Pythagoreanism can be defined in a number of ways.

(1) Pythagoreanism is the philosophy of the ancient Greek philosopherPythagoras (ca. 570–ca. 490 BCE), which prescribed a highly structured wayof life and espoused the doctrine of metempsychosis (transmigration ofthe soul after death into a new body, human or animal).

(2) Pythagoreanism is the philosophy of a group of philosophers activein the fifth and the first half of the fourth century BCE, whomAristotle refers to as “the so-called Pythagoreans” and towhom Plato also refers. Aristotle’s expression, “so-calledPythagoreans,” suggests both that at his time this group ofthinkers was commonly called Pythagoreans and, at the same time, callsinto question the actual connection between these thinkers andPythagoras himself. Aristotle ascribes no specific names to thesePythagoreans, but the philosophy which he assigns to them is verysimilar to what is found in the fragments ofPhilolaus of Croton (ca. 470–ca. 390 BCE). Thus, Philolaus and hissuccessor Eurytus are likely to have been the most prominent of thesePythagoreans. Philolaus posits limiters and unlimiteds as firstprinciples and emphasizes the role of number in understanding thecosmos. Aristotle also identifies a distinct group of these so-calledPythagoreans who formulated a set of basic principles known as thetable of opposites. Plato’s sole reference to Pythagoreans citestheir search for the numerical structure of contemporary music and isprobably an allusion toArchytas (ca. 420–ca. 350 BCE), who, as far as the evidence allows us tosee, is the first great mathematician in the Pythagorean tradition.Starting from the system of Philolaus he developed his ownsophisticated account of the world in terms of mathematicalproportion.

(3) Many other sixth-, fifth- and fourth-century thinkers are labeledPythagoreans in the Greek tradition after the fourth century BCE. Bythe late fourth century CE many of the most prominent Greekphilosophers including Parmenides, Plato and Aristotle come to becalled Pythagoreans, with no historical justification. There arenonetheless a number of thinkers of the fifth and fourth century BCE,who can legitimately be called Pythagoreans, although often little isknown about them except their names. The most important of thesefigures is Hippasus. What criterion should be used to identify anearly figure as a Pythagorean is controversial and there is debateabout individual cases. Fourth-century evidence shows thatPythagoreanism gave an unusually large role to women for an ancientphilosophical school. It is likely that the Pythagorean communitiesthat practiced a way of life that they traced back to Pythagoras diedout in the middle of the fourth century BCE.

(4) The last manifestation of Pythagoreanism, Neopythagoreanism, hasbeen the most influential. Neopythagoreanism is not a unified schoolof thought but rather a tendency, stretching over many centuries, toview Pythagoras, with no historical justification, as the central andoriginal figure in the whole Greek philosophical tradition. ThisPythagoras is often thought to have received his philosophy as adivine revelation, which had been given even earlier to wise men ofthe ancient Near East such as the Persian Magi, the Hebrews (Moses inparticular), and the Egyptian priests. All Greek philosophy afterPythagoras, insofar as it may be true, is seen as derived from thisrevelation. Thus, Plato’s and Aristotle’s ideas are viewedas derived from Pythagoras (with the mediation of other earlyPythagoreans). Many pseudepigrapha are produced in later times inorder to provide the Pythagorean “originals” on whichPlato and Aristotle drew. Some strands of the Neopythagorean traditionemphasize Pythagoras as master metaphysician, who supposedlyoriginated what are, in fact, the principles of Plato’s latermetaphysics, the one and the indefinite dyad. Other Neopythagoreanscelebrate Pythagoras as the founder of thequadrivium ofmathematical sciences (arithmetic, geometry, astronomy and music),while still others portray him as a magician or as a religious expertand sage, upon whom we should model our lives. Neopythagoreanismprobably began already in the second half of the fourth century BCEamong Plato’s first successors in the Academy, but particularlyflourished from the first century BCE until the end of antiquity.Neopythagoreanism has close connections to Middle and Neoplatonism andfrom the time of Iamblichus (4th c. CE) is largely absorbed intoNeoplatonism. It was the Neopythagorean version of Pythagoreanism thatdominated in the Middle Ages and Renaissance.

• 1. The Philosophy of Pythagoras
• 2. The Most Prominent Pythagoreans of the Fifth and Fourth Century
  • 2.1 Philolaus
  • 2.2 Eurytus
  • 2.3 Aristotle’s “So-called” Pythagoreans
  • 2.4 The Pythagoreans of the Table of Opposites
  • 2.5 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 Pythagorean Women
  • 3.4 Hippasus and Other Fifth-century Pythagoreans:acusmatici and mathêmatici
  • 3.5 The Fourth Century: Aristoxenus, the Last of the Pythagoreans, and the Pythagorists
  • 3.6 Timaeus, Ocellus, Hicetas and Ecphantus
  • 3.7 Plato and Pythagoreanism
• 4. Neopythagoreanism
  • 4.1 Origins in the Early Academy: Speusippus, Xenocrates and Heraclides in Contrast to Aristotle and the Peripatetics
  • 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, TheGolden Verses and Apollonius of Tyana
• 5. Pythagoreanism in the Middle Ages and Renaissance
  • 5.1 Boethius/Nicomachus, Calcidius, Macrobius and the Middle Ages
  • 5.2 The Renaissance: Ficino, Pico, Reuchlin, Copernicus and Kepler
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Philosophy of Pythagoras

See the entry onPythagoras.

2. The Most Prominent Pythagoreans of the Fifth and Fourth Century

2.1 Philolaus

See the entry onPhilolaus.

2.2 Eurytus

In the ancient sources, Eurytus is most frequently mentioned in thesame breath as Philolaus, and he is probably the student of Philolaus(Iamblichus,VP 148, 139). Aristoxenus (4th c. BCE) presentsPhilolaus and Eurytus as the teachers of the last generation ofPythagoreans (Diogenes Laertius VIII 46) and Diogenes Laertius reportsthat Plato came to Italy to meet Philolaus and Eurytus after the deathof Socrates (III 46). In order to be the pupil of Philolaus, who wasborn around 470, and teach the last generation of Pythagoreans around400, Eurytus would need to be born between 450 and 440. The sourcesare 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 266and 267). It may be that the Eurytus from Metapontum is a differentEurytus. It is possible that Archytas studied with Eurytus, sinceTheophrastus (Aristotle’s successor in the Lyceum) citesArchytas as the source for the one testimony we have about thephilosophy of Eurytus (Metaph. 6a 19–22). In thecatalogue of Pythagoreans at the end of Iamblichus’On thePythagorean Life (267), Eurytus appears between Philolaus andArchytas in the list of Pythagoreans from Tarentum, which may thussuggest that he was regarded as the pupil of Philolaus and a teacherof Archytas.

According to Theophrastus (Metaph. 6a 19–22), Eurytusarranged pebbles in a certain way in order to show the number whichdefined things in the world, such as a man or a horse. Aristotlerefers to the same practice (Metaph. 1092b8 ff.), andAlexander provides commentary on the Aristotelian passage(CAG I. 827.9). Aristotle introduces Eurytus as someone whoregarded numbers as causes of substances by being the points thatbound spatial magnitudes. He says that Eurytus made likenesses of theshapes of things in the natural world with pebbles and thus determinedthe number which belongs to each thing by the number of pebblesrequired. Scholars often treat Eurytus’ procedure as puerile andhave sometimes not taken him seriously (Kahn 2001, 33), or suggestedthat Theophrastus is ironical in his presentation (e.g., Zhmud 2012,410–411). There is, however, no obvious irony inTheophrastus’ remarks. He, in fact, presents Eurytus verypositively as someone who showed in detail how specific parts of thecosmos arose out of basic principles, in contrast to other thinkers,who posit basic principles but do not go very far in explaining howthe world arises from those principles. This positive presentation mayreflect Theophrastus’ source, Archytas, who perhaps saw Eurytusas attempting to carry out Philolaus’ project of determining thenumbers that give us knowledge of things in the world (Huffman 2005,55; see also Netz 2014, 173–178).

How are we, then, to understand Eurytus’ procedure? It does notseem plausible to suppose that he simply drew a picture or an outlinedrawing of a man or a horse and then counted the number of pebblesrequired to make the outline (Riedweg 2005, 86) or fill in thepicture, since the number would vary with the size of the drawing andthe size of the pebbles. A large picture of a man would require manymore pebbles than a small one, so that it would seem arbitrary whichnumber to associate with man. This interpretation treats Eurytus as amosaicist and is largely derived from Alexander’s testimony.Aristotle’s presentation supports another interpretation. Hedraws a parallel with those who arrange numbers of pebbles intoshapes, such as a triangle or a square. This suggests that Eurytus hadobserved that, e.g., any three points in a plane determine a triangleand any four a quadrilateral. He may then have drawn the generalconclusion that any shape or structure was determined by a uniquenumber of points and tried to represent these by setting out thenecessary number of pebbles. Thus, the complex structure of athree-dimensional object such as the human body would require a largenumber of points, but the number of points required to determine ahuman being could be expected to be unique and to differ from thenumber that determined any other object in the natural world, such asa horse (Kirk and Raven 1957, 313 ff.; Guthrie 1962, 273 ff.; Barnes1982, 390–391; Cambiano 1998; Betegh 2014b, 89). It is importantto note that nothing in these reports suggests that Eurytus thoughtthat things were composed of numbers or that he regarded the pointsthat defined a given thing as atoms of which things were made, as hassometimes been supposed (Cornford 1922–1923, 10–11).Instead, he is best understood as making a bold attempt to show thatthe structure of all things is determined by number and thus toprovide specifics for Philolaus’ general thesis that all thingsare known through number. Another approach is to argue that noreference is being made to creating a picture out of pebbles. Thepebbles refer instead to counters on an abacus, which the Greeks usedfor calculations. In this case Eurytus can be supposed to have startedby identifying certain basic numerical properties with features of theworld and then deriving the number of man or horse throughcalculations using the abacus (Netz 2014, 173–178).

2.3 Aristotle’s “So-called” Pythagoreans

Aristotle refers to the Pythagoreans frequently in his extant works,especially in theMetaphysics. There are several puzzlesabout these references. First, his usual practice is to refer to thePythagoreans as a group rather than naming individuals. He mentionsPhilolaus and Eurytus by name only once each and Archytas four times.Yet, the basic Pythagorean system which he describes in most detail inMetaphysics 1.5 shows such strong similarities to thefragments of Philolaus that Philolaus must be the primary source(Huffman 1993, 28–94, Schofield 2012, 147), although somescholars emphasize that Aristotle clearly did use other sources(Primavesi 2012, 255) and even that Philolaus, while perhaps the acmeof Pythagorean philosophy, might not have represented mainstreamPythagoreanism thus explaining why Aristotle refers to thePythagoreans as a group rather than singling out Philolaus (McKirahan2013). Second, he frequently refers to the Pythagoreans that hediscusses as the “so-called” Pythagoreans. Why does he addthe qualifying phrase “so-called?” This phrase indicatesnot that these are false Pythagoreans in contrast to some other truePythagoreans but rather that this is the standard way of referring tothese people, it is what people call them; but the phrase alsoindicates that Aristotle has reservations about the name. Aristotle isexpressing his doubts about how or whether these figures are connectedto Pythagoras himself, whom Aristotle regards as a wonder-workingfounder of a way of life rather than as participating in the traditionof Presocratic cosmology (Huffman 1993, 31–34. This view iscriticized by Álvarez Salas 2021, who argues that Aristotleincludes Pythagoras in his plural references to the Pythagoreans andtreats him as part of the tradition of Presocratic cosmology and notjust as a wonder-worker). It could also be that it is the very varietyof sources that Aristotle is using that leads him to recognize thatthere are quite different stages in the develpment of Pythagoreanismand hence to wonder in what sense a figure like Philolaus who is atthe end of that development should still be called a Pythagorean(Primavesi 2012).

The biggest puzzle, however, concerns the philosophical system thatAristotle assigns to the Pythagoreans. For the purposes of hisdiscussion in theMetaphysics, he treats most Pythagoreans asadopting a mainstream system in contrast to another group ofPythagoreans whose system is based on the table of opposites (seesection 2.4). The central thesis of the mainstream system is stated intwo basic ways: the Pythagoreans say that things are numbers or thatthey are made out of numbers. In his most extended account of thesystem inMetaphysics 1.5, Aristotle says that thePythagoreans were led to this view by noticing more similaritiesbetween things and numbers than between things and the elements, suchas fire and water, adopted by earlier thinkers. The Pythagoreans thusconcluded that things were or were made of numbers and that theprinciples of numbers, the odd and the even, are principles of allthings. The odd is limited and the even unlimited. Aristotlecriticizes the Pythagoreans for being so enamored of numerical orderthat they imposed it on the world even where it was not suggested bythe phenomena. Thus appearances suggested that there were nineheavenly bodies orbiting in the heavens but, since they regarded tenas the perfect number, they supposed that there must be a tenthheavenly body, the counter-earth, which we cannot see. Later,Aristotle is also critical of the Pythagoreans for employingprinciples that do not derive from the sensible world, i.e.,mathematical principles, even though all their efforts were directedat explaining the physical world (Metaphysics 989b29). Howcan they explain features of physical bodies such as weight or motionusing principles which have no weight and do not move(990a8–990a16)? Indeed, it becomes clear that Aristotleinterpreted the Pythagorean cosmogony as starting out by constructingthe number one. The one then draws in the unlimited and produces therest of the number series and evidently the cosmos at the same time.The number one and the other numbers from 1 to 10 are conceived of asphysical entities (Metaphysics 1091a13–18). The puzzleis that Aristotle’s description makes clear that he is basicallydescribing Philolaus’ system (e.g., the counter-earth, limit andunlimited, the generation of a one), yet a number of his centralassertions are flatly contradicted by the surviving fragments ofPhilolaus. Most importantly, Philolaus never says that things arenumbers or are made out of numbers. For Philolaus things are composedof limiters and unlimiteds held together by harmony (Frs. 1, 2 and 6)and unlimiteds appear to include physical things like fire and breath(Fr. 7, Aristotle Fr. 201). Numbers and the odd and the even do play aprominent role in Philolaus (Frs. 4–5), but there is no hintthat they are understood as physical entites. Instead number has anepistemological role: all things are known through number (Fr. 4). Howare we to explain this tension between what Aristotle reports and thefragments of Philolaus? One approach is to recognize that Aristotle isnot giving a historical report of what the Pythagoreans said but aninterpretation of what he found in Philolaus and others. He does notin fact know of any text in which the Pythagoreans said that thingswere numbers or were made of numbers. Instead this is a conclusiondrawn by Aristotle; it is his summary statement of what thePythagorean system amounts to. That this is what Aristotle is doing issuggested by another passage in theMetaphysics where hestarts out by flatly stating that the Pythagoreans say that all thingsare numbers but then goes on to add “at least they applymathematical theories to bodies as if they (the bodies) consisted ofthose numbers” (Metaphysics 1083b16). The “atleast” and “as if” show that Aristotle is drawing aninference rather than referring to any explicit statement by thePythagoreans that things are numbers. Thus for Philolaus there areanalogies between numbers and things and numbers give us knowledge ofthings but Aristotle mistakenly takes this to be equivalent to sayingthat things are numbers or are made of numbers. Another approach is toargue that Aristotle was right that Philolaus and other Pythagoreansthought of the number one and other numbers as physical entities. Theone constructed in Philolaus Fr. 7 is not just the primal physicalunity but also the number one (Schofield 2012). At the oppositeextreme, Zhmud argues that Aristotle has essentially invented thisPythagorean system with little regard for what any actual Pythagoreanssaid in order to serve as background for his account of Plato’stheory of principles (2012a, 438, 394–414). Another approachtries to mitigate the differences between Philolaus and Aristotle andsuggests that Aristotle’s emphasis on number was derived fromPythagorean numerology that was independent of Philolaus but that wascombined with material from Philolaus as a result of Aristotle’sdecision to present one mainstream Pythagorean system (Primavesi2014).

