Pythagoreanism (Stanford Encyclopedia of Philosophy
Pythagoreanism

First published Wed Mar 29, 2006

Pythagoreanism can be defined in a number of ways.

(1) Pythagoreanism is the philosophy of the ancient Greek philosopher Pythagoras (ca. 570-ca. 490 BC), which prescribed a highly structured way of life and espoused the doctrine of metempsychosis (transmigration of the soul after death into a new body, human or animal).

(2) Pythagoreanism is the philosophy of a group of philosophers active in the fifth and the first half of the fourth century BC, whom Aristotle refers to as “the so-called Pythagoreans” and to whom Plato also refers. Aristotle's expression, “so-called Pythagoreans,” suggests both that at his time this group of thinkers was commonly called Pythagoreans and, at the same time, calls into question the actual connection between these thinkers and Pythagoras himself. Aristotle ascribes no specific names to these Pythagoreans, but the philosophy which he assigns to them is very similar to what is found in the fragments of Philolaus of Croton (ca. 470-ca. 390 BC). Thus, Philolaus and his successor Eurytus are likely to have been the most prominent of these Pythagoreans. Philolaus posits limiters and unlimiteds as first principles and emphasizes the role of number in understanding the cosmos. Aristotle also identifies a distinct group of these so-called Pythagoreans who formulated a set of basic principles known as the table of opposites. Plato's sole reference to Pythagoreans cites their search for the numerical structure of contemporary music and is probably an allusion to Archytas (ca. 420-ca. 350 BC), who is the first great mathematician in the Pythagorean tradition. Starting from the system of Philolaus he developed his own sophisticated account of the world in terms of mathematical proportion.

(3) Many other fifth- and fourth-century thinkers are labeled Pythagoreans in the Greek tradition after the fourth century BC. By the late fourth century AD many of the most prominent Greek philosophers including Parmenides, Plato and Aristotle come to be called Pythagoreans, with no historical justification. There are nonetheless a number of thinkers of the fifth and fourth century BC, who can legitimately be called Pythagoreans, although often little is known about them except their names. The most important of these figures is Hippasus.

(4) The last manifestation of Pythagoreanism, Neopythagoreanism, has been the most influential. Neopythagoreanism is not a unified school of thought but rather a tendency, stretching over many centuries, to view Pythagoras, with no historical justification, as the central and original figure in the whole Greek philosophical tradition. This Pythagoras is often thought to have received his philosophy as a divine revelation, which had been given even earlier to wise men of the ancient Near East such as the Persian Magi, the Hebrews (Moses in particular), and the Egyptian priests. All Greek philosophy after Pythagoras, insofar as it may be true, is seen as derived from this revelation. Thus, Plato's and Aristotle's ideas are viewed as derived from Pythagoras (with the mediation of other early Pythagoreans). Many pseudepigrapha are produced in later times in order to provide the Pythagorean “originals” on which Plato and Aristotle drew. Some strands of the Neopythagorean tradition emphasize Pythagoras as master metaphysician, who supposedly originated what are, in fact, the principles of Plato's later metaphysics, the one and the indefinite dyad. Other Neopythagoreans celebrate Pythagoras as the founder of the quadrivium of mathematical sciences (arithmetic, geometry, astronomy and music), while still others portray him as a magician or as a religious expert and sage, upon whom we should model our lives. Neopythagoreanism began already in the second half of the fourth century BC among Plato's first successors in the Academy, but particularly flourished from the first century BC until the end of antiquity. Neopythagoreanism has close connections to Middle and Neoplatonism and from the time of Iamblichus (4th c. AD) is largely absorbed into Neoplatonism. It was the Neopythagorean version of Pythagoreanism that dominated in the Middle Ages and Renaissance.

1. The Philosophy of Pythagoras

2. Aristotle's and Plato's Pythagoreans

2.1 Philolaus

2.2 Eurytus

2.3 The Pythagoreans of the Table of Opposites

2.4 Archytas

3. Other Pythagoreans of the Sixth, Fifth and Fourth Centuries

3.1 The Catalogue of Pythagoreans in Iamblichus' On the Pythagorean Life: Who Counts as a Pythagorean?

3.2 The Earliest Pythagoreans: Brontinus, Theano, etc.

3.3 Hippasus and Other Fifth-century Pythagoreans

3.4 The Fourth Century: Aristoxenus, the Last of the Pythagoreans, and the Pythagorists

3.5 Timaeus, Ocellus, Hicetas and Ecphantus

4. Neopythagoreanism

4.1 Origins in the Early Academy: Speusippus, Xenocrates and Heraclides

4.2 The Pythagorean Pseudepigrapha

4.3 Neopythagorean Metaphysics: Eudorus, Moderatus and Numenius

4.4 Neopythagorean Mathematical Sciences: Nicomachus, Porphyry and Iamblichus

4.5 Pythagoras and Pythagoreans as Religious Experts, Magicians and Moral Exemplars: Pythagoreanism in Rome, The Golden Verses and Apollonius of Tyana

5. Pythagoreanism in the Middle Ages and Renaissance

5.1 Boethius/Nicomachus and the Middle Ages

5.2 The Renaissance: Ficino, Pico, Reuchlin, Copernicus and Kepler

Bibliography

Other Internet Resources

Related Entries

1. The Philosophy of Pythagoras

See the SEP entry on Pythagoras.

2. Aristotle's and Plato's Pythagoreans

2.1 Philolaus

See the SEP entry on Philolaus.

2.2 Eurytus

In the ancient sources, Eurytus is most frequently mentioned in the same breath as Philolaus, and he is probably the student of Philolaus (Iamblichus, VP 148, 139). Aristoxenus (4th c. BC) presents Philolaus and Eurytus as the teachers of the last generation of Pythagoreans (Diogenes Laertius VIII 46) and Diogenes Laertius reports that Plato came to Italy to meet Philolaus and Eurytus after the death of Socrates (III 46). In order to be the pupil of Philolaus, who was born around 470, and teach the last generation of Pythagoreans around 400, Eurytus would need to be born between 450 and 440. The sources are very confused as to which S. Italian city he was from, Croton (Iamblichus, VP 148), Tarentum (Iamblichus, VP 267; Diogenes Laertius VIII 46) or Metapontum (Iamblichus, VP 266 and 267). It may be that the Eurytus from Metapontum is a different Eurytus. It is possible that Archytas studied with Eurytus, since Theophrastus (Aristotle's successor in the Lyceum) cites Archytas as the source for the one testimony we have about the philosophy of Eurytus (Metaph. 6a 19-22). In the catalogue of Pythagoreans at the end of Iamblichus' On the Pythagorean Life (267), Eurytus appears between Philolaus and Archytas in the list of Pythagoreans from Tarentum, which may thus suggest that he was regarded as the pupil of Philolaus and a teacher of Archytas.

According to Theophrastus (Metaph. 6a 19-22), Eurytus arranged pebbles in a certain way in order to show the number which defined things in the world, such as a man or a horse. Aristotle refers to the same practice (Metaph. 1092b8 ff.), and Alexander provides commentary on the Aristotelian passage (CAG I. 827.9). Aristotle introduces Eurytus as someone who regarded numbers as causes of substances by being the points that bound spatial magnitudes. He says that Eurytus made likenesses of the shapes of things in the natural world with pebbles and thus determined the number which belongs to each thing by the number of pebbles required. Scholars often treat Eurytus' procedure as puerile and have sometimes not taken him seriously (Kahn 2001, 33), or suggested that Theophrastus is ironical in his presentation. There is, however, no obvious irony in Theophrastus' remarks. He, in fact, presents Eurytus very positively as someone who showed in detail how specific parts of the cosmos arose out of basic principles, in contrast to other thinkers, who posit basic principles but do not go very far in explaining how the world arises from those principles. This positive presentation may reflect Theophrastus' source, Archytas, who perhaps saw Eurytus as attempting to carry out Philolaus' project of determining the numbers that give us knowledge of things in the world (Huffman 2005, 55).

How are we, then, to understand Eurytus' procedure? It does not seem plausible to suppose that he simply drew a picture or an outline drawing of a man or a horse and then counted the number of pebbles required to make the outline (Riedweg 2005, 86) or fill in the picture, since the number would vary with the size of the drawing and the size of the pebbles. A large picture of a man would require many more pebbles than a small one, so that it would seem arbitrary which number to associate with man. Aristotle's presentation supports another interpretation. He draws a parallel with those who arrange numbers of pebbles into shapes, such as a triangle or a square. This suggests that Eurytus had observed that, e.g., any three points in a plane determine a triangle and any four a quadrilateral. He may then have drawn the general conclusion that any shape or structure was determined by a unique number of points and tried to represent these by setting out the necessary number of pebbles. Thus, the complex structure of a three-dimensional object such as the human body would require a large number of points, but the number of points required to determine a human being could be expected to be unique and to differ from the number that determined any other object in the natural world, such as a horse (Kirk and Raven 1957, 313 ff.; Guthrie 1962, 273 ff.; Barnes 1982, 390-391; Cambiano 1998). It is important to note that nothing in these reports suggests that Eurytus thought that things were composed of numbers or that he regarded the points that defined a given thing as atoms of which things were made, as has sometimes been supposed (Cornford 1922-1923, 10-11). Instead, he is best understood as making a bold attempt to show that the structure of all things is determined by number and thus to provide specifics for Philolaus' general thesis that all things are known through number.

2.3 The Pythagoreans of the Table of Opposites

At Metaphysics 986a22, after presenting his account of the philosophy of “the so-called” Pythagoreans (985b23), which has strong connections to the philosophy of Philolaus, Aristotle turns to “others of this same group” and assigns to them what is commonly known as the table of opposites (the opposites arranged according to column [kata sustoichian]). These Pythagoreans presented the principles of reality as consisting of ten pairs of opposites:

limit unlimited

odd even

unity plurality

right left

male female

rest motion

straight crooked

light darkness

good bad

square oblong

Aristotle then contrasts these Pythagoreans with Alcmaeon of Croton, who said that the majority of human things come in pairs, and praises the Pythagoreans for carefully defining the pairs of opposites both in number and character, whereas Alcmaeon seemed to present a randomly selected and ill-defined group of opposites. Aristotle suggests that either Alcmaeon was influenced by these Pythagoreans or they by him. Aristotle was thus not sure of the date of these Pythagoreans but seems to entertain the idea that they either lived a little before Alcmaeon or a little after, which would make them active anywhere from the late 6th to the mid 5th century. Aristotle's manner of introducing these Pythagoreans suggests that they are distinct from Philolaus and his pupil Eurytus, but it is not possible to be more specific about their identity. It is possible that Aristotle only knows of the table through oral transmission and that there were no specific names attached to it.

