Ancient Philosophy

Pre-Socratics

Pythagoras

Pythagoras, whose influence in both ancient and modern times will be the subject of this chapter, stands as one of the most significant intellectual figures ever to have lived on earth—both in moments of wisdom and in those of folly. The foundations of mathematics, understood in terms of deductive proof, originate with Pythagoras. For him, mathematics was intimately intertwined with a particular form of mysticism. The impact of mathematics on philosophy, partially associated with this philosopher's name, has since been both a profound blessing and a devastating curse.

Let us begin with what little is known about his life. He was born on the island of Samos, and his flourishing activity dates to around 532 BCE. Some claim he was the son of a wealthy citizen named Mnesarch, while others believe he was the son of the god Apollo; I leave it to the reader to choose between these two opposing accounts. During Pythagoras's time, the island of Samos was ruled by the tyrant Polycrates, an old scoundrel who possessed countless riches and a vast fleet.

Samos was a commercial rival of Miletus; its merchants ventured so far westward that they reached Tartessus in Spain, renowned for its mines. Polycrates became the tyrant of Samos around 535 BCE and ruled until 515 BCE. He was not much tormented by pangs of conscience, having disposed of two brothers who initially shared in his tyranny, and he largely employed his fleet for piracy. Polycrates took advantage of Miletus's recent loss of independence due to Persian conquest. To thwart any further Persian expansion westward, Polycrates allied himself with the Pharaoh of Egypt, Amasis. However, when the Persian king Cambyses directed his full energy toward conquering Egypt, Polycrates deduced that the Persians would likely prevail and switched his allegiance to their side. He sent his fleet, manned by his political opponents, against Egypt, but the crews mutinied and turned back to Samos to attack him. Polycrates triumphed over them, but ultimately fell victim to his treachery, as a Persian satrap in Sardis presented himself as a potential rebel against the "Great King" and offered Polycrates substantial sums for assistance. Polycrates journeyed to the mainland for discussions, where he was captured and crucified.

Polycrates patronized the arts and adorned Samos with remarkable public works. The poet Anacreon served as his court poet. However, Pythagoras was not fond of Polycrates's rule and thus left Samos. It is said that Pythagoras traveled to Egypt (which seems plausible), where he gleaned much wisdom, but it is definitively known that he ultimately settled in Croton, a city located in Southern Italy.

The Greek cities in Southern Italy, like Samos and Miletus, were thriving and wealthy. Moreover, they were not threatened by the Persians. The two largest cities in Southern Italy were Sybaris and Croton. Sybaris became proverbial for its luxury; at its peak, its population, according to Diodorus, reached nearly 300,000, though this figure is undoubtedly exaggerated. Croton was almost equal in size to Sybaris. Both cities engaged in the importation of Ionian goods into Italy, which were consumed partly there and partly re-exported from the western coast to Gaul and Spain. Various Greek cities in Italy fiercely warred against one another. When Pythagoras arrived in Croton, the city had just suffered a defeat by Locri. However, shortly after Pythagoras's arrival, Croton achieved a complete victory in the war against Sybaris, which was subsequently utterly destroyed (510 BCE). Sybaris maintained close trade connections with Miletus. Croton became renowned for its medical school; a certain Democedes from Croton became the court physician to Polycrates and later to Darius.

In Croton, Pythagoras established a society of his disciples that wielded influence in the city for some time. However, the citizens eventually turned against Pythagoras, and he relocated to Metapontum (also in Southern Italy), where he died. Soon, Pythagoras became a mythical figure; wonders and magical abilities were attributed to him, and he was regarded as the founder of a school of mathematics. Thus, two opposing traditions contend around his name, making it difficult to extract the truth.

Pythagoras is one of the most intriguing and enigmatic figures in history. Not only do traditional representations of his activities present an almost inseparable blend of truth and falsehood, but even in their simplest and least contentious form, these representations portray a rather strange character that Pythagoras embodied. Pythagoras can be succinctly characterized as embodying traits reminiscent of both Einstein and Mrs. Eddy. He established a religion whose principal tenets revolved around the doctrine of the transmigration of souls and the sinfulness of consuming beans. Pythagoreanism manifested itself in a particular religious order that intermittently seized control of the state, instituting the rule of its saints. Yet those who were not reborn into this new faith longed for beans and eventually rebelled.