2.4 The Pythagoreans of the Table of Opposites

AtMetaphysics 986a22, after presenting his account of thephilosophy of “the so-called” Pythagoreans (985b23), whichhas strong connections to the philosophy of Philolaus, Aristotle turnsto “others of this same group” and assigns to them what iscommonly known as the table of opposites (the opposites arrangedaccording to column [kata sustoichian]). These Pythagoreanspresented the principles of reality as consisting of ten pairs ofopposites:

Aristotle then contrasts these Pythagoreans with Alcmaeon of Croton,who said that the majority of human things come in pairs, and praisesthe Pythagoreans for carefully defining the pairs of opposites both innumber and character, whereas Alcmaeon seemed to present a randomlyselected and ill-defined group of opposites. Aristotle suggests thateither Alcmaeon was influenced by these Pythagoreans or they by him.Aristotle was thus not sure of the date of these Pythagoreans butseems to entertain the idea that they either lived a little beforeAlcmaeon or a little after, which would make them active anywhere fromthe late 6th to the mid 5th century. Aristotle’s manner ofintroducing these Pythagoreans suggests that they are distinct fromPhilolaus and his pupil Eurytus and perhaps earlier (Schofield 2012:156), but it is not possible to be more specific about their identity.It is possible that Aristotle only knows of the table through oraltransmission and that there were no specific names attached to it.

The table shows a strong normative slant by including good in onecolumn and bad in the other. In contrast, while Philolaus posits thefirst two opposites in the table, limit and unlimited, as firstprinciples, there is no suggestion in the extant fragments ofPhilolaus that limit was good and unlimited bad. Opposites played alarge role in most Presocratic philosophical systems. The Pythagoreanswho posited the table of opposites differed from other early Greekphilosophers not only in the normative view of the opposites but alsoby including strikingly abstract pairs such as straight and crookedand odd and even, in contrast to the more concrete opposites such ashot and cold, which are typical elsewhere in early Greek philosophy.Goldin (2015) argues that the table embodies the associations ofconcepts that formed the basis for the Pythagorean way of life andthat Aristotle recognized that the table was a valuable early attemptto explain the world, although one that failed because it did notidentify relationships of priority and posteriority among theprinciples. Similar tables of opposites appear in the Academy(Aristotle,Metaph. 1093b11;EN 1106b29 referring toSpeusippus; Simplicius inCAG IX. 247. 30ff.), and Aristotlehimself seems at times to adopt such a table (Metaph. 1004b27ff.;Phys. 201b25). Later Platonists and Neopythagoreans willcontinue to develop these tables (see Burkert 1972a, 52, n. 119 for alist). The table of opposites thus provides one of the clearest casesof continuity between early Pythagoreanism and Platonism. Zhmud arguesthat the table has little to do with early Pythagoreanism and islargely a product of the Academy (2012: 449–452) and Burkertthought the table was a mixture of Academic and Pythagorean elements(1972: 51–52) but Aristotle’s discussion of it inconnection with Alcmaeon clearly shows that he regarded it asbelonging to the fifth-century and it is implausible to suppose thathe confused the work of his contemporaries in the Academy withPythagorean ideas that were developed over a century earlier. Goldinargues that we must accept Aristotle’s evidence that somePythagoreans arranged principles in columns even if we cannot be surethey identified specifically the ten pairs listed by Aristotle (2015:173). It may well be that the similarity between this Pythagoreantable of opposites and later Academic versions led to theNeopythagorean habit, starting already in the early Academy, ofmistakenly assigning the fundamental pair of opposites inPlato’s late metaphysics, the one and the indefinite dyad, backto Pythagoras (see on Neopythagoreanism below).

2.5 Archytas

See the entry onArchytas.

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. CE) endswith a catalogue of 218 Pythagorean men organized by city followed bya list of 17 of the most famous Pythagorean women. Of these 235Pythagoreans, 145 appear nowhere else in the ancient tradition. Thisimpressive list of names shows the wide impact of Pythagoreanism inthe fifth and fourth centuries BCE. To what extent is it reliable? Along line of scholars has argued that the catalogue has closeconnections to and is likely to be based on Aristoxenus in the fourthcentury BCE and is thus a reasonably accurate reflection of earlyPythagoreanism rather than a creation of the later Neopythagoreantradition (Rohde 1871–1872, 171; Diels 1965, 23;Timpanaro-Cardini 1958–1964, III 38 ff.; Burkert 1972a, 105, n.40; Zhmud 2012b, 235–244). This is up to a point a reasonableconclusion, since it is hard to see who would have been better placedthan Aristoxenus to have such detailed information.

The arguments connecting Aristoxenus to the catalogue are notunassailable, however, and it is likely that the list has been alteredin transmission, so that it cannot simply be accepted as the testimonyof Aristoxenus (Huffman 2008a). No names on the list can be positivelyassigned to a date later than Aristoxenus, but this would be likely tobe true, even if the list were compiled at a later date, sincePythagoreanism appears to have largely died out for the two centuriesimmediately following Aristoxenus’ death. Thus, Iamblichus doesnot mention any Pythagorean who can be positively dated after the timeof Aristoxenus anywhere else inOn the Pythagorean Lifeeither. Scholars have also argued that Iamblichus cannot have composedthe catalogue, since he mentions some 18 names that do not appear inthe catalogue. This argument would only work, if Iamblichus were acareful and systematic author, which the repetitions andinconsistencies inOn the Pythagorean Life show that he wasnot. While it is unlikely that Iamblichus composed the catalogue fromscratch, it is perfectly possible that he edited it in a number ofways, while not feeling compelled to make it consistent witheverything he says elsewhere in the text. There are some peculiaritiesof the catalogue that suggest a connection to Aristoxenus. Philolausand Eurytus are listed not under Croton but under Tarentum, just asthey are in one of the Fragments of Aristoxenus (Fr. 19 Wehrli =Diogenes Laertius VIII 46). On the other hand, some features of thecatalogue are inconsistent with what we know of Aristoxenus.Aristoxenus’ teacher, Xenophilus, who is identified as from theThracian Chalcidice in the Fragments of Aristoxenus (Frs. 18 and 19Wehrli), is identified as from Cyzicus in the catalogue. Moreover, thelegendary figure, Abaris, is included in the catalogue and even saidto be from the mythical Hyperborea, whereas Aristoxenus is usuallyseen as resolutely trying to rationalize the Pythagorean tradition.Thus, while Aristoxenus is quite plausibly taken to be the author ofthe core of the catalogue, it is likely that additions, omissions, andvarious changes have been made to the original document and hence itis impossible to be sure, in most cases, whether a given name has theauthority of Aristoxenus behind it or not.

The catalogue includes several problematic names, such as Alcmaeon,Empedocles, Parmenides and Melissus. Alcmaeon was active in Crotonwhen the Pythagoreans flourished there, but Aristotle explicitlydistinguishes Alcmaeon from the Pythagoreans and scholarly consensusis that he is not a Pythagorean (see the entry onAlcmaeon). Most scholars would agree that Empedocles was heavily influenced byPythagoreanism; in the later tradition fragments of Empedocles areroutinely cited to support the Pythagorean doctrines of metempsychosisand vegetarianism (e.g., Sextus Empiricus,AdversusMathematicos IX 126–30). On the other hand, both in theancient and in the modern world, Empedocles is not usually labeled aPythagorean, because, whatever the initial Pythagorean influences, hedeveloped a philosophical system that was his own originalcontribution. Parmenides is again not usually identified as aPythagorean in either the ancient or modern tradition and, althoughscholars have speculated on Pythagorean influences on Parmenides,there is little that can be identified as overtly Pythagorean in hisphilosophy. The reason for Parmenides’ inclusion in thecatalogue is pretty clearly the tradition that his alleged teacherAmeinias was a Pythagorean (Diogenes Laertius IX 21). There is noreason to doubt this story, but it gives us no more reason to callParmenides a Pythagorean than to call Plato a Socratic or Aristotle aPlatonist. It would appear that Melissus was included on the listbecause he was regarded in turn as the pupil of Parmenides. Inclusionin the catalogue thus need not indicate that a figure lived aPythagorean way of life or that he adopted metaphysical principlesthat were distinctively Pythagorean; he need only have had contactwith a Pythagorean teacher. It is possible that Aristoxenus includedParmenides and Melissus on the list for these reasons or that he hadbetter reasons for including them (e.g., evidence that they lived aPythagorean life), but it is precisely famous names such as these thatwould be likely to have been added to the list in later times, andthey may well not have appeared in Aristoxenus’ catalogue atall.

Zhmud (2012a, 109–134) has argued that it begs the question touse a doctrinal criterion to identify Pythagoreans. We need to firstidentify Pythagoreans and then see what their doctrines are.Aristoxenus’ catalogue of Pythagoreans as preserved inIamblichus is the crucial source. We should take the Pythagoreans onthis list whom we can identify (the overwhelming majority are justnames for us) and study their interests and activities in order toarrive at a picture of early Pythagoreanism. Of the 235 names thereare only 15 about whom we know anything significant. Some of these arenon-controversial (Hippasus, Philolaus, Eurytus and Archytas).However, Zhmud puts particular emphasis on a series of figures nottypically regarded as Pythagoreans, e.g., Democedes, Alcmaeon, Iccus,Menestor,and Hippon. The range of interests of these figures leads himto conclude that there is no one characteristic that is shared by allPythagoreans and that Wittgestein’s concept of a familyresemblance should be employed to describe Pythagoreanism. Moreover,his reliance on figures like Alcmaeon and Menestor leads him to thesurprising conclusion that natural science and medicine were moreimportant than mathematics for the philosophical views of earlyPythagoreans (2012a, 23). The foundation for this view of earlyPythagoreanism is problematic since the scholarly consensus is thatAlcmaeon was not a Pythagorean and it is also far from certain thatMenestor was a Pythagorean (see below). As argued above,Iamblichus’ catalogue cannot be used mechanically as a guaranteethat a given figure was a Pythagorean, because we cannot be sure thatit always reflects Aristoxenus. What criteria should then be used?

First, anyone identified as a Pythagorean by an early sourceuncontaminated by the Neopythagorean glorification of Pythagoras (seebelow) can be regarded as a Pythagorean. This would include sourcesdating before the early Academy (ca. 350 BCE), where Neopythagoreanismhas its origin, and Peripatetic sources contemporary with the earlyAcademy (ca. 350–300 BCE, e.g., Aristotle, Aristoxenus andEudemus), who, under the influence of Aristotle, defined themselves inopposition to the Academic view of Pythagoras.

Second, a doctrinal criterion is applicable. Anyone who espouses thephilosophy assigned to the Pythagoreans by Aristotle can be regardedas a Pythagorean, although Aristotle presents that philosophy under aninterpretation that must be taken into account. It is important thatthe use of such a doctrinal criterion be limited to quite specificdoctrines such as limiters and unlimiteds as first principles and thecosmology that includes the counter-earth and central fire.Particularly to be avoided is the assumption that any earlymathematician or any early figure who assigns mathematical ideas arole in the cosmos is a Pythagorean. Mathematicians such as Theodorusof Cyrene (who is included in Iamblichus’ catalogue) andHippocrates of Chios (who is not) are not treated as Pythagoreans inthe early sources such as Plato, Aristotle and Eudemus, and there isaccordingly no good reason to call them Pythagoreans. Similarly, thesculptor, Polyclitus of Argos, stated that “the good comes to be… through many numbers,” (Fr. 2 DK), but no ancientsource calls him a Pythagorean (Huffman 2002). As Burkert hasemphasized, mathematics is a Greek and not just a specificallyPythagorean passion (1972a, 427).

Third, anyone universally (or almost universally) called a Pythagoreanby later sources, and whom early sources do not treat as independentof Pythagoreanism, explicitly or implicitly, can be regarded as aPythagorean. This would include figures embedded in the biographicaltradition about Pythagoras and the early Pythagoreans, such as thehusband and wife, Myllias and Timycha.

