SOURCE: Richard D. McKirahan, Jr., "Pythagoras of Samos and the Pythagoreans," in Philosophy before Socrates: An Introduction with Texts and Commentary, Hackett Publishing Company, Inc., 1994, pp. 79-115.

[In the following essay, McKirahan presents an overview of Pythagorean thought on issues of religion, mathematics, number theory, and cosmology, citing contemporaneous sources as evidence for his statements.]

Pythagoras' Life and the Pythagorean Movement

Although details of Pythagoras' life and work are unclear, even mysterious, the following brief account is widely accepted. Born on the island of Samos c. 570, he left c. 530 on account of disagreement with the policies of the tyrant¹ Polycrates. At this time or before, he visited Egypt and Babylonia. He settled in Croton, a Greek city in southern Italy, where political life was based on associations or clubs. A Pythagorean association soon came to prominence, bringing Croton to increased military and economic importance. This association was characterized by certain religious and philosophical views, and is frequently called a school or brotherhood. Pythagoras and his followers are said to have governed the state so well that it was truly an aristocracy ("government of the best").² Similar associations were formed in other Greek cities in south Italy. Pythagorean power in Croton lasted unbroken for twenty years, but c. 500 many leading Pythagoreans were murdered in a revolution. Pythagoras himself escaped to Metapontum, where he died.

Pythagoras was far more than a politician. A religious and moral reformer, he inaugurated a way of life that made the Pythagorean associations distinctive and exclusive. His followers were devoted to his sayings, which they collected, memorized, and passed down. He was a charismatic figure who became the subject of legends: he killed a poisonous snake by biting it; a river hailed him by name; he made predictions; he appeared simultaneously in two different places; he had a golden thigh. The people of Croton addressed him as Hyperborean³ Apollo.⁴ Pythagoreans identified three types of rational beings: gods, humans, and beings like Pythagoras.⁵

The Pythagorean movement did not end with the Founder's death. The revolution in Croton and Pythagoras' death were only temporary setbacks, and the Pythagorean associations' political importance in southern Italy increased in the first half of the fifth century. However, mid-century saw anti-Pythagorean uprisings throughout the area. At Croton, a house where the Pythagoreans were gathered was set on fire and all but two were burned alive. Their meeting houses elsewhere were destroyed too, their leaders killed, and the whole region was severely affected. Afterwards the character of the movement changed. Some fled to mainland Greece. Those who stayed were centered in Rhegium, but some time later, perhaps about 400 B.C., almost all left Italy, with the notable exception of Archytas, who became an able monarch at Tarentum, where Plato visited him in the early fourth century. The Pythagorean movement effectively died out in the fourth century, as the scattered remnants of this persecution were unable or unwilling to organize and establish active Pythagorean centers again.

Even so, the influence of Pythagoras continued throughout antiquity. Philosophically its most important legacy is the strong stamp it left on Plato's thought, as found notably in the myths of the afterlife at the end of Gorgias, Phaedo, and Republic, in the cosmology of the Timaeus…, in Plato's belief in the importance of mathematics, and possibly in some fundamental aspects of his theory of Ideas.

Later on, "Neopythagoreans" from the mid-first century B.C. to the third century A.D. emphasized the religious, superstitious, and numerological aspects of Pythagoreanism, and followed some of Plato's successors, from three-hundred years before, in combining Pythagorean ideas with elements of Plato's thought. These Neopythagoreans followed the common ancient practice of ascribing their own doctrines to the Founder in order to gain authority for their views, which they regarded as implicit in or extensions of his teachings. Neopythagorean beliefs were absorbed from the third century A.D. by the Neoplatonists, and it is to Neoplatonist writings based largely on Neopythagorean works that most of our information about Pythagoras is due.

Sources

Information about Pythagoras and Pythagorean philosophy in our period presents special difficulties. There are few contemporary or near-contemporary references. The Pythagorean influence present in many of Plato's dialogues is of some help, but cannot be the basis of a detailed historical treatment because of the difficulties in distinguishing Pythagorean ideas from Platonic developments of them. Aristotle gives valuable information about Pythagorean doctrines, but rarely mentions Pythagoras, more frequently speaking of "those who are called Pythagoreans" or "the Italians," as though unwilling to attribute the doctrines he reports to Pythagoras himself. Moreover, Aristotle's information is hard to interpret since he is out of sympathy, even impatient, with Pythagorean doctrines, which do not fit well into his own system. Although Neopythagorean and Neoplatonic writings provide abundant materials on Pythagoras' life and teachings, they are mostly unhistorical and worthless for reconstructing the thought of Pythagoras and his followers in the fifth century.

We are no better off as regards original writings. Pythagoras wrote nothing and neither did his early followers. There are many references to Pythagorean secrecy, an unsurprising feature of a religious brotherhood, and reports of the evil end that befell one early follower for revealing a secret, a discovery in geometry.⁶ The earliest Pythagoreans for whom there are authentic fragments are Philolaus (who lived in the second half of the fifth century and perhaps as late as 380) and Archytas (early fourth century).

The following account of the early Pythagorean school, which is based mainly on Aristotle's discussions and on works on Pythagorean philosophy by two of Aristotle's pupils,⁷ stresses the philosophical and scientific elements of Pythagoreanism rather than the religious and political, but the many facets of Pythagorean thought are closely linked, and it is important to bear in mind the fundamental religious strand that pervades them all….

Early Source Material

Most of the contemporary and near-contemporary evidence on Pythagoras and fifth-century Pythagoreanism is found in the following passages.

Xenophanes mocks Pythagoras' belief in the transmigration of souls.

Heraclitus, whose life overlapped Pythagoras', comments sarcastically about Pythagoras and others.

9.2 Much learning ["polymathy"] does not teach insight. Otherwise it would have taught Hesiod and Pythagoras and moreover Xenophanes and Hecataeus.

(Heraclitus, DK 22B40)

9.3 Pythagoras the son of Mnesarchus practiced inquiry (HISTORIE) more than all other men, and making a selection of these writings constructed his own wisdom, polymathy, evil trickery.

(Heraclitus, DK 22B129)

Ion of Chios (born c. 490), in describing Pherecydes, a sixth-century mythographer and author of a Theogony, says:

Herodotus (485/4-c. 430) reports the story that Greeks in the region of the Black Sea told of Pythagoras' Thracian slave Zalmoxis. I paraphrase.

9.5 After being set free, he [Zalmoxis] returned to Thrace. He decided to civilize his compatriots, who were primitive and stupid, since through his contact with Greeks and especially with Pythagoras, who was "not the weakest sophist [wise man] of the Greeks," he had become acquainted with the Ionian way of life and with more profound sorts of people than could be found among the Thracians. "He built a hall in which he received and feasted leading Thracians, and taught them the better view, that neither he nor his guests nor any of their descendants would die, but would come to a place where they would live forever and have all good things." To convince them of his teaching, Zalmoxis disappeared for three years, living in a secret underground chamber, while everyone thought him dead. In the fourth year he reappeared above ground, and since then the Thracians believe in immortality.

(Herodotus, Histories 4.95 = DK 14,2 [the quoted words are an adaptation of Godley's translation])

Empedocles, who adopted some Pythagorean beliefs, describes Pythagoras⁸ in this way.

Plato says of Pythagoras:

9.7 Is Homer said to have been during his life a guide in education for people who delighted in associating with him and passed down to their followers a Homeric way of life? Pythagoras himself was greatly admired for this, and his followers even nowadays name a way of life Pythagorean and are conspicuous among others.

(Plato, Republic 10 600a–b = DK 14,10)

Herodotus says Egyptian religious customs forbid people to wear wool into temples and to be buried in woolen clothing, and he links those practices with the Pythagoreans.

9.8 The Egyptians agree in this with those called Orphics … and with the Pythagoreans; for it is likewise unholy for anyone who takes part in these rites to be buried in woolen garments.

(Herodotus, Histories 2.81 = DK 14,1)

Plato, while outlining the mathematically based curriculum of his philosopher kings, says that the Pythagoreans assert that astronomy and harmony are sister sciences, and that they

9.9 … measure audible concords against one another and look for numbers, but do not ascend to the level of considering problems generally and asking which numbers are concordant, which are not, and why.

(Plato, Republic 7 530d-31c, not in DK)

Interestingly, the contemporary references (9.1-9.3) are ambivalent at best. Heraclitus' references to Pythagoras' wide-ranging knowledge and his intellectual curiosity (HISTORIE⁹) are evidence that Pythagoras engaged in the kind of inquiry pursued by the Milesians.¹⁰ (Samos is only a few miles from Miletus.) As Heraclitus criticizes many of his predecessors but not the Milesian philosophers,¹¹ the charges in 9.2 and 9.3 may be addressed to aspects of Pythagoras' thought that differ from Milesian-style investigations. The charge of lacking insight may mean only that he did not see eye to eye with Heraclitus, but there may be something specific in the claims that his wide knowledge is plagiarized or based on the ideas of others and is "evil trickery."¹² Heraclitus may be expressing his contempt for Pythagoras' mystical, religious views or possibly for some physical doctrines which he found seriously wrong.

