Many Ionians believed the underlying harmony of the universe to be accessible through observation and experiment, the method that dominates science today. However, Pythagoras employed a very different method. He taught that the laws of Nature could be deduced by pure thoughts. He and his followers were not fundamentally experimentalists.* They were mathematicians. And they were thoroughgoing mystics. According to Bertrand Russell, in a perhaps uncharitable passage, Pythagoras ‘founded a religion, of which the main tenets were the transmigration of souls and the sinfulness of eating beans. His religion was embodied in a religious order, which, here and there, acquired control of the State and established a rule of the saints. But the unregenerate hankered after beans, and sooner or later rebelled.’
* Although there were a few welcome exceptions. The Pythagorean fascination with whole-number ratios in musical harmonies seems clearly to be based on observation, or even experiment on the sounds issued from plucked strings. Empedocles was, at least in part, a Pythagorean. One of Pythagoras’ students, Alcmaeon, is the first person known to have dissected a human body; he distinguished between arteries and veins, was the first to discover the optic nerve and the eustachian tubes, and identified the brain as the seat of the intellect (a contention later denied by Aristotle, who placed intelligence in the heart, and then revived by Herophilus of Chalcedon). He also founded the science of embryology. But Alcmaeon’s zest for the impure was not shared by most of his Pythagorean colleagues in later times.
The Pythagoreans delighted in the certainty of mathematical demonstration, the sense of a pure and unsullied world accessible to the human intellect, a Cosmos in which the sides of right triangles perfectly obey simple mathematical relationships. It was in striking contrast to the messy reality of the workaday world. They believed that in their mathematics they had glimpsed a perfect reality, a realm of the gods, of which our familiar world is but an imperfect reflection. In Plato’s famous parable of the cave, prisoners were imagined tied in such a way that they saw only the shadows of passersby and believed the shadows to be real – never guessing the complex reality that was accessible if they would but turn their heads. The Pythagoreans would powerfully influence Plato and, later, Christianity.
They did not advocate the free confrontation of conflicting points of view. Instead, like all orthodox religions, they practiced a rigidity that prevented them from correcting their errors. Cicero wrote:
In discussion it is not so much weight of authority as force of argument that should be demanded. Indeed, the authority of those who profess to teach is often a positive hindrance to those who desire to learn; they cease to employ their own judgment, and take what they perceive to be the verdict of their chosen master as settling the question. In fact I am not disposed to approve the practice traditionally ascribed to the Pythagoreans, who, when questioned as to the grounds of any assertion that they advanced in debate, are said to have been accustomed to reply ‘The Master said so,’ ‘the Master’ being Pythagoras. So potent was an opinion already decided, making authority prevail unsupported by reason.
The Pythagoreans were fascinated by the regular solids, symmetrical three-dimensional objects all of whose sides are the same regular polygon. The cube is the simplest example, having six squares as sides. There are an infinite number of regular polygons, but only five regular solids. (The proof of this statement, a famous example of mathematical reasoning, is given in Appendix 2.) For some reason, knowledge of a solid called the dodecahedron having twelve pentagons as sides seemed to them dangerous. It was mystically associated with the Cosmos. The other four regular solids were identified, somehow, with the four ‘elements’ then imagined to constitute the world: earth, fire, air and water. The fifth regular solid must then, they thought, correspond to some fifth element that could only be the substance of the heavenly bodies. (This notion of a fifth essence is the origin of our word quintessence.) Ordinary people were to be kept ignorant of the dodecahedron.