This last criterion is more subjective than the first two anddifficult cases arise. The fifth-century botanist Menestor (DK I 375)is discussed by Theophrastus and called one of “the old naturalphilosophers” (CP VI 3.5) with no mention of anyPythagoreanism. In this case, the inclusion of a Menestor inIamblichus’ catalogue is not enough reason to regardTheophrastus’ 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 Hippasusas a Pythagorean in the later tradition and his deep involvement inthe biography of early Pythagoreanism, show that he should be regardedas a Pythagorean (on Hippasus, see section 3.4 below). Thefifth-century figure Hippo (DK I 385), who is derided by Aristotle andpaired with Thales as positing water as the first principle(Metaph. 984a3), is a particularly difficult case. An Hippois listed in Iamblichus’ catalogue under Samos and Censorinustells us that Aristoxenus assigned Hippo to Samos rather thanMetapontum (DK I 385.4–5). This makes it look as if Aristoxenusmay be responsible for including Hippo in the catalogue. Burkert hasalso tried to demonstrate connections between Hippo’s philosophyand that of the Pythagoreans (1972a, 290, n. 62). On the other hand,neither Aristotle nor Theophrastus nor any of the Aristoteliancommentators call him a Pythagorean and the doxographers describe thisHippo as from Rhegium (e.g., Hippolytus in DK I 385.17). It is thusnot clear whether we are dealing with one person or two people namedHippo and it is doubtful that the Hippo discussed by the Peripateticswas a Pythagorean (Zhmud regards Hippo as well as Menestor andTheodorus as Pythagoreans — 2012a, 126–128). Those figuresof the sixth, fifth and fourth century who have the best claim to beconsidered Pythagoreans will be discussed in the followingsections.

3.2 The Earliest Pythagoreans: Brontinus, Theano, etc.

In the standard collection of the fragments and testimonia of thePresocratics, Cercops, Petron, Brontinus, Hippasus, Calliphon,Democedes, and Parmeniscus are listed as older Pythagoreans (DK I105–113). Hippasus, who is the most important of these figures,will be discussed separately below (sect. 3.4). Of the rest onlyBrontinus, Calliphon and Parmeniscus appear in Iamblichus’catalogue.

Brontinus is presented as either the husband or father of Theano (seesection 3.3 below). Brontinus (DK I 106–107) is elsewhere saidto have had a wife Deino and to be either from Metapontum or Croton.Little is known about him, but his existence appears to be confirmedby Alcmaeon, writing in the late sixth or early fifth century, whoaddresses his book to a Brontinus along with Leon and Bathyllus (Fr. 1DK). The latter two may also be Pythagoreans, since a Leon is listedunder Metapontum and a Bathylaus (sic) under Posidonia, inIamblichus’ catalogue.

The elusive connection between Orphism and Pythagoreanism rears itshead with Brontinus. In late antiquity there was a consensus thatPythagoras himself had been initiated into the Orphic mysteries andderived much of his philosophy from Orphism (Proclus,Commentaryon Plato’s Timaeus, 3.168.8). Authors of the fifth centuryBCE know of no such initiation and often indicate that the influencewent the other way by reporting that Pythagoras was, in fact, theauthor of supposed Orphic texts (Ion of Chios as reported in Diog.Laert. 8.8). Similarly, the fourth-century author, Epigenes, reportsthat Brontinus is supposed to be the real author of two workscirculating in the name of Orpheus (West 1983, 9 ff.). In the end itis impossible to determine the relationship between Pythagoreanism andOrphism because of the difficulty of defining either movementprecisely (see Betegh 2014a).

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

To Petron (DK I 106) is ascribed the startling doctrine that there are183 worlds arranged in a triangle, but he is only known from a passagein Plutarch, is not called a Pythagorean there and is probably aliterary fiction (Guthrie 1962, 322–323; Burkert 1972a,114).

A Parmeniscus (DK I 112–113) is called a Pythagorean by DiogenesLaertius (IX 20) and may be the same as the Parmiskos listed underMetapontum in Iamblichus’ catalogue. Athenaeus reports that aParmeniscus of Metapontum lost the ability to laugh after descendinginto the cave of Trophonius, only to recover it in a temple on Delos,where the surviving inventory of the temple of Artemis records adedication of a cup by a Parmiskos (Burkert 1972a, 154).

There no good reason to think that Democedes (DK I 110–112), thephysician from Croton, was himself a Pythagorean, although he had somePythagorean connections. He is famous from Herodotus’ account(III 125 ff.) of his service to the tyrant, Polycrates, and thePersian king, Darius. One late source names him a Pythagorean (DK I112.21). Iamblichus mentions a Pythagorean named Democedes, who wasinvolved in the political turmoil surrounding the conspiracy of Cylonagainst the Pythagoreans, but it is far from clear that this was thephysician (VP 257–261). Herodotus never calls Democedesa 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, ispresented as an associate of Pythagoras by Hermippus (DK I 111.36 ff.)and appears in Iamblichus’ catalogue, so it is reasonable toregard him as a Pythagorean, although we know nothing more of him. Itis reported (Herodotus III 137) that Democedes married the daughter ofthe Olympic victor, Milon, who was the Pythagorean, whose house wasused as a meeting place (Iamblichus,VP 249). It wasundoubtedly because Democedes came from Croton at roughly the timewhen Pythagoras was prominent there and because of the Pythagoreanconnections of his father and father-in-law that late sources came tolabel Democedes himself a Pythagorean. For an argument that Democedeswas a Pythagorean see Zhmud 2012a, 120.

3.3 Pythagorean Women

Women were probably more active in Pythagoreanism than any otherancient philosophical movement. The evidence is not extensive but issufficient to give us a glimpse of their role. At the end of thecatalogue of Pythagoreans in Iamblichus’On the PythagoreanLife, after the list of 218 male Pythagoreans, the names of 17Pythagorean women are given (VP 267). Since this list islikely to be based on the work of Aristoxenus, it probably representswhat Aristoxenus learned from fourth-century Pythagoreans, although wecannot, of course, be certain that some names were not inserted intothe list after the time of Aristoxenus (see section 3.1 above andDutsch 2020, 43–51 for a new sceptical reading of thiscatalogue). Eleven are identified as the wife, daughter or sister of aman but seven are simply identified by their region or city-state oforigin, although the Echecrateia of Phlius listed seems likely to beconnected to the Echecrates of Phlius who appears in Plato’sPhaedo. We know nothing else about most of the names on thelist and thus cannot be sure in individual cases whether they belongto the sixth, fifth or fourth century. For a speculativereconstruction of the role of women in the Pythagorean society seeRowett (2014, 122–123), but this reconstruction partly dependson the speech that Iamblichus reports Pythagoras gave to the women ofCroton upon his arrival (VP 54–57); however, whilePythagoras did give speeches to different groups, including women, thetext of the speech in Iamblichus is probably a later fabrication(Burkert 1972a, 115). The Pythagoreans put particular emphasis onmarital fidelity on the part of both men and women (Gemelli Marciano2014, 145). There is also no reliable evidence for any writings bythese women, although in the later tradition works were forged in thenames of some of them and of other Pythagorean women not on the list(see Pellò 2022 and section 4.2 below).

The most famous name on the list is Theano who is here called the wifeof Brontinus but who is elsewhere treated as either the wife, daughteror pupil of Pythagoras (Diogenes Laertius VIII 42; Burkert 1972a,114). The role of women in early Pythagoreanism and the centrality ofTheano is further attested by Aristoxenus’ contemporary,Dicaearchus, who reports that Pythagoras had as followers not just menbut also women and that one of these, Theano, became famous (Fr. 40Mirhday = Porphyry,VP 19). It is striking that Dicaearchusdoes not identify her as the wife of either Brontius or Pythagoras butsimply as a follower of Pythagoras. In the later tradition a number ofworks were forged in her name (see section 4.2 below), but we havelittle reliable evidence about her (see Thesleff 1965, 193–201,for testimonia and texts; Delatte 1922, 246–249; Montepaone1993; and Macris 2016). The second most famous name on the list isTimycha who, when ten months pregnant, reportedly bit off her owntongue so that she could not, under torture, reveal Pythagoreansecrets to the tyrant Dionysius (Iamblichus,VP189–194). This story goes back to Neanthes, writing in the latefourth or early third century and may rely on local Pythagoreantradition (Schorn 2014, 310). See also Macris 2016.

3.4 Hippasus and Other Fifth-century Pythagoreans:acusmatici and mathêmatici

Hippasus is a crucial figure in the history of Pythagoreanism, becausethe tradition about him suggests that even in the fifth century therewas debate within the Pythagorean tradition itself as to whetherPythagoras was largely important as the founder of a set of rules tofollow in living one’s life or whether his teaching also had amathematical and scientific dimension. Hippasus was probably fromMetapontum (Aristotle,Metaph. 984a7; Diogenes Laertius VIII84), although Iamblichus says there was controversy as to whether hewas from Metapontum or Croton (VP 81), and he is listed underSybaris in Iamblichus’ catalogue (VP 267). He isconsistently portrayed as a rebel in the Pythagorean tradition, in onecase a democratic rebel who challenged the aristocratic Pythagoreanleadership in Croton (Iamb.VP 257), but more commonly as thethinker who initiated Pythagorean study of mathematics and the naturalworld.

It is in this latter role that he is connected with the split betweentwo groups in ancient Pythagoreanism, theacusmatici (whoemphasized rules for living one’s life, including varioustaboos) and themathêmatici (who emphasized study ofmathematics and the natural world). Each group claimed to be the truePythagoreans. Our knowledge of this split comes from Iamblichus, whounfortunately presents two contradictory versions, with the resultthat Hippasus is sometimes said to be one of themathêmatici and sometimes one of theacusmatici. Burkert has convincingly shown that the correctversion is that reported by Iamblichus atDe Communi MathematicaScientia 76.19 ff. (1972a, 192 ff.). According to this account,theacusmatici denied that themathêmaticiwere Pythagoreans at all, saying that their philosophy derived fromHippasus instead. Themathêmatici for their part, whilerecognizing that theacusmatici were Pythagoreans of a sort,argued that they themselves were Pythagoreans in a truer sense.Hippasus’ supposed innovations, they said, were in factplagiarisms from Pythagoras himself. Themathêmaticiexplained that, upon Pythagoras’ arrival in Italy, the leadingmen in the cities did not have time to learn the sciences and theproofs of what Pythagoras said, so that Pythagoras just gave theminstructions on how to act, without explaining the reasons. Theyounger men, who did have the leisure to devote to study, learned themathematical sciences and the proofs. The former group were the firstacusmatici, who learned the oral instructions of Pythagorason how to live (theacusmata = “things heard”),while the latter group were the firstmathêmatici.Hippasus was thus closely connected to themathêmaticiin this split in Pythagoreanism but ended up being disavowed by bothsides. For an attempt to further characterize themathêmatici see Horky 2013. For more discussion of theacusmata see section 4.3 of the SEP article onPythagoras.

It is difficult to be sure of Hippasus’ dates, but he istypically regarded as active in the first half of the fifth centuryand perhaps early in that period (Burkert 1972a, 206). The split inPythagoreanism may have occurred after the main period of his work andwas perhaps connected to the attacks on the Pythagorean societies byoutsiders around 450 BCE (Burkert 1972a, 207), but certainty is notpossible. Zhmud (2012a, 169–195) has argued that the split is aninvention of the later tradition, appearing first in Clement ofAlexandria and disappearing after Iamblichus. He also notes that theterm acusmata appears first in Iamblichus (On thePythagorean Life 82–86) and suggests that it also is acreation of the later tradition. He admits that the Pythagorean maximsdid exist earlier, as the testimony of Aristotle shows, but they wereknown assymbola, were originally very few in number and weremainly a literary phenomenon rather than being tied to people whoactually practiced them. The consensus view, which accepts the split,is based on Burkert’s argument that Iamblichus’account ofthe split between theacusmatici andmathêmatici can be shown to be derived from Aristotle(1972a, 196). Burkert later reaffirmed this position, although with alittle less confidence, asserting that the Aristotelian provenance ofthe text is “as obvious as it is unprovable” (1998, 315).Indeed the description of the split in what is likely to be theoriginal version (Iamblichus,On General Mathematical Science76.16–77.18) uses language in describing the Pythagoreans thatis almost an Aristotelian signature, “There are two forms of theItalian philosophy which is called Pythagorean” (76.16).Aristotle famously describes the Pythagoreans as “those calledPythagoreans” and also as “the Italians” (e.g.,Mete. 342b30,Cael. 293a20). Thus, Aristotle remainsthe most likely source. One might also argue against the split on thegrounds that there are no individuals in the historical record thatcan be confidently identified asacusmatici. Since theacusmatici were neither original nor full-time philosophers,however, and simply preserved the oral taboos handed down byPythagoras, it is not surprising that they are not singled out forattention in the sources. Only a relatively small number of the namesin Iamblichus’ catalogue can certainly be identified asmathêmatici and most of the others, particularly the145 individuals whose names are only known from the catalogue, arelikely to beacusmatici, who to a greater or lesser degreefollowed the Pythagoreanacusmata, but left no other trace oftheir activity. In addition, a number of other Pythagoreans of thefifth and fourth century, who figure in anecdotes about thePythagorean life are likely to beacusmatici (see below).

Hippasus is the first figure in the Pythagorean tradition who can withsome confidence be identified as a natural philosopher, mathematicianand music theorist. His connections are as much with figures outsidethe Pythagorean tradition as those within it. This independence mayexplain why neither Aristotle nor the doxographical tradition labelhim a Pythagorean, but he is too deeply embedded in the traditionsabout early Pythagoreanism for there to be any doubt that he was insome sense a Pythagorean. Aristotle pairs Hippasus with Heraclitus aspositing fire as the primary element (Metaph. 984a7) and thispairing is repeated in the doxography that descends from Theophrastus(DK I 109. 5–16), according to which Hippasus also said that thesoul was made of fire. Philolaus, who was probably two generationslater than Hippasus, might have been influenced by Hippasus instarting his cosmology with the central fire (Fr. 7). For Philolaus,however, the central fire is a compound of limiter and unlimited,whereas Hippasus is presented as a monist and does not start fromPhilolaus’ fundamental opposition between limiters andunlimiteds.

There are only a few other assertions about the cosmology of Hippasusand most of these seem to be the result of Peripatetic attempts toclassify him, such as the assertions that he makes all things fromfire by condensation and rarefaction and dissolves all things intofire, which is the one underlying nature and that he and Heraclitusregarded the universe as one, (always) moving and limited in extent(DK I 109.8–10). More intriguing is the claim that he thoughtthere was “a fixed time for the change of the cosmos”(Diogenes Laertius VIII 84), which might be a reference to a doctrineof eternal recurrence, according to which events exactly repeatthemselves at fixed periods of time. This doctrine is attestedelsewhere for Pythagoras (Dicaearchus in Porphyry,VP 19).Our information about Hippasus is sketchy, because he evidently didnot write a book. Demetrius of Magnesia (1st century BCE) reports thatHippasus left nothing behind in writing (Diogenes Laertius VIII 84)and this is in accord with the tradition that Philolaus was the firstPythagorean to write a book.