9.1, 9.4, 9.5, and 9.6 associate Pythagoras with a belief in an afterlife. The soul upon death might enter the body of a lower animal or a human. 9.4 suggests that one's fate after death is a reward or punishment for one's character in the previous life, and perhaps refers to the later attested view that the reward for an outstandingly good life is eternal happiness untouched by the need for rebirth. 9.5 is a patent attempt by some Greeks¹³ to claim a Greek origin for a native Thracian belief (Zalmoxis was a Thracian god, not a human). The portrait of Zalmoxis as an imposter may be meant to reflect negatively on Pythagoras himself. The remark in 9.7 on Pythagoras' disciples and way of life illustrates the special nature of the Pythagorean "brotherhood," whose principal beliefs, many of them bound closely to the belief in reincarnation, were traced back to the Founder. Moreover, the strong Pythagorean in terest in mathematics and in a mathematical approach to harmonics and astronomy (attested to by Plato in 9.9 and by Aristotle in 9.19), which goes back at least to the mid-fifth century, may have originated with Pythagoras himself.

Immortality and Reincarnation

The religious message of Pythagoras is based on the doctrine of the immortality of the individual soul, which is recounted in the following passage along with other related beliefs.

9.10 First he declares that the soul is immortal; then that it changes into other kinds of animals; in addition that things that happen recur at certain intervals, and nothing is absolutely new; and that all things that come to be alive must be thought akin. Pythagoras seems to have been the first to introduce these opinions into Greece.

(Porphyry, Life of Pythagoras 19 = DK 14,8a)

The final statement in 9.10 is disputed,¹⁴ but the views mentioned are certainly Pythagorean. In declaring the soul immortal, Pythagoras obliterated the barrier the Olympian religion placed between humans and gods. "Immortal" is, for the Greeks, tantamount to "divine."

The doctrine is a development of ideas of the Milesian philosophers, who made their originative substances immortal and divine, and held that divinity was widespread in the KOSMOS. According to Anaximenes (6.6), the human soul is composed of the divine originative substance, air. It is no great leap to infer that the soul is immortal. The Pythagoreans gave special importance to breath in their cosmogony (9.27, 9.28), and some Pythagoreans identified the soul with air.

9.11 Some of them [the Pythagoreans] declared that the soul is the motes in the air, and others that it is what makes the motes move.

(Aristotle, On the Soul 1.2 404a 17 = DK 58B40)

Another Pythagorean view of the soul mentioned by Aristotle (though not identified by him as Pythagorean) is that it is a HARMONIA. This theory also appears in Plato's Phaedo, where it is presented by a pupil of the Pythagorean Philolaus and endorsed by another Pythagorean, Echecrates. However, in the Phaedo it is used to disprove the soul's immortality: if the soul is a HARMONIA of the parts of the body, when the body becomes seriously ill the HARMONIA, and therefore the soul, is destroyed.¹⁵ But if the HARMONIA theory of the soul implies that the soul is not immortal, how can it be reconciled with Pythagorean philosophy? One solution is to suppose that the original Pythagorean idea was that the soul, like the whole universe, is a HARMONIA of numbers (cf. 9.36).¹⁶ As such it does not suffer the decay and alteration that affect the body, but is the same in kind as the divine KOSMOS. Insofar as it is polluted or tainted by its association with the body, it contains an element of discord which the Pythagorean way of life aims to remove, thus restoring it to a state of perfect order (KOSMOS).

Pythagoras represents a new and radical challenge to the Olympian tradition. To promote each human, or part of each human, to the level of the gods simultaneously devalues the gods and their worship and raises the importance of our care for ourselves, or more precisely for our selves, where the self is the soul as opposed, say, to the body. Moreover, this doctrine is not anthropocentric. Not only human souls are at stake; all living things possess souls. Only thus can transmigration of souls take place. Our concern is for all ensouled things, with whom we are in a literal sense related.

The doctrines of the immortality and transmigration of souls imply a major restructuring of values. Our interests, even our egoistic interests, now extend beyond our selves and beyond this lifetime. Further, if what we do and how we live in this life affect our soul's next incarnation, as 9.4 suggests, then we have strong prudential reasons to choose certain actions and ways of life over others. The Pythagorean way of life (9.7) aimed to improve the soul and to attain for it the best possible destiny, which consists either in attaining the best of reincarnations or in complete freedom from the necessity of continued rebirth through reunion with the divine universal soul.¹⁷

The following passages say more about this doctrine.

9.12 The Egyptians were the first to declare this doctrine too, that the human soul is immortal, and each time the body perishes it enters into another animal as it is born. When it has made a circuit of all terrestrial, marine, and winged animals, it once again enters a human body as it is born. Its circuit takes three-thousand years. Some Greeks have adopted this doctrine, some earlier and some later, as if it were peculiar to them. I know their names, but do not write them.

(Herodotus, Histories 2.123 = DK 14,1)

9.13 Heraclides of Pontus says that Pythagoras said the following about himself. Once he had been born Aethalides and was believed to be the son of Hermes. When Hermes told him to choose whatever he wanted except immortality, he asked to retain both alive and dead the memory of what happened to him…. Afterwards he entered into Euphorbus and was wounded by Menelaus. Euphorbus said that once he had been born as Aethalides and received the gift from Hermes, and told of the migration of his soul and what plants and animals it had belonged to and all it had experienced in Hades. When Euphorbus died his soul entered Hermotimus, who, wishing to provide evidence, went to Branchidae, entered the sanctuary of Apollo, and showed the shield Menelaus had dedicated. (He said that when Menelaus was sailing away from Troy he dedicated the shield to Apollo.) The shield had already rotted away and only the ivory facing was preserved. When Hermotimus died, it [the soul] became Pyrrhus the Delian fisherman, and again remembered everything…. When Pyrrhus died it became Pythagoras and remembered all that has been said.

(Diogenes Laertius, Lives of the Philosophers 8.4–5 = DK 14,8)

The Greeks Herodotus infuriatingly refuses to name in 9.12 are thought to include the Pythagoreans. 9.13, which is attributed to a good source,¹⁸ differs from 9.12 in important points. According to 9.13 but not 9.12, the soul spends time in Hades as well as in living things. In 9.13 but not 9.12, the soul sometimes animates plants in addition to animals and humans. In 9.12 but not in 9.13 the soul occupies all animals in between human incarnations, as if all souls have the same fate. Finally, the 3000-year span between successive human incarnations in 9.12 is incompatible with the three human incarnations of Pythagoras' soul which 9.13 places after the Trojan War.¹⁹ Some of the discrepancies may stem from the fact that 9.12 claims to be giving an account not of Pythagorean beliefs but of Egyptian ones (although the Egyptians, who had an elaborate doctrine of the afterlife, did not believe in transmigration). The Greeks referred to allegedly borrowed beliefs from the Egyptians, but there is no guarantee that they did not alter them. In any case, neither passage proves that the Pythagorean belief in reincarnation involved rewards and punishments for previous lives. Still, the likelihood is great that it did. First, there is the evidence of 9.4. Also, Empedocles, who was influenced by Pythagoreanism, held that the best sort of animal for a soul²⁰ to occupy is a lion, and the best sort of plant a laurel (14.131), and that the best souls become outstanding men and even blessed gods (14.132, 14.133). Poems from the early fifth century, which may have been written for people with Pythagorean beliefs, refer to judgment after death leading to rewards in subsequent lives for outstanding success in this one²¹ and to everlasting happiness in the Isles of the Blest as the reward of "all those who have had the courage to keep their soul completely away from unjust deeds for three stays in each place [on earth and in the underworld]."²² Moreover, later Pythagoreans held these beliefs.

This evidence makes it plausible that early Pythagoreans believed not only that the soul is immortal and passes into one living being after another (directly or after a time in Hades), but also that some incarnations are preferable to others and the next kind of being a soul will inhabit is determined by a postmortem judgment of its previous life. These beliefs formed the basis of the Pythagorean religion and way of life.

Prohibition on Killing; Dietary Restrictions

Many features of the Pythagorean life can be understood from this perspective. The aim of life is to ensure a good future for the soul. Vegetarianism, prominent in the Pythagorean life, results from the belief in transmigration and the kinship of all living things. Bluntly put,²³ what you kill and eat for dinner may have the soul of your dear departed mother or father. More generally, since all living beings are related, it is an equal offense to kill anything, without reference to the possibility that its soul might once have ensouled a human. If it is alive, it is at least a distant relative. Any killing is tantamount to murder; eating animals amounts to cannibalism. Empedocles developed this idea in much greater detail²⁴ than can be attributed to the Pythagoreans from early sources, but there is no reasonable doubt that violating this prohibition was the premier form of injustice which merited punishment after death.

There are difficulties about this doctrine, which amounts to a rationalization of the instinctive prephilosophical Greek horror of incurring pollution by bloodshed. First, if all living things are related, killing and eating plant life (possibly including fruits and vegetables, which can, loosely speaking, grow into plants) should be prohibited too, so that Pythagoreans could eat only a very few things, such as milk and honey; yet there was no general ban on vegetable foods. Quite likely only some plants, such as laurels,²⁵ were thought to have souls. Second, there is conflicting evidence about the prohibition on eating meat, some sources declaring that all meat was prohibited, others that only certain kinds were, still others denying that any such prohibition existed. A sensible approach to this contradictory information is to see in it traces of variations in Pythagoreanism, whose initial ban on all meat ceased to be observed rigorously in the fifth-century diaspora, when there were only scattered remnants of the Pythagoreans and hence the likelihood of local deviations from the original norm.

Another notorious practice of the Pythagoreans was their refusal to eat beans. The amount of ancient speculation about this dietary aberration proves that the custom was found odd and that there was no obvious reason for it. We are told that beans were banned because their flatulent tendency disturbs our sleep and our mental tranquillity, because they resemble testicles, or the gates of Hades, or the shape of the universe, or because they are used in allotting political offices (a reference to antidemocratic Pythagorean politics), or because if buried in manure they take on a human shape, or because their stems are hollow so that they are connected directly to the underworld, and so on.²⁶ On a plausible recent interpretation,²⁷ Pythagoras introduced this prohibition because eating beans can be bad for health: some people grow ill upon eating the fava beans which are common in south Italy, so the ban on beans might be a practical expedient, not a ritual abstention.