The table shows a strong normative slant by including good in one column and bad in the other. In contrast, while Philolaus posits the first two opposites in the table, limit and unlimited, as first principles, there is no suggestion in the extant fragments of Philolaus that limit was good and unlimited bad. Opposites played a large role in most Presocratic philosophical systems. The Pythagoreans who posited the table of opposites differed from other early Greek philosophers not only in the normative view of the opposites but also by including strikingly abstract pairs such as straight and crooked and odd and even, in contrast to the more concrete opposites such as hot and cold, which are typical elsewhere in early Greek philosophy. Similar tables of opposites appear in the Academy (Aristotle, Metaph. 1093b11; EN 1106b29 referring to Speusippus; Simplicius in CAG IX. 247. 30ff.), and Aristotle himself seems at times to adopt such a table (Metaph. 1004b27 ff.; Phys. 201b25). Later Platonists and Neopythagoreans will continue to develop these tables (see Burkert 1972a, 52, n. 119 for a list). The table of opposites thus provides one of the clearest cases of continuity between early Pythagoreanism and Platonism. It may well be that the similarity between this Pythagorean table of opposites and later Academic versions led to the Neopythagorean habit, starting already in the early Academy, of mistakenly assigning the fundamental pair of opposites in Plato's late metaphysics, the one and the indefinite dyad, back to Pythagoras (see on Neopythagoreanism below).

2.4 Archytas

See the SEP entry on Archytas.

3. Other Pythagoreans of the Sixth, Fifth and Fourth Centuries

3.1 The Catalogue of Pythagoreans in Iamblichus' On the Pythagorean Life: Who Counts as a Pythagorean?

Iamblichus' On the Pythagorean Life (4th c. AD) ends with a catalogue of 218 Pythagorean men organized by city followed by a list of 17 of the most famous Pythagorean women. Of these 235 Pythagoreans, 145 appear nowhere else in the ancient tradition. This impressive list of names shows the wide impact of Pythagoreanism in the fifth and fourth centuries BC. To what extent is it reliable? A long line of scholars has argued that the catalogue goes back to Aristoxenus in the fourth century BC and is thus a reasonably accurate reflection of early Pythagoreanism rather than a creation of the later Neopythagorean tradition (Rohde 1871-1872, 171; Diels 1965, 23; Timpanaro-Cardini 1958-1964, III 38 ff.; Burkert 1972a, 105, n. 40; Zhmud 1997, 67-69). This is up to a point a reasonable conclusion, since it is hard to see who would have been better placed than Aristoxenus to have such detailed information.

The arguments connecting Aristoxenus to the catalogue are not unassailable, however, and it is likely that the list has been altered in transmission, so that it cannot simply be accepted as the testimony of Aristoxenus. No names on the list can be positively assigned to a date later than Aristoxenus, but this would be likely to be true, even if the list were compiled at a later date, since Pythagoreanism appears to have largely died out for the two centuries immediately following Aristoxenus' death. Thus, Iamblichus does not mention any Pythagorean who can be positively dated after the time of Aristoxenus anywhere else in On the Pythagorean Life either. Scholars have also argued that Iamblichus cannot have composed the catalogue, since he mentions some 18 names that do not appear in the catalogue. This argument would only work, if Iamblichus were a careful and systematic author, which the repetitions and inconsistencies in On the Pythagorean Life show that he was not. While it is unlikely that Iamblichus composed the catalogue from scratch, it is perfectly possible that he edited it in a number of ways, while not feeling compelled to make it consistent with everything he says elsewhere in the text. There are some peculiarities of the catalogue that suggest a connection to Aristoxenus. Philolaus and Eurytus are listed not under Croton but under Tarentum, just as they are in one of the Fragments of Aristoxenus (Fr. 19 Wehrli = Diogenes Laertius VIII 46). On the other hand, some features of the catalogue are inconsistent with what we know of Aristoxenus. Aristoxenus' teacher, Xenophilus, who is identified as from the Thracian Chalcidice in the Fragments of Aristoxenus (Frs. 18 and 19 Wehrli), is identified as from Cyzicus in the catalogue. Moreover, the legendary figure, Abaris, is included in the catalogue and even said to be from the mythical Hyperborea, whereas Aristoxenus is usually seen as resolutely trying to rationalize the Pythagorean tradition. Thus, while Aristoxenus is quite plausibly taken to be the author of the core of the catalogue, it is likely that additions, omissions, and various changes have been made to the original document and hence it is impossible to be sure, in most cases, whether a given name has the authority of Aristoxenus behind it or not.

The catalogue includes several problematic names, such as Alcmaeon, Empedocles, Parmenides and Melissus. Alcmaeon was active in Croton when the Pythagoreans flourished there, but Aristotle explicitly distinguishes Alcmaeon from the Pythagoreans and scholarly consensus is that he is not a Pythagorean (see the entry on Alcmaeon). Most scholars would agree that Empedocles was heavily influenced by Pythagoreanism; in the later tradition fragments of Empedocles are routinely cited to support the Pythagorean doctrines of metempsychosis and vegetarianism (e.g., Sextus Empiricus, Adversus Mathematicos IX 126-30). On the other hand, both in the ancient and in the modern world, Empedocles is not usually labeled a Pythagorean, because, whatever the initial Pythagorean influences, he developed a philosophical system that was his own original contribution. Parmenides is again not usually identified as a Pythagorean in either the ancient or modern tradition and, although scholars have speculated on Pythagorean influences on Parmenides, there is little that can be identified as overtly Pythagorean in his philosophy. The reason for Parmenides' inclusion in the catalogue is pretty clearly the tradition that his alleged teacher Ameinias was a Pythagorean (Diogenes Laertius IX 21). There is no reason to doubt this story, but it gives us no more reason to call Parmenides a Pythagorean than to call Plato a Socratic or Aristotle a Platonist. It would appear that Melissus was included on the list because he was regarded in turn as the pupil of Parmenides. Inclusion in the catalogue thus need not indicate that a figure lived a Pythagorean way of life or that he adopted metaphysical principles that were distinctively Pythagorean; he need only have had contact with a Pythagorean teacher. It is possible that Aristoxenus included Parmenides and Melissus on the list for these reasons or that he had better reasons for including them (e.g. evidence that they lived a Pythagorean life), but it is precisely famous names such as these that would be likely to have been added to the list in later times, and they may well not have appeared in Aristoxenus' catalogue at all.

Zhmud (1997, 67-74) has argued that it begs the question to use a doctrinal criterion to identify Pythagoreans. We need to first identify Pythagoreans and then see what their doctrines are. His alternative, however, to regard as Pythagoreans those figures identified by reliable ancient sources runs into difficulties as well, as is shown by Iamblichus' catalogue, which Zhmud does accept as reliable, but which is problematic for the reasons just given. A great deal of circumspection is required in determining what constitutes a reliable ancient source, when it comes to Pythagoreanism. What criteria should then be used in order to identify a given figure as a Pythagorean?

First, anyone identified as a Pythagorean by an early source uncontaminated by the Neopythagorean glorification of Pythagoras (see below) can be regarded as a Pythagorean. This would include sources dating before the early Academy (ca. 350 BC), where Neopythagoreanism has its origin, and Peripatetic sources contemporary with the early Academy (ca. 350-300 BC, e.g. Aristotle, Aristoxenus and Eudemus), who, under the influence of Aristotle, defined themselves in opposition to the Academic view of Pythagoras.

Second, a doctrinal criterion is applicable. Anyone who espouses the philosophy assigned to the Pythagoreans by Aristotle can be regarded as a Pythagorean. It is important that the use of such a doctrinal criterion be limited to quite specific doctrines such as limiters and unlimiteds as first principles and the cosmology that includes the counter-earth and central fire. Particularly to be avoided is the assumption that any early mathematician or any early figure who assigns mathematical ideas a role in the cosmos is a Pythagorean. Mathematicians such as Theodorus of Cyrene (who is included in Iamblichus' catalogue) and Hippocrates of Chios (who is not) are not treated as Pythagoreans in the early sources such as Plato, Aristotle and Eudemus, and there is accordingly no good reason to call them Pythagoreans. Similarly, the sculptor, Polyclitus of Argos, stated that “the good comes to be … through many numbers,” (Fr. 2 DK), but no ancient source calls him a Pythagorean (Huffman 2002). As Burkert has emphasized, mathematics is a Greek and not just a specifically Pythagorean passion (1972a, 427).

Third, anyone universally (or almost universally) called a Pythagorean by later sources, and whom early sources do not treat as independent of Pythagoreanism, explicitly or implicitly, can be regarded as a Pythagorean. This would include figures embedded in the biographical tradition about Pythagoras and the early Pythagoreans, such as the husband and wife, Myllias and Timycha.

This last criterion is more subjective than the first two and difficult cases arise. The fifth-century botanist Menestor (DK I 375) is discussed by Theophrastus and called one of “the old natural philosophers” (CP VI 3.5) with no mention of any Pythagoreanism. In this case, the inclusion of a Menestor in Iamblichus' catalogue is not enough reason to regard Theophrastus' Menestor as a Pythagorean. On the other hand, although Aristotle treats Hippasus separately from the Pythagoreans, as he does Archytas, the almost universal identification of Hippasus as a Pythagorean in the later tradition and his deep involvement in the biography of early Pythagoreanism, show that he should be regarded as a Pythagorean (on Hippasus, see section 3.3 below). The fifth-century figure Hippo (DK I 385), who is derided by Aristotle and paired with Thales as positing water as the first principle (Metaph. 984a3), is a particularly difficult case. An Hippo is listed in Iamblichus' catalogue under Samos and Censorinus tells us that Aristoxenus assigned Hippo to Samos rather than Metapontum (DK I 385.4-5). This makes it look as if Aristoxenus may be responsible for including Hippo in the catalogue. Burkert has also tried to demonstrate connections between Hippo's philosophy and that of the Pythagoreans (1972a, 290, n. 62). On the other hand, neither Aristotle nor Theophrastus nor any of the Aristotelian commentators call him a Pythagorean and the doxographers describe this Hippo as from Rhegium (e.g., Hippolytus in DK I 385.17). It is thus not clear whether we are dealing with one person or two people named Hippo and it is doubtful that the Hippo discussed by the Peripatetics was a Pythagorean (Zhmud regards Hippo as well as Menestor and Theodorus as Pythagoreans — 1997, 69-74). Those figures of the sixth, fifth and fourth century who have the best claim to be considered Pythagoreans will be discussed in the following sections.

3.2 The Earliest Pythagoreans: Brontinus, Theano, etc.

In the standard collection of the fragments and testimonia of the Presocratics, Cercops, Petron, Brontinus, Hippasus, Calliphon, Democedes, and Parmeniscus are listed as older Pythagoreans (DK I 105-113). Hippasus, who is the most important of these figures, will be discussed separately below (sect. 3.3). Of the rest only Brontinus, Calliphon and Parmeniscus appear in Iamblichus' catalogue.