Hippasus originates the early Pythagorean tradition of scientific andmathematical analysis of music, which reaches its culmination inArchytas a century later. The correspondence between the centralmusical concords of the octave, fifth, and fourth and the whole numberratios 2 : 1, 3 : 2 and 4 : 3 is reflected in theacusmata(Iamblichus,VP 82) and was thus probably already known byPythagoras. This correspondence was central to Philolaus’conception of the cosmos (Fr. 6a). Although the later tradition triedto assign the discovery to Pythagoras himself (Iamblichus,VP115), the method described in the story would not in fact have worked(Burkert 1972a, 375–376). Hippasus is the first person to whomis assigned an experiment demonstrating these correspondences that isscientifically possible. Aristoxenus (Fr. 90 Wehrli = DK I 109. 31ff.) reports that Hippasus prepared four bronze disks of equaldiameters, whose thicknesses were in the given ratios, and it is truethat, if free hanging disks of equal diameter are struck, the soundproduced by, e.g., a disk half as thick as another will be an octaveapart from the sound produced by the other disk (Burkert 1972a, 377).Hippasus, thus, may be the first person to devise an experiment toshow that a physical law can be expressed mathematically (Zhmud 2012a,310).

Another text associates Hippasus with Lasus of Hermione in an attemptto demonstrate the correspondence by filling vessels with liquid inthe appropriate ratios. It is less clear whether this experiment wouldhave worked as described (Barker 1989, 31–32). Lasus wasprominent in Athens in the second half of the sixth century at thetime of the Pisistratid tyranny and was thus probably a generationolder than Hippasus. There is no indication that Lasus was aPythagorean and this testimony suggests that the discovery of andinterest in the mathematical basis of the concordant musical intervalswas not limited to the Pythagorean tradition. Lasus and Hippasus aresometimes said to have been the first to put forth the influential butmistaken thesis that the pitch of a sound depended on the speed withwhich it travels, but it is far more likely that Archytas originatedthis view. In the later tradition Hippasus is reported to have rankedthe musical intervals in terms of degrees of concordance, making theoctave the most concordant, followed by the fifth, octave + fifth,fourth and double octave (Boethius,Mus. II 19).

Finally, Iamblichus associates Hippasus with the history of thedevelopment of the mathematics of means (DK I 110. 30–37), whichare important in music theory, but Iamblichus’ reports areconfused. It is likely that Hippasus worked only with the threeearliest means (the arithmetic, geometric and subcontrary/harmonic)and that the changing of the name of the subcontrary mean to theharmonic mean should be ascribed to Archytas rather than Hippasus(Huffman 2005, 179–173).

The most romantic aspect of the tradition concerning Hippasus is thereport that he drowned at sea in punishment for the impiety of makingpublic and giving a diagram of the dodecahedron, a figure with twelvesurfaces each in the shape of a regular pentagon (Iamblichus,VP 88). This is best understood as reflecting some sort ofmathematical analysis of the dodecahedron by Hippasus, but it isimplausible in terms of the history of Greek mathematics to supposethat he carried out a strict construction of the dodecahedron, whichalong with the other four regular solids is most likely to have firstreceived rigorous treatment by Theaetetus in the fourth century BCE(Mueller 1997, 277; Waterhouse 1972; Sachs1917, 82). Nor is it clearwhy public presentation of technical mathematical analysis shouldcause a scandal, since few people would understand it. The most likelyexplanation is that the dodecahedron was a cult object for thePythagoreans (dodecahedra in stone and bronze have been found datingback to prehistoric times) and that it was because of these religiousconnections that Hippasus’ public work on the mathematicalaspects of the solid was seen as impious (Burkert 1972a, 460).

Another late story, which appears first in Plutarch, reports a scandalwhich 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, theperson who first spread knowledge of the existence of the irrationalwas punished by drowning (Junge and Thomson 1930, 63–64).Iamblichus knows two different versions of the story, one according towhich the malefactor was banished and a tomb was erected for him,signifying his expulsion from the community (VP 246), butanother according to which he was punished by drowning as was theperson (not specifically said to be Hippasus here) who revealed thedodecahedron (VP 247). Modern scholars have tried to combinethe two stories and suppose that Hippasus discovered the irrationalthrough his work on the dodecahedron (von Fritz 1945). This is purespeculation, however, since neither does any ancient source connectHippasus to the discovery of the irrational nor does any source relatethe discovery of the irrational to the dodecahedron (Burkert 1972a,459). Some scholars nonetheless credit Hippasus with the discovery ofirrationality (Zhmud 2012a, 274–278).

Some have argued that Hippasus was an important figure for the earlyAcademy to whom Academic doctrines were ascribed in order give themhis authority and even that he might be the Prometheus mentioned byPlato as handing down the method from the gods in thePhilebus (Horky 2013). However, there is no explicit mentionof Hippasus by any member of the Academy and he is a minor figure infourth-century accounts of early Greek philosophy (e.g., Aristotle) soit is hard to see what authority he could give to Academic views.

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

The name, but not too much more, is known of a number of other fifthcentury figures, who with varying degrees of probability may beconsidered Pythagoreans. To the beginning of the fifth century belongsAmeinias the teacher of Parmenides (Diogenes Laertius VIII 21). Theathlete and trainer, Iccus of Tarentum, is listed in Iamblichus’catalogue, but none of the other sources, including Plato, call him aPythagorean. In the later tradition, he was famous for the simplicityof his life and “the dinner of Iccus” was proverbial forplain fare. Plato praises his self control and reports that he touchedneither women nor boys while training. (Laws 839e; seeProtagoras 316d and DK I 216. 11 ff.).

Some scholars have treated the Sicilian comic poet Epicharmus as aPythagorean and argued that the growing argument which appears in afragment of controversial authenticity ascribed to him in DiogenesLaertius (3.11) is thus Pythagorean in origin (Horky 2013,131–140). However, no fifth- or fourth-century source identifiesEpicharmus as a Pythagorean and he does not appear in the catalogue ofIamblichus. The earliest explicit mention of him as a Pythagorean isin Plutarch (Numa 9) in the first century CE. There is nocompelling evidence that the reference to Epicharmus as a Pythagoreanin Iamblichus’On the Pythagorean Life 266 derives fromthe fourth-century historian Timaeus as Horky proposes (2013, 116).Burkert suggests that the information on Didorus in 266 might derivefrom Timaeus (1972, 203–204) but Iamblichus regularly combinesmaterial from a number of sources so that neither Burkert nor mostscholars regard the passage as a whole as deriving from Timaeus(Schorn 2014 only mentions VP 254–264 as having material fromTimaeus). Epicharmus has also been thought to be a Pythagorean becausethe growing argument which he uses for comic effect uses pebbles torepresent numbers and refers to odd and even numbers. However, neitherof the features is peculiarly Pythagorean; the concept of odd and evennumbers belongs to Greek mathematics in general and not just to thePythagoreans and the use of counters (pebbles) on an abacus is thestandard way in which Greeks manipulated numbers (Netz 2014, 178; cf.Burkert’s doubts that there is anything Pythagorean in theEpicharmus fragment 1972a, 438). Most scholars regardEpicharmus’ Pythagoreanism as a creation of the later tradition(Zhmud 2012a, 118 and 2019b, 138–140; Riedweg 2005, 115; Kahn2001, 87).

There is no reason to regard the physician Acron of Acragas as aPythagorean, as Zhmud does (1997, 73; he appears to have changed hismind in 2012a, 116). Acron is a contemporary of Empedocles and isconnected 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 ofAristoxenus (Aristoxenus, Fr. 22 Wehrli), but it is pure speculationthat Aristoxenus labeled him a Pythagorean; Euryphon the Cnidiandoctor of the fifth century, who was not a Pythagorean, also appearsin the papyrus. Acron’s father’s name was Xenon, and aXenon appears in Iamblichus’ catalogue, but he is listed as fromLocri and not Acragas, so again this is not good evidence that Acronwas a Pythagorean.

The Pythagorean Paron (DK I 217. 10–15) is probably a fictionresulting from a misreading of Aristotle (Burkert 1972a, 170).Aristotle reports the expression of a certain Xuthus, that “theuniverse would swell like the ocean,” if there were not voidinto which parts of the universe could withdraw, when compressed(Physics 216b25). Simplicius says, on unknown grounds, thatthis Xuthus was a Pythagorean, and scholars have speculated that hewas responding to Parmenides (DK I. 376. 20–26; Kirk and Raven1957, 301–302; Barnes 1982, 616).

Aristoxenus reports that two Tarentines, Lysis and Archippus, were thesole survivors when the house of Milo in Croton was burned, during ameeting of the Pythagoreans, by their enemies (Iamblichus,VP250). A later romantic version in Plutarch (On the Sign ofSocrates 583a) has it that Lysis and Philolaus were the twosurvivors, but it appears that the famous name of Philolaus has beensubstituted for Archippus, about whom nothing else is known.Aristoxenus goes on to say that Lysis left southern Italy and wentfirst to Achaea in the Peloponnese before finally settling in Thebes,where the famous Theban general, Epaminondas, became his pupil andcalled him father. In order to be the teacher of Epaminondas in theearly fourth century, Lysis must have been born no earlier than about470. Thus the conflagration that he escaped as a young man must havebeen part of the attacks on the Pythagoreans around 450, rather thanthose that occurred around 500, when Pythagoras himself was stillalive. The later sources often conflate these two attacks on thePythagoreans (Minar 1942, 53). Nothing is known of the philosophy ofLysis, but it seems probable that he should be regarded as one of theacusmatici, since his training of Epaminondas appears to haveemphasized a way of life rather than mathematical or scientificstudies (Diodorus Siculus X 11.2) and Epaminondas’ use of thename father for Lysis suggests a cult association (Burkert 1972a,179). In the later tradition, Lysis became quite famous as the authorof a spurious letter (Thesleff 1965, 111; cf. Iamblichus,VP75–78) rebuking a certain Hipparchus for revealing Pythagoreanteachings to the uninitiated (see on the Pythagorean pseudepigraphabelow, sect. 4.2).

Zopyrus of Tarentum is mentioned twice, in a treatise on siege-enginesby Biton (3rd or 2nd century BCE), as the inventor of an advanced formof the type of artillery known as the belly-bow (Marsden 1971,74–77). Zopyrus’ bow used a winch to pull back the stringand hence could shoot a six-foot wooden missile 4.5 inches thick(Marsden 1969, 14). It is not implausible to suppose that this is thesame Zopyrus as is listed in Iamblichus’ catalogue ofPythagoreans under Tarentum (Diels 1965, 23), although Biton does notcall him a Pythagorean. The traditional dating for Zopyrus puts him inthe first half of the fourth century (Marsden 1971, 98, n. 52), butKingsley has convincingly argued that he was in fact active in thelast quarter of the fifth century, when he designed artillery forCumae and Miletus (1995, 150 ff.). In a famous passage, Diodorusreports that in 399 BCE Dionysius I, the tyrant of Syracuse, gatheredtogether skilled craftsmen from Italy, Greece and Carthage in order toconstruct artillery for his war with the Carthaginians (XIV 41.3). Itseems not unlikely that Zopyrus was one of those who came from Italy.There is no reason to suppose, however, as Kingsley (1995, 146) andothers do, that Zopyrus’ interest in mechanics was connected tohis Pythagoreanism or that there was a specifically Pythagorean schoolof mechanics in Tarentum (Huffman 2005, 14–17).

It is controversial whether this Zopyrus of Tarentum is the same asZopyrus of Heraclea, who is not called a Pythagorean in the sources,but who is reported in late sources to have written three Orphicpoems,The Net,The Robe andThe Krater,which probably dealt with the structure of human beings and the earth(West 1983, 10 ff.). This Zopyrus could be from the Heraclea closelyconnected to Tarentum, but he might also be from the Heraclea on theBlack Sea. A late source connects Zopyrus of Heraclea with Pisistratusin the 6th century (West 1983, 249), which would mean that he couldnot be the same as Zopyrus of Tarentum in the late 5th century. On theother hand, Orphic writings are assigned to a number of otherPythagoreans, and it is not impossible that the same person hadinterests both in Orphic mysticism and mechanics. Kingsley supposesthat the myth at the end of Plato’sPhaedo is based inminute detail on Zopyrus’Krater or an intermediaryreworking of it (1995, 79–171), and tries to connect specificfeatures of the myth to Zopyrus’ interest in mechanics (1995,147–148), but the parallel which he detects between theoscillation of the rivers in the mythic account of the underworld andthe balance of opposing forces used in a bow is too general to becompelling. The connection between Zopyrus and thePhaedo ishighly conjectural and must remain so, as long as there are nofragments of theKrater, with which to compare thePhaedo.

A harmonic theorist named Simus is accused of having plagiarized oneof seven pieces of wisdom inscribed on a bronze votive offering, whichwas dedicated in the temple of Hera on Pythagoras’ native islandof Samos, by Pythagoras’ supposed son Arimnestus (Duris of Samosin 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, likeHippasus, stole some of the master’s teaching for his own glory.There is, however, no obvious connection between the two individualsnamed Simus except the name. Most scholars have thus treated Simus asif he were a harmonic theorist in competition with and independent ofthe Pythagorean tradition (Burkert 1972a, 449–450; West 1992, 79and 240; Wilamowitz 1962, II 93–94).

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

Iamblichus describes an ‘arithmetical method’ known as thebloom of Thymaridas (In Nic. 62), and elsewhere discusses twopoints of terminology in Thymaridas, including his definition of themonad 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 beearlier than Plato and shows connections to Diophantus in the thirdcentury CE (see Burkert 1972a, 442, n. 92 for a summary of thescholarship). There is also a Thymaridas in the biographicaltradition, who may or may not be the same individual. In a highlysuspect passage in Iamblichus, Thymarides is listed as a pupil ofPythagoras himself (VP 104) and a Thymaridas of Paros appearsin Iamblichus’ catalogue and is mentioned in one anecdote(VP 239). There is also a worrisome connection to thepseudo-Pythagorean literature. A Thymaridas of Tarentum is presentedin an anecdote (Iamblichus,VP 145) as arguing that peopleshould wish for what the gods give them rather than praying that thegods give them what they want, a sentiment that is also found in agroup of three treatises forged in Pythagoras’ name (DiogenesLaertius VIII 9). The anecdote is drawn from Androcydes’ work onthe Pythagoreansymbola or taboos. If this work could bedated to the fourth century, it would confirm an early date forThymaridas, but all that is certain is that Androcydes’ work wasknown in the first century BCE and thus that the anecdote originatedbefore that date (Burkert 1972a, 167). It seems rash, given thisconfused evidence, to follow Zhmud and regard Thymaridas as a youngercontemporary or pupil of Archytas (2012a, 131). For more on Thymaridassee Macris 2016.

