When the 1825 revolution failed, five leaders were hanged and the others exiled to Siberia. Perhaps there was consolation in recalling that Pythagoras, their iconic ancient model, had – at least in the mythology they thought they knew – been forced to flee in ignominy from a city he had tried to introduce to a better way of life.
Beyond revolutionary circles, other literature of the nineteenth century remembered Pythagoras. The poet Percy Bysshe Shelley wrote a piece praising the vegetarian ‘Pythagorean Diet’, and Leo Tolstoy chose to follow it. Louisa May Alcott knew her readers would need no explanation when she wrote in Jo’s Boys that ‘Grandpa March cultivated the little mind with the tender wisdom of a modern Pythagoras, not tasking it with long, hard lessons, parrot-learned, but helping it to unfold as naturally and beautifully as sun and dew help roses bloom.’ Honoré de Balzac attributed the saying ‘no man is known until he dies’ to Pythagoras. Pythagoras was one of the ghosts present in Charles Dickens’ The Haunted House and also made an appearance in The Pickwick Papers.
Also in the nineteenth century, the belief continued that the concept of the mathematical structure of the universe had originated with the Pythagoreans. The economist William Stanley Jevons wrote: ‘Not without reason did Pythagoras represent the world as ruled by number. Into almost all our acts of thought number enters, and in proportion as we can define numerically we enjoy exact and useful knowledge of the universe.’11
One Pythagorean ideal began to come into its own in a way it could not have done earlier. The assumption that there was unity to the universe had already become one of the pillars on which science rested, but not until the nineteenth century did the knowledge and the instruments begin to be available that would allow scientists to explore the question whether this assumption was valid, or whether, like the music of the spheres, it was best relegated to the realm of poetic metaphor. The idea that there is unity to nature emerged strongly in the work of three men who had a particularly significant impact on the future of science.
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 towards 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.
[1]The three-man cell was used again in Vietnamese communism, in Algeria in the 1950s, and in the USSR in the late 1960s.
CHAPTER 18
Janus Face
Twentieth Century
In 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 analysed, 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 analysing 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 b
een 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.
Bertrand Russell
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.’
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 towards the ultimate level of knowledge, towards 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 equalled 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 sympathised 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.[1] 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.[2] 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.
Pythagorus Page 34