In love with whole numbers, the Pythagoreans believed all things could be derived from them, certainly all other numbers. A crisis in doctrine arose when they discovered that the square root of two (the ratio of the diagonal to the side of a square) was irrational, that it cannot be expressed accurately as the ratio of any two whole numbers, no matter how big these numbers are. Ironically this discovery (reproduced in Appendix 1) was made with the Pythagorean theorem as a tool. ‘Irrational’ originally meant only that a number could not be expressed as a ratio. But for the Pythagoreans it came to mean something threatening, a hint that their world view might not make sense, which is today the other meaning of ‘irrational.’ Instead of sharing these important mathematical discoveries, the Pythagoreans suppressed knowledge of the square root of two and the dodecahedron. The outside world was not to know.* Even today there are scientists opposed to the popularization of science: the sacred knowledge is to be kept within the cult, unsullied by public understanding.
* A Pythagorean named Hippasus published the secret of the ‘sphere with twelve pentagons’, the dodecahedron. When he later died in a shipwreck, we are told, his fellow Pythagoreans remarked on the justice of the punishment. His book has not survived.
The Pythagoreans believed the sphere to be ‘perfect’, all points on its surface being at the same distance from its center. Circles were also perfect. And the Pythagoreans insisted that planets moved in circular paths at constant speeds. They seemed to believe that moving slower or faster at different places in the orbit would be unseemly; noncircular motion was somehow flawed, unsuitable for the planets, which, being free of the Earth, were also deemed ‘perfect.’
The pros and cons of the Pythagorean tradition can be seen clearly in the life’s work of Johannes Kepler (Chapter 3). The Pythagorean idea of a perfect and mystical world, unseen by the senses, was readily accepted by the early Christians and was an integral component of Kepler’s early training. On the one hand, Kepler was convinced that mathematical harmonies exist in nature (he wrote that ‘the universe was stamped with the adornment of harmonic proportions’); that simple numerical relationships must determine the motion of the planets. On the other hand, again following the Pythagoreans, he long believed that only uniform circular motion was admissible. He repeatedly found the observed planetary motions could not be explained in this way, and repeatedly tried again. But unlike many Pythagoreans, he believed in observations and experiment in the real world. Eventually the detailed observations of the apparent motion of the planets forced him to abandon the idea of circular paths and to realize that planets travel in ellipses. Kepler was both inspired in his search for the harmony of planetary motion and delayed for more than a decade by the attractions of Pythagorean doctrine.
A disdain for the practical swept the ancient world. Plato urged astronomers to think about the heavens, but not to waste their time observing them. Aristotle believed that: ‘The lower sort are by nature slaves, and it is better for them as for all inferiors that they should be under the rule of a master …. The slave shares in his master’s life; the artisan is less closely connected with him, and only attains excellence in proportion as he becomes a slave. The meaner sort of mechanic has a special and separate slavery.’ Plutarch wrote: ‘It does not of necessity follow that, if the work delight you with its grace, the one who wrought it is worthy of esteem.’ Xenophon’s opinion was: ‘What are called the mechanical arts carry a social stigma and are rightly dishonoured in our cities.’ As a result of such attitudes, the brilliant and promising Ionian experimental method was largely abandoned for two thousand years. Without experiment, there is no way to choose among contending hypotheses, no way for science to advance. The antiempirical taint of the Pythagoreans survives to this day. But why? Where did this distaste for experiment come from?
An explanation for the decline of ancient science has been put forward by the historian of science, Benjamin Farrington: The mercantile tradition, which led to Ionian science, also led to a slave economy. The owning of slaves was the road to wealth and power. Polycrates’ fortifications were built by slaves. Athens in the time of Pericles, Plato and Aristotle had a vast slave population. All the brave Athenian talk about democracy applied only to a privileged few. What slaves characteristically perform is manual labor. But scientific experimentation is manual labor, from which the slaveholders are preferentially distanced; while it is only the slaveholders – politely called ‘gentle-men’ in some societies – who have the leisure to do science. Accordingly, almost no one did science. The Ionians were perfectly able to make machines of some elegance. But the availability of slaves undermined the economic motive for the development of technology. Thus the mercantile tradition contributed to the great Ionian awakening around 600 B.C., and, through slavery, may have been the cause of its decline some two centuries later. There are great ironies here.