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

Aristoxenus (ca. 375– ca. 300 BCE) is most famous as a musictheorist and as a member of the Lyceum, who was disappointed not be tonamed Aristotle’s successor (Fr. 1 Wehrli). In his early years,however, he was a Pythagorean, and he is one of the most importantsources for early Pythagoreanism. He wrote five works onPythagoreanism, although it is possible that some of these titles arealternative names for the same work:The Life of Pythagoras,On Pythagoras and His Associates,On the PythagoreanLife,Pythagorean Precepts and aLife ofArchytas. None of these works have survived intact, but portionsof them were preserved by later authors (Wehrli 1945). Aristoxenus isa valuable source because, as a member of the Lyceum, he is free ofthe distorted image of Pythagoras propagated during his lifetime byPlato’s successors in the Academy (see below, sect. 4.1) andbecause of his unique connections to Pythagoreanism.

He was born in Tarentum during the years when the most importantPythagorean of the fourth century, Archytas, was the leading publicfigure and his father, Spintharus, had connections to Archytas (Fr. 30Wehrli). When Aristoxenus left Tarentum, as a young man, andeventually came to Athens (ca. 350), his first teacher was Xenophilus,a Pythagorean. Then he went on to become the pupil of Aristotle (Fr. 1Wehrli). Some modern scholars are skeptical of Aristoxenus’testimony, seeing his denial that there was a prohibition on eatingbeans and his assertion that Pythagoras was not a vegetarian andparticularly enjoyed eating young pigs and tender kids (Fr. 25 =Gellius IV 11), as attempts to make Pythagoreanism more rational thanit was (Burkert 1972a, 107, 180). On the other hand, hisLife ofArchytas is not a simple panegyric; Archytas’ foibles arerecognized and his opponents are given a fair hearing. On Aristoxenusas a source for Pythagoreanism see most recently Zhmud 2012b andHuffman 2014b, 285–295.

Perhaps Aristoxenus’ most interesting work on Pythagoreanism isthePythagorean Precepts, which is known primarily throughsubstantial excerpts preserved by Stobaeus (Frs. 33–41 Wehrli).This work does not mention any Pythagoreans by name but presents a setof ethical precepts that “they” (i.e. the Pythagoreans)proposed concerning the various stages of human life, education, andthe proper place of sexuality and reproduction in human life. Thereare also analyses of concepts important in ethics, such as desire andluck. Given Aristoxenus’ background, thePrecepts wouldappear to be invaluable evidence for Pythagorean ethics in the firsthalf of the fourth century, when Aristoxenus was studyingPythagoreanism. They might be expected to partially embody the viewsof his teacher Xenophilus. The standard scholarly view of this work,however, is that Aristoxenus plundered Platonic and Aristotelian ideasfor the glory of the Pythagoreans (Wehrli 1945, 58 ff.; Burkert 1972a,107–108). There are serious difficulties with the standard view,however (Huffman 2019). The analysis of luck that was supposedly takenfrom Aristotle is, in fact, in sharp conflict with Aristotle’sview (Mills 1982) and appears to be one of the views Aristotle wasattacking. While thePrecepts do have similarities topassages in Plato and Aristotle, they are at a very high level ofgenerality and are shared with passages in other fifth and fourthcentury authors, such as Xenophon and Thucydides; it is thedistinctively Platonic and Aristotelian features that are missing.

ThePrecepts are thus best regarded as what they appear onthe surface to be, an account of Pythagorean ethics of the fourthcentury. This ethical system shows a similarity to a conservativestrain of Greek ethics, which is also found in Plato’sRepublic, but has its own distinctive features (Huffman2019). The central outlook of thePrecepts is a distrust ofbasic human nature and an emphasis on the necessity for supervision ofall aspects of human life (Fr. 35 Wehrli). The emphasis on order inlife is so marked that thestatus quo is preferred to what isright (Fr. 34). The Pythagoreans were particularly suspicious ofbodily desire and analyzed the ways in which it could lead peopleastray (Fr. 37). There are strict limitations on sexual desire and thepropagation of children (Fr. 39). Despite the best efforts ofhumanity, however, many things are outside of human control, so thePythagoreans examined the impact of luck on human life (Fr. 41).

Aristoxenus is a source for the famous story of the two Pythagoreanfriends Damon and Phintias, which was set during the tyranny ofDionysius II in Syracuse (367–357). As a test of theirfriendship Dionysius falsely accused Phintias of plotting against himand sentenced him to death. Phintias asked time to set his affairs inorder, and Dionysius was amazed when Damon took his place, while hedid so. Phintias showed his equal devotion to his friend by showing upon time for his execution. Dionysius cancelled the execution and askedto become a partner in their friendship but was refused (Iamblichus,VP 234; Porphyry,VP 59–60; Diodorus X4.3).

In Diodorus’ version, Phintias is presented as actually engagedin a plot against Dionysius 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 theirfriendship, which occurs in both versions, is harder to understand ifthere really had been a plot (see Burkert 1972a, 104). There are twoother considerations. First, Aristoxenus cites Dionysius II himself ashis source, whereas it is unclear what source Diodorus used. Second,it is far from clear that Aristoxenus would object to the Pythagoreansplotting against a tyrant. Thus, there are good reasons for regardingAristoxenus’ version as more accurate.

Cleinias and Prorus are another pair of Pythagorean friends, whosestory may have been told by Aristoxenus (Iamblichus,VP 127),although they were not friends in the usual sense. Cleinias, who wasfrom Tarentum, knew nothing of Prorus of Cyrene other than that he wasa Pythagorean, who had lost his fortune in political turmoil. On thesegrounds alone he went to Cyrene, taking the money to restoreProrus’ fortunes (Iamblichus,VP 239; Diodorus X 4.1).Nothing else is known of Prorus, although some pseudepigrapha wereforged in his name (Thesleff 1965, 154.13). It appears that Cleiniaswas a contemporary of Plato, since Aristoxenus reports that he and anotherwise unknown Pythagorean, Amyclas, persuaded Plato not to burnthe 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 hesupposedly refused to punish when angry (VP 198) and, whenangered, calmed himself by playing the lyre (Athenaeus XIV 624a).Asked when one should resort to a woman he said “when onehappens to want especially to be harmed” (Plutarch,Moralia 654b). Several pseudepigrapha appear inCleinias’ 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 werepersecuted by the tyrant Dionysius II of Syracuse, but Timycha showedher loyalty and courage by biting off her tongue and spitting it inthe tyrant’s face, rather than risk divulging Pythagoreansecrets under torture.

None of the Pythagoreans mentioned in the previous four paragraphsappear to have to have anything to do with the sciences or naturalphilosophy. Since their Pythagoreanism consists exclusively in theirway of life, they are best regarded as examples of theacusmatici. Many scholars have regarded Diodorus of Aspendusin Pamphylia (southern Asia Minor), as an important example of whatthe Pythagoreanacusmatici were like in the first half of thefourth century (Burkert 1972a, 202–204). Diodorus is primarilyknown through a group of citations preserved by Athenaeus (IV 163c-f),which describe him as a vegetarian who was outfitted in an outlandishway, some features of which later became characteristic of the Cynics,e.g., long hair, long beard, a shabby cloak, a staff andbeggar’s rucksack (cf. Diogenes Laertius VI 13). The historianTimaeus (350–260), however, casts doubt on Diodorus’credentials as a Pythagorean saying that “he pretended to haveassociated with the Pythagoreans” and Sosicrates, anotherhistorian (2nd century BCE; fragments in Jacoby) says that hisoutlandish dress was his own innovation, since before thisPythagoreans had always worn white clothing, bathed and wore theirhair according to fashion (Athenaeus IV 163e ff.). Iamblichus, theother major source for Diodorus outside Athenaeus, also treatsDiodorus with reserve, saying that he was accepted by the leader ofthe Pythagorean school at the time, one Aresas, because there were sofew members of the school. He continues, perhaps again withdisapproval, to report that Diodorus returned to Greece and spreadabroad the Pythagorean oral teachings.

These sources clearly suggest that Diodorus was anything but a typicalPythagorean, even of theacusmatic variety. Burkert hasargued that this reflects a bias of sources such as Aristoxenus, whowanted to make Pythagoreanism appear reasonable and emphasized theversion of Pythagoreanism practiced by themathêmaticirather than theacusmatici. In support of this conclusion, heargues that the two earliest sources present Diodorus as a Pythagoreanwithout any qualifications (1972a, 204). It is important to lookcarefully at those sources, however. First, neither is a philosopheror a historian, who might be expected to give a careful presentationof Diodorus. The oldest is a lyre player named Stratonicus (died 350BCE), who was famous for his witticisms, and the other, Archestratus(fl. 330 BCE), wrote a book entitledThe Life of Luxury,which focused on culinary delights. Such sources might be expected toaccept typical stories that went around about Diodorus without anyclose analysis.

In the case of our earliest source, Stratonicus, there is, moreover,once again evidence suggesting that Diodorus was not regarded as atypical Pythagorean. In describing Diodorus’ relationship toPythagoras, Stratonicus does not use a typical word for student ordisciple, but rather the same word (pelatês) that Platoused in theEuthyphro to describe the day-laborer who died atthe hands of Euthyphro’s father. Diodorus is thus beingpresented sarcastically as a hired hand in the Pythagorean tradition,which is very much in accord with the later presentations of him as apoor man’s Pythagoras on the fringes of Pythagoreanism. Thus,rather than accusing the sources of bias against Diodorus, it seemsbetter to accept their almost universal testimony that he was not atypicalacusmatic but rather a marginal figure, who usedPythagoreanism in part to try to gain respectability for his owneccentric 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 ofthe fourth century (see Burkert 1972a, 198, n. 25 for the evidence and200, n. 41 for the dating). The most important of the fragments ofthese comedies that deal with the Pythagorists are collected byAthenaeus (IV 160f ff) and Diogenes Laertius (VIII 37–38). Theterm “Pythagorist” is usually negative in the comicwriters (Arnott 1996, 581–582) and picks out people who sharesome of the same extreme ascetic lifestyle as Diodorus. A fragment ofAntiphanes describes someone as eating “nothing animate, as ifPythagorizing” (Fr. 133 Kassel and Austin = Athenaeus IV 161a).InThe Pythagorizing Woman, Alexis presents the vegetariansacrificial feast that is customary for the Pythagoreans as includingdried figs, cheese and olive cakes, and reports that the Pythagoreanlife 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 theacusmata (Arnott 1996, 583), e.g., the lack of bathing may bea joke based on theacusma that forbids the Pythagoreans fromusing the public baths (Iamblichus,VP 83), Antiphanes (fr.158) satirizes theacusmata’s bizarre list of foodsthat can be eaten (D.L. 8.19) by describing his Pythagoreans assearching for sea orach, and the silence or sullenness ascribed to thePythagoreans in comedy accords not just with theacusmata butwith early testimony about the Pythagoreans in Isocrates(Busiris 29) and Dicaearchus (Fr. 40 Mirhady). A fragment ofAristophon’sPythagorist suggests that this asceticlife was based on poverty rather than philosophical scruple and that,if you put meat and fish in front of these Pythagorists, they wouldgobble 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 eatsdogs, and he is a Pythagorean,” to which the response is,“yes, but he kills them first and so they are not stillanimate” (Fr. 223 + Athenaeus 161b). Epicharides and some othernamed figures may well be Athenians who are satirized by beingassigned a Pythagorean life (Athenaeus 2006, 272). Another fragment ofAristophon’sPythagorist reports that the Pythagoreanshave a far different existence in the underworld than others, in thatthey feast with Hades because of their piety, but this just occasionsthe remark that Hades is an unpleasant god to enjoy the company ofsuch 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 entitledThePeople of Tarentum, which, although they may not have beenprimarily about Pythagoreans, featured depictions of them (Arnott1996, 625–626). In this case, the Pythagoreans are againsatirized 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 ofAristophanes’ treatment of Socrates in theClouds.

Given the fragmentary nature of the evidence, it is unclear whetherthese ascetic Pythagoreans who engage in argument are the same as thePythagorists in the other comedies, who are characterized by theirfilth and eccentric appearance. Certainly the latter are morereminiscent of Diodorus of Aspendus, while the former might be closerto what we know of someone like Cleinias. In the first half of thethird century, the poet Theocritus still preserves a memory of thesePythagorists as “pale and without shoes” (XIV 5). Thescholiast to the passage testifies to the continuing controversy aboutthe Pythagorists by drawing a distinction between Pythagoreans whogive every attention to their body and Pythagorists who are filthy(although another scholion reports that others say the opposite, seeArnott 1996, 581). A passage in Iamblichus (VP 80) similarlyargues 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 andthe Pythagorists as legitimate Pythagoreans of the acusmatic stamp,whose eccentricities are perhaps a little exaggerated in comedy. Theextensive evidence from antiquity which argues that they were not truePythagoreans is interpreted as bias on the part of conservativePythagoreans of thehyper-mathêmatici sort, such asAristoxenus, who wanted to disassociate themselves and Pythagoreanismin general from such strange people. This is a possible interpretationof the evidence, but, as the evidence for Diodorus shows, it is alsoquite possible that Diodorus and the more extreme Pythagoristsdepicted in comedy were in fact people with whom few Pythagoreanseither of themathêmatici or theacusmaticiwanted to associate themselves. Many religious movements have aradical fringe, and there is little reason to think thatPythagoreanism should differ in this regard. In connection with histhesis that the acusmata were a literary phenomenon and thatno one lived a life in accordance with them Zhmud argues that thePythagorists of comedy are a creation of the comic stage and do notprovide evidence for Pythagoreans living a life governed byacusmata (2012a, 175–183). It is true that many of thefeatures of the Pythagorists are shared with Socrates as presented intheClouds (subtle arguments, plain food, filthy clothes).Zhmud suggests that vegetarianism was added to this stock picture ofthe philosopher to give a Pythagorean color and that thisvegetarianism was derived solely from the eccentric figure of Diodorusof Aspendus. However, as noted above there are more connections to theacusmata than just vegetarianism and it is hard to believethat the repeated jokes at the expense of those living a Pythagoreanlife had no correlate in reality other than Diodorus.

