The Music of Pythagoras (46 page)

When the Danish physicist and chemist Hans Christian Oersted wrote his doctoral thesis about a book by Immanuel Kant called
The Metaphysical Foundations of Knowledge
, he was already convinced that all experience could be accounted for by a correct understanding of the forces of nature, and that the forces of nature were actually not many
forces but one. Kant had suggested there were two basic forces, but Oersted decided to push forward with the certainty that light, heat, chemical affinity, electricity, and magnetism were all different faces of “one primordial power.” In 1820 he discovered electromagnetism, having “adhered to the opinion, that the magnetical effects are produced by the same powers as the electrical . . . not so much led to this by the reasons commonly alleged for this opinion, as by the philosophical principle, that all phenomena are produced by the same original power.”

Michael Faraday was another early-nineteenth-century scientist who undertook a lifelong search for ways in which the forces of nature are unified. He began his professional life as a chemist and discovered several new organic compounds. As had been true of Linnaeus’ numerous previously unknown species, those discoveries might have been taken to indicate a
lack
of unity, but instead they expanded awareness of what was out there to be unified. A tally of Faraday’s most notable contributions included producing an electric current from a magnetic field, showing the relationship between chemical bonding and electricity, and discovering the effect of magnetism on light.

Michael Faraday

Faraday’s work was the experimental foundation—and also a large part of the theoretical foundation—for the work of James Clerk Maxwell later in the century. Maxwell’s electromagnetic field theory achieved the full unification of electricity and magnetism. The “electromagnetic force” would enter the twentieth century as one of four basic forces of nature. Maxwell’s equations, based in turn on Faraday’s study of electric and magnetic lines of force, would also be instrumental in setting a scientific trajectory toward the linking of mass and energy in Einstein’s special theory of relativity. Science at the turn of the twentieth century was well on the way to finding the unity of nature that Pythagoreans had so fervently believed in. Paradoxically, Maxwell’s work also provided a vision of reality with problems that would be resolved in the twentieth century by quantum theory. And quantum theory, in its turn, would cause a crisis of faith in the rationality of the universe, a crisis on a scale with that perhaps caused by the ancient Pythagorean discovery of incommensurability.

Twentieth Century

I
N THE TWENTIETH CENTURY
, two major books appeared that highlighted humanity’s debt to Pythagoras and the Pythagoreans. “Debt to Pythagoras” might seem to imply that there is something positive for which to thank Pythagoras and his followers, and one of the authors, Arthur Koestler, certainly believed there was. Bertrand Russell, on the other hand, insisted that most of Pythagoras’ influence had been negative. Their two accounts constitute an excellent example of how taking off one pair of glasses and putting on another can change the view in astounding ways.
¹

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.