AKOUSMATIKOI and MATHEMATIKOI

9.14 There are two kinds of the Italian philosophy called Pythagorean since two types of people practiced it, the AKOUSMATIKOI and the MATHEMATIKOI. Of these, the AKOUSMATIKOI were admitted to be Pythagoreans by the others, but they did not recognize the MATHEMATIKOI, but claimed that their pursuits were not those of Pythagoras, but of Hippasus…. The philosophy of the AKOUSMATIKOI consists of unproved and unargued AKOUSMATA to the effect that one must act in appropriate ways, and they also try to preserve all the other sayings of Pythagoras as divine dogma. These people claim to say nothing of their own invention, and say that to make innovations would be wrong. But they suppose that the wisest of their number are those who have got the most AKOUSMATA.

(Iamblichus, Life of Pythagoras 81,82 = DK 18,2, 58C4)

The AKOUSMATIKOI (the word derives from AKOUSMA, "thing heard") learned and accepted Pythagoras' sayings simply on the strength of Pythagoras' having said them, but refused to recognize continued mathematical and scientific research as part of the Founder's intentions. In contrast, the MATHEMATIKOI (from MATHEMA, "learning" or "studying," not specifically mathematical learning and studying, although the study these Pythagoreans pursued was largely mathematical) promoted the scientific studies Pythagoras allegedly began, while acknowledging the religious side of Pythagoreanism. This split, between the religious, conservative, dogmatic AKOUSMATIKOI and the scientific, progressive, intellectually active MATHEMATIKOI, resembles the sectarianism often found in the early history of religious movements.

The AKOUSMATA

Some Pythagorean practices are called AKOUSMATA. Their role is described thus.

9.15 All the AKOUSMATA referred to in this way fall under three headings, (a) Some indicate what something is, (b) others indicate what is something in the greatest degree, and (c) others what must or must not be done, (a) The following indicate what something is. What are the Isles of the Blest? Sun and Moon. What is the oracle at Delphi? The tetractys, which is the harmony in which the Sirens sing, (b) Others indicate what is something in the greatest degree. What is most just? To sacrifice. What is the wisest? Number, and second wisest is the person who assigned names to things. What is the wisest thing in our power? Medicine. What is most beautiful? Harmony.

(Iamblichus, Life of Pythagoras 82 = DK 58C4) (continuation of 9.14)

Examples of the third type of AKOUSMATA are found in the following passages.

9.16 not to pick up which had fallen, to accustom them not to eat self-indulgently or because it fell on the occasion of someone's death … not to touch a white rooster, because it is sacred to the Month and is a suppliant. It is a good thing, and is sacred to the Month because it indicates the hours, and white is of the nature of good, while black is of the nature of evil … not to break bread, because friends long ago used to meet over a single loaf just as foreigners still do, and not to divide what brings them together. Others with reference to the judgment in Hades, others say that it brings cowardice in war, and still others that the whole universe begins from this.

(Aristotle, fr. 195 [Rose], quoted in Diogenes Laertius, Lives of the Philosophers 8.34 ff. = DK 58C3)

Some of the justifications in 9.16 are moral precepts (behave with moderation, respect the gods), but others reek of prephilosophical ways of thought. Still others point to an aspect of Pythagoreanism that remains to be discussed, the study of the KOSMOS with the aid of mathematics.

So far, Pythagoreanism hardly deserves space in a treatment of early Greek philosophy. The religious side is in many ways the antithesis of the rational approach to nature. Not only does it contain superstitions and other taboos, it makes no attempt to justify or systematize them. The AKOUSMATIKOI followed Pythagoras differently from the way Anaximenes followed Thales and Anaximander. They aimed to preserve his ideas, not to criticize or enlarge them. Their acceptance of the AKOUSMATA unproved and unargued (9.14) is unphilosophical and unscientific.

The MATHEMATIKOI

The situation is different with the MATHEMATIKOI, whose ideas deserve a place in any study of Greek thought. Still, their work was carried out against the religious backdrop sketched above, and after looking at some of their main ideas I shall suggest some points of contact between the two faces of Pythagoreanism.

The scientific side of Pythagoreanism marks a new approach to understanding the world. Number takes precedence over matter, mathematical accounts of phenomena are preferred to descriptions in terms of physical constituents, and perhaps definitions and proofs begin to take the place of the Ionians' "likely stories" in explaining the relations among things.

The Concordant Intervals

The starting point for these developments was the Pythagorean discovery that concordant musical intervals can be expressed mathematically. The musical intervals of the octave (C-C), fifth (C-G), and fourth (C-F) were basic to Greek music. In the seven-stringed lyre, four of the strings were tuned to pitches separated by these intervals (e.g., C, F, G, C) and the other three were put at different pitches depending on the "mode" desired. In a lyre the strings all have the same length: it is clear that the higher notes come from the tauter strings, but there is no obvious numerical relation between pitch and tension. In a monochord, a single-stringed instrument with a movable bridge, changing the position of the bridge changes the pitch produced by plucking or bowing the string, which remains under the same tension. There are a limitless number of possible positions the bridge can have, and so an unlimited number of possible pitches. When the bridge is placed exactly halfway between the fixed ends of the string, the note produced is an octave higher than that produced by the entire length of the string. This is the case no matter how long the string is, what the string is made of, or how taut it is (as long as it is taut enough to produce a tone). The essence of the octave is the numerical ratio 2:1, not the actual length or material involved in making the sound. Since the intervals of the fifth and fourth are also expressible in terms of the ratios of small whole numbers (3:2 and 4:3, respectively) music appears to result from the imposition, by means of number, of order and limit on the unlimited continuum of possible tones…. It is difficult to imagine how wonderful and surprising it must have been to learn that fundamental features of music could be expressed numerically. After all, we are used to expressing qualitative notions in quantitative, numerical terms. We measure and count color, sound, weight, and speed in wavelengths, grams, and feet per second. In fact, we regard quantitative treatment as one of the hallmarks of science. This discovery, which was possibly made by Pythagoras himself, was the first time any quality was reduced to a quantity, and so it stands at the beginning of this aspect of our scientific tradition. Also, within Pythagorean thought, the discovery had important effects on mathematics, cosmology, and the doctrine of the soul.²⁸

Before taking these topics up, however, I shall introduce two concepts that are central to them: KOSMOS and HARMONIA. KOSMOS, a word that it is said Pythagoras was first to apply to the universe, has two basic meanings: orderly arrangement and ornament. It combines the notions of regularity, tidiness, and arrangement on the one hand with beauty, perfection, and positive moral value. The Ionians had already treated the world as a KOSMOS, but the Pythagoreans enlarged and deepened this idea to apply to the mathematical structure and religious significance which they found in the world around them, HARMONIA, from which our word "harmony" comes, originally meant a fitting together, connection, or joint. Later it meant the string of a lyre, and then a way of stringing the lyre, i.e., a tuning or scale. The essence of the order in the world, the Pythagoreans believed, is located in the connections of its parts, i.e., KOSMOS depends on HARMONIA, especially on HARMONIA based on number. This doctrine was first applied to musical HARMONIA, but was later extended more widely….

The tetractys [a triangular figure composed of ten points] was called "the harmony in which the Sirens sing" and was mystically identified with the oracle at Delphi (see 9.15). The following passage mentions some of its other associations.

9.18 The tetractys is a certain number, which being composed of the four first numbers produces the most perfect number, ten. For one and two and three and four come to be ten. This number is the first tetractys, and is called the source of ever flowing nature since according to them the entire KOSMOS is organized according to HARMONIA, and HARMONIA is a system of three concords—the fourth, the fifth, and the octave—and the proportions of these three concords are found in the aforementioned four numbers.

(Sextus Empiricus, Against the Mathematicians 7.94-95, not in DK)

We will come back to the statement that the KOSMOS is arranged according to HARMONIA in discussing Pythagorean cosmology, but the general nature of Pythagorean thought can be gathered from this passage. The concordant musical intervals are accounted for in terms of the numbers one, two, three, and four. These are assumed to explain the structure of the universe, and a particular way of exhibiting them takes on a sacred character, as does their sum. The kind of thought found here works by association rather than logical analysis. It moves too fast and too far for our taste, although no faster or farther than earlier presocratic philosophers had done.

Pythagorean Number Theory

The Greek word for number is ARITHMOS, and ARITHMETKE, the study of numbers, corresponds to what mathematicians how call number theory rather than to arithmetic, the method of calculating with numbers. Perhaps because of their discovery of the numerical basis of music, the Pythagoreans pursued—and may have initiated—the study of numbers, their different kinds and properties, and their principles. The following sketch of Pythagorean number theory includes some developments which probably come from a later period than ours (the Neopythagorean period), but which are in the spirit of early Pythagoreanism.