Perhaps the best way to evaluate the complicated evidence forfourth-century Pythagoreanism is to conclude that there were threemain groups, each of which admitted some variation. There weremathêmatici such as Archytas who did serious researchin the mathematical disciplines and natural philosophy but who alsolived an ascetic life that emphasized self-control and avoidance ofbodily pleasure. Other Pythagoreans such as Cleinias or Xenophilus mayhave done no work in the sciences but lived a Pythagorean life, whichwas similar to that of Archytas and followed principles similar tothose set out in Aristoxenus’Pythagorean Precepts.They may have observed some mild dietary restrictions and may besimilar to the figures satirized inThe Men of Tarentum aseating a simple diet but still engaged in subtle arguments. There wasprobably a continuum of people in this category with some followingmore or different sets of theacusmata than others. Finallythere are the Pythagorean hippies such as Diodorus and thePythagorists, who ostentatiously live a life in accord with some oftheacusmata, but who take such an extreme interpretation ofthem 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 ThracianChalcidice (Aristoxenus’ teacher), and four Pythagoreans fromPhlius: Phanton, Echecrates, Diocles and Polymnastus. ThesePythagoreans are further identified as the pupils of Philolaus andEurytus. Little more is known of Xenophilus beyond his living for morethan 105 years (DK I 442–443). The Pythagoreans from Phlius arejust names except Echecrates (DK I 443), to whom Phaedo narrates,evidently in Phlius, the events of Socrates’ last day inPlato’sPhaedo. Socrates’ interlocutors in thePhaedo, Simmias and Cebes, are often regarded asPythagoreans, because they are said to have been pupils of Philolauswhen he was in Thebes. They are also shown to be pupils of Socrates,however, and it is unclear that their connection to Philolaus was anycloser than their connection to Socrates. They are not listed inIamblichus’ catalogue as Pythagoreans; Diogenes Laertiusincludes them with other followers of Socrates (II 124–125).Echecrates might have been born around 420 and thus be a young man atthe dramatic date of thePhaedo. Aristoxenus’ assertionthat these were the last of the Pythagoreans would then suggest thatPythagoreanism died out around 350, when Echecrates was an oldman.

Riedweg says that this claim is “demonstrably untrue”pointing to a Pythagorean, Lycon, who criticized Aristotle’ssupposed extravagant way of life and to the Pythagorists discussedabove (2005, 106). This seems slender evidence upon which to be socritical of Aristoxenus. Virtually nothing is known of Lycon, andAristocles (1st-2nd c. CE), who recounts the criticism of Aristotle,says that Lycon “called himself a Pythagorean,” thusexpressing some sort of reservation about his credentials (DK I445–446). Aristoxenus’ assertion is probably to beunderstood as a general claim that, with the deaths of thePythagoreans from Phlius around the middle of the fourth century,Pythagoreanism as an active movement was dead. This would becompatible with a few individuals still claiming to be Pythagoreansafter 350.

This is not inconsistent with the existence of a few isolatedindividuals, who still claim to be Pythagoreans. Certainly, from theevidence available to modern scholars, Aristoxenus’ claim islargely true. From about 350 BCE until about 100 BCE, there is aradical drop in evidence for individuals who call themselvesPythagoreans. Iamblichus (In Nic. 116.1–7) appears todate the Pythagoreans Myonides and Euphranor, who worked on themathematics of means, after the time of Eratosthenes (285–194BCE) and hence to the second century BCE or later (Burkert 1972a,442), but Iamblichus’ history of the means is very confused andthey might belong to the rise of Neopythagoreanism in the firstcenturies BCE and CE. Kahn (2001, 83) sees a hint of Pythagorean cultactivity in the spuriousPythagorean Memoirs, which must datesometime before the first half of the first century BCE, when they arequoted by Alexander Polyhistor (see section 4.2 below). A few otherPythagorean 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 completelydead between 350 and 100 (see further below, sect. 3.5), but fewindividual Pythagoreans or organized groups of Pythagoreans can beidentified in this period.

3.6 Timaeus, Ocellus, Hicetas and Ecphantus

The names Timaeus of Locri and Ocellus of Lucania are famous as theauthors of the two most influential Pythagorean pseudepigrapha (seebelow, sect. 4.2). In his catalogue of Pythagoreans, Iamblichus listsan Ocellus under Lucania and two men named Timaeus, neither underLocri. The later forgery of works attributed to Timaeus and Ocellusdoes not of course mean that Pythagoreans of these names did notexist, and it is possible that the Timaeus of Locri who is the mainspeaker in Plato’sTimaeus was an historical Timaeus(some have thought Plato uses him as a mask for Archytas, however). Ifthey really did exist, however, nothing is known about them, since allother reports in the ancient tradition are likely to be based onPlato’sTimaeus or the spurious works in theirname.

Some scholars have argued that Hicetas and Ecphantus, both ofSyracuse, were not historical figures at all but rather characters indialogues written by Heraclides of Pontus, a fourth-century member ofthe Academy. By a misunderstanding, they came to be treated ashistorical Pythagoreans in the doxographical tradition (see Guthrie1962, 323 ff. for references). This theory arose because both Hicetasand Ecphantus are said to have made the earth rotate on its axis,while the heavens remained fixed, in order to explain astronomicalphenomena, and, in one report, Heraclides is paired with Ecphantus ashaving adopted this view (Aetius III 13.3 =DK I 442.23). In additionEcphantus is assigned a form of atomism (DK I 442.7 ff.) similar tothat assigned to Heraclides (Fr. 118–121 Wehrli). It is notuncommon in the doxographical tradition for a report of the form“x and y believe z” to mean that “y, as reported byx, believes z,” so it is suggested that in this case“Heraclides and Ecphantus” means “Ecphantus aspresented by Heraclides.” There is a serious problem with thisingenious theory. The doxographical reports about Hicetas andEcphantus ultimately rely on Theophrastus (Cicero mentionsTheophrastus by name at DK I 441.27), and it is implausible thatTheophrastus would treat characters invented by his oldercontemporary, Heraclides, as historical figures. Theophrastus didaccept the Academic glorification of Pythagoras (see onNeopythagoreanism below, sect. 4.1), but this provides no grounds forsupposing that he accepted a character in a dialogue as a historicalperson (pace Burkert 1972a, 341).

The testimonia for Hicetas are meager and contradictory (DK I441–442). He appears to have argued that the celestial phenomenaare best explained by assuming that all heavenly bodies are stationaryand that the apparent movement of the stars and planets is the resultof the earth’s rotation around its own axis. He may also havefollowed Philolaus in positing a counter-earth, opposite the earth onthe other side of a central fire, although, if he did, it is unclearhow he would have explained why it and the central fire are notvisible from the rotating earth. In Philolaus’ system thecentral fire remains invisible because the earth orbits the centralfire as it rotates on its axis, thus keeping one side of the earthalways turned away from the central fire. A little more is known aboutEcphantus (DK I 442). He too is said to have believed that the earthmoved, not by changing its location (as Philolaus proposed, in makingthe earth and counter-earth revolve around the central fire: seeSection 4.2 of the entry onPhilolaus), but by rotating on its axis.

Copernicus was inspired by these testimonia about Hicetas andEcphantus, as well as those about Philolaus, to consider the motion ofthe earth (see below, sect. 5.2). Ecphantus developed his own originalform of atomism. He is best understood as reacting to and developingthe views of Democritus. He agreed with Democritus 1) “thathuman beings do not grasp true knowledge of the things that are, butdefine them as they believe them to be” (DK I 442.7–8; cf.Democritus Frs. 6–10) and 2) that all sensible things arise fromindivisible first bodies and void. He differs from Democritus,however, in supposing that atoms are limited rather than unlimited innumber and that there is just one cosmos rather than many. As inDemocritus, atoms differ in shape and size, but Ecphantus adds power(dynamis) as a third distinguishing factor. He explainsatomic motion not just in terms of weight and external blows, as theatomists did, but also by a divine power, which he called mind orsoul, so that “the cosmos was composed of atoms but organized byprovidence” (DK I 442.21–22). It is because of this divinepower that the cosmos is spherical in shape. This unique sphericalcosmos is reminiscent of Plato’sTimaeus, but the restof Ecphantus’ system differs enough from Plato that there is noquestion of its being a forgery based on theTimaeus. Onetestimony says that he was the first to make Pythagorean monadscorporeal, thus differing from the fifth-century Pythagoreansdescribed by Aristotle, who do not seem to have addressed the questionof whether numbers were physical entities or not.

It is difficult to be sure of the date of either Hicetas or Ecphantus.Since, however, both seem to be influenced by Philolaus’ idea ofa moving earth and since Ecphantus appears to be developing theatomism of Democritus, it is usually assumed that they belong to thefirst half of the fourth century (Guthrie 1962, 325–329).Hicetas does not appear in Iamblichus’ catalogue. There is anEcphantus in the catalogue, but he is listed under Croton rather thanSyracuse, so it cannot be certain whether he is the Ecphantusdescribed in the doxography.

3.7 Plato and Pythagoreanism

There is currently a very wide range of opinions about therelationship of Plato to Pythagoreanism. Many scholars both ancientand modern have thought that Plato was very closely tied toPythagoreanism. In the biography of Pythagoras read by Photius in the9th century CE (Bibl. 249) Plato is presented as a member ofthe Pythagorean school. He is the pupil of Archytas and the ninthsuccessor to Pythagoras himself. If this were true then Plato wouldcertainly be the most illustrious early Pythagorean after Pythagorashimself. Some modern scholars, while not going this far, have seen theconnections between Plato and the Pythagoreans to be very closeindeed. Thus, A. E. Taylor in his great commentary on theTimaeus says that his main thesis is that “the teachingof Timaeus [in Plato’sTimaeus] can be shown to be indetail exactly what we should expect from an fifth-century ItalianPythagorean” (1928, 11), although Taylor does not regard theseas Plato’s own teachings at the time. Guthrie in his famoushistory of ancient philosophy commented that Pythagorean and Platonicphilosophy were so close that it is difficult to separate them (1975,35). Recently it has been argued that Plato was so steeped inPythagoreanism that he structured his dialogues by counting numbers oflines and placing important passages at points in the dialogue thatcorrespond to important ratios in Pythagorean harmonic theory(Kennedy, 2010 and 2011). Thus, the vision of the form of beautyappears 3/4 of the way through theSymposium by line countand the ratio 3 : 4 corresponds to the central musical interval of thefourth. There are, however, serious questions about the methodologyused (Gregory 2012) and it is a serious problem both that no one inthe ancient world reports that Plato used such a practice and that themiddle of the dialogue, which corresponds to the most concordantmusical interval, the octave (2:1), does not usually contain the mostphilosophically important content. Another approach sees Plato asengaged with and heavily influenced by Pythagorean ideas in passageswhere the Pythagoreans are not specifically mentioned in dialoguessuch as theCratylus (401b11–d7) andPhaedo(101b10–104c9) (Horky 2013). The problem is that in contrast tothePhilebus, where the connection to Philolaus is clear (seebelow), the connections to the Pythagoreans in these passages are tooindirect or general (e.g., the concepts odd and even and the number 3in thePhaedo passage are not unique to the Pythagoreans) tobe very convincing and partly depend on the doubtful assumption thatEpicharmus was a Pythagorean (see section 3.4 above). The central textfor many of those who see Plato as closely tied to Pythagoreanism isAristotle’s comment inMetaphysics 1.6 that Plato“followed these men (i.e. the Pythagoreans according to thesescholars) in most respects” (987a29–31). In contrast tothese attempts to connect Plato closely to Pythagoreanism, most recentPlatonic scholars seem to think Pythagoreanism of little importancefor Plato. Thus two prominent handbooks to Plato’s thought(Kraut and Ebrey 2022; Benson 2006) and another book of essays devotedspecifically to theTimaeus, (Mohr and Sattler 2010) hardlymention the Pythagoreans at all.

In recent studies of the topic that lie somewhere between theseextremes, one approach is to argue that there is clear Pythagoreaninfluence on Plato but that its scope is much more limited than oftenassumed (Huffman 2013). Plato explicitly mentions Pythagoras and thePythagoreans only one time each in the dialogues and this providesprima facie evidence that Pythagorean influence was notextensive. Moreover, atMetaphysics 987a29–31 the“these men” that Aristole says Plato follows in mostrespects may not be the Pythagoreans but the Presocratics in general.Aristotle’s presentation as a whole mainly attests toPythagorean influence only on Plato’s late theory of principles.It is often assumed that Plato owes his mathematical conception of thecosmos and his belief in the immortality and transmigration of thesoul to Pythagoreanism (Kahn 2001, 3–4). However, the role ofPythagoreanism in Greek mathematics has been overstated and whilePlato had contacts with mathematicians who were Pythagoreans likeArchytas, the most prominent mathematicians in the dialogues,Theodorus and Theaetetus, are not Pythagoreans. It is thus a seriousmistake to assume that any mention of mathematics in Plato suggestsPythagorean influence. The same is true of the immortality andtransmigration of the soul in Plato, which are often assumed to bederived from Pythagoreanism. Some have also thought that Platonicmyths and especially the myth at the end of thePhaedo drawheavily on Pythagoreanism (Kingsley 1995, 79–171). However, mostof the contexts in which Plato mentions the immortality of the soulincluding the Platonic myths, suggest that he is thinking of mysterycults and the Orphics rather than the Pythagoreans (Huffman 2013,243–254). On the other hand, in thePhilebus (16c-17a)Plato gives clear acknowledgement of the debt he owes to men beforehis time who posit limit and unlimited as basic principles. Thefragments of Philolaus and Aristotle’s reports on Pythagoreanismmake clear that this is a reference to Philolaus and the Pythagoreans.The principles of limit and unlimited are clearly connected toPlato’s one and indefinite dyad and it is precisely theseprinciples of Plato that Aristotle connects most closely toPythagoreanism (Metaph. 987b25–32). Thus Plato’sevidence coheres with Aristotle’s to suggest that Pythagoreanismexerted considerable influence on Plato’s late theory ofprinciples. It is also true that specific aspects of Plato’smathematical view of the world are owed to the Pythagoreans, e.g., theworld soul in theTimaeus is constructed according to thediatonic scale that is prominent in Philolaus (Fr. 6a). However, mostof theTimaeus is not derived from Pythagoreanism and some ofit in fact conflicits with Pythagoreanism (e.g., Archytas famouslyargued that the universe was unlimited while Plato’s inlimited). The same is true for Plato as a whole. Isolated ideas suchas the one and the dyad and the structure of the world soul show heavyPythagorean influence, but there is no evidence that Pythagoreanismplayed a central role in the development of the core of Plato’sphilosophy (e.g., the theory of forms).

