The Music of Pythagoras (47 page)

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.
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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.
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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.
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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, from 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.
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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.”
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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.”

While Russell and his colleagues recognized there were questions they could not answer, they preferred to leave them unanswered rather than cling to what they felt were foolish and misleading “answers,” or believe there are “higher” sources of answers:

The pursuit of truth, when it is profound and genuine, requires also a kind of humility which has some affinity to submission to the will of God. The universe is what it is, not what I choose that it should be. Towards facts, submission is the only rational attitude, but in the realm of ideals there is nothing to which to submit.
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Reading that, one cannot avoid the conclusion that Russell was far more ambivalent about the issue of “discovery” versus “invention” than he was willing to admit.

Though he deplored the way mathematics had been “misused” in other areas, Russell believed that what he was insisting philosophy do—utilize logical analysis, adopt methods of science, and try to base its conclusions on impersonal, disinterested observations and inferences—should be applied in all spheres of human activity. This would bring about a decrease of fanaticism and an increase in sympathy and mutual understanding. He attempted, with scant success, to apply logical analysis to fields such as metaphysics, epistemology, ethics, and political theory, making (ironically) what was arguably a “Pythagorean” leap of faith that what seemed to be a good idea in one area of experience would be a good idea in all.

Russell decried yet another aspect of the Pythagorean legacy: The Pythagoreans lived by an ethic that held the contemplative life in high esteem and had bequeathed to the future something he called “the contemplative ideal.” In the fable about the people at the Olympic Games, Pythagoras and his followers were in the third group, those who had come to watch. These “onlookers” celebrated not practical but “disinterested” science—in other words, they were disengaged from the world of buying, selling, and competing, able to view the whole scene with greater objectivity—thinking that their roles as independent observers placed them in a better position on the path of escape from the eternal circle of the transmigration of souls. Russell contrasted this view with a modern set of values that sees the players on the field as superior to mere spectators, and that admires politicians, financiers, and those who govern the state, the “competitors in the game,” above those who keep to the sidelines and watch and make wise observations.

Nevertheless, said Russell, the elevated status of the “gentlemanly
on-looker” who does not dirty his hands has endured, and this began in ancient Croton, was carried forward with the Greek idea of genius, then with the monks and scholars of the church, and later with the academic university life. He criticized all these, including “saints and sages,” who, except for a few activists, had lived on “slave labor,” “or at any rate upon the labor of men whose inferiority is unquestioned.” It is these “gentlemen,” these “spectators at the Games,” he lamented, who have given us pure mathematics, and that contribution has meant, for them, prestige and success in theology, ethics, and philosophy, because pure mathematics is generally regarded as a “useful activity.” Russell did not mention that he himself was one of these gentlemen he was criticizing—literally so, for he was born into the British nobility, studied at the University of Cambridge and became a fellow of Trinity College there, and spent most of his life as an academic and writer. However, he did, certainly, become one of the activists as well.