An important step in understanding a subject is to classify and to define the subject as a whole (in this case, number) and its subdivisions (the types of numbers). The Pythagoreans were first to distinguish and define odd and even numbers, prime and composite numbers, square numbers, and cube numbers. They also made some classifications no longer used, such as triangular and oblong numbers. These numbers, as well as squares

and cubes, are called after the shapes into which an appropriate number of units, or pebbles, can be arranged. In thinking of 9 as a square number because 9 dots can be arranged in a certain pattern [a square composed of three rows of three points]…. Pythagoreans have in mind a spatial representation no longer present when we write 9 = 3² and think of "squaring" as the algebraic operation of multiplying something by itself. They also defined a number N as triangular, pentagonal, etc., if N dots can be arranged to form an equilateral triangle, pentagon, etc. The tetractys of the decad represents the fourth triangular number, 10. Similarly they defined a number N as oblong if N dots can be arranged to form a rectangular figure with sides differing by one, i.e., if there is a number M such that N = M(M+ 1). The smallest oblong number is 2 (= 1 × 2), the next is 6, then 12, etc.

This manner of representing numbers spatially makes it possible to see and to prove certain numerical relations. Consider square numbers. Beginning from 1, which for some purposes was not considered a number, we form the first square number by placing three dots around it in the form of a "gnomon" (literally, "carpenter's square"). The next square number is formed by wrapping another gnomon around the previous one…. This time the gnomon has five dots. In this way it is evident that each square number results from adding successive odd numbers, and that we reach the next square number by adding the next odd number to the last square number reached. If N = 1 + 3 + … +M, then N is a square number, and the next square number = N + (M + 2). Another result is represented by the formula (N + 1) = N₂ + 2N + 1. Consider the gnomon used in moving from 3² to 4² …. 4² = 3² plus the gnomon, and the gnomon can be divided into three parts, of which two have 3 units each, and the other is a single unit, i.e. 4² = 3² + (2 × 3) + 1, and this relation holds for any pair of successive square numbers.

This style of proof is ingenious, but has limitations. Most obviously it is restricted to whole numbers. In addition, it is not a general proof, but a series of par ticular cases, though the general principle is correct, and the proof can be made respectable if put into the form known as mathematical induction.

By extending the notion of a gnomon the Pythagoreans spoke of wrapping gnomons around triangular, pentagonal, oblong, etc., numbers and so could prove other results. One important result is that just as the sequence of square numbers is generated by adding successive odd numbers, so the sequence of oblong numbers is generated by adding successive even numbers…. Since omitting the left column yields square numbers, the sequence of oblong numbers differs from the sequence of square numbers by a unit at each stage…. The numbers forming the "sides" of square numbers always have the same ratio. We would write 2² = 2 × 2, 3² = 3 × 3, … N² = N×N, and 2:2 = 3:3 = … = N:N. On the other hand, the numbers forming the "sides" of successive oblong numbers have a different ratios: 2:1, 3:2, 4:3, … This fact leads to the doctrine of the principles of number.

Principles of Number

A prominent feature of Ionian thought is the desire to identify a small number of principles from which the world is constructed or out of which it has grown: the world depends on a small number of principles and can be accounted for in terms of them. The Pythagoreans believed that number is fundamental to the world, that somehow the world can be accounted for in terms of number. But there is no end to the number of numbers. They therefore needed to account for numbers in terms of a small number of principles, so that these principles ultimately serve as the principles for all things.

Several Aristotelian testimonia are relevant here.

9.19 At the same time as these [Leucippus and Democritus] and before them, those called Pythagoreans took hold of mathematics and were the first to advance that study, and being brought up in it, they believed that its principles are the principles of all things that are. Since numbers are naturally first among these, and in numbers they thought they observed many likenesses to things that are and that come to be … and since they saw the attributes and ratios of musical scales in numbers, and other things seemed to be made in the likeness of numbers in their entire nature, and numbers seemed to be primary in all nature, they supposed the elements of numbers to be the elements of all things that are.

(Aristotle, Metaphysics 1.5 985b23–986a2 = DK 58B4)

9.20 The elements of number are the even and the odd, and of these the latter is limited and the former unlimited. The One is composed of both of these (for it is both even and odd) and number springs from the One; and numbers, as I have said, constitute the whole universe.

(Aristotle, Metaphysics 1.5 986a17–21 = DK 58B5)

9.21 The Pythagoreans similarly posited two principles, but added something peculiar to themselves, not that the limited and the unlimited are distinct natures like fire or earth or something similar, but that the unlimited itself and the One itself are the substance of what they are predicated of. This is why they call number the substance of all things.

(Aristotle, Metaphysics 1.5 987a13–19 = DK 58B8)

9.22 They say that the unlimited is the even. For when this is surrounded and limited by the odd it provides things with the quality of unlimitedness. Evidence of this is what happens with numbers. For when gnomons are placed around the one, and apart, in the one case the shape is always different, and in the other it is always one.

(Aristotle, Physics 3.4 203a10–15 = DK 58B28)

9.22 refers to the gnomon-wrapping described above and shows how even is linked with unlimited, and odd with one. The successive figures formed by wrapping gnomons with odd numbers of dots "around the one" are all square and so have the same shape, whereas in the sequence of oblong figures formed by wrapping gnomons with even numbers of dots around the two,²⁹ no two shapes are the same (since the ratios of their sides are different).

9.19 says that the Pythagoreans thought the elements of numbers are the elements of all things, and 9.20 identifies the elements of number as the even and the odd, which compose the One, from which springs number, of which the universe is composed. The generation implied by 9.20 is:

even and odd → the One → number → the universe

On the other hand, 9.21 does not mention even and odd, but makes it clear that the limited and the unlimited are the two principles, and associates unity with the limited, whereas 9.20 makes the One composed of both odd and even. Here the generation seems to be:

unlimited and limited (= the One) → number → all things

These passages leave much unclear, and their inconsistencies are particularly puzzling since Aristotle is likely to have been well informed and the passages occur in the same book, a mere page away from one another. As we shall see, Aristotle is quite out of sympathy with the Pythagoreans and cannot make sense of their view that number is the principle of the KOSMOS. Still, 9.22 establishes a connection between even and unlimited and between odd, one, and limited or limit, and we can understand in a general way the desire to find a way of generating numbers out of a small number of principles, though it is unclear whether this generation takes place in time (so that there is a time when even and odd exist, but not yet the numbers) or represents not a temporal priority but one of another kind.

Geometry

We have already seen that Thales had an interest in geometry, though his contribution is unclear. The claim in 9.19 that the Pythagoreans were "first to advance" the study of mathematics is frequently understood to refer to their contributions to geometry, although their interest in number theory and their belief that numbers are the key to the KOSMOS would be enough to justify the statement even if they had done nothing to advance geometry. The Pythagorean theorem is one of the most famous geometrical propositions, and ancient sources attribute its discovery to Pythagoras himself.³⁰ But Pythagoras did not discover Euclid's proof (which could not have been worked out before the late fifth century), and if all he did was to state the theorem, he may have learned it when he visited Babylon, where it was known, though not proved.

In evaluating the Pythagoreans' contributions to geometry, three pieces of evidence are crucial: first and earliest, Aristotle's statement in 9.19; second, the history of mathematics composed by Aristotle's pupil Eudemus, of which an important extract is preserved in Proclus' Commentary on the First Book of Euclid's Elements; and third, attributions of particular geometrical results to the Pythagoreans. The difficulty here is that Aristotle is positive but vague, Eudemus gives no prominence to Pythagoras or anyone known to be a Pythagorean, and the specific attributions, while showing that Eudemus and others did acknowledge a Pythagorean contribution to geometry, do not provide enough evidence to understand the nature and extent of Pythagorean geometry.

Two questions in particular remain open. Did the Pythagoreans discover the incommensurability of the side and diagonal of the square, and did they invent the notions of mathematical proof and the arrangement of theorems into a deductive system?

The first question, which many have thought important both for Pythagoreanism and for the history of mathematics, asks whether the Pythagoreans discovered that if a square has sides of length A and diagonals of length B, then there are no whole numbers M, N such that A:B = M:N. We would express this fact by saying that the ratio A:B is irrational, or equivalently that √2 is irrational. This discovery preceded the work of Theodorus of Cyrene,³¹ who extended it in the late fifth century, proving in effect which square roots are irrational as far as √17.³² Euclid presents two different theories of proportion, one of which works only for "rational magnitudes" (those that can be represented as the ratio of whole numbers) and one of which works for "irrational magnitudes" (those that cannot be so represented) as well. The obvious inference is that the former theory was worked out before the discovery of the irrational, and the latter, which is due to the fourthcentury mathematical genius Eudoxus, was developed afterwards. Some attribute the earlier theory to the Pythagoreans (recall their interest in the ratios of whole numbers in connection with the musical intervals), and hold in addition that the discovery of the irrational caused a crisis for the Pythagoreans since it showed that the world could not be entirely accounted for through whole numbers. Some declared that Hippasus, an early Pythagorean, perished at sea through divine retribution for revealing this discovery, as if divulging the existence of the irrational and making public the shortcomings of the Pythagorean numerical conception of the world was a great scandal.

The second question is equally important for the history of mathematics and philosophy. What distinguishes Greek from Babylonian and Egyptian mathematics is the notion and prominence of proofs. The earlier cultures developed methods of doing arithmetic and calculating areas, and the Babylonians had some interest in the relations of numbers that was not entirely practical, but it was the Greeks who discovered and developed the idea of showing how one fact follows from others, and arranging facts into a logically ordered system. This is the practice of Euclid's Elements "(c. 300 B.C.), parts of which are familiar to anyone who has studied geometry. Euclid was by no means the first person to prove theorems. Eudemus names several writers of "Elements" before Euclid, beginning with the fifth-century mathematician Hipparchus, and the practice of proving theorems must have been well established before Hipparchus, who will have arranged and systematized existing theorems and proofs. To whom, then, do we owe the discovery of proofs? Did the idea of proof come from mathematics, or did it start elsewhere? And why did it arise at all?