Here are some precepts of the Pythagorean order:

1. Refrain from consuming beans.
2. Do not pick up what has fallen.
3. Do not touch a white rooster.
4. Do not break bread.
5. Do not step over a crossbar.
6. Do not stir the fire with iron.
7. Do not take a bite from a whole loaf.
8. Do not pluck a wreath.
9. Do not sit on a measure of a quart.
10. Do not eat hearts.
11. Do not walk on the main road.
12. Do not allow swallows to nest under the roof.
13. When taking a pot from the fire, do not leave a mark in the ash, but stir the ash.
14. Do not look into a mirror near the fire.
15. When rising from bed, roll up the bedding and smooth out the impressions left by your body.

All these rules pertain to primitive notions of taboo.

Cornford, in "From Religion to Philosophy," posits that, in his view, "the Pythagorean school represents the main current in the mystical tradition, which we set against the scientific trend." He regards Parmenides, whom he sees as "the discoverer of logic," as a "branch of Pythagoreanism, and Plato himself—as a person who found in Italian philosophy the primary source of his inspiration." He asserts that Pythagoreanism was a reform movement within Orphism, and Orphism was a reform movement within the cult of Dionysus. Throughout history, the opposition between the rational and the mystical first emerges among the Greeks as a dichotomy between the Olympic gods and those other, less civilized deities, which bear greater resemblance to primitive beliefs that are the subject of anthropological study. In this division, Pythagoras stands on the side of mysticism, although his mysticism was of a specifically intellectual kind. Pythagoras attributed to himself a semi-divine status and apparently declared, "Rational beings are divided into [three kinds]: humans, gods, and beings resembling Pythagoras." "All systems inspired by Pythagoras," Cornford asserts, "strive for transcendence; they refer all values to an unseen unity in God and denounce the visible world as false and illusory, as a murky medium in which the rays of divine light refract and disperse amidst darkness and mist."

As Dicaearchus observes, Pythagoras taught that, firstly, the soul is immortal; secondly, it transmigrates into other forms of animals; thirdly, all that has ever occurred happens again after certain periods of time, and there is absolutely nothing new; and fourthly, that all living beings should be regarded as relatives of one another. It is said that Pythagoras, much like Saint Francis, preached before animals.

In the society he organized, both men and women were accepted on equal terms; all members of the community shared property and led a common way of life. Likewise, scientific and mathematical discoveries were regarded as collective endeavors and were mystically attributed to Pythagoras even after his death. Hippasus of Metapontum, who violated this rule, suffered shipwreck due to divine wrath provoked by his impiety.

But what relevance does all this have to mathematics? Through ethics, mathematics becomes intertwined with the glorification of the contemplative life. Barnett summarizes this ethics as follows: “We are wayfarers in this world, our body is the tomb of the soul; nonetheless, we must not seek to exit this world through suicide, for we are all in the hands of God, He is our shepherd, and without His command, we have no right to leave this world. There are three kinds of people in this world, comparable to three categories of individuals attending the Olympic Games. The lowest class consists of those who come to buy and sell; the next higher class is comprised of those who compete. However, best of all are those who come simply to watch. Thus, disinterested science is the most crucial means of purification; and a person who dedicates himself to science is a true philosopher, as he liberates himself most fully from the ’cycle of birth.’”

Changes in the meanings of words can often be quite enlightening. I previously spoke of the word “orgy”; now I wish to consider the word “theory.” This term was originally Orphic, interpreted by Cornford as “passionate and sympathetic contemplation.” In this state, Cornford states, “the observer identifies with the suffering God, dies with his death, and is reborn alongside his resurrection.” Pythagoras understood “passionate and sympathetic contemplation” as intellectual contemplation, which we also engage in during mathematical inquiry. Thus, through Pythagoreanism, the word “theory” gradually acquired its present meaning, yet for all those inspired by Pythagoras, it retained an element of ecstatic revelation. This may seem strange to those who have reluctantly and minimally studied mathematics in school, but for those who have experienced the intoxicating joy of unexpected understanding that mathematics brings to its lovers from time to time, the Pythagorean perspective seems entirely natural, even if it does not correspond with reality. It might easily appear that the empirical philosopher is a slave to the material under investigation, but the pure mathematician, like the musician, is a free creator of his own world of ordered beauty.