A second approach is to argue that, while it is true that not allmentions of mathematics or all mentions of the transmigration of thesoul derive from Pythagoreanism, nonetheless a central system of valuethat appears early in Plato’s work and persists to the end isderived from Pythagoreanism (Palmer 2014). Already in theGorgias Plato argues that principles of order and correctnesswhich are found in the cosmos and explain its goodness also governhuman relations. Socrates here puts forth a much more definiteconception of the good than in earlier dialogues. His complaint thatCallicles pays no attention to the role played by orderliness andself-control and neglects geometrical equality (507e6–508a8)mirrors the emphasis on organization and calculation in contemporaryPythagorean texts such as Archytas Fr. 3 and Aristoxenus’Pythagorean Precepts Fr. 35. It thus appears that“Socrates’” new insight into the good inGorgias derives from Plato’s contact with thePythagoreans after the death of the historical Socrates. Plato neverabandons this Pythagorean conception of value and it can be tracedthrough thePhaedo andRepublic to late dialoguessuch as theTimaeus, where the cosmos is embued withprinciples of mathematical order, andPhilebus, where thehighest value is assigned to measure (66a). The question is whetherthis emphasis on measure and order is uniquely Pythagorean inorigin.

4. Neopythagoreanism

Neopythagoreanism is characterized by the tendency to see Pythagorasas the central and original figure in the development of Greekphilosophy, to whom, according to some authors (e.g. Iamblichus,VP 1), a divine revelation had been given. This revelationwas often seen as having close affinities to the wisdom of earliernon-Greeks such as the Hebrews, the Magi and the Egyptians. Because ofthe belief in the centrality of the philosophy of Pythagoras, laterphilosophy was regarded as simply an elaboration of the revelationexpounded by Pythagoras; it thus became the fashion to father theviews of later philosophers, particularly Plato, back onto Pythagoras.Neopythagoreans typically emphasize the role of number in the cosmosand treat the One and Indefinite Dyad as ultimate principles goingback to Pythagoras, although these principles in fact originate withPlato. The origins of Neopythagoreanism are probably to be foundalready in Plato’s school, the Academy, in the second half ofthe fourth century BCE. There is evidence that Plato’ssuccessors, Speusippus and Xenocrates, both presented Academicspeculations arising in part from Plato’s later metaphysics asthe work of Pythagoras, who lived some 150 years earlier. After adecline in interest in Pythagoreanism for a couple of centuries,Neopythagoreanism emerged again and developed further starting in thefirst century BCE and extending throughout the rest of antiquity andinto the middle ages and Renaissance. During this entire period, it isthe Neopythagorean construct of Pythagoras that dominates, a constructthat has only limited contact with early Pythagoreanism; there islittle interest in an historically accurate presentation of Pythagorasand his philosophy. In reading the following account ofNeopythagoreanism, it may be helpful to refer to theChronological Chart of Sources for Pythagoras, in the entry on Pythagoras.

4.1 Origins in the Early Academy: Speusippus, Xenocrates and Heraclides in Contrast to Aristotle and the Peripatetics

The evidence for Speusippus, Plato’s successor as head of theAcademy, is fragmentary and second hand, so that certainty ininterpretation is hardly possible. In one passage, however, he assignsnot just Plato’s principles, the one and the dyad, to “theancients,” who in context seem likely to be the Pythagoreans(although Sedley 2021a, 17 suggests that the reference is toParmenides), but also a development of the Platonic system accordingto which the one was regarded as beyond being (Fr. 48 Tarán; seeBurkert 1972a, 63–64; Dillon 2003, 56–57). Some scholarsreject this widely held view on the grounds that this fragment ofSpeusippus is spurious (Zhmud 2012a, 424—425, who cites otherscholars; Tarán 1981, 350ff.; for a response see Dillon 2014, 251)and if this were true it would seriously weaken the case for supposingthat Neopythagoreanism began already in the Academy. Speusippus alsowrote a bookOn Pythagorean Numbers (Fr. 28 Tarán), whichbuilds on ideas attested for the early Pythagoreans (e.g., ten as theperfect number, although Zhmud regards the perfection of ten as aPlatonic rather than a Pythagorean doctrine 2012a, 404–09, andSpeusippus’ book as the first work of arithmology, which only inthe first century BCE is ascribed to the Pythagoreans [2016]). Wecannot be sure, however, either that the title goes back to Speusippusor that he assigned all ideas in it to the Pythagoreans. Aristotletwice cites agreement between Speusippus and the Pythagoreans(Metaph. 1072b30 ff.;EN 1096b5–8), whichmight suggest that Speusippus himself had identified the Pythagoreansas his predecessors in these areas. Speusippus and Xenocrates deniedthat the creation of the universe in Plato’sTimaeusshould be understood literally; when the view that the cosmos was onlycreated in thought and not in time is assigned to Pythagoras in thelater doxography (Aëtius II 4.1 — Diels 1958, 330), itcertainly looks as if an idea which had its origin in theinterpretation of Plato’sTimaeus in the Academy isbeing assigned back to Pythagoras (Burkert 1972a, 71). The evidence isnot sufficient to conclude that Speusippus routinely assigned Platonicand Academic ideas to the Pythagoreans (Tarán 1981, 109), but thereis enough evidence to suggest that he did so in some cases. Sedley2021b argues that a famous mosaic from Pompeii portrays Speusppus asdistracted from Platonic teaching by Pythagoreanism as represented bythe figure of Archytas.

Speusippus’ successor as head of the Academy, Xenocrates, mayactually have followed some version of the Pythagorean way of life,e.g., he was apparently a vegetarian, refused to give oaths, wasprotective of animals and followed a highly structured daily regimen,setting aside time for silence (Dillon 2003, 94–95 and 2014,254–257; Burkert, however, argues that he rejectedmetempsychosis [1972a, 124]). Horky 2013b argues thatXenocrates’ account of the relation between Pythagoreanism andPlatonism influenced Theophrastus but Sedley 2021a and 2021b distancesXenocrates from Pythagoreanism. Xenocrates wrote a book entitledThings Pythagorean, the contents of which are unfortunatelyunknown (Diogenes Laertius IV 13). In the extant fragments of hiswritings, he refers to Pythagoras by name once, reporting that“he discovered that the musical intervals too did not ariseapart from number” (Fr. 9 Heinze). Several doctrines ofXenocrates are also assigned to Pythagoras in the doxographicaltradition, e.g., the definition of the soul as “a number movingitself,” which Burkert (1972a, 64–65) argues thatXenocrates may have developed on the basis of Plato’sTimaeus (Plutarch,On the Generation of the Soul1012d; Aëtius IV 2.3–4). This suggests that Xenocrates,like Speusippus, may have assigned his own teachings back toPythagoras or at least treated Pythagoras as his precursor in such away that it was easy for others to do so (Dillon 2003, 153–154;Zhmud [2012a, 55 and 426–427] disputes this interpretation).

Yet another member of the early Academy, Heraclides of Pontus(Gottschalk 1980), in a series of influential dialogues, furtherdeveloped 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 LaertiusProem 12 and Cicero,Tusc. V 3.8). Although some scholarshave tried to find a kernel of truth in the story (e.g., Riedweg 2005,90 ff., for a response see Huffman 2008b), its definition of thephilosopher as one who seeks wisdom rather than possessing it isregarded by many scholars as a Socratic/Platonic formulation, whichHeraclides, in his dialogue, is assigning to Pythagoras as part of aliterary fiction (Burkert 1960 and 1972a, 65). Heraclides also assignsto Pythagoras a definition of happiness as “the knowledge of theperfection of the numbers of the soul” (Fr. 44 Wehrli), in whichagain the Platonic account of the numerical structure of the soul intheTimaeus appears to be fathered on Pythagoras. Otherfragments show Heraclides’ further fascination with thePythagoreans. He developed what would become one of the canonicalaccounts of Pythagoras’ previous incarnations (Fr. 89 Wehrli).Perhaps on the basis of the Pythagorean Philolaus’ astronomicalsystem, he developed the astronomical theory, later to be championedby Copernicus, according to which the apparent daily motion of the sunand stars was to be explained by the rotation of the earth (Frs.104–108; see on Hicetas and Ecphantus above, sect. 3.6). For adifferent view of Heraclides’ relation to the Pythagoreans seeZhmud 2012a, 427–432.

In contrast to the fascination with and glorification of Pythagoras inthe Academy after Plato’s death, Aristotle did not treatPythagoras as part of the philosophical tradition at all. In thesurveys of his predecessors in his extant works, Aristotle does notinclude Pythagoras himself and he evidently presented him in his lostspecial treatises on the Pythagoreans only as a wonder-worker andfounder of a way of life. While Aristotle did acknowledge closeconnections between Plato’s late theory of principles (One andIndefinite Dyad) and fifth-century Pythagoreans, he also sharplydistinguished Plato from the Pythagoreans on a series of importantpoints (Metaph. 987b23 ff.), perhaps in response to theAcademy’s tendency to assign Platonic doctrines to Pythagoras.Aristotle’s students Eudemus, in his histories of arithmetic,geometry and astronomy and Meno, in his history of medicine, followAristotle’s practice of not mentioning Pythagoras himself,referring to individual Pythagoreans such as Philolaus or to thePythagoreans as a group. Eudemus assigns the Pythagoreans a number ofimportant contributions to the sciences but does not give them thedecisive or foundational role found in the Neopythagorean tradition.Aristotle’s pupils Dicaearchus (Porphyry,VP 19) andAristoxenus do mention Pythagoras but this is because they arefocusing on the Pythagorean way of life and the history of thePythagorean communities. Neither assign to Pythagoras or thePythagoreans the characteristics of Neopythagoreanism. Aristoxenus isone of the most important and extensive sources for Pythagoreanism(see 3.5 above). He presents Pythagoras and the Pythagoreans in apositive manner but avoids the hagiography and extravagant claims ofthe later Neopythagorean tradition. The standard view is that he triesto emphasize the rational as opposed to the religious side ofPythagoras (e.g. Burkert 1972a, 200–205), but several fragmentsdo highlight the religious aspect of Pythagoras’ work, assigninghim the doctrine of metempsychosis (fr. 12) and associating him withthe Chaldaean Zaratas (Fr. 13) and the Delphic oracle (Fr. 15). It isonly by rejecting the authenticity of such fragments (as does Zhmud2012a, 88–91) that Aristoxenus’ account is purged ofreligious elements. Dicaearchus’ account of Pythagoreas is alsousually viewed as positive. He is supposed to have presentedPythagoras as the model of the practical life as opposed to thecontemplative life (Jaeger 1948, 456; Kahn 2001, 68). However,Dicaearchus presents a very sarcastic account of Pythagoras’rebirths according to which he was reborn as the beautiful prostituteAlco (Fr. 42) and careful reading of his other accounts of Pythagorassuggests that he may have presented him as a charismatic charlatan whobewitched his hearers (Fr. 42) and was seen as a threat to theestablished laws of the state and hence was refused entrance by suchcity-states as Locri (Fr. 41a). Thus, Aristoxenus and Dicaearchus wereas divided in their interpretation of Pythagoras as were Heraclitusand Empedocles in earlier centuries. The Peripatetic tradition as awhole is in strong contrast, then, with the Academy insofar as itemphasizes Pythagoreans rather than Pythagoras himself. WhenPythagoras is mentioned, it is mostly in connection with the way oflife, and interpretations range from positive to strongly satiricalbut in either case avoid the hagiography of the Neopythagoreantradition.

It is then one of the great paradoxes of the ancient Pythagoreantradition that Aristotle’s successor, Theophrastus, evidentlyaccepted the Academic lionization of Pythagoras, and identifiesPlato’s one and the indefinite dyad as belonging to thePythagoreans (Metaph. 11a27 ff.), although Aristotle isemphatic 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 laterdoxographical tradition, it may be that Theophrastus is responsiblefor the Neopythagorean Pythagoras of the Academy dominating the laterdoxography, the Pythagoras who originated the one and the indefinitedyad (Aëtius I 3. 8), but it may also be that the Pythagoreansections of the doxography were rewritten in the first century BCE,under the influence of the Neopythagoreanism of that period (Burkert1972a, 62; Zhmud 2012a, 455).

The standard view has thus been that the Academy was the origin ofNeopythagoreanism with its glorification of Pythagoras and itstendency to assign mature Platonic views back to Pythagoras and thePythagoreans. At the very least, most scholars agree that the earlyAcademy was heavily influenced by the Pythagoreans (Bonazzi 2023, 12,n. 35). Aristotle and the Peripatetics on the other hand diminish therole of Pythagoras himself and, while noting connections between Platoand the Pythagoreans, carefully distinguish Pythagorean tenets fromPlatonism. Zhmud has recently put forth a challenge to this viewarguing the situation is almost the reverse: the Academy in generalregards Pythagoras and Pythagoreans favorably but does not assignmature Platonic views to them, it is rather Aristotle who ties Platoclosely to the Pythagoreans (2012a, 415–456).