One suggestion³³ is that rigorous proofs are more likely to have arisen in connection with negative than with positive results. With positive results, say that the sum of the angles of a triangle is equal to two right angles, you can "see" that they hold by inspecting a few cases, perhaps with the help of cutting or folding the figure. This sort of procedure makes no pretense to rigor, but it does establish the result (at least for reasonably obvious facts in arithmetic and geometry). With negative results it is quite different. If you want to show that if A,B are the side and diagonal of a square, then there are no whole numbers M,N such that A:B = M:N, you cannot examine all possible pairs of whole numbers and show that none of them has the property in question. If your only way of showing that a proposition is true is to examine specific cases, you can never prove this result, only that none of the pairs so far considered has the property. Establishing a negative result in arithmetic or geometry therefore requires a proof, and in fact such problems, perhaps even the one involving the side and diagonal of the square, may have given rise to the idea of proving theorems.

It is further claimed that the idea of rigorous proof was first developed in philosophy and then applied to mathematics.³⁴ On this view, the sophisticated arguments of the Eleatic philosophers Parmenides and Zeno,³⁵ who proved negative results (e.g., that there is no motion), came first and mathematical proofs came second. However, our evidence does not allow us to say for certain whether the priority goes to mathematics or to philosophy. It is safe to say that deductive arguments were first used in both fields in the first half of the fifth century, and it is likely that they originated in south Italy, where both the city of Elea and the Pythagorean movement were located. Thus, a Pythagorean connec tion is possible, but cannot be proved.³⁶

It is nevertheless certain that the Pythagoreans were interested in geometry. Their interest in reducing things to numbers naturally led them to do the same for geometry. The relations between the two fields are seen in the definitions of the basic entity in each field, the unit and the point. (A unit is a point lacking position, and a point is a unit having position.³⁷) Numbers are pluralities of units, and lines, planes, and solids are determined by pluralities of points. One way in which one-, two-, and three-dimensional space depends on points is indi cated in the following fragment of Pythagorean Numbers, a work by Plato's nephew Speusippus.

9.23 formulas for lines, surfaces, and solids; for one is a point, two a line, three a triangle, and four a pyramid, and all these are primary and the starting points for the other figures of each kind.

(Speusippus, fr. 4 [Lang] = DK 44A13)

Here we are to think of the number ten as represented in the tetractys of the decad and so composed of the numbers one, two, three, and four in that order. Two points determine a straight line, three points not in a straight line mark the corners of a triangle, and four points not in the same plane the vertices of a pyramid. These are the simplest one-, two-, and three-dimensional figures…. Again we see how Pythagorean reductionism works. In each situation we treat the minimal case and that is supposed to take care of the more complicated cases. How it does so is vague; there is no reason to suppose that, for example, all plane figures (including ones with curved sides) are triangles or can be formed out of or approximated by triangle.³⁸ Moreover, we see again the "associative" nature of Pythagorean thought,³⁹ for geometrical points are different from arithmetical units, and straight lines are determined by two points in a different way from that in which the number two is composed of two units.

Generation of the Physical World

Having in some way generated geometrical figures out of numbers, the Pythagoreans' next task was to generate the physical world.

9.24 From the unit and the indefinite dyad spring numbers; from numbers, points; from points, lines; from lines, plane figures; from plane figures, solid figures; from solid figures, sensible bodies, the elements of which are four: fire, water, earth, and air; these elements interchange and turn into one another completely, and combine to produce a universe [KOSMOS] animate, intelligent, spherical, with the earth at its center, the earth itself being spherical, and inhabited round about.

(Alexander Polyhistor, Pythagorean Notebooks, quoted in Diogenes Laertius, Lives of the Philosophers 8.25 = DK 58Bla, tr. after Hicks)

The reference to the four elements may make this theory later than Empedocles, who seems to have been the first to base a physical theory on them, but even so, it is likely to describe a genuinely Pythagorean theory. Another passage says more about the geometrical structure of the four elements.

9.25 There being five solid figures called the mathematical solids, Pythagoras says that earth is made from the cube, fire from the pyramid, air from the octahedron, water from the icosahedron, and from the dodecahedron is made the sphere of the whole.

(Aetius 2.6.5 = DK 44A15)

The shapes mentioned are the five regular solids, the only geometrical solids in which all edges are equal, all faces are congruent and equilateral, and all vertices form equal solid angles … The cube has six faces that are square; the pyramid, octahedron, and icosahedron have for faces four, eight, and twenty equilateral triangles respectively; and the dodecahedron has twelve regular pentagons. Clearly the dodecahedron is not a sphere, although it and all the others can be inscribed in a sphere (all their vertices touch the surface of the smallest sphere that encloses them). 9.25 may allude to making balls by cutting twelve pieces of leather into pentagons, sewing them together and then stuffing them so that they fill out to a spherical shape (like soccer balls), but the details of the theory remain obscure.⁴⁰ Nev-ertheless, the attempt to account for the physical nature of the universe in terms of basic kinds of matter and to analyze these in terms of the small, definite, finite number of simple geometrical bodies is thoroughly Pythagorean in nature.

Alongside this "mathematical reductionist" approach we find traces of a cosmogony which makes the origin of the KOSMOS analogous to the generation of numbers.

9.26 When the unit had been constructed—whether from planes or surfaces or seed or from something they are at a loss to specify—the nearest parts of the unlimited at once began to be drawn in and limited by the limit.

(Aristotle, Metaphysics 14.3 1091a15–18 = DK 58B26)

9.27 The Pythagoreans also said that void exists, and enters the universe from the unlimited breath, the universe being supposed in fact to inhale the void, which distinguishes things. For void is that which separates and distinguishes things that are next to each other. This happens first in numbers; the void divides their nature.

(Aristotle, Physics 4.6 213b22–27 = DK 58B30)

9.28 The universe is unique, and from the unlimited it draws in time, breath, and void, which distinguishes the places of separate things.

(Aristotle, fr. 201 [Rose] = DK 58B30)

This account has both early and late elements. The idea that unlimited breath surrounds the KOSMOS recalls Anaximenes…, and the picture of the KOSMOS growing by inhaling this breath is at home among early Ionian ideas, while the conception of this breath as void cannot antedate Parmenides and may show influence of the fifth-century Atomists. The overall picture is that the universe is formed by the imposition of limit on the unlimited. Limit, determinacy, definiteness, and number are associated with order and intelligibility. As the musical scale is formed by imposing determinate numerical relations on the indefinite and continuous spectrum of sound, and numbers are generated when the determinate unit (representing limit) imposes order on the "indefinite dyad" (representing the unlimited [9.24]), the KOSMOS too is formed when the unit (representing limit) operates on the unlimited. Order begins in the center of the universe and expands by assimilating unordered, unlimited stuff into the ordered universe.

The KOSMOS, being ordered and limited, is finite in extent, and hence must have a physical boundary and a geometrical shape. Thus the Pythagoreans could put to use the dodecahedron, which was left out when shapes were assigned to the elements. An important feature of the order in the KOSMOS is that its parts are separate, distinct from one another. The void keeps things apart and performs an analogous function in the ordered realm of discrete, whole numbers, separating each from the rest and guaranteeing to each its identity and uniqueness. 9.28 adds that time was also drawn in from the unlimited. The idea here is that the unlimited has a temporal as well as a spatial aspect, and the KOSMOS has both spatial and temporal order which are imposed in analogous ways by the limiting principle.

The Pythagorean cosmogony is different from the Ionian ones—so different, in fact, that it is hard to believe that the KOSMOS that results is the world around us. At the least it seems that the account fails to address a number of crucial issues. As the following passage shows, Aristotle, who wrote a (no longer extant) treatise on the Pythagoreans, and so must have had access to relevant materials, shares these feelings.

9.29 Those called Pythagoreans use stranger principles and elements than the natural philosophers do. The reason is that they did not take their principles from perceptible things … yet everything they discuss and treat has to do with nature; for they generate the heaven and observe what happens regarding its parts, its attributes and the events in it, and use up the principles and causes on these, as if they agreed with the others, the natural philosophers, that what exists is precisely all that is perceptible and contained by what they call the heaven…. However, they say nothing about how there can be motion if limit and unlimited and odd and even are the only things assumed, or how without motion and change there can be generation and destruction, or the behavior of the bodies that move through the heavens.

(Aristotle, Metaphysics 1.8 989b29–990a12 = DK 58B22)

It is possible, then, that what the Pythagoreans said really was unclearly or incompletely stated, so perhaps the broad sketch given above of how they founded their KOSMOS on numbers and on the principle of imposing limit on the unlimited is as far as it is reasonable to go.

Pythagorean Cosmology

The Pythagorean account of the KOSMOS contains three noteworthy features: its rejection of the geocentric picture, the role of the number ten, and the harmony of the spheres.

9.30 Although most say that the earth is situated at the center … those in Italy called Pythagoreans assert the contrary opinion. For they declare that fire is at the center and the earth is one of the stars and by being carried in a circle round the center it causes night and day. Further, opposite to this one they construct another earth which they name "counter-earth." In this they are not inquiring for theories and causes with a view to the phenomena, but are forcing the phenomena to fit certain theories and opinions of their own, and trying to bring them into line. Many others agree that the earth should not be put at the center, finding reliability on the basis not of the phenomena but rather of their theories. For they believe that the most honorable thing deserves to have the most honorable region, and that fire is more honorable than earth, and that the limit is more honorable than what is intermediate, and the extremity and the center are limits. So, reasoning from these premises they think that not it but fire is situated at the center of the sphere. Moreover, the Pythagoreans call the fire occupying this region Zeus' guardhouse because the most important part of the universe should be the best guarded, and the center is most important, as if "center" had a single meaning and the center of the spatial extension and of the thing itself were also the natural center. But just as in animals the center of the animal is not the same as the center of its body, we must suppose the same to hold concerning the whole heaven.