Brontinus is presented as either the husband or father of Theano, who is treated as either the wife, daughter or pupil of Pythagoras (Diogenes Laertius VIII 42; Burkert 1972a, 114). In the later tradition a number of works were forged in the name of Theano, but we have little reliable evidence about her (see Thesleff 1965, 193-201, for testimonia and texts; Delatte 1922, 246-249; and Montepaone 1993). Brontinus (DK I 106-107) is elsewhere said to have had a wife Deino and to be either from Metapontum or Croton. The elusive connection between Orphism and Pythagoreanism rears its head with Brontinus, since the fourth-century author, Epigenes, reports that Brontinus is supposed to be the real author of two works circulating in the name of Orpheus (West 1983, 9 ff.). Little else is known about him, but his existence appears to be confirmed by Alcmaeon, writing in the late sixth or early fifth century, who addresses his book to a Brontinus along with Leon and Bathyllus (Fr. 1 DK). The latter two may also be Pythagoreans, since a Leon is listed under Metapontum and a Bathylaus (sic) under Posidonia, in Iamblichus' catalogue.

Cercops (DK I 105-106) is an even more obscure figure who is, again according to Epigenes, the supposed Pythagorean author of Orphic texts (West 1983, 9, 248), although Burkert doubts that he was a Pythagorean (1972a, 130).

To Petron (DK I 106) is ascribed the startling doctrine that there are 183 worlds arranged in a triangle, but he is only known from a passage in Plutarch, is not called a Pythagorean there and is probably a literary fiction (Guthrie 1962, 322-323; Burkert 1972a, 114; Zhmud 1997, 69-70).

A Parmeniscus (DK I 112-113) is called a Pythagorean by Diogenes Laertius (IX 20) and may be the same as the Parmiskos listed under Metapontum in Iamblichus' catalogue. Athenaeus reports that a Parmeniscus of Metapontum lost the ability to laugh after descending into the cave of Trophonius, only to recover it in a temple on Delos, where the surviving inventory of the temple of Artemis records a dedication of a cup by a Parmiskos (Burkert 1972a, 154).

There no good reason to think that Democedes (DK I 110-112), the physician from Croton, was himself a Pythagorean, although he had some Pythagorean connections. He is famous from Herodotus' account (III 125 ff.) of his service to the tyrant, Polycrates, and the Persian king, Darius. One late source names him a Pythagorean (DK I 112.21). Iamblichus mentions a Pythagorean named Democedes, who was involved in the political turmoil surrounding the conspiracy of Cylon against the Pythagoreans, but it is far from clear that this was the physician (VP 257-261). Herodotus never calls Democedes a Pythagorean nor do any other of the later sources (e.g., Aelian, Athenaeus, the Suda), nor does he appear in Iamblichus' catalogue. A Calliphon, who could be Democedes' father, is presented as an associate of Pythagoras by Hermippus (DK I 111.36 ff.) and appears in Iamblichus' catalogue, so it is reasonable to regard him as a Pythagorean, although we know nothing more of him. It is reported (Herodotus III 137) that Democedes married the daughter of the Olympic victor, Milon, who was the Pythagorean, whose house was used as a meeting place (Iamblichus, VP 249). It was undoubtedly because Democedes came from Croton at roughly the time when Pythagoras was prominent there and because of the Pythagorean connections of his father and father-in-law that late sources came to label Democedes himself a Pythagorean.

3.3 Hippasus and Other Fifth-century Pythagoreans

Hippasus is a crucial figure in the history of Pythagoreanism, because the tradition about him suggests that even in the fifth century there was debate within the Pythagorean tradition itself as to whether Pythagoras was largely important as the founder of a set of rules to follow in living one's life or whether his teaching also had a mathematical and scientific dimension. Hippasus was probably from Metapontum (Aristotle, Metaph. 984a7; Diogenes Laertius VIII 84), although Iamblichus says there was controversy as to whether he was from Metapontum or Croton (VP 81), and he is listed under Sybaris in Iamblichus' catalogue (VP 267). He is consistently portrayed as a rebel in the Pythagorean tradition, in one case a democratic rebel who challenged the aristocratic Pythagorean leadership in Croton (Iamb. VP 257), but more commonly as the thinker who initiated Pythagorean study of mathematics and the natural world.

It is in this latter role that he is connected with the split between two groups in ancient Pythagoreanism, the acusmatici (who emphasized rules for living one's life, including various taboos) and the mathêmatici (who emphasized study of mathematics and the natural world). Each group claimed to be the true Pythagoreans. Our knowledge of this split comes from Iamblichus, who unfortunately presents two contradictory versions, with the result that Hippasus is sometimes said to be one of the mathêmatici and sometimes one of the acusmatici. Burkert has convincingly shown that the correct version is that reported by Iamblichus at De Communi Mathematica Scientia 76.19 ff. (1972a, 192 ff.). According to this account, the acusmatici denied that the mathêmatici were Pythagoreans at all, saying that their philosophy derived from Hippasus instead. The mathêmatici for their part, while recognizing that the acusmatici were Pythagoreans of a sort, argued that they themselves were Pythagoreans in a truer sense. Hippasus' supposed innovations, they said, were in fact plagiarisms from Pythagoras himself. The mathêmatici explained that, upon Pythagoras' arrival in Italy, the leading men in the cities did not have time to learn the sciences and the proofs of what Pythagoras said, so that Pythagoras just gave them instructions on how to act, without explaining the reasons. The younger men, who did have the leisure to devote to study, learned the mathematical sciences and the proofs. The former group were the first acusmatici, who learned the oral instructions of Pythagoras on how to live (the acusmata = “things heard”), while the latter group were the first mathêmatici. Hippasus was thus closely connected to the mathêmatici in this split in Pythagoreanism but ended up being disavowed by both sides.

It is difficult to be sure of Hippasus' dates, but he is typically regarded as active in the first half of the fifth century (Burkert 1972a, 206). The split in Pythagoreanism may have occurred after the main period of his work and was perhaps connected to the attacks on the Pythagorean societies by outsiders around 450 BC (Burkert 1972a, 207), but certainty is not possible. Zhmud (1992) has argued that the split is an invention of the later tradition, partially on the grounds that there are no individuals that can be confidently identified as acusmatici. Since the acusmatici were neither original nor full-time philosophers, however, and simply preserved the oral taboos handed down by Pythagoras, it is not surprising that they are not singled out for attention in the sources. Only a relatively small number of the names in Iamblichus' catalogue can certainly be identified as mathêmatici and most of the others, particularly the 145 individuals whose names are only known from the catalogue, are likely to be acusmatici, who to a greater or lesser degree followed the Pythagorean acusmata, but left no other trace of their activity. In addition, a number of other Pythagoreans of the fifth and fourth century, who figure in anecdotes about the Pythagorean life are likely to be acusmatici (see below). Burkert originally argued that Iamblichus' account of the split between the acusmatici and mathêmatici could be shown to be derived from Aristotle (1972a, 196), and he still maintains this view, although with a little less confidence, in his most recent work (1998, 315). It is certainly true that there are striking similarities between the language of the passage in Iamblichus and Aristotle's usage (compare “the so-called Pythagorean philosophy” at Iamblichus, Comm. Math. Sc. 76.16 and Aristotle, Metaph. 989b29 etc.), so that Aristotle remains the most likely source.

Hippasus is the first figure in the Pythagorean tradition who can with some confidence be identified as a natural philosopher, mathematician and music theorist. His connections are as much with figures outside the Pythagorean tradition as those within it. This independence may explain why neither Aristotle nor the doxographical tradition label him a Pythagorean, but he is too deeply embedded in the traditions about early Pythagoreanism for there to be any doubt that he was in some sense a Pythagorean. Aristotle pairs Hippasus with Heraclitus as positing fire as the primary element (Metaph. 984a7) and this pairing is repeated in the doxography that descends from Theophrastus (DK I 109. 5-16), according to which Hippasus also said that the soul was made of fire. Philolaus, who was probably two generations later than Hippasus, might have been influenced by Hippasus in starting his cosmology with the central fire (Fr. 7 Huffman). For Philolaus, however, the central fire is a compound of limiter and unlimited, whereas Hippasus is presented as a monist and does not start from Philolaus' fundamental opposition between limiters and unlimiteds.

There are only a few other assertions about the cosmology of Hippasus and most of these seem to be the result of Peripatetic attempts to classify him, such as the assertions that he makes all things from fire by condensation and rarefaction and dissolves all things into fire, which is the one underlying nature and that he and Heraclitus regarded the universe as one, (always) moving and limited in extent (DK I 109.8-10). More intriguing is the claim that he thought there was “a fixed time for the change of the cosmos” (Diogenes Laertius VIII 84), which might be a reference to a doctrine of eternal recurrence, according to which events exactly repeat themselves at fixed periods of time. This doctrine is attested elsewhere for Pythagoras (Dicaearchus in Porphyry, VP 19). Our information about Hippasus is sketchy, because he evidently did not write a book. Demetrius of Magnesia (1st century BC) reports that Hippasus left nothing behind in writing (Diogenes Laertius VIII 84) and this is in accord with the tradition that Philolaus was the first Pythagorean to write a book (Huffman 1993, 15).

Hippasus originates the early Pythagorean tradition of scientific and mathematical analysis of music, which reaches its culmination in Archytas a century later. The correspondence between the central musical concords of the octave, fifth, and fourth and the whole number ratios 2 : 1, 3 : 2 and 4 : 3 is reflected in the acusmata (Iamblichus, VP 82) and was thus probably already known by Pythagoras. This correspondence was central to Philolaus' conception of the cosmos (Fr. 6a Huffman). Although the later tradition tried to assign the discovery to Pythagoras himself (Iamblichus, VP 115), the method described in the story would not in fact have worked (Burkert 1972a, 375-376). Hippasus is the first person to whom is assigned an experiment demonstrating these correspondences that is scientifically possible. Aristoxenus (Fr. 90 Wehrli = DK I 109. 31 ff.) reports that Hippasus prepared four bronze disks of equal diameters, whose thicknesses were in the given ratios, and it is true that, if free hanging disks of equal diameter are struck, the sound produced by, e.g., a disk half as thick as another will be an octave apart from the sound produced by the other disk (Burkert 1972a, 377).

Another text associates Hippasus with Lasus of Hermione in an attempt to demonstrate the correspondence by filling vessels with liquid in the appropriate ratios. It is less clear whether this experiment would have worked as described (Barker 1989, 31-32). Lasus was prominent in Athens in the second half of the sixth century at the time of the Pisistratid tyranny and was thus probably a generation older than Hippasus. There is no indication that Lasus was a Pythagorean and this testimony suggests that the discovery of and interest in the mathematical basis of the concordant musical intervals was not limited to the Pythagorean tradition. Lasus and Hippasus are sometimes said to have been the first to put forth the influential but mistaken thesis that the pitch of a sound depended on the speed with which it travels, but it is far more likely that Archytas originated this view (Huffman 2005, 138-139). In the later tradition Hippasus is reported to have ranked the musical intervals in terms of degrees of concordance, making the octave the most concordant, followed by the fifth, octave + fifth, fourth and double octave (Boethius, Mus. II 19; see Huffman 2005, 433).

Finally, Iamblichus associates Hippasus with the history of the development of the mathematics of means (DK I 110. 30-37), which are important in music theory, but Iamblichus' reports are confused. It is likely that Hippasus worked only with the three earliest means (the arithmetic, geometric and subcontrary/harmonic) and that the changing of the name of the subcontrary mean to the harmonic mean should be ascribed to Archytas rather than Hippasus (Huffman 2005, 170-173).

