, second, for the belief that mathematicians were discovering mathematical truth, not inventing it. Because numbers are eternal, not existing in time, it was possible to conceive of numbers and mathematics as “God’s thoughts,” and just there, said Russell, rooted in Pythagoreanism, was Plato’s idea that God is “a geometer.” A sort of “rational” religion had come to dominate mathematics and mathematical method.

  Russell was willing to concede one positive outcome from the Pythagorean doctrine of a universe undergirded with rationality and mathematical order: It had led people to be dissatisfied with movements in the heavens that were irregular and complicated, as they appear to a naive observer. Such a messy situation was not “what a Pythagorean creator would have chosen,” and that puzzle had led astronomers like Ptolemy, and later Copernicus and Kepler, to propose systems that an orderly designer would have preferred.

  Russell wrote The History of Western Philosophy before the discovery of the scribal tablets that showed that the “Pythagorean” theorem was known long before Pythagoras. Justifiably, he was confident in calling the Pythagorean theorem the “greatest discovery of Pythagoras.” He sympathized with the misfortune of the Pythagoreans, the discovery of incommensurability. He had reason to be sympathetic, for during his lifetime several discoveries occurred that seemed to undermine his own efforts, in the same way that the discovery of incommensurability had traditionally undermined Pythagorean faith that the world was based on rational numerical relationships. One of the discoveries was “Russell’s paradox.” He was trying to set mathematics on a better track by seeking to found it on logic, with one true mathematical statement implying the next. However, a true statement sometimes implies more than one next statement. Sometimes it implies two statements that contradict one another.* That paradox was no trivial snag. Russell wrote a letter about it to the German mathematician and logician Gottlob Frege, who received it as he was completing the second volume of a treatise on the logical foundations of arithmetic that had taken twelve years of painstaking work. Frege responded by adding the following sad words to his book:

  A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter from Mr. Bertrand Russell as the work was nearly through the press.4

  Russell spent some time in his chapter on Pythagoras considering the problem of incommensurability. He thought that the square root of 2, being the simplest form of the problem, was the “first irrational number to be discovered” and that it was known to early Pythagoreans who had found the following ingenious method for approximating its value.* Suppose you have drawn an isosceles triangle, the one Plato used in his Meno, which contains the problem of incommensurability. Russell thought it was while studying this triangle that the Pythagoreans came upon the problem, so let us follow his thinking.

  First, review the problem. The Pythagorean theorem says that the square of Side A plus the square of Side B will equal the square of Side C. Say that Side A measures 1 inch. Side B also measures 1 inch. The square of 1 is 1. So the square of Side A plus the square of Side B (1 + 1) equals 2. If the Pythagorean theorem is correct, the square of Side C must likewise be 2, but what is the length of Side C? You cannot find out if you cannot calculate the square root of 2. Here is how Russell suggested the Pythagoreans might have approximated it:

  Make two columns: Column A and B, and let each begin with the number 1.

  A B

  1 1

  To get the next pair of numbers:

  For Column A, add the first A and B (1 + 1).

  For Column B, double the first A and add the first B (2 + 1)

  A B

  1 1

  2 3

  Continue using the same method of getting the next pair of numbers, always using the two previous numbers as your “former A and B,” and you soon have:

  A B

  1 1

  2 3

  5 7

  12 17

  29 41

  70 99

  For each pair the following is true: 2A squared minus B squared equals either 1 or minus 1. In each case, B divided by A is close to the square root of 2, and the farther down the chart you move, the closer it is to the square root of 2, though it never quite gets there because the square root of 2 is not a rational number. Would this have satisfied the Pythagoreans? One cannot help thinking that for people who believed they had found complete rationality and simplicity in the universe, it would have been poor consolation.

  Russell in great part credited Pythagoras with linking philosophy with geometry and mathematics, with the result that geometry and mathematics had been an influence on philosophy and theology ever since—an influence Russell regarded as “both profound and unfortunate.” In geometry, as Euclid and other Greeks established it, and as it is still taught today, one does not begin in a void, thinking nothing true unless proved. There are statements that are not proved but are “self-evident” (or at least seem to be), called axioms. Some bit of self-evident truth must be there as the starting place. That may seem a shaky foundation to build on, but many generations have managed to accept it and proceed. Beginning with the axioms, the next step is to use deductive reasoning to arrive at things that may not be at all self-evident, called theorems. Axioms and theorems are supposed to be true about actual space; they are something that could be experienced. In other words, by taking something self-evident and using deductive thinking it is possible to discover things that are true of the actual world.

  Russell had no argument with this line of thinking in geometry. His regret was that it been applied to other areas. The American Declaration of Independence, for example, declared, “We hold these truths to be self-evident,” on the assumption that there are, indeed, things having nothing to do with geometry or mathematics that are so clearly true that no sane person would question them. The words “self-evident” were one of Benjamin Franklin’s changes in the draft of the Declaration. Thomas Jefferson had written, “We hold these truths to be sacred and undeniable,” a less down-to-earth version of the same idea. The point was that everyone could proceed from there without looking back. But could they?

  Russell was not really trying to undermine Franklin, but he was disgruntled that the process by which geometry is done had been co-opted not only by brilliant rebels but by theologians. Thomas Aquinas had used it in arguments for the existence of God. His arguments did not start from nothing, but rather from “first principles.” In fact, what Aquinas meant by “science” was a body of knowledge that has “first principles” or “givens.” Again, Russell blamed the Pythagoreans: “Personal religion is derived from ecstasy, theology from mathematics; and both are to be found in Pythagoras.” The Pythagorean marriage of mathematics and theology had polluted the religious philosophy of Greece, then the Middle Ages, and so on through Immanuel Kant and beyond. In his essay “How to Read and Understand History,” Russell lamented,

  There was a serpent in the philosophic paradise, and his name was Pythagoras. From Pythagoras this outlook descended to Plato, from Plato to Christian theologians, f
rom them, in a new form, to Rousseau and the romantics and the myriad purveyors of nonsense who flourish wherever men and women are tired of the truth.5

  Russell identified some characteristics of what he saw as a blending of religion and reasoning, of “moral aspiration with logical admiration of what is timeless,” in Plato, Augustine, Thomas Aquinas, Descartes, Spinoza, and Kant. Their offenses were belief in insight or intuition as a valid route to knowledge, a route distinct from analytic intellectual processes; denial of the reality of time and the passage of time in the ultimate scheme of things; belief in a unity of all things and a resistance to any fragmentation of our knowledge of the world. This “philosophical mysticism”—a term used not by Russell but coined by the physicist John Barrow—according to Russell “distinguished the intellectualized theology of Europe from the more straightforward mysticism of Asia.”6 However, he believed it was a much earlier form of Eastern mysticism that had entered, through Orphism, into Pythagoreanism, in which fertile ground it had taken root to develop into the intellectualized but still partly mystical theology of Europe.

  Russell was not a lone voice. He was one of the founders of a school of thought called logical analysis, an effort “to eliminate Pythagoreanism from the principles of mathematics,” ridding it of “mysticism” and “metaphysical muddles.” He and those who joined him in this movement refused to indulge in what they saw as “falsification of logic to make mathematics appear mystical, and the practice of passing off, as authentic intuitions of reality, prejudices about what is real.” Russell also tried to put logic to work in an attempt to clarify issues in philosophy, making “logical analysis the main business of philosophy,” rejecting any notion that moral considerations have a place in philosophy or that philosophy might either prove or disprove the truth of religious doctrine. Philosophy, stripped of its “dogmatic pretensions,” would nevertheless “not cease to suggest and inspire a way of life.”