(Aristotle, On the Heaven 2.13 293a18–b8 = DK 58B36)

9.31 Philolaus says that there is fire in the middle around the center, which he calls the hearth of the universe and the house of Zeus and the mother of the gods, and the altar, bond, and measure of nature. Moreover, he says that what surrounds the universe at the furthest extreme is another fire. The center is by nature first. Around it ten divine bodies dance—after the sphere of the fixed stars, the five planets; after them, the sun; beneath it, the moon; beneath it, the earth; beneath it, the counter-earth; after them all, the fire of the hearth keeping its position around the center.

(Aetius 2.7.7. = DK 44A16)

9.32 They supposed … the entire heaven to be a HARMONIA and a number. And all the characteristics of numbers and HARMONI AI (plural of HARMONIA) they found corresponding to the attributes and parts of the heaven and to the entire ordering, they collected and made them fit. If anything was missing anywhere they eagerly filled in the gaps to make their entire system coherent. For example, since they think the number ten is something perfect and encompasses the entire nature of numbers, they declare that the bodies that move in the heaven are also ten. But since only nine are visible, they invent the counter-earth as the tenth.

(Aristotle, Metaphysics 1.5 986a2–12 = DK 58B4) (continuation of 9.19)

The KOSMOS consists of a fiery center orbited by the counter-earth, earth, moon, sun, Mercury, Venus, Mars, Jupiter, Saturn, and the fixed stars.⁴¹ 9.32 objects to the Pythagoreans' reason for positing the counter-earth, though modern physicists, who are used to positing the existence of entities on the basis of theory, might be somewhat sympathetic. Some ancient sources, including perhaps Aristotle himself, say that the Pythagoreans used the counter-earth to account for lunar eclipses.⁴²

The Pythagoreans were the first to remove the earth from the center of the KOSMOS, and their reasons as reported in 9.30 and 9.31 are not astronomical, but religious ("house of Zeus") and metaphysical ("the center is the most important"). It is not surprising that this idea was not adopted even by all Pythagoreans (9.24 shows that some Pythagoreans placed the earth at the center of the universe) or by most other ancient astronomers, who retained the traditional geocentric view.⁴³ However, the proponents of this theory defended it against the astronomical objection that the circular appearing orbits of the heavenly bodies imply that the earth is in the center of the KOSMOS.

9.33 Since the earth's surface is not in fact the center, but is distant from the center by its whole hemisphere [i.e., radius], the Pythagoreans feel no difficulty in supposing that although we do not occupy the center the phenomena are the same as if the earth were at the center. For they hold that even on the current view [that the earth is at the center] there is nothing to show that we are distant from the center by half the earth's diameter.

(Aristotle, On the Heaven 2.13 293b25–30, not in DK)

Nevertheless, the claim that some have maintained, that the Pythagoreans discovered that the earth is a planet, overstates the case, seeing that they held the false view that the earth (as well as the sun, moon, etc.) all go round a central fire, rather than round the sun. Nevertheless, the fact that they could conceive of the earth's not being at the center is an important advance, and from this perspective "the identification of the central fire with the sun is a detail in comparison."⁴⁴

The Pythagorean doctrine of the music of the spheres was based on several basic features of Pythagoreanism—harmonics, cosmology, and mathematics—and caught the fancy of literary authors in later antiquity and the Renaissance. The clearest and most sober account is given by Aristotle.⁴⁵

9.34 Although the assertion that a harmony arises from the motion of the heavenly bodies since the sounds that are produced are concordant, is expressed cleverly and remarkably by its proponents, it does not contain the truth. For some think a sound must be produced when bodies of such great size are in motion, since it happens with bodies on earth too which do not have so great a bulk and do not move with so great speed. And when the sun and moon and the stars which are so great in number and size move so quickly, there must be a noise overwhelming in loudness. Assuming these things and that the speeds, which depend on the distances, have the ratios of the concords, they declare that the sound of the stars in circular motion is harmonious. But since it appeared illogical that we do not hear this sound, they declare that the reason is that the sound is present to us from birth, and so is not evident in contrast to the opposing silence, for noise and silence are recognized by contrast to one another. And so the same thing happens to humans as to bronzesmiths: as a result of habituation there seems to be no difference.

(Aristotle, On the Heaven 2.9 290b12–29 = DK 58B35)

This gives some meaning to the statement that "they supposed the whole heaven to be an attunement and a number" (9.32). The pitches of the various notes correspond to their speeds, which depend on their distances (from the central fire). Indeed the Pythagoreans are said to have been "first to discover the order of the positions of the planets."⁴⁶

Opposites

The importance of the notion of opposition, already present in Anaximander and Anaximenes, continues in later philosophers, including the Pythagoreans, some of whom developed it in a distinctive way.

9.35 Others of this same school declare that there are ten principles arranged in parallel columns:

limit	unlimited
odd	even
one	plurality
right	left
male	female
at rest	moving
straight	bent
light	darkness
good	evil
square	oblong

This is how Alcmaeon of Croton too seems to have understood things, and either he took this theory from them or they from him…. He says that most human matters are pairs, identifying as the oppositions not definite ones like the Pythagoreans … but the Pythagoreans described how many and what the oppositions are.

(Aristotle, Metaphysics 1.5 986a22-b2 = DK 58B5) (continuation of 9.20)

The table of opposites contains twenty opposites but ten principles, each pair counting as one principle. The table manifests interest in a wide range of aspects of the world, including moral values, which accords with the Pythagoreans' use of numbers to account for features of the physical universe and also for qualities like justice (9.36).

The table displays many Pythagorean ideas. First, the number of pairs of basic opposites is ten. Second, the prominence of limit and unlimited, followed by odd and even, recalls the accounts of the generation of number in 9.19-9.22. One and plurality, lined up respectively with limit and unlimited, recall another account of the generation of number and of the KOSMOS (9.24). Square and oblong bring to mind the properties of square and oblong numbers discussed previously. The remaining pairs of opposites are diverse and not in all cases clearly related.

From the point of view of logic, each pair seems intended to consist of mutually exclusive items. Some pairs seem intended to exhaust their fields of application (all animals are either male or female, all whole numbers are either odd or even) and some do not (some numbers are neither square nor oblong). Some items admit degrees (moving, bent), while others do not. From the point of view of Pythagorean metaphysics, some of the pairs are basic (odd and even, cf. 9.19 and 9.20; alternatively, limit and unlimited, cf. 9.21), and some are derivative (one and plurality, cf. 9.20). But the table leaves some important issues open. No effort is made to distinguish the types of opposition involved, and there is no explanation of the way in which these opposites are principles or of why these particular pairs of opposites are chosen instead of those which figure conspicuously in earlier cosmologies, such as dense and rare, hot and cold, or wet and dry. Indeed, if odd and even (or limit and unlimited) are the principles of all things, how can there be any other principles?

Things and Numbers

The Pythagoreans extended their program of accounting for phenomena in terms of number in surprising directions.

9.36 In numbers they thought they observed many resemblances to the things that are and that come to be … such and such an attribute of numbers being justice, another being soul and intellect, another being decisive moment, and similarly for virtually all other things … since all other things seemed to be made in the likeness of numbers in their entire nature.

(Aristotle, Metaphysics 1.5 985b28-33 = DK 58B4)

Further insight into this aspect of Pythagoreanism is found in the following passages, of which the first comments on 9.36.

9.37 They supposed that requital and equality were characteristic of justice and found these features in numbers, and so declared that justice was the first number that is equal-times-equal…. They said that decisive moment is the number seven, since things which are natural appear to have their decisive moments of fulfillment in birth and growth by sevens. Humans, for example. They are born in the seventh month and teethe in as many months, and reach adolescence in the second span of seven years and get a beard in the third…. They said that marriage is the number five, because marriage is the union of male and female, and according to them the odd is male and the even is female, and this number is the first which has its origin from two, the first even number, and three, the first odd…. They declared intellect and essence to be the one, since he spoke of the soul as the intelligence. They said that because it is stable and similar in every way and sovereign, the intelligence is the unity and one.

(Alexander, Commentary on Aristotle 's Metaphysics 38.10-39.20, not in DK)

9.38 Concerning what things are, they began to make statements and definitions, but treated the matter too simply. For they would define superficially and thought that the first thing an indicated term applies to was the essence of the thing, as if one were to suppose that double and the number two are the same because two is the first thing double applies to. But surely to be double and to be two are not the same; otherwise one thing will be many—a consequence they actually draw.

(Aristotle, Metaphysics 1.5 987a20-27 = DK 58B8)

Some of these cases reveal the reductionist reasoning found elsewhere, but the association of "decisive moment" with the number seven (9.37) is based on the wildest sort of speculative association, which is found in Neopythagorean treatments of other numbers.⁴⁷ One Pythagorean extended this approach to concrete substances.

9.39 Eurytus assigned what was the number of what, e.g., this is the number of a human, that is the number of a horse, like those who bring numbers into triangular and square figures, fashioning with pebbles the forms of plants.