The most romantic aspect of the tradition concerning Hippasus is the report that he drowned at sea in punishment for the impiety of making public and giving a diagram of the dodecahedron, a figure with twelve surfaces each in the shape of a regular pentagon (Iamblichus, VP 88). This is best understood as reflecting some sort of mathematical analysis of the dodecahedron by Hippasus, but it is implausible in terms of the history of Greek mathematics to suppose that he carried out a strict construction of the dodecahedron, which along with the other four regular solids is most likely to have first received rigorous treatment by Theaetetus in the fourth century BC (Mueller 1997, 277; Waterhouse 1972; Sachs1917, 82). Nor is it clear why public presentation of technical mathematical analysis should cause a scandal, since few people would understand it. The most likely explanation is that the dodecahedron was a cult object for the Pythagoreans (dodecahedra in stone and bronze have been found dating back to prehistoric times) and that it was because of these religious connections that Hippasus' public work on the mathematical aspects of the solid was seen as impious (Burkert 1972a, 460).

Another late story, which appears first in Plutarch, reports a scandal which arose when knowledge of irrational magnitudes was revealed, without specifying any punishment for the one who revealed it (Numa 22). In Pappus' later version of the story, the person who first spread knowledge of the existence of the irrational was punished by drowning (Junge and Thomson 1930, 63-64). Iamblichus knows two different versions of the story, one according to which the malefactor was banished and a tomb was erected for him, signifying his expulsion from the community (VP 246), but another according to which he was punished by drowning as was the person (not specifically said to be Hippasus here) who revealed the dodecahedron (VP 247). Modern scholars have tried to combine the two stories and suppose that Hippasus discovered the irrational through his work on the dodecahedron (von Fritz 1945). This is pure speculation, however, since neither does any ancient source connect Hippasus to the discovery of the irrational nor does any source relate the discovery of the irrational to the dodecahedron (Burkert 1972a, 459).

The other major Pythagoreans of the fifth century were Philolaus and Eurytus, who are discussed above.

The name, but not too much more, is known of a number of other fifth century figures, who with varying degrees of probability may be considered Pythagoreans. To the beginning of the fifth century belongs Ameinias the teacher of Parmenides (Diogenes Laertius VIII 21). The athlete and trainer, Iccus of Tarentum, is listed in Iamblichus' catalogue, but none of the other sources, including Plato, call him a Pythagorean. In the later tradition, he was famous for the simplicity of his life and “the dinner of Iccus” was proverbial for plain fare. Plato praises his self control and reports that he touched neither women nor boys while training. (Laws 839e; see Protagoras 316d and DK I 216. 11 ff.).

There is no reason to regard the physician Acron of Acragas as a Pythagorean, as Zhmud does (1997, 73). Acron is a contemporary of Empedocles and is connected to him in the doxographical tradition (DK I 283. 1-9; Diogenes Laertius VIII 65). No ancient source calls him a Pythagorean. His name appears in a very lacunose papyrus along with the name of Aristoxenus (Aristoxenus, Fr. 22 Wehrli), but it is pure speculation that Aristoxenus labeled him a Pythagorean; Euryphon the Cnidian doctor of the fifth century, who was not a Pythagorean, also appears in the papyrus. Acron's father's name was Xenon, and a Xenon appears in Iamblichus' catalogue, but he is listed as from Locri and not Acragas, so again this is not good evidence that Acron was a Pythagorean.

The Pythagorean Paron (DK I 217. 10-15) is probably a fiction resulting from a misreading of Aristotle (Burkert 1972a, 170). Aristotle reports the expression of a certain Xuthus, that “the universe would swell like the ocean,” if there were not void into which parts of the universe could withdraw, when compressed (Physics 216b25). Simplicius says, on unknown grounds, that this Xuthus was a Pythagorean, and scholars have speculated that he was responding to Parmenides (DK I. 376. 20-26; Kirk and Raven 1957, 301-302; Barnes 1982, 616).

Aristoxenus reports that two Tarentines, Lysis and Archippus, were the sole survivors when the house of Milo in Croton was burned, during a meeting of the Pythagoreans, by their enemies (Iamblichus, VP 250). A later romantic version in Plutarch (On the Sign of Socrates 583a) has it that Lysis and Philolaus were the two survivors, but it appears that the famous name of Philolaus has been substituted for Archippus, about whom nothing else is known. Aristoxenus goes on to say that Lysis left southern Italy and went first to Achaea in the Peloponnese before finally settling in Thebes, where the famous Theban general, Epaminondas, became his pupil and called him father. In order to be the teacher of Epaminondas in the early fourth century, Lysis must have been born no earlier than about 470. Thus the conflagration that he escaped as a young man must have been part of the attacks on the Pythagoreans around 450, rather than those that occurred around 500, when Pythagoras himself was still alive. The later sources often conflate these two attacks on the Pythagoreans (Minar 1942, 53). Nothing is known of the philosophy of Lysis, but it seems probable that he should be regarded as one of the acusmatici, since his training of Epaminondas appears to have emphasized a way of life rather than mathematical or scientific studies (Diodorus Siculus X 11.2) and Epaminondas' use of the name father for Lysis suggests a cult association (Burkert 1972a, 179). In the later tradition, Lysis became quite famous as the author of a spurious letter (Thesleff 1965, 111; cf. Iamblichus, VP 75-78) rebuking a certain Hipparchus for revealing Pythagorean teachings to the uninitiated (see on the Pythagorean pseudepigrapha below, sect. 4.2).

Zopyrus of Tarentum is mentioned twice, in a treatise on siege-engines by Biton (3rd or 2nd century BC), as the inventor of an advanced form of the type of artillery known as the belly-bow (Marsden 1971, 74-77). Zopyrus' bow used a winch to pull back the string and hence could shoot a six-foot wooden missile 4.5 inches thick (Marsden 1969, 14). It is not implausible to suppose that this is the same Zopyrus as is listed in Iamblichus' catalogue of Pythagoreans under Tarentum (Diels 1965, 23), although Biton does not call him a Pythagorean. The traditional dating for Zopyrus puts him in the first half of the fourth century (Marsden 1971, 98, n. 52), but Kingsley has convincingly argued that he was in fact active in the last quarter of the fifth century, when he designed artillery for Cumae and Miletus (1995, 150 ff.). In a famous passage, Diodorus reports that in 399 BC Dionysius I, the tyrant of Syracuse, gathered together skilled craftsmen from Italy, Greece and Carthage in order to construct artillery for his war with the Carthaginians (XIV 41.3). It seems not unlikely that Zopyrus was one of those who came from Italy. There is no reason to suppose, however, as Kingsley (1995, 146) and others do, that Zopyrus' interest in mechanics was connected to his Pythagoreanism or that there was a specifically Pythagorean school of mechanics in Tarentum (Huffman 2005, 14-17).

It is controversial whether this Zopyrus of Tarentum is the same as Zopyrus of Heraclea, who is not called a Pythagorean in the sources, but who is reported in late sources to have written three Orphic poems, The Net, The Robe and The Krater, which probably dealt with the structure of human beings and the earth (West 1983, 10 ff.). This Zopyrus could be from the Heraclea closely connected to Tarentum, but he might also be from the Heraclea on the Black Sea. A late source connects Zopyrus of Heraclea with Pisistratus in the 6th century (West 1983, 249), which would mean that he could not be the same as Zopyrus of Tarentum in the late 5th century. On the other hand, Orphic writings are assigned to a number of other Pythagoreans, and it is not impossible that the same person had interests both in Orphic mysticism and mechanics. Kingsley supposes that the myth at the end of Plato's Phaedo is based in minute detail on Zopyrus' Krater or an intermediary reworking of it (1995, 79-171), and tries to connect specific features of the myth to Zopyrus' interest in mechanics (1995, 147-148), but the parallel which he detects between the oscillation of the rivers in the mythic account of the underworld and the balance of opposing forces used in a bow is too general to be compelling. The connection between Zopyrus and the Phaedo is highly conjectural and must remain so, as long as there are no fragments of the Krater, with which to compare the Phaedo.

A harmonic theorist named Simus is accused of having plagiarized one of seven pieces of wisdom inscribed on a bronze votive offering, which was dedicated in the temple of Hera on Pythagoras' native island of Samos, by Pythagoras' supposed son Arimnestus (Duris of Samos in Porphyry, VP 3). There is a Simus listed under Posidonia (Paestum in S. Italy) in Iamblichus' catalogue of Pythagoreans, so that DK treated him as a Pythagorean (I 444-445) who, like Hippasus, stole some of the master's teaching for his own glory. There is, however, no obvious connection between the two individuals named Simus except the name. Most scholars have thus treated Simus as if he were a harmonic theorist in competition with and independent of the Pythagorean tradition (Burkert 1972a, 449-450; Zhmud 1997, 194; West 1992, 79 and 240; Wilamowitz 1962, II 93-94).

What exactly he stole is very unclear. He is said to have removed seven pieces of wisdom from the monument and put forth one of them as his own. This is perhaps best understood as meaning that he took an inscribed piece of metal from the dedicated object, perhaps a cauldron (see Wilamowitz 1962, II 94). The inscription will have included all seven pieces of wisdom, but Simus chose to publish only one of them as his own, the other six being thus lost. The piece of wisdom he put forth as his own is called a kanôn (“rule”). West takes this as a reference to the monochord, which was called the kanôn, used to determine and illustrate the numerical ratios, which were related to the concordant intervals (1992, 240). Since, however, the kanôn seems to have been something inscribed on the dedication, along with six other pieces of wisdom, it is perhaps better to assume that the kanôn was a description of a set of ratios determining a scale (Burkert 1972a, 455; Wilamowitz 1962, 94). There must have been a scale in circulation associated with the name of Simus. The story that Duris reports is then an attempt by the Pythagoreans to claim this scale as, in fact, the work of Pythagoras or his son, which Simus plagiarized. Duris wrote in the first part of the third century BC, so Simus has to be earlier than that. If the son of Pythagoras really made the dedication in the temple, this would have occurred in the fifth century, but it is unclear how much later than that Simus' kanôn became known. West dates him to the fifth century, whereas DK places him in the fourth.

Iamblichus describes an ‘arithmetical method' known as the bloom of Thymaridas (In Nic. 62), and elsewhere discusses two points of terminology in Thymaridas, including his definition of the monad as “limiting quantity” (In Nic. 11 and 27). Some scholars have dated Thymaridas to the time of Plato or before, but others argue that the terminology assigned to him cannot be earlier than Plato and shows connections to Diophantus in the third century AD (see Burkert 1972a, 442, n. 92 for a summary of the scholarship). There is also a Thymaridas in the biographical tradition, who may or may not be the same individual. In a highly suspect passage in Iamblichus, Thymaridas is listed as a pupil of Pythagoras himself (VP 104) and a Thymaridas of Paros appears in Iamblichus' catalogue and is mentioned in one anecdote (VP 239). There is also a worrisome connection to the pseudo-Pythagorean literature. A Thymaridas of Tarentum is presented in an anecdote (Iamblichus, VP 145) as arguing that people should wish for what the gods give them rather than praying that the gods give them what they want, a sentiment that is also found in a group of three treatises forged in Pythagoras' name (Diogenes Laertius VIII 9). The anecdote is drawn from Androcydes' work on the Pythagorean symbola or taboos. If this work could be dated to the fourth century, it would confirm an early date for Thymaridas, but all that is certain is that Androcydes' work was known in the first century BC and thus that the anecdote originated before that date (Burkert 1972a, 167). It seems rash, given this confused evidence, to follow Zhmud and regard Thymaridas as a younger contemporary or pupil of Archytas (1997, 74).