4.2 The Pythagorean Pseudepigrapha

Although the origins of Neopythagoreanism are thus found in the fourthcentury BCE, the figures more typically labeled Neopythagoreans belongto the upsurge in interest in Pythagoreanism that begins in the firstcentury BCE and continues through the rest of antiquity. Beforeturning to these Neopythagoreans, it is important to discuss anotheraspect of the later Pythagorean tradition, the Pythagoreanpseudepigrapha. Many more writings forged in the name of Pythagorasand other Pythagoreans have survived than genuine writings. Most ofthe pseudepigrapha themselves only survive in excerpts quoted byanthologists such as John of Stobi, who created a collection of Greektexts for the edification of his son in early fifth century CE. Themodern 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 theseworks were created. No one answer to these questions will fit all ofthe treatises. Most scholars (e.g., Burkert 1972b, 40–44;Centrone 1990, 30–34, 41–44 and 1994) have chosen Rome orAlexandria between 150 BCE and 100 CE as the most likely time andplace for these compositions, since there was a strong resurgence ofinterest in Pythagoreanism in these places at these times (see below).Thesleff’s view that the majority were composed in the thirdcentury BCE in southern Italy (1961 and 1972, 59) has found lessfavor. Centrone argues convincingly that a central core of thepseudepigrapha were forged in the first centuries BCE and CE inAlexandria, because of their close connection to Eudorus and Philo,who worked in Alexandria in that period (Centrone 2014a). For anoverview of the Pythagorean pseudepigrapha see Centrone 2014a andMoraux 1984, 605–683.

A number of motives probably led to the forgeries. The existence ofavid collectors of Pythagorean books such as Juba, King of Mauretania(see below), and the scarcity of authentic Pythagorean texts will haveled to forgeries to sell for profit to the collectors. Other shortletters or treatises may have originated as exercises for students inthe rhetorical schools (e.g., the assignment might have been to writethe letter that Archytas wrote to Dionysius II of Syracuse asking thatPlato be freed; see Diogenes Laertius III 21–22). The contentsof the treatises suggest, however, that the primary motivation was toprovide the Pythagorean texts to support the Neopythagorean position,first adumbrated in the early Academy, that Pythagoras was the sourceof all that is true in the Greek philosophical tradition. Thepseudepigrapha show the Pythagoreans anticipating the mostcharacteristic ideas of Plato and Aristotle. Most of the treatises arecomposed in the Doric dialect (spoken in Greek S. Italy) but, apartfrom that concession to verisimilitude, there is little other attemptto make them appear to be archaic documents that anticipated Plato andAristotle. Instead, Plato’s and Aristotle’s philosophicalpositions are stated in a bald fashion using the exact Platonic andAristotelian terminology. In many cases, however, this glorificationof Pythagoras may not have been the final goal. The ancient authorityof Pythagoras was sometimes used to argue for a specificinterpretation of Plato, often an interpretation that showed Plato ashaving anticipated and having responded to criticisms of Aristotle.For example, in defense of the interpretation of Plato’sTimaeus, which defends Plato against Aristotle’scriticisms by claiming that the creation of the world in theTimaeus is metaphorical, a Platonist could point to theforged treatise of Timaeus of Locri which does present the generationas metaphorical but which can also be regarded as Plato’ssource. These pseudo-Pythagorean treatises are adopting the samestrategy as Eudorus of Alexandria and thus may be more important fordebates within later Platonism than for Pythagoreanismper se(Bonazzi 2013). Given these motivations for the pseudepigrapha, it isno surprise that there is little in them that has any connection togenuine early Pythagoreanism. All that is Pythagorean are the names ofthe authors (which are derived in large part from Aristoxenus’works on the Pythagoreans), the Doric dialect in which the works arewritten and a few general Pythagorean concepts such as harmony. Thephilosophical content is mostly derived from the Platonic andAristotelian tradition and shows no awareness of the actual works ofearly Pythagoreans such as Archytas and Philolaus (see Zhmud2019a).

One plausible explanation of the sudden proliferation of Pythagoreanpseudepigrapha in the first century BCE and first century CE is thereappearance of Aristotle’s esoteric writings in the middle ofthe first century BCE (Kalligas 2004, 39–42). In those treatisesPlato is presented as adopting a pair of principles, the one and theindefinite dyad, which are not obvious in the dialogues, but whichAristotle compares to the Pythagorean principles limit and unlimited(e.g.,Metaph. 987b19–988a1). Aristotle can be read,although probably incorrectly, as virtually identifying Platonism andPythagoreanism in these passages. Thus, Pythagorean enthusiasts mayhave felt emboldened by this reading of Aristotle to create thesupposed original texts upon which Plato drew. They may also havefound support for this in Plato’s making the south-ItalianTimaeus his spokesman in the dialogue of the same name. It is thus notsurprising that the most famous of the pseudepigrapha is the treatisesupposedly written by this Timaeus of Locri (Marg 1972), which hassurvived complete and which is clearly intended to represent theoriginal document on which Plato drew, although it, in fact, alsoresponds to criticisms made of Plato’s dialogue in the firstcouple of centuries after it was written (Ryle 1965, 176–178).The treatise of Timaeus of Locri is first mentioned by Nicomachus inthe second century CE (Handbook 11) and is thus commonlydated to the first century CE. Another complete short treatise (13pages in Thesleff) isOn the Nature of the Universesupposedly by the Pythagorean Ocellus (Harder 1966), which haspassages that are almost identical to passages in Aristotle’sOn Generation and Corruption. Since Ocellus’ work isfirst mentioned by the Roman polymath, Varro, scholars have dated itto the first half of the first century BCE. Although Plato was ingeneral more closely associated with the Pythagorean tradition thanAristotle, a significant number of Pythagorean pseudepigrapha follow‘Ocellus’ in drawing on Aristotle (see Karamanolis 2006,133–135).

It is likely that in some cases letters were forged in order toauthenticate these forged treatises. Thus a correspondence betweenPlato and Archytas dealing with the acquisition of the writings ofOcellus (Diogenes Laertius VIII 80–81) may be intended tovalidate the forgery in Ocellus’ name (Harder 1966, 39ff). Aletter from Lysis to Hipparchus (Thesleff 1965, 111–114), whichenjoyed considerable fame in the later tradition and is quoted byCopernicus, urges that the master’s doctrines not be presentedin public to the uninitiated and recounts Pythagoras’daughter’s preservation of his “notebooks”(hypomnêmata) in secrecy, although she could have soldthem for much money (see Riedweg 2005, 120–121). Burkert (1961,17–28) has argued that this letter was forged to authenticatethe “Pythagorean Notes” from which Alexander Polyhistor(1st century BCE) derived his influential account of Pythagoreanism(Diogenes Laertius VIII 24–36 — see the end of thissection and for Alexander see section 4.5 below). While some ofPythagoras’ teachings were undoubtedly secret, many were not,and the claim of secrecy in the letter of Lysis is used to explainboth the previous lack of early Pythagorean documents and the recent“discovery” of what are in reality forged documents, suchas the notebooks.

There are fewer forged treatises in Pythagoras’ name than in thename of other Pythagoreans and they are a very varied group suggestingdifferent origins. Callimachus, in the third century BCE, knew of aspurious astronomical work circulating in Pythagoras’ name(Diogenes Laertius IX 23) and there may have been a similar workforged in the second century (Burkert 1961, 28–42). A group ofthree books,On Education,On Statesmanship andOn Nature, were forged in Pythagoras’ name sometimebefore the second century BCE (Diogenes Laertius VIII 6 and 9; Burkert1972a, 225). Heraclides Lembus, in the second century BCE, knew of atleast six other works in Pythagoras’ name, all of which musthave been spurious, including aSacred Discourse (DiogenesLaertius VIII 7). The thesis that the historical Pythagoras wrote aSacred Discourse should be rejected (Burkert 1972a, 219).There was also a spurious treatise on the magical properties of plantsand theGolden Verses, which are discussed further below(sect. 4.5). On the spurious treatises assigned to Pythagoras seeCentrone 2014a, 316–318.

Archytas appears to have been the most popular name in which to forgetreatises, undoubtedly because of his connections to Plato and hisfame in the first centuries BCE and CE, when the Pythagoreanpseudepigrapha arose (Centrone 2021, 122–127). Archytas was seenas the crucial connection between Pythagoreanism and Plato and hissuccessor Aristotle. Some 45 pages are devoted to pseudo-Archytantreatises in Thesleff’s collection as compared to 30 pages forPythagoras. The most famous of the pseudo-Archytan texts isTheWhole System of Categories, which, along withOnOpposites, represents the attempt to claim Aristotle’ssystem of categories for the Pythagoreans. The pseudo-Archytan workson categories are very frequently cited by the commentators onAristotle’sCategories (e.g., Simplicius and Syrianus)and were regarded as authentic by them, but in fact includemodifications made to Aristotle’s theory in the first centuryBCE and probably were composed in that century (Szlezak 1972). Anothertreatise,On Principles, is full of Aristotelian terminologysuch as “form,” “substance,” and “whatunderlies”;On Intelligence and Perception contains aparaphrase of the divided line passage in Plato’sRepublic. There are also a series of pseudepigrapha on ethicsby Archytas and other authors (Centrone 1990. For more on the Archytanpseudepigrapha see the SEP article onArchytas). Philolaus, the third most famous Pythagorean after Pythagoras andArchytas, also turns up as the author of several spurious treatises,but a number of the forgeries were in the names of obscure orotherwise unknown Pythagoreans. Thus, Callikratidas and Metopos arepresented as anticipating Plato’s doctrine of the tripartitesoul and as using Plato’s exact language to articulate it(Thesleff 1965, 103.5 and 118.1–4). Although there areindications that some ancient scholars had doubts about theauthenticity of the pseudo-Pythagorean texts, for the most part theysucceeded in their purpose all too well and were accepted as genuinetexts on which Plato and Aristotle drew.

Although the pseudepigrapha are too varied to admit of one origin,Centrone has recently argued that a core group of pseudepigrapha doappear to be part of a single project (2014a). They are written inDoric Greek (the dialect used in southern Italy where the Pythagoreansflourished) in order to give them the appearance of authenticity andshare a common style. There are some twenty-five treatises belongingto this group and they include some of the most famous pseudepigrapha,including the work by ps.-Timaeus that was supposed to bePlato’s model, ps.-Archytas’ works on categories andps.-OcellusOn the Universe. These treatises espouse the samebasic system and seem designed to cover all the basic fields ofknowledge. The system is based on theory of principles in which God isthe supreme entity above a pair of principles, one of which is limitedand the other unlimited, and which are identified with Aristotelianform and matter. This system is very similar to what is found inEudorus, a Platonist working in Alexandria in the fist cenutury BCE.Starting from these principles a common system is then developed whichapplies to theology, cosmology, ethics, and politics. The connectionsto Eudorus and to Philo who also worked in Alexandria, very muchsuggest that this group of treatises was developed as a coherentproject in Alexandria sometime in the first century BCE or the firstcentury CE. A number of the pseudepigrapha were forged in the names ofobscure Pythagoreans such as Theages or Metopus. Obviously suchobscure authors can give little authority to the texts but it may bethat the goal of composing texts espousing the same basic system inthe names of a wide range of authors was to show the unity of theschool (Centrone 2021, 120–121). One idiosyncratic view arguesthat the philosophical system of the pseudepigrapha did not arisearound figures like Eudorus in the first century BCE but derives inpart from a genuine tradtion of Hellenistic Pythagoreanism (Horky2023, 20), but the evidence for this is meagre.

One important group of Pythagorean pseudepigrapha are those forged inthe names of Pythagorean women. These texts had been seriouslyneglected by scholars until recently. Pomeroy 2013 provides someuseful commentary but has serious drawbacks (see Centrone 2014b andBrodersen 2014). Huizenga 2013 is a reliable guide but Dutsch 2020provides what is by far the most insightful treatment of the figure ofthe Pythagorean woman in (mostly later) antiquity as well asilluminating readings of the texts themselves. Many of the texts arecollected in Thesleff 1965 under the names Theano, Periktione,Melissa, Myia and Phintys and taken together occupy about 15 pages oftext. To Periktione are assigned two fragments from a treatiseOnthe Harmony of a Woman. Periktione is the name of Plato’smother and it is probable that hers is the famous name in which theseworks were forged. Two further fragments fromOn Wisdom arealso assigned to her. These fragments show a strong similarity tofragments from a treatise with identical title by Archytas and arelikely to have been assigned to Periktione by mistake. Two fragmentsfrom a workOn the Temperance of a Woman are assigned toPhintys. For Theano, the most famous Pythagorean woman (see 3.3above), one fragment of a workOn Piety is preserved as wellas the titles of several other works, numerous apophthegms and anumber of letters. On Theano in the pseudepigraphal tradition seeHuizenga 2013, 96–117 and Dutsch 2020. Melissa and Myia arerepresented by one letter each. Although a few of the texts deal withmore universal philosophical topics (see Pellò 2022) most ofthe works focus on female virtue, proper marital conduct, andpractical issues such as how to choose a wet nurse and how to dealwith slaves. The advice is quite conservative, stressing obedience toone’s husband, chastity and temperance. There is little that isspecifically Pythagorean and the connections are clearest withStoicism (Dutsch 2020, 139). Since the authors are pseudonymous it isimpossible to be sure whether they were in fact written by women usingfemale pseudonyms or men using female pseudonyms (Huizenga 2013, 116).In the case of the letters Städele’s edition (1980) is tobe preferred to Thesleff (1965). The letters of Melissa and Myia alongwith three letters of Theano are often found together in themanuscript tradition and may have come to be seen as offering acurriculum for the moral training of women (Huizenga 2013 and Dutsch2020, 173–212). Due to the dearth of preserved writings by womenfrom the ancient world some have been tempted to suppose that thewritings are genuine works by the named authors. However, asdemonstrated above, Pythagorean pseudepigrapha were very widespreadand more common than genuine Pythagorean works. In such a context theonus of proof is on someone who wants to show that a work is genuine.The content of the writings by Pythagorean women is simply too generalto make a convincing case that a specific writing could only have beenwritten by the supposed author rather than by a later forger. In fact,the writings by women fit the pattern of the rest of thepseudepigrapha very well. They are generally forged in the name offamous Pythagorean women, whose names give authority to the adviceimparted (Huizenga 2013, 117). How better could one impart force toadvice to women than to assign that advice to women who belonged tothe philosophical school that gave most prominence to women? Thepseudepigrapha written in the names of Pythagorean women probablymostly date to the first centuries BCE and CE like the otherPythagorean pseudepigrapha, but certainty is not possible.

One of the most discussed treatises among the pseudepigrapha are thePythagorean Notes, which were excerpted by AlexanderPolyhistor in the first century BCE, who was in turn quoted byDiogenes Laertius in hisLife of Pythagoras (VIII24–33). Thus theNotes date before the middle