(Aristotle, Metaphysics 14.5 1092b10-13 = DK 45,3)

9.40 For example, suppose the number 250 is the definition of human being…. After positing this, he would take 250 pebbles, some green, some black, others red, and generally pebbles of all colors. Then he smeared a wall with lime and drew a human being in outline … and then fastened some of these pebbles in the drawn face, others in the hands, others elsewhere, and he completed the drawing of the human being there represented by means of pebbles equal to the units which he declared define human being. As a result of this procedure he would state that just as the particular sketched human being is composed of, say, 250 pebbles, so a real human being is defined by so many units.

(Alexander, Commentary on Aristotle's Metaphysics 827.9-19 = DK 45,3)

In a procedure only distantly related to the reasoning given in 9.37, Eurytus displays the number of a human being by placing pebbles on his diagram, thus showing that they are the smallest number that can fill in the shape of a human.

The Pythagoreans founded their account of the KOSMOS on numbers. But how did they think numbers are related to things? Aristotle makes several remarks on this subject (also 9.36).

9.41 Because they noticed that many attributes of numbers belong to sensible objects, the Pythagoreans held that existing things are numbers—not separate numbers, but composed of numbers [literally, out of numbers]. Why so? Because the attributes of numbers are found in HARMONIA, in the heaven and in many other things.

(Aristotle, Metaphysics 14.3 1090a20-25, not in DK)

9.42 In making physical bodies (things possessing lightness and weight) out of numbers (which possess neither) they seem to be speaking about a different heaven and different bodies, not sensible ones.

(Aristotle, Metaphysics 14.3 1090a32-35, not in DK)

9.43 They supposed the elements of numbers to be the elements of all existing things.

(Aristotle, Metaphysics 1.5 986al-2 = DK 58B4)

Aristotle expresses the Pythagorean view in four ways: (a) things are identical with numbers; (b) things are composed of numbers; (c) things resemble numbers; (d) the principles of numbers are the principles of all things. Some of these claims are puzzling. Regarding interpretation (a), even though four points determine the vertices of the simplest geometrical solid, how can the tetrahedron be identical with the number four? Likewise there may be a reason to associate justice with the first square number, but how can justice be the number four? Also identity is a transitive relation, but how can justice be a tetrahedron? Aristotle makes this objection too.

9.44 If all things must share in number, many things must turn out to be the same, and the same number must belong to one thing and to another…. Therefore, if the same number had belonged to certain things, these would have been the same as one another, since they would have had the same form of number.

(Aristotle, Metaphysics 14.6 1093al-13 = DK 58B27)

Interpretation (b) involves other difficulties. If figures and physical objects are composed of numbers, then numbers must have size and weight. Thus, Aristotle:

9.45 The Pythagoreans say that there is one kind of number, the mathematical kind, only it is not separate, but they hold that sensible substances are constituted out of it. For they construct the entire heaven out of numbers, only not units, but they suppose the units to possess magnitude. But they seem to be at a loss about how the first one possessing magnitude was constituted.

(Aristotle, Metaphysics 13.6 1080b 16-21 = DK 58B9)

9.46 In one way the Pythagoreans' approach has fewer problems than the previously mentioned ones, but in another way it has others of its own. Making number not separate removes many of the impossibilities, but it is impossible for bodies to be composed of numbers and for this number to be mathematical. For it is not true to speak of indivisible magnitudes, but even if this were very much the case, units, anyway, do not possess magnitude. How can magnitude be made up of indivisible things? But arithmetical number is made of units. And they say that existing things are number. Anyway they apply theorems to bodies as if they were composed of those numbers.

(Aristotle, Metaphysics 13.8 1083b8-19 = DK 58B10)

From these passages it appears that at least some Pythagoreans said that things are numbers. In Greek as in English, the statement "this stick is wood" can mean either (a) "this stick is identical with a particular piece of wood" or (b) "this stick is made of wood." (a) is a statement of identity and (b) is a statement of composition. Identity statements are transitive: if this stick is identical with a particular piece of wood and is also identical with a particular branch of the oak tree in my back yard, then that piece of wood is that branch. Composition statements are not transitive: if this stick is made of wood and that table is also made of wood, it does not follow that this stick is made of that table, or that table is made of this stick, nor does the identity statement "this stick is that table" follow. Both these uses of "is" are so familiar that we take such statements in the appropriate way without reflection. When Anaximenes says "everything is air" we take him to be making a composition statement, not an identity statement. There is no difficulty because Anaximenes' assertion is intelligible as a theory about the composition of the world, whereas the Pythagorean claim that things are numbers, if taken as a composition statement, is absurd. Although air is a material substance and so the right kind of thing for other things to be made of, numbers are different. Aristotle shares our discomfort at the thought that "things are numbers" could mean that they are composed of numbers. But if it is not a composition statement, it is natural to interpret it as an identity statement. But here too, we run into serious objections—in fact the objections Aristotle raises, the consequence that many different things will wind up with the same number and so are identical.

Aristotle criticizes Pythagorean philosophy on the grounds that it leads to absurd consequences, and he is surely correct if the theory asserts that numbers are identical with things or that things are composed of numbers. However, interpretation (c), that things resemble numbers, is not open to these objections and also has links with central features of Pythagorean thought.⁴⁸ There are many ways in which things may resemble numbers. 9.37 points out some ways in which qualities such as justice can be thought of as resembling numbers, i.e., by having some of the same properties as a particular number. More generally, numbers, geometrical figures, the physical KOSMOS, and musical scales are generated similarly: all come to be when limit is imposed on the unlimited. All are instances of order, perhaps even of sequential order, which exists in different realms. And they all have numerical aspects that are basic: the number of sides of a triangle, the number and distances of the heavenly bodies, the ratios of the lengths of strings. Moreover, the analysis of the generation of all these things in terms of limit being imposed on the unlimited gives a clear sense to interpretation (d), the principles of number are the principles of all things.

The Pythagoreans believed that number is fundamental to all things, that the basic features of all things are numerical, that numerical considerations are basic in understanding all things, that all things are generated in a similar way to numbers. These statements are all ways of claiming primacy for numbers, but they are different ways. The Pythagoreans noticed all these ways, but instead of keeping them distinct gath ered them together into a single thought. One way of expressing the point is to say that they did not think that number is fundamental in many distinct and perhaps unrelated ways, some of which apply here and others there, but simply thought that number is fun damental and looked for evidence to support this claim. The difference is important. The Pythagoreans piled up evidence without calling attention to how different the bits of evidence are. They were not interested in analyzing different ways numbers are primary, only in establishing that numbers are in fact primary. They formulated their thesis vaguely, to accommodate the different relations they found between things and numbers, and to judge by the different ways Aristotle states their doctrine, they phrased it differently on different occasions. Also, to judge by Aristotle's criticisms, their vague notion of priority does not stand up to analysis, but as soon as the questions are put "in what way are numbers primary?" and "in what way are all things numbers?" it becomes necessary to specify once again all the different ways: different things are numbers, or imitate numbers, or resemble numbers, or are generated in the same way as numbers, etc.

These problems arise for the Pythagoreans because they based their physical system on numbers. How numbers are basic to the universe and things around us is less straightforward a matter than how a substance like air is, and the Ionian background offered little help towards drawing the necessary distinctions and analyzing connections at a sufficiently abstract level to identify the issues involved or offer a philosophically satisfactory account. What does it mean, for example, to say that the One is generated out of odd and even, or that the universe is composed of numbers, or that justice is the number four? What notions of generation, composition, and identity are in play—and if these are not precisely the notions in play, what relations are meant?

In fact, the Pythagoreans probably could not express their ideas accurately, given the state of the Greek language and the primitive state of philosophical analysis in their time. In the fifth century, Greek lacked most of the philosophical vocabulary needed to distinguish between sameness and resemblance (the same Greek word HOMOIOS meant both "same" and "similar"), identity and composition (the two uses of "is" discussed above), or origin and metaphysical structure. (In Greek, to say that one thing [A] is or comes "out of another [B] can mean that what was once B is now A, or that A is made up of B, or that A depends on B, or that A can be analyzed into B.) These ambiguities need to be resolved before statements like the ones the Pythagoreans made about number can be fully understood, but nothing in earlier philosophy encouraged Pythagoras or his early followers to make fine distinctions. In fact, the philo sophical work needed for the task was not undertaken before Socrates, Plato, and Aristotle, whose evident frustration with the Pythagoreans reflects the intellectual distance that separates him from ideas formulated only two or three generations before.

The Unity of Pythagoreanism

A general problem for understanding the Pythagoreans is why a religious movement dedicated to purifying the soul should have promoted mathematics and the study of the KOSMOS. In other words, how are the two sides of the movement related? Do they form a unity? I believe they do, and the connection between them may go back to the Founder. Other Greek cults promised their devotees immortality, but how can some souls be immortal while others are mortal, and how can attending religious rites make souls immortal? Milesian speculation on the nature of the KOSMOS and the composition of things including souls pointed to the view that souls are made up of the basic stuff of the universe and so, immortal. The issue is thus not how to gain immortality, but how best to use it. Pythagoras taught that the best and most important thing to do is to purify the soul, to rid it of pollution and disorder, because pure souls have the best afterlife, and perhaps ultimately attain a kind of divinity.

Distinctive to Pythagoreanism is idea that purification is not achieved solely by ritual means. It requires more than abstaining from meat and beans and more than obedience to the AKOUSMATA. It also requires eliminating the disorder which affects our soul when we lack clear knowledge of the KOSMOS. For the Pythagoreans (more precisely, the MATHEMATIKOI, this clear knowledge is not simply a matter of parroting a set of beliefs, saying a catechism of fixed doctrine without understanding. It involves the study of mathematics and the KOSMOS. The numerical basis of the KOSMOS implies that the KOSMOS is comprehensible to humans, and the knowledge of it which benefits our soul demands thought and understanding. Our soul becomes orderly (KOSMIOS) when it understands the order (KOSMOS) in the universe.⁴⁹ This is the inspiration that underlay the developments in Pythagorean thought and which gives the Pythagoreans much common ground with their Ionian predecessors as well as with their successors in mathematics, science, and philosophy.