3.4 The Fourth Century: Aristoxenus, the Last of the Pythagoreans, and the Pythagorists

Aristoxenus (ca. 375- ca. 300 BC) is most famous as a music theorist and as a member of the Lyceum, who was disappointed not be to named Aristotle's successor (Fr. 1 Wehrli). In his early years, however, he was a Pythagorean, and he is one of the most important sources for early Pythagoreanism. He wrote four works on Pythagoreanism: On Pythagoras and His Associates, On the Pythagorean Life, Pythagorean Precepts and a Life of Archytas. None of these works have survived intact, but portions of them were preserved by later authors (Wehrli 1945). Aristoxenus is a valuable source because, as a member of the Lyceum, he is free of the distorted image of Pythagoras propagated during his lifetime by Plato's successors in the Academy (see below, sect. 4.1) and because of his unique connections to Pythagoreanism.

He was born in Tarentum during the years when the most important Pythagorean of the fourth century, Archytas, was the leading public figure and his father, Spintharus, had connections to Archytas (Fr. 30 Wehrli). When Aristoxenus left Tarentum, as a young man, and eventually came to Athens (ca. 350), his first teacher was the Pythagorean, Xenophilus, before he went on to become the pupil of Aristotle (Fr. 1 Wehrli). Some modern scholars are skeptical of Aristoxenus' testimony, seeing his denial that there was a prohibition on eating beans and his assertion that Pythagoras was not a vegetarian and particularly enjoyed eating young pigs and tender kids (Fr. 25 = Gellius IV 11), as attempts to make Pythagoreanism more rational than it was (Burkert 1972a, 107, 180). On the other hand, his Life of Archytas is not a simple panegyric; Archytas' foibles are recognized and his opponents are given a fair hearing (Huffman 2005, 4-5, 289-290).

Perhaps Aristoxenus' most interesting work on Pythagoreanism is the Pythagorean Precepts, which is known primarily through substantial excerpts preserved by Stobaeus (Frs. 33-41 Wehrli). This work does not mention any Pythagoreans by name but presents a set of ethical precepts that “they” (i.e. the Pythagoreans) proposed concerning the various stages of human life, education, and the proper place of sexuality and reproduction in human life. There are also analyses of concepts important in ethics, such as desire and luck. Given Aristoxenus' background, the Precepts would appear to be invaluable evidence for Pythagorean ethics in the first half of the fourth century, when Aristoxenus was studying Pythagoreanism. They might be expected to partially embody the views of his teacher Xenophilus. The standard scholarly view of this work, however, is that Aristoxenus plundered Platonic and Aristotelian ideas for the glory of the Pythagoreans (Wehrli 1945, 58 ff.; Burkert 1972a, 107-108). There are serious difficulties with the standard view, however. The analysis of luck that was supposedly taken from Aristotle is, in fact, in sharp conflict with Aristotle's view (Mills 1982) and appears to be one of the views Aristotle was attacking. While the Precepts do have similarities to passages in Plato and Aristotle, they are at a very high level of generality and are shared with passages in other fifth and fourth century authors, such as Xenophon and Thucydides; it is the distinctively Platonic and Aristotelian features that are missing.

The Precepts are thus best regarded as what they appear on the surface to be, an account of Pythagorean ethics of the fourth century. This ethical system shows a similarity to a conservative strain of Greek ethics, which is also found in Plato's Republic, but has its own distinctive features. The central outlook of the Precepts is a distrust of basic human nature and an emphasis on the necessity for supervision of all aspects of human life (Fr. 35 Wehrli). The emphasis on order in life is so marked that the status quo is preferred to what is right (Fr. 34). The Pythagoreans were particularly suspicious of bodily desire and analyzed the ways in which it could lead people astray (Fr. 37). There are strict limitations on sexual desire and the propagation of children (Fr. 39). Despite the best efforts of humanity, however, many things are outside of human control, so the Pythagoreans examined the impact of luck on human life (Fr. 41).

Aristoxenus is a source for the famous story of the two Pythagorean friends Damon and Phintias, which was set during the tyranny of Dionysius II in Syracuse (367-357). As a test of their friendship Dionysius falsely accused Phintias of plotting against him and sentenced him to death. Phintias asked time to set his affairs in order, and Dionysius was amazed when Damon took his place, while he did so. Phintias showed his equal devotion to his friend by showing up on time for his execution. Dionysius cancelled the execution and asked to become a partner in their friendship but was refused (Iamblichus, VP 234; Porphyry, VP 59-60; Diodorus X 4.3).

In Diodorus' version, Phintias is presented as actually engaged in a plot against Dionysus and some argue that Aristoxenus' version is an attempt to whitewash the Pythagoreans (Riedweg 2005, 40). On the other hand, Dionysius' eagerness to join in their friendship, which occurs in both versions, is harder to understand if there really had been a plot (see Burkert 1972a, 104). There are two other considerations. First, Aristoxenus cites Dionysius II himself as his source, whereas it is unclear what source Diodorus used. Second, it is far from clear that Aristoxenus would object to the Pythagoreans plotting against a tyrant. Thus, there are good reasons for regarding Aristoxenus' version as more accurate.

Cleinias and Prorus are another pair of Pythagorean friends, whose story may have been told by Aristoxenus (Iamblichus, VP 127), although they were not friends in the usual sense. Cleinias, who was from Tarentum, knew nothing of Prorus of Cyrene other than that he was a Pythagorean, who had lost his fortune in political turmoil. On these grounds alone he went to Cyrene, taking the money to restore Prorus' fortunes (Iamblichus, VP 239; Diodorus X 4.1). Nothing else is known of Prorus, although some pseudepigrapha were forged in his name (Thesleff 1965, 154.13). It appears that Cleinias was a contemporary of Plato, since Aristoxenus reports that he and an otherwise unknown Pythagorean, Amyclas, persuaded Plato not to burn the books of Democritus, on the grounds that it would do no good, since they were already widely known (Diogenes Laertius IX 40). Cleinias was involved in several other anecdotes. Like Archytas he supposedly refused to punish when angry (VP 198) and, when angered, calmed himself by playing the lyre (Athenaeus XIV 624a). Asked when one should resort to a woman he said “when one happens to want especially to be harmed” (Plutarch, Moralia 654b). Several pseudepigrapha appear in Cleinias' name as well.

Myllias of Croton and his wife Timycha appear in Iamblichus' catalogue and are known from a famous anecdote of uncertain origin, which is preserved by Iamblichus (VP 189 ff.). They were persecuted by the tyrant Dionysius II of Syracuse, but Timycha showed her loyalty and courage by biting off her tongue and spitting it in the tyrant's face, rather than risk divulging Pythagorean secrets under torture.

None of the Pythagoreans mentioned in the previous four paragraphs appear to have to have anything to do with the sciences or natural philosophy. Since their Pythagoreanism consists exclusively in their way of life, they are best regarded as examples of the acusmatici. Many scholars have regarded Diodorus of Aspendus in Pamphylia (southern Asia Minor), as an important example of what the Pythagorean acusmatici were like in the first half of the fourth century (Burkert 1972a, 202-204). Diodorus is primarily known through a group of citations preserved by Athenaeus (IV 163c-f), which describe him as a vegetarian who was outfitted in an outlandish way, some features of which later became characteristic of the Cynics, e.g. long hair, long beard, a shabby cloak, a staff and beggar's rucksack (cf. Diogenes Laertius VI 13). The historian Timaeus (350-260), however, casts doubt on Diodorus' credentials as a Pythagorean saying that “he pretended to have associated with the Pythagoreans” and Sosicrates, another historian (2nd century BC; fragments in Jacoby) says that his outlandish dress was his own innovation, since before this Pythagoreans had always worn white clothing, bathed and wore their hair according to fashion (Athenaeus IV 163e ff.). Iamblichus, the other major source for Diodorus outside Athenaeus, also treats Diodorus with reserve, saying that he was accepted by the leader of the Pythagorean school at the time, one Aresas, because there were so few members of the school at the time. He continues, perhaps again with disapproval, to report that Diodorus returned to Greece and spread abroad the Pythagorean oral teachings.

These sources clearly suggest that Diodorus was anything but a typical Pythagorean, even of the acusmatic variety. Burkert has argued that this reflects a bias of sources such as Aristoxenus, who wanted to make Pythagoreanism appear reasonable and emphasized the version of Pythagoreanism practiced by the mathêmatici rather than the acusmatici. In support of this conclusion, he argues that the two earliest sources present Diodorus as a Pythagorean without any qualifications (1972a, 204). It is important to look carefully at those sources, however. First, neither is a philosopher or a historian, who might be expected to give a careful presentation of Diodorus. The oldest is a lyre player named Stratonicus (died 350 BC), who was famous for his witticisms, and the other, Archestratus (fl. 330 BC), wrote a book entitled The Life of Luxury, which focused on culinary delights. Such sources might be expected to accept typical stories that went around about Diodorus without any close analysis.

In the case of our earliest source, Stratonicus, there is, moreover, once again evidence suggesting that Diodorus was not regarded as a typical Pythagorean. In describing Diodorus' relationship to Pythagoras, Stratonicus does not use a typical word for student or disciple, but rather the same word (pelatês) that Plato used in the Euthyphro to describe the day-laborer who died at the hands of Euthyphro's father. Diodorus is thus being presented sarcastically as a hired hand in the Pythagorean tradition, which is very much in accord with the later presentations of him as a poor man's Pythagoras on the fringes of Pythagoreanism. Thus, rather than accusing the sources of bias against Diodorus, it seems better to accept their almost universal testimony that he was not a typical acusmatic but rather a marginal figure, who used Pythagoreanism in part to try to gain respectability for his own eccentric lifestyle.

Individuals known as “Pythagorists,” i.e. Pythagorizers, are ridiculed by writers of Greek comedy, such as Alexis, Antiphanes, Aristophon, and Cratinus the younger, in the middle and second half of the fourth century (see Burkert 1972a, 198, n. 25 for the evidence and 200, n. 41 for the dating). The most important of the fragments of these comedies that deal with the Pythagorists are collected by Athenaeus (IV 160f ff) and Diogenes Laertius (VIII 37-38). The term “Pythagorist” is usually negative in the comic writers (Arnott 1996, 581-582) and picks out people who share some of the same extreme ascetic lifestyle as Diodorus. A fragment of Antiphanes describes someone as eating “nothing animate, as if Pythagorizing” (Fr. 133 Kassel and Austin = Athenaeus IV 161a). In The Pythagorizing Woman, Alexis presents the vegetarian sacrificial feast that is customary for the Pythagoreans as including dried figs, cheese and olive cakes, and reports that the Pythagorean life entailed “scanty food, filth, cold, silence, sullenness, and no baths” as well as drinking water instead of wine (Frs. 201-202 = Athenaeus IV 161c and III 122f).