In Barnett’s depiction of Pythagorean ethics, it is interesting to note its opposition to contemporary assessments. For instance, at a football match, those who think in a modern manner consider players far more important than mere spectators. These individuals similarly regard the state: they admire politicians who are competitors in the political game more than those who are merely onlookers. This reevaluation of values is connected to a shift in the social system: the warrior, the noble, the plutocrat, and the dictator each possess their own norms of good and truth. The philosophical theory of the noble type has endured for quite some time because it is linked with the Greek genius, because the virtue of contemplation received theological endorsement, and because the ideal of knowledge of disinterested truth was equated with academic life. The noble must be defined as a member of a society of equals who live off the fruits of servile labor or, at the very least, the fruits of labor performed by individuals whose lower status is beyond doubt. It is essential to note that both the saint and the sage fit this definition, as these individuals lead lives more contemplative than active.

Contemporary definitions of truth, such as those offered by pragmatism or instrumentalism, which are more practical than contemplative doctrines, are products of industrialism in contrast to aristocratism. Whatever we may think of a social system that tolerates slavery, we owe the nobles, in the aforementioned sense, to pure mathematics. The ideal of the contemplative life, since it led to the establishment of pure mathematics, became a source of useful activity. This circumstance enhanced the prestige of the ideal itself, granting it success in theology, ethics, and philosophy—success that might otherwise have eluded it.

Thus stands the case with the explanation of the two aspects of Pythagoras's activity: Pythagoras as a religious prophet and Pythagoras as a pure mathematician. In both regards, his influence is immeasurable, and these two facets were not as independent as they might appear to modern consciousness.

At their inception, most sciences were tied to certain forms of false beliefs that conferred fictitious value upon them. Astronomy was linked to astrology, chemistry to alchemy. Mathematics, however, was connected to a more refined type of delusion. Mathematical knowledge appeared certain, precise, and applicable to the real world; moreover, it seemed that this knowledge was acquired through pure reflection without resorting to observation. Thus, it began to be thought that it provided us with an ideal of knowledge against which mundane empirical knowledge seemed inadequate. Based on mathematics, the assumption arose that thought is superior to sensation, intuition above observation. If the sensory world does not conform to mathematical frameworks, then so much the worse for that sensory world. Thus, through various means, attempts were made to discover methods of investigation that closely align with the mathematical ideal. The concepts resulting from this endeavor became sources of many erroneous views in metaphysics and epistemology. This form of philosophy begins with Pythagoras.

As is well known, Pythagoras claimed that “all things are numbers.” If this proposition is interpreted in a modern light, it seems logically nonsensical. Yet what Pythagoras understood by this assertion is not entirely without meaning. Pythagoras discovered that numbers hold great significance in music; the connection he established between music and arithmetic is still evoked today in mathematical expressions such as “harmonic mean” and “harmonic progression.” In his view, numbers, akin to those on dice or cards, possess form. We still speak of the squares and cubes of numbers, and we owe these terms to Pythagoras. Pythagoras also spoke of rectangular, triangular, and pyramidal numbers, among others. These were numbers of handfuls of pebbles (or, more naturally for us, numbers of handfuls of fractions) required to create a form. Pythagoras evidently believed that the world is composed of atoms, that bodies are constructed from molecules, which in turn consist of atoms arranged in various forms. Thus, he hoped to establish arithmetic as a scientific foundation in physics, just as in aesthetics.

The proposition that the sum of the squares of the sides of a right triangle adjacent to the right angle equals the square of the third side—the hypotenuse—was the greatest discovery of Pythagoras or his immediate disciples. The Egyptians knew that a triangle with sides measuring 3, 4, or 5 is a right triangle, but evidently, the Greeks were the first to notice that 32+42=523^2 + 4^2 = 5^232+42=52 and, based on this observation, discovered the proof of the general theorem.

Unfortunately for Pythagoras, his theorem immediately led to the discovery of incommensurability, a phenomenon that undermined his entire philosophy. In an isosceles right triangle, the square of the hypotenuse equals twice the square of either leg. Let us suppose each leg measures one inch; what then is the length of the hypotenuse? Assume its length is t/n inches. Thus, we have m2/n2=2m^2/n^2 = 2m2/n2=2. If t and n share a common factor, we may divide them by it. In such a case, at least one of t or n must be odd. However, noting that since m2=2n2m^2 = 2n^2m2=2n2, it follows that m2m^2m2 is even, and consequently, t is even while n is odd. Hence, let us assume t=2pt = 2pt=2p. We then find 4p2=2n24p^2 = 2n^24p2=2n2; therefore, n2=2p2n^2 = 2p^2n2=2p2, which implies n is even, contradicting our assumption. Thus, the hypotenuse cannot be expressed as a fractional number t/pt/pt/p. This proof is fundamentally the same as the one presented by Euclid in Book X.