Conclusion

Pythagoreanism was a two-faced movement that combined primitive ingredients with ideas still current today into a doctrine at home in the presocratic period. Its breadth of interest is typified by its concern with the individual soul on the one hand and with the structure of the universe on the other, and is represented by the ten pairs of fundamental opposites. The Pythagoreans shared with their Ionian predecessors an interest in the physical world and the goal of explaining it through a small number of basic principles, as well as the confidence to base a theory on a breathtaking generalization from a limited range of evidence. Different was their proclamation of the fundamental importance of number in the world. Instead of basing other things on a material substance such as water or air, they explained them in terms of numbers. For this they are given credit for recognizing the importance of the quantitative aspects of phenomena and for the first reduction of quality to quantity (in their numerical account of the concordant musical intervals). On the other hand, the clear distinction between quantity and quality was not made until Aristotle, and in the absence of this and other relevant philosophical distinctions the Pythagoreans literally did not know what they were doing. However, their mathematical explorations made a lasting contribution. They were concerned to define mathematical concepts and invented the field of number theory. They were also involved in the development of geometry, and it is possible, but no more than that, that they created the notion of mathematical proof. Their cosmology is a blend of their mathematics, their musical theory, their religious ideas, and their mystical numerology. In its details it is noteworthy for removing the earth from the center of the universe and for postulating the harmony of the spheres. The main philosophical interest of their discussion of the universe is in its account of the origin, in which the KOSMOS resembles number, geometrical figures, and the musical intervals by being the product of the imposition of limit on the unlimited. Their failure to distinguish between the nature of numbers and the nature of material objects, however, leaves them open to charges that their cosmogony attempts the impossible, to make numbers the physical constituents of material things. Their doctrines of the soul's immortality, its rebirth into different living things, and the possibility of its ultimate release into a better existence have practical implications for how Pythagoreans should live their lives. The beliefs that a living being is composed of a body and a soul and that the soul is more important than the body would have an important legacy in ethical and metaphysical as well as religious thought. Finally, the bold conception of the universe in all its aspects—includ ing the living and nonliving, the cosmological, mu sical and mathematical, and the ethical—as an intelligible, ordered whole, in a word a KOSMOS, was the ultimate basis of their thought and life.

Notes

¹ For this term, see p. 361 [of Philosophy before Socrates'].

² Diogenes Laertius, Lives of the Philosophers 8.3 (not in DK).

³ Literally, "beyond the North Wind." Several myths associate Apollo with this distant place.

⁴ Aristotle fr. 191 (Rose), Aelian Varia Historia 2.26 (both = DK 14,7).

⁵ Aristotle, fr. 192 (Rose) (= DK 14,7).

⁶ See p. 99 [of Philosophy before Socrates].

⁷ These pupils are Dicaearchus and Aristoxenus of Tarentum, a friend of the group known as the last generation of Pythagoreans, who are identified as pupils of Philolaus. Their works are excerpted in Neoplatonic works.

⁸ Most interpreters believe that these verses praise Pythagoras, but there was doubt even in antiquity. According to Diogenes Laertius, Lives of the Philosophers 8.54 (= DK 31A1), some held that they describe Arsenides.

⁹ See p. 75 [of Philosophy before Socrates].

¹⁰ Some believe that 9.2 pairs Pythagoras with Hesiod as opposed to Xenophanes and Hecataeus, the former as religious thinkers, the latter as representing the new ways of thought, and that the writings 9.3 refers to are writings of the Orphic sects who believed in an afterlife and the immortality of the soul. This is the interpretation of W. Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Mass., 1972; first German ed., 1962).

¹¹ See 10.5, 10.16, 10.71 and discussion on p. 131 [in Philosophy before Socrates].

¹² The word is also rendered "worthless artifice" (Guthrie, HGP, vol. 1, p. 157), "artful knavery" (KRS, p. 217), and "imposture" (Burnet, Early Greek Philosophy, p. 134).

¹³ Not including Herodotus himself, who is sceptical about the whole tale.

¹⁴ Many give priority to Orphism. (See n. 83, n. 10.)

¹⁵ Plato, Phaedo 85e-86d.

¹⁶ Guthrie, HGP, vol. 1, pp. 306-19. Guthrie discounts the idea that the soul is a HARMONIA of bodily parts as an intrusion from fifth-century medical writers like Alcmaeon who believed that health is a balance, or HARMONIA, of the elements or parts of the body.

¹⁷ A case for the latter view is made on the basis of little evidence by Guthrie, HGP, vol. 1, p. 203.

¹⁸ Heraclides of Pontus was a pupil of Plato and a contemporary of Aristotle, and had a special interest in the Pythagorean movement.

¹⁹ The date of the Trojan War was disputed in antiquity, but it was usually put at about 1200 B.C.

²⁰ Empedocles speaks of the DAIMON instead of the soul. For the equivalence of the two notions in Empedocles, see [below].

²¹ Pindar, fr. 133 (not in DK) = KRS passage 410, p. 317.

²² Pindar, Olympians 2.56-77 (not in DK) = KRS passage 284, p. 286.

²³ See Empedocles, 14.137.

²⁴ See p. 286 [in Philosophy before Socrates].

²⁵ See p. 87 [in Philosophy before Socrates].

²⁶ These and other explanations are discussed by Guthrie, HGP, vol. 1, pp. 184-85.

²⁷ R. Brumbaugh and J. Schwartz, "Pythagoras and Beans. A Medical Explanation," Classical World 73 (1980): 421-22.

²⁸ For the soul as HARMONIA, see p. 85 [in Philosophy before Socrates].

²⁹ The text says merely "and apart." Perhaps this refers to a diagram Aristotle drew as he gave the lectures for which the Physics is the notes. What is "apart" is the extra row of dots, which when added to the square numbers makes oblong numbers.

³⁰ Diogenes Laertius, Lives of the Philosophers 8.12 (not in DK); Porphyry, Life of Pythagoras 36 (not in DK); Athenaeus, Table Talk 10.13 (not in DK). But it is far from certain that Pythagoras actually made the discovery. See T. L. Heath, History of Greek Mathematics, vol. 1 (Oxford, 1921), pp. 144-49.

³¹ Iamblichus, Life of Pythagoras 36 (not in DK), asserts that Theodorus was a Pythagorean.

³² Plato, Theaetetus 147d (= DK 43,4).

³³ A. Szabo, The Beginnings of Greek Mathematics (Dordrecht and Boston, 1978), pp. 185-216.

³⁴Ibid., pp. 216-20.

³⁵ See chaps. 11, 12.

³⁶ It cannot be excluded that the inspiration for deductive proofs came from the Ionian geometrical tradition that originated with Thales and which made important contributions well into the fourth century, even though this method of argument was not characteristic of early Ionian philosophy. See Guthrie, HGP, vol. 1, pp. 218-19 for a statement of the pro-Ionian, anti-Pythagorean view. B. L. vander Waerden advocates the view that Thales invented the notion of mathematical proof (Science Awakening [Groningen, 1954], pp. 87-90).

³⁷ Aristotle, Metaphysics 13.9 1084b26-27 (not in DK) and On the Soul 1.5 409a6 (not in DK).

³⁸ This last result and the fact that all plane figures with straight sides can be broken up into triangles were known to Euclid, and so may have been known to the Pythagoreans, but they do not seem to be the inspiration for this generation of lines, planes, and solids out of points.

³⁹ See [above].

⁴⁰ This view of the four elements and the shape of the KOSMOS is also found in Plato (Plato, Timaeus 53c-57c, not in DK), and it is disputed whether Plato owed these ideas to the Pythagoreans or whether later writers wrongly ascribed to the Pythagoreans Plato's theory.

⁴¹ The outer planets are invisible to the naked eye and were only discovered in 1781 (Uranus), 1846 (Neptune), and 1930 (Pluto).

⁴² Aristotle, On the Heaven 2.13 293b21-25 (not in DK), Aetius 2.29.4 (= DK 58B36).

⁴³ An important ancient exception to this view was the theory of Aristarchus of Samos (first half of the third century) who hypothesized that "the fixed stars and the sun remain unmoved and that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit" (Archimedes, The Sand-Reckoner 4-5, not in DK).

⁴⁴ Burnet, Early Greek Philosophy, p. 299.

⁴⁵ This doctrine first appears in Plato (Plato, Republic 10 616b-617d, not in DK).

⁴⁶ Eudemus, quoted in Simplicius, Commentary on Aristotle's On the Heaven 471.5-6 (= DK 12A19); Guthrie, HGP, vol. 1, p. 298.

⁴⁷ Iamblichus, Theologoumena Arithmeticae, contains much of this kind of fanciful material. Some of the material in 9.37 may be due to Neopythagorean sources too.

⁴⁸ The Pythagorean doctrine that things resemble numbers is a probable forerunner of Plato's doctrine that sensible things resemble or imitate Forms. Many of the philosophical problems inherent in the Pythagorean conceptions of the relations between things and numbers—issues of identity, resemblance, and predication—also arise for Plato, who struggles with them in such dialogues as Phaedo, Parmenides, and Sophist.

⁴⁹ Plato, Republic 6 500c, (not in DK). The idea is nicely developed in Guthrie, HGP, vol. 1, pp. 206-12.