A number of these characteristics can be connected to the acusmata (Arnott 1996, 583), e.g. the lack of bathing may depend on the acusma that forbids the Pythagoreans to use the public baths (Iamblichus, VP 83). A fragment of Aristophon's Pythagorist suggests that this ascetic life was based on poverty rather than philosophical scruple and that, if you put meat and fish in front of these Pythagorists, they would gobble them down (Fr. 9 = Athenaeus IV 161e). In a fragment of Alexis, after the speaker reports that the Pythagoreans eat nothing animate, he is interrupted by someone who objects that “Epicharides eats dogs, and he is a Pythagorean,” to which the response is, “yes, but he kills them first and so they are not still animate” (Fr. 223 + Athenaeus 161b). Another fragment of Aristophon's Pythagorist reports that the Pythagoreans have a far different existence in the underworld than others, in that they feast with Hades because of their piety, but this just occasions the remark that Hades is an unpleasant god to enjoy the company of such filthy wretches (Fr. 12 = Diogenes Laertius VIII 38).

Both Alexis (Fr. 223 = Athenaeus IV 161b) and Cratinus the younger (Fr. 7 = Diogenes Laertius VIII 37) wrote plays entitled The People of Tarentum, which, although they may not have been primarily about Pythagoreans, featured depictions of them (Arnott 1996, 625-626). In this case, the Pythagoreans are again satirized for their simple diet, bread and water (which is called “prison fare”), and for drinking no wine. In these plays, however, the Pythagoreans are also presented as feeding on “subtle arguments” and “finely honed thoughts” and as pestering others with them, in a way that is reminiscent of Aristophanes' treatment of Socrates in the Clouds.

Given the fragmentary nature of the evidence, it is unclear whether these ascetic Pythagoreans who engage in argument are the same as the Pythagorists in the other comedies, who are characterized by their filth and eccentric appearance. Certainly the latter are more reminiscent of Diodorus of Aspendus, while the former might be closer to what we know of someone like Cleinias. In the first half of the third century, the poet Theocritus still preserves a memory of these Pythagorists as “pale and without shoes” (XIV 5). The scholiast to the passage testifies to the continuing controversy about the Pythagorists by drawing a distinction between Pythagoreans who give every attention to their body and Pythagorists who are filthy (although another scholion reports that others say the opposite, see Arnott 1996, 581). A passage in Iamblichus (VP 80) similarly argues that the Pythagoreans were the true followers of Pythagoras, while the Pythagorists just emulated them.

In recent scholarship, the tendency has been to regard Diodorus and the Pythagorists as legitimate Pythagoreans of the acusmatic stamp, whose eccentricities are perhaps a little exaggerated in comedy. The extensive evidence from antiquity which argues that they were not true Pythagoreans is interpreted as bias on the part of conservative Pythagoreans of the hyper-mathêmatici sort, such as Aristoxenus, who wanted to disassociate themselves and Pythagoreanism in general from such strange people. This is a possible interpretation of the evidence, but, as the evidence for Diodorus shows, it is also quite possible that Diodorus and the more extreme Pythagorists depicted in comedy were in fact people with whom few Pythagoreans either of the mathêmatici or the acusmatici wanted to associate themselves. Many religious movements have a radical fringe, and there is little reason to think that Pythagoreanism should differ in this regard.

Perhaps the best way to evaluate the complicated evidence for fourth-century Pythagoreanism is to conclude that there were three main groups, each of which admitted some variation. There were mathêmatici such as Archytas who did serious research in the mathematical disciplines and natural philosophy but who also lived an ascetic life that emphasized self-control and avoidance of bodily pleasure. Other Pythagoreans such as Cleinias or Xenophilus may have done no work in the sciences but lived a Pythagorean life, which was similar to that of Archytas and followed principles similar to those set out in Aristoxenus' Pythagorean Precepts. They may have observed some mild dietary restrictions and may be similar to the figures satirized in The Men of Tarentum as eating a simple diet but still engaged in subtle arguments. There was probably a continuum of people in this category with some following more or different sets of the acusmata than others. Finally there are the Pythagorean hippies such as Diodorus and the Pythagorists, who ostentatiously live a life in accord with some of the acusmata, but who take such an extreme interpretation of them as to be regarded as eccentrics by most Pythagoreans.

Diogenes Laertius reports, evidently on the authority of Aristoxenus, that the last Pythagoreans were Xenophilus from the Thracian Chalcidice (Aristoxenus' teacher), and four Pythagoreans from Phlius: Phanton, Echecrates, Diocles and Polymnastus. These Pythagoreans are further identified as the pupils of Philolaus and Eurytus. Little more is known of Xenophilus beyond his living for more than 105 years (DK I 442-443). The Pythagoreans from Phlius are just names except Echecrates (DK I 443), to whom Phaedo narrates, evidently in Phlius, the events of Socrates last day in Plato's Phaedo. Socrates' interlocutors in the Phaedo, Simmias and Cebes, are often regarded as Pythagoreans, because they are said to have been pupils of Philolaus when he was in Thebes. They are also shown to be pupils of Socrates, however, and it is unclear that their connection to Philolaus was any closer than their connection to Socrates. They are not listed in Iamblichus' catalogue as Pythagoreans; Diogenes Laertius includes them with other followers of Socrates (II 124-125). Echecrates might have been born around 420 and thus be a young man at the dramatic date of the Phaedo. Aristoxenus' assertion that these were the last of the Pythagoreans would then suggest that Pythagoreanism died out around 350, when Echecrates was an old man.

Riedweg says that this claim is “demonstrably untrue” pointing to a Pythagorean, Lycon, who criticized Aristotle's supposed extravagant way of life and to the Pythagorists discussed above (2005, 106). This seems slender evidence upon which to be so critical of Aristoxenus. Virtually nothing is known of Lycon, and Aristocles (1st-2nd c. AD), who recounts the criticism of Aristotle, says that Lycon “called himself a Pythagorean,” thus expressing some sort of reservation about his credentials (DK I 445-446). Aristoxenus' assertion is probably to be understood as a general claim that, with the deaths of the Pythagoreans from Phlius around the middle of the fourth century, Pythagoreanism as an active movement was dead.

This is not inconsistent with the existence of a few isolated individuals, who still claim to be Pythagoreans. Certainly, from the evidence available to modern scholars, Aristoxenus' claim is largely true. From about 350 BC until about 100 BC, there is a radical drop in evidence for individuals who call themselves Pythagoreans. Iamblichus (In Nic. 116.1-7) appears to date the Pythagoreans Myonides and Euphranor, who worked on the mathematics of means, after the time of Eratosthenes (285-194 BC) and hence to the second century BC or later (Burkert 1972a, 442), but Iamblichus' history of the means is very confused and they might belong to the rise of Neopythagoreanism in the first centuries BC and AD. Kahn (2001, 83) sees a hint of Pythagorean cult activity in the spurious Pythagorean Memoirs, which must date sometime before the first half of the first century BC, when they are quoted by Alexander Polyhistor. A few other Pythagorean pseudepigrapha appear in the period (see further below, sect. 4.2), although it is unclear what sort of Pythagorean community, if any, was associated with them. Pythagoreanism is not completely dead between 350 and 100 (see further below, sect. 3.5), but few individual Pythagoreans or organized groups of Pythagoreans can be identified in this period.

3.5 Timaeus, Ocellus, Hicetas and Ecphantus

The names Timaeus of Locri and Ocellus of Lucania are famous as the authors of the two most influential Pythagorean pseudepigrapha (see below, sect. 4.2). In his catalogue of Pythagoreans, Iamblichus lists an Ocellus under Lucania and two men named Timaeus, neither under Locri. The later forgery of works attributed to Timaeus and Ocellus does not of course mean that Pythagoreans of these names did not exist, and it is possible that the Timaeus of Locri who is the main speaker in Plato's Timaeus was an historical Timaeus (some have thought Plato uses him as a mask for Archytas, however). If they really did exist, however, nothing is known about them, since all other reports in the ancient tradition are likely to be based on Plato's Timaeus or the spurious works in their name.

Some scholars have argued that Hicetas and Ecphantus, both of Syracuse, were not historical figures at all but rather characters in dialogues written by Heraclides of Pontus, a fourth-century member of the Academy. By a misunderstanding, they came to be treated as historical Pythagoreans in the doxographical tradition (see Guthrie 1962, 323 ff. for references). This theory arose because both Hicetas and Ecphantus are said to have made the earth rotate on its axis, while the heavens remained fixed, in order to explain astronomical phenomena, and, in one report, Heraclides is paired with Ecphantus as having adopted this view (Aetius III 13.3 =DK I 442.23). In addition Ecphantus is assigned a form of atomism (DK I 442.7 ff.) similar to that assigned to Heraclides (Fr. 118-121 Wehrli). It is not uncommon in the doxographical tradition for a report of the form “x and y believe z” to mean that “y, as reported by x, believes z,” so it is suggested that in this case “Heraclides and Ecphantus” means “Ecphantus as presented by Heraclides.” There is a serious problem with this ingenious theory. The doxographical reports about Hicetas and Ecphantus ultimately rely on Theophrastus (Cicero mentions Theophrastus by name at DK I 441.27), and it is implausible that Theophrastus would treat characters invented by his older contemporary, Heraclides, as historical figures. Theophrastus did accept the Academic glorification of Pythagoras (see on Neopythagoreanism below, sect. 4.1), but this provides no grounds for supposing that he accepted a character in a dialogue as a historical person (pace Burkert 1972a, 341).

The testimonia for Hicetas are meager and contradictory (DK I 441-442). He appears to have argued that the celestial phenomena are best explained by assuming that all heavenly bodies are stationary and that the apparent movement of the stars and planets is the result of the earth's rotation around its own axis. He may also have followed Philolaus in positing a counter-earth, opposite the earth on the other side of a central fire, although, if he did, it is unclear how he would have explained why it and the central fire are not visible from the rotating earth. In Philolaus' system the central fire remains invisible because the earth orbits the central fire as it rotates on its axis, thus keeping one side of the earth always turned away from the central fire. A little more is known about Ecphantus (DK I 442). He too is said to have believed that the earth moved, not by changing its location (as Philolaus proposed, in making the earth and counter-earth revolve around the central fire: see Section 4.2 of the entry on Philolaus), but by rotating on its axis.