This demonstration indicates that, regardless of the unit of length chosen, there exist segments that are not in a precise numerical relationship with this unit. In other words, there are no two whole numbers m and n such that the segment in question, taken m times, equals the unit of length, taken n times. This conclusion led Greek mathematicians to consider the necessity of developing geometry independently of arithmetic. Certain passages in Plato's dialogues reveal that, in his time, an interpretation of geometry separate from arithmetic was accepted; this principle reached its culmination with Euclid. In Book II, Euclid geometrically proves much that we would find more natural to demonstrate algebraically, such as the identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2. Euclid deemed this method essential precisely because of the challenges posed by the incommensurability of magnitudes. The same observation can be made regarding Euclid's interpretation of proportion in Books V and VI. The entirety of Euclid’s system is outstandingly logical, anticipating the mathematical rigor of the conclusions drawn by 19th-century mathematicians. Given the lack of an adequate arithmetic theory of incommensurable magnitudes, Euclid’s method was the best possible approach within the realm of geometry. When Descartes reintroduced coordinates into geometry, thus reinstating arithmetic’s supremacy, he presumed that the resolution of the problem of incommensurability was entirely feasible, although no such solution had yet been found in his time.

The influence of geometry on philosophy and the scientific method has been profound. The geometry established by the Greeks begins with axioms that are self-evident (or presumed to be so) and, through deductive reasoning, arrives at theorems that are quite distant from self-evidence. It is asserted that both axioms and theorems hold true in relation to actual space, which is something given in experience. Thus, it appears possible to make discoveries pertinent to the real world using deduction, based on what is self-evident. This perspective has significantly influenced philosophers from Plato and Kant to many others who fell between them. When the Declaration of Independence states, "We hold these truths to be self-evident," it follows the model of Euclid. The doctrine of natural rights, prevalent in the 18th century, seeks Euclidean axioms within the realm of politics.

The form of Newton's "Principia," despite its universally acknowledged empirical material, is entirely shaped by Euclid's influence. Theology, in its most precise scholastic forms, owes its style to the same source. Personal religion stems from ecstasy, while theology arises from mathematics; both can be traced back to Pythagoras.

I believe that mathematics is the principal source of faith in eternal and precise truth, as well as in the supersensible intelligible world. Geometry deals with perfect circles, yet no sensory object is perfectly round; no matter how meticulously we apply our compass, circles will always be imperfect and irregular to some degree. This suggests that every precise reflection engages with an ideal that stands in opposition to sensory objects. It is natural to take another step forward and assert that thought is nobler than sensation, and that the objects of thought are more real than those of sensory perception. Mystical doctrines regarding the relationship between time and eternity also find support in pure mathematics, for mathematical objects, such as numbers (if they are indeed real), are eternal and timeless. Such eternal objects may in turn be interpreted as thoughts of God. Hence arises the Platonic doctrine that God is a geometer, as well as Sir James Jeans' conception of God engaging in arithmetic pursuits. Since the time of Pythagoras, especially through Plato, rationalist religion, standing in contrast to the religion of revelation, has been profoundly influenced by mathematics and the mathematical method.

The synthesis of mathematics and theology that began with Pythagoras characterizes the religious philosophy of Greece, the Middle Ages, and the Modern period, extending up to Kant. Before Pythagoras, Orphism was akin to Asian mystical religions. Yet for Plato, Augustine, Aquinas, Descartes, Spinoza, and Kant, a close intertwining of religion and reasoning, moral inspiration and logical admiration for what is timeless, is characteristic—an amalgamation that originates with Pythagoras and distinguishes the intellectualized theology of Europe from the more overt mysticism of Asia. Only in very recent times has it become possible to clearly articulate the error of Pythagoras. I know of no other individual who has wielded such influence in the realm of thought as Pythagoras. I assert this because what appears to be Platonism, upon closer examination, essentially reveals itself to be Pythagoreanism. From Pythagoras onward, we encounter the entire concept of an eternal world, accessible to intellect yet inaccessible to the senses. Were it not for him, Christians would not have taught of Christ as the Word; were it not for him, theologians would not have sought logical proofs for the existence of God and the immortality of the soul. All of this is given in a concealed form in Pythagoras. How it became manifest will be demonstrated in the following discourse.