Pythagoras and Pythagoreans

Pythagoras and Pythagoreans

Pythagoras

The great scientists from Thales to Democritus and Anaxagoras have usually been described in history of philosophy books as “Prescoratics”, as if their main function was to hold the philosophical fort until the advent of Socrates, Plato, and Aristotle and perhaps influence them a little. Instead, the old Ionians represent a different and largely contradictory tradition, one in much better accord with modern science. That their influence was felt powerfully for only two or three centuries in as irreparable loss for all those human beings who lived between the Ionian Awakening and the Italian Renaissance.

Perhaps the most influential person ever associated with Samos was Pythagoras, a contemporary of Polycrates in the sixth century B.C. In fact, the sixth century B.C. was a time of remarkable intellectual and spiritual ferment across the planet. Not only was it the time of Thales, Anaximander, Pythagoras and others in Ionia, but also the time of the Egyptian Pharaoh Necho who caused Africa to be circumnavigated, of Zoroaster in Persia, Confucius and Lao-tse in China, the Jewish prophets in Israel, Egypt and Babylon, and Gautama Buddha in India. It is hard to think these activities altogether unrelated.

According to local tradition, Pythagoras lived for a time in a cave on the Samian Mount Kerkis, and was the first person in the history of the world to deduce that the Earth is a sphere. Perhaps he argued by analogy with the Moon or the Sun, or noticed the curve shadow of the Earth on the Moon during a lunar eclipse, or recognized that when ships live Samos and reside over the horizon, their masts disappear last.

Pythagoras world. Photo by Elena

He or his disciples discovered the Pythagorean theorem : the sum of the squares or the shorter sides of a right triangle equals the square of the longer side. Pythagoras did not simply enumerate examples of this theorem; he developed a method of mathematical deduction to prove the thing generally.

The modern tradition of mathematical argument, essential to all of science, owes much to Pythagoras. It was he who first used the word Cosmos to denote a well ordered and harmonious universe, a world amenable to human understanding.

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 thought. He and his followers were not fundamentally experimentalists. They were mathematicians. And they were thoroughgoing mystics.

Could the Laws of Nature be deduced by pure thought? I doubt it, but who knows (Quotations from Megan Jorgensen). Image: © Elena

Pythagoreans

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 os the State and established the rule of the saints. But the unregenerate hankered after beans, and sooner or later rebelled.

Although there were a few welcome exceptions. The Pythagoras 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.

Pythagoreans. Photo by Elena

The Pythagoreans delighted in the certainty of mathematical demonstration, the sense of a pure and usullied 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 passerby 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 judgement, 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 Pythagorians, who, when questioned as to the ground 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.

Authority must not prevail unsupported by reason, but is always does. (Quotations from Megan Jorgensen). Image: © Elena

Kepler and Pythagoreans

The pros and cons of the Pythagorean tradition can be seen clearly in the life’s work of Johannes Kepler. 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 that the observed planetary motions could not be explained in this way, and repeatedly tried again. But unlike many Pythagoreans, he believed in observation 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 realise that the planets travel in ellipses. Kepler was both inspired in the search for the harmony of planetary motion and delayed for more than a decade by the attractions of Pythagorean doctrine.

Kepler's World. Photo by Elena

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 hypothesis, 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?

Where does the distaste for experiment come from? Image: Blue Mosaic © Elena

Divine Mathematics

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, exists). 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 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 V2 cannot be expressed accurately as the ratio of any two whole numbers, no matter how big these numbers are. Ironically this discovery 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 the knowledge of V2 and the dodecahedron. The outside world was not to know.

The sacred knowledge is to be kept within the cult, unsullied by public understanding. Image: © Megan Jorgensen (Elena)

(In fact, 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).

Even today there are scientists opposed to the popularisation of science: the sacred knowledge is to be kept within the cult, unsullied by public understanding.

The Pythagoreans believed the sphere to be “perfect”, all point on its surface being at the same distance from the 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”.