Copernicus was inspired by these testimonia about Hicetas and Ecphantus, as well as those about Philolaus, to consider the motion of the earth (see below, sect. 5.2). Ecphantus developed his own original form of atomism. He is best understood as reacting to and developing the views of Democritus. He agreed with Democritus 1) “that human beings do not grasp true knowledge of the things that are, but define them as they believe them to be” (DK I 442.7-8; cf. Democritus Frs. 6-10) and 2) that all sensible things arise from indivisible first bodies and void. He differs from Democritus, however, in supposing that atoms are limited rather than unlimited in number and that there is just one cosmos rather than many. As in Democritus, atoms differ in shape and size, but Ecphantus adds power (dynamis) as a third distinguishing factor. He explains atomic motion not just in terms of weight and external blows, as the atomists did, but also by a divine power, which he called mind or soul, so that “the cosmos was composed of atoms but organized by providence” (DK I 442.21-22). It is because of this divine power that the cosmos is spherical in shape. This unique spherical cosmos is reminiscent of Plato's Timaeus, but the rest of Ecphantus' system differs enough from Plato that there is no question of its being a forgery based on the Timaeus. One testimony says that he was the first to make Pythagorean monads corporeal, thus differing from the fifth-century Pythagoreans described by Aristotle, who do not seem to have addressed the question of whether numbers were physical entities or not (Huffman 1993, 61 ff.).

It is difficult to be sure of the date of either Hicetas or Ecphantus. Since, however, both seem to be influenced by Philolaus' idea of a moving earth and since Ecphantus appears to be developing the atomism of Democritus, it is usually assumed that they belong to the first half of the fourth century (Guthrie 1962, 325-329). Hicetas does not appear in Iamblichus' catalogue. There is an Ecphantus in the catalogue, but he is listed under Croton rather than Syracuse, so it cannot be certain whether he is the Ecphantus described in the doxography.

4. Neopythagoreanism

Neopythagoreanism is characterized by the tendency to see Pythagoras as the central and original figure in the development of Greek philosophy, to whom, according to some authors (e.g. Iamblichus, VP 1), a divine revelation had been given. This revelation was often seen as having close affinities to the wisdom of earlier non-Greeks such as the Hebrews, the Magi and the Egyptians. Because of the belief in the centrality of the philosophy of Pythagoras, later philosophy was regarded as simply an elaboration of the revelation expounded by Pythagoras; it thus became the fashion to father the views of later philosophers, particularly Plato, back onto Pythagoras. The origins of Neopythagoreanism are probably to be found already in Plato's school, the Academy, in the second half of the fourth century BC. There is evidence that Plato's successors, Speusippus and Xenocrates, both presented Academic speculations arising in part from Plato's later metaphysics as the work of Pythagoras, who lived some 150 years earlier. After a decline in interest in Pythagoreanism for a couple of centuries, Neopythagoreanism emerged again and developed further starting in the first century BC and extending throughout the rest of antiquity and into the middle ages and Renaissance. During this entire period, it is the Neopythagorean construct of Pythagoras that dominates, a construct that has only limited contact with early Pythagoreanism; there is little interest in an historically accurate presentation of Pythagoras and his philosophy. In reading the following account of Neopythagoreanism, it may be helpful to refer to the Chronological Chart of Sources for Pythagoras, in the SEP entry on Pythagoras.

4.1 Origins in the Early Academy: Speusippus, Xenocrates and Heraclides

The evidence for Speusippus, Plato's successor as head of the Academy, is fragmentary and second hand, so that certainty in iterpretation is hardly possible. In one passage, however, he assigns not just Plato's principles, the one and the dyad, to “the ancients,” who in context seem likely to be the Pythagoreans, but also a development of the Platonic system according to which the one was regarded as beyond being (Fr. 48 Taran; see Burkert 1972a, 63-64; Dillon 2003, 56-57; for doubts see Taran 1981, 350 ff.). Speusippus also wrote a book On Pythagorean Numbers (Fr. 28 Taran), which builds on ideas clearly attested for the early Pythagoreans (e.g. numbers conceived as arrangements of units and ten as the perfect number). We cannot be sure, however, either that the title goes back to Speusippus or that he assigned all ideas in it to the Pythagoreans. Aristotle twice cites agreement between Speusippus and the Pythagoreans (Metaph. 1072b30 ff.; EN 1096b5-8), which might suggest that Speusippus himself had identified the Pythagoreans as his predecessors in these areas. Speusippus and Xenocrates denied that the creation of the universe in Plato's Timaeus should be understood literally; when the view that the cosmos was only created in thought and not in time is assigned to Pythagoras in the later doxography (Aëtius II 4.1 — Diels 1958, 330), it certainly looks as if an idea which had its origin in the interpretation of Plato's Timaeus in the Academy is being assigned back to Pythagoras (Burkert 1972a, 71). The evidence is not sufficient to conclude that Speusippus routinely assigned Platonic and Academic ideas to the Pythagoreans (Taran 1981, 109), but there is enough evidence to suggest that he did so in some cases.

Speusippus' successor as head of the Academy, Xenocrates, may actually have followed some version of the Pythagorean way of life, e.g. he was apparently a vegetarian, refused to give oaths, was protective of animals and followed a highly structured daily regimen, setting aside time for silence (Dillon 2003, 94-95; Burkert, however, argues that he rejected metempsychosis [1972a, 124]). He wrote a book entitled Things Pythagorean, the contents of which are unfortunately unknown (Diogenes Laertius IV 13). In the extant fragments of his writings, he refers to Pythagoras by name once, reporting that “he discovered that the musical intervals too did not arise apart from number” (Fr. 9 Heinze). Several doctrines of Xenocrates are also assigned to Pythagoras in the doxographical tradition, e.g. the definition of the soul as "a number moving itself," which Xenocrates clearly developed on the basis of Plato's Timaeus (Plutarch, On the Generation of the Soul 1012d; Aëtius IV 2.3-4). This suggests that Xenocrates, like Speusippus, may have assigned his own teachings back to Pythagoras or at least treated Pythagoras as his precursor in such as way that it was easy for others to do so (Burkert 1972a, 64-65; Dillon 2003, 153-154).

Yet another member of the early Academy, Heraclides of Pontus (Gottschalk 1980), in a series of influential dialogues, further developed the presentation of Pythagoras as the founder of philosophy. In the dialogue, On the Woman Who Stopped Breathing, Pythagoras is presented as the inventor of the word “philosophy” (Frs. 87-88 Wehrli = Diogenes Laertius Proem 12 and Cicero, Tusc. V 3.8). Although some scholars have tried to find a kernel of truth in the story (e.g. Riedweg 2005, 90 ff.), its definition of the philosopher as one who seeks wisdom rather than possessing it is regarded by many scholars as a Socratic/Platonic formulation, which Heraclides, in his dialogue, is assigning to Pythagoras as part of a literary fiction (Burkert 1960 and 1972a, 65). Heraclides also assigns to Pythagoras a definition of happiness as “the knowledge of the perfection of the numbers of the soul” (Fr. 44 Wehrli), in which again the Platonic account of the numerical structure of the soul in the Timaeus appears to be fathered on Pythagoras. Other fragments show Heraclides' further fascination with the Pythagoreans. He developed what would become one of the canonical accounts of Pythagoras' previous incarnations (Fr. 89 Wehrli). Perhaps on the basis of the Pythagorean Philolaus' astronomical system, he developed the astronomical theory, later to be championed by Copernicus, according to which the apparent daily motion of the sun and stars was to be explained by the rotation of the earth (Frs. 104-108; see on Hicetas and Ecphantus above, sect. 3.5).

In contrast to the fascination with and glorification of Pythagoras in the Academy after Plato's death, Aristotle did not treat Pythagoras as part of the philosophical tradition at all. In the surveys of his predecessors in his extant works, Aristotle does not include Pythagoras himself and he evidently presented him in his lost special treatises on the Pythagoreans only as a wonder-worker and founder of a way of life. While Aristotle did acknowledge some connections between Plato and fifth-century Pythagoreans, he also sharply distinguished Plato from the Pythagoreans on a series of important points (Metaph. 987b23 ff.), perhaps in response to the Academy's tendency to assign Platonic doctrines to Pythagoras. Aristotle's students Dicaearchus (Porphyry, VP 19) and Aristoxenus also took a much more sober view of Pythagoras's achievements.

It is then one of the great paradoxes of the ancient Pythagorean tradition that Aristotle's successor, Theophrastus, evidently accepted the Academic lionization of Pythagoras, and identifies Plato's one and the indefinite dyad as Pythagorean basic principles (Metaph. 11a27 ff.), although Aristotle is emphatic that this pair of principles in fact belong to Plato (Metaph. 987b25-27). Since Theophrastus' work, Tenets in Natural Philosophy, was the basis of the later doxographical tradition, however, it was the Neopythagorean Pythagoras of the Academy that dominated the later tradition, the Pythagoras who originated the one and the indefinite dyad (Aëtius I 3. 8) as well as a number of other Platonic and early Academic doctrines.

4.2 The Pythagorean Pseudepigrapha

Although the origins of Neopythagoreanism are thus found in the fourth century BC, the figures more typically labeled Neopythagoreans belong to the upsurge in interest in Pythagoreanism that begins in the first century BC and continues through the rest of antiquity. Before turning to these Neopythagoreans, it is important to discuss another aspect of the later Pythagorean tradition, the Pythagorean pseudepigrapha. Many more writings forged in the name of Pythagoras and other Pythagoreans have survived than genuine writings. Most of the pseudepigrapha themselves only survive in excerpts quoted by anthologists such as John of Stobi, who created a collection of Greek texts for the edification of his son in early fifth century AD. The modern edition of these Pythagorean pseudepigrapha by Thesleff (1965) runs to some 245 pages.

There is much uncertainly as to when, where, why and by whom these works were created. No one answer to these questions will fit all of the treatises. Most scholars (e.g. Burkert 1972b, 40-44; Centrone 1990, 30-34, 41-44 and 1994) have chosen Rome or Alexandria between 150 BC and 100 AD as the most likely time and place for these compositions, since there was a strong resurgence of interest in Pythagoreanism in these places at these times (see below). Thesleff's view that the majority were composed in the third century BC in southern Italy (1961 and 1972, 59) has found less favor.

A number of motives probably led to the forgeries. The existence of avid collectors of Pythagorean books such as Juba, King of Mauretania (see below), and the scarcity of authentic Pythagorean texts will have led to forgeries to sell for profit to the collectors. Other short letters or treatises may have originated as exercises for students in the rhetorical schools (e.g. the assignment might have been to write the letter that Archytas wrote to Dionysius II of Syracuse asking that Plato be freed; see Diogenes Laertius III 21-22). The contents of the treatises suggest, however, that the primary motivation was to provide the Pythagorean texts to support the Neopythagorean position, first adumbrated in the early Academy, that Pythagoras was the source of all that is true in the Greek philosophical tradition. The pseudepigrapha show the Pythagoreans anticipating the most characteristic ideas of Plato and Aristotle. Most of the treatises are composed in the Doric dialect (spoken in Greek S. Italy) but, apart from that concession to verisimilitude, there is little other attempt to make them appear to be archaic documents that anticipated Plato and Aristotle. Instead, Plato's and Aristotle's philosophical positions are stated in a bald fashion using the exact Platonic and Aristotelian terminology.

The most famous of these pseudepigrapha is the treatise supposedly written by Timaeus of Locri (Marg 1972), the principal speaker of Plato's Timaeus, which has survived complete and which is clearly intended to represent the original document on which Plato drew, although it, in fact, also responds to criticisms made of Plato's dialogue in the first couple of centuries after it was written (Ryle 1965, 176-178). The treatise of Timaeus of Locri is first mentioned by Nicomachus in the second century AD (Handbook 11) and is thus commonly dated to the first century AD. Another complete short treatise (13 pages in Thesleff) is On the Nature of the Universe supposedly by the Pythagorean Ocellus (Harder 1966), which has passages that are almost identical to passages in Aristotle's On Generation and Corruption. Since Ocellus' work is first mentioned by the Roman polymath, Varro, scholars have dated it to the first half of the first century BC.

It is likely that in some cases letters were forged in order to authenticate these forged treatises. Thus a correspondence between Plato and Archytas dealing with the acquisition of the writings of Ocellus (Diogenes Laertius VIII 80-81) may be intended to validate the forgery in Ocellus' name (Harder 1966, 39ff). A letter from Lysis to Hipparchus (Thesleff 1965, 111-114), which enjoyed considerable fame in the later tradition and is quoted by Copernicus, urges that the master's doctrines not be presented in public to the uninitiated and recounts Pythagoras' daughter's preservation of his “notebooks” (hypomnêmata) in secrecy, although she could have sold them for much money (see Riedweg 2005, 120-121). Burkert (1961, 17-28) has argued that this letter was forged to authenticate the “Pythagorean notebooks” from which Alexander Polyhistor (1st century BC) derived his influential account of Pythagoreanism (Diogenes Laertius VIII 36 — for Alexander see section 4.5 below). While some of Pythagoras' teachings were undoubtedly secret, many were not, and the claim of secrecy in the letter of Lysis is used to explain both the previous lack of early Pythagorean documents and the recent “discovery” of what are in reality forged documents, such as the notebooks.

Callimachus, in the third century BC, knew of a spurious astronomical work circulating in Pythagoras' name (Diogenes Laertius IX 23) and there may have been a similar work forged in the second century (Burkert 1961, 28-42). A group of three books, On Education, On Statesmanship and On Nature, were forged in Pythagoras' name sometime before the second century B.C. (Diogenes Laertius VIII 6 and 9; Burkert 1972a, 225). Heraclides Lembus, in the second century BC, knew of at least six other works in Pythagoras' name, all of which must have been spurious. There was also a spurious treatise on the magical properties of plants and the Golden Verses, which are discussed further below (sect. 4.5).

Archytas appears to have been an even more popular name in which to forge treatises than Pythagoras himself, since some 45 pages are devoted to pseudo-Archytan treatises in Thesleff's collection as compared to 30 pages for Pythagoras (see Huffman 2005, 595 ff. for the pseudo-Archytan treatises). The most famous of the pseudo-Archytan texts is The Whole System of Categories, which, along with On Opposites, represents the attempt to claim Aristotle's system of categories for the Pythagoreans. The pseudo-Archytan works on categories are very frequently cited by the commentators on Aristotle's Categories (e.g. Simplicius and Syrianus) and were regarded as authentic by them, but in fact include modifications made to Aristotle's theory in the first century BC and probably were composed in that century (Szlezak 1972). Another treatise, On Principles, is full of Aristotelian terminology such as “form,” “substance,” and “what underlies”; On Intelligence and Perception contains a paraphrase of the divided line passage in Plato's Republic. There are also a series of pseudepigrapha on ethics by Archytas and other authors (Centrone 1990). Philolaus, the third most famous Pythagorean after Pythagoras and Archytas, also turns up as the author of several spurious treatises (Huffman 1993, 341 ff.), but a number of the forgeries were in the names of obscure or otherwise unknown Pythagoreans. Thus, Callikratidas and Metopos are presented as anticipating Plato's doctrine of the tripartite soul and as using Plato's exact language to articulate it (Thesleff 1965, 103.5 and 118.1-4). Although there are indications that some ancient scholars had doubts about the authenticity of the pseudo-Pythagorean texts, for the most part they succeeded in their purpose all too well and were accepted as genuine texts on which Plato and Aristotle drew.

“Neopythagorean” is a modern label, which overlaps with two other modern labels, “Middle Platonist” and “Neoplatonist,” so that a given figure will be called a Neoplatonist or Middle Platonist by some scholars and a Neopythagorean by others. There are several different strands in Neopythagoreanism. One strand focuses on Pythagoras as a master metaphysician. In this guise he is presented as the author of a theory of principles, which went even beyond the principles of Plato's later metaphysics, the one and the indefinite dyad, and which shows similarities to the Neoplatonic system of Plotinus. The first Neopythagorean in this sense is Eudorus of Alexandria, who was active in the middle and later part of the first century BC. He evidently presented his own innovations as the work of the Pythagoreans (Dillon 1977, 119). According to Eudorus, the Pythagoreans posited a single supreme principle, known as the one and the supreme god, which is the cause of all things. Below this first principle are a second one, which is also called the monad, and the indefinite dyad. These latter two are Plato's principles in the unwritten doctrines, but Eudorus says they are properly speaking elements rather than principles (Simplicius, in Phys., CAG IX 181. 10-30). The system of principles described by Eudorus also appears in the pseudo-Pythagorean writings (e.g. pseudo-Archytas, On Principles; Thesleff 1965, 19) and it is hard to be certain in which direction the influence went (Dillon 1977, 120-121). Two generations after Eudorus, another Alexandrian, the Jewish thinker Philo, used a Pythagorean theory of principles, which is similar to that found in Eudorus, and Pythagorean number symbolism in order to give a philosophical interpretation of the Old Testament (Kahn 2001, 99-104; Dillon 1977, 139-183).

Moderatus of Gades (modern Cadiz in Spain), who was active in the first century AD, shows similarities to Eudorus in his treatment of Pythagorean principles. Plutarch explicitly labels him a Pythagorean and presents his follower, Lucius, as living a life in accord with the Pythagorean taboos, known as symbola or acusmata (Table Talk 727b). It is thus tempting to assume that Moderatus too lived a Pythagorean life (Dillon 1977, 345). His philosophy is only preserved in reports of other thinkers, and it is often difficult to distinguish what belongs to Moderatus from what belongs to the source.

He wrote a comprehensive eleven volume work entitled Lectures on Pythagoreanism from which Porphyry quotes in sections 48-53 of his Life of Pythagoras. In this passage, Moderatus argues that the Pythagoreans used numbers as a way to provide clear teaching about bodiless forms and first principles, which cannot be expressed in words. In another excerpt, he describes a Pythagorean system of principles, which appears to be developed from the first two deductions of the second half of Plato's Parmenides. In this system there are three ones: the first one which is above being, a second one which is identified with the forms and which is accompanied by intelligible matter (i.e. the indefinite dyad) and a third one which is identified with soul. The first two ones show connections to Eudorus' account of Pythagorean first principles; the whole system anticipates central ideas of the most important Neoplatonist, Plotinus (Dillon 1977, 346-351; Kahn 2001, 105-110).

Moderatus was a militant Neopythagorean, who explicitly charges that Plato, Aristotle and members of the early academy claimed as their own the most fruitful aspects of Pythagorean philosophy with only small changes, leaving for the Pythagoreans only those doctrines that were superficial, trivial and such as to bring discredit on the school (Porphyry, VP 53). These trivial doctrines have been thought to be the various taboos preserved in the symbola, but, since his follower Lucius is explicitly said to follow the symbola, it seems unlikely that Moderatus was critical of them. The charge of plagiarism might suggest that Moderatus was familiar with the pseudo-Pythagorean treatises, which appear to have been forged in part to show that Pythagoras had anticipated the main ideas of Plato and Aristotle (see Kahn 2001, 105).

It is with Numenius (see Dillon 1977, 361-379 and Kahn 2001, 118-133, to whom the following account is indebted), who flourished ca. 150 AD in Apamea in northern Syria (although he may have taught at Rome), that Neopythagoreanism has the clearest direct contact with the great Neoplatonist, Plotinus. Porphyry reports that Plotinus was, in fact, accused of having plagiarized from Numenius and that, in response, Amelius, a devotee of Numenius' writings and follower of Plotinus, wrote a treatise entitled Concerning the Difference Between the Doctrines of Plotinus and Numenius (Life of Plotinus 3 and 17). The third century Platonist, Longinus, to a degree describes Plotinus himself as a Neopythagorean, saying that Plotinus developed the exegesis of Pythagorean and Platonic first principles more clearly than his predecessors, who are identified as Numenius, his follower Cronius, Moderatus and Thrasyllus, all Neopythagoreans (Porphyry, Life of Plotinus 20).

Numenius is regularly described as a Pythagorean by the sources that cite his fragments such as Eusebius (e.g. Fr. 1, 4b, 5 etc. Des Places). He presents himself as returning to the teaching of Plato and the early Academy. That teaching is in turn presented as deriving from Pythagoras. Plato is described as “not better than the great Pythagoras but perhaps not inferior to him either” (Fr. 24 Des Places). For Numenius a true philosopher adheres to the teaching of his master, and he wrote a polemical treatise, directed particularly at the skeptical New Academy, with the title On the Revolution of the Academics against Plato (Fr. 24 Des Places). Numenius presents the Pythagorean philosophy to which Plato adhered as ultimately based on a still earlier philosophy, which can be found in Eastern thinkers such as the Magi, Brahmans, Egyptian priests and the Hebrews (Fr. 1 Des Places). Thus, Numenius was reported to have asked “What else is Plato than Moses speaking Greek?” (Fr. 8 Des Places).

Numenius presents his own doctrine of matter, which is clearly developed out of Plato's Timaeus, as the work of Pythagoras (Fr. 52 Des Places). Matter in its disorganized state is identified with the indefinite dyad. Numenius argues that for Pythagoras the dyad was a principle independent of the monad; later thinkers, who tried to derive the dyad from the monad (he does not name names but Eudorus, Moderatus and the Pythagorean system described by Alexander Polyhistor fit the description), were thus departing from the original teaching. In emphasizing that the monad and dyad are independent principles, Numenius is indeed closer to the Pythagorean table of opposites described by Aristotle and to Plato's unwritten doctrines. Since it is in motion, disorganized matter must have a soul, so that the world and the things in it have two souls, one evil derived from matter and one good derived from reason. Numenius avoids complete dualism in that reason does have ultimate dominion over matter, thus making the world as good as possible, given the existence of the recalcitrant matter.

The monad, which is opposed to the indefinite dyad, is just one of three gods for Numenius (Fr. 11 Des Places), who here follows Moderatus to a degree. The first god is equated with the good, is simple, at rest and associates only with itself. The second god is the demiurge, who by organizing matter divides himself so that a third god arises, who is either identified with the organized cosmos or its animating principle, the world soul (Dillon 1977, 366-372). Numenius is famous for the striking images by means of which he elucidated his philosophy, such as the comparison of the helmsman, who steers his ship by looking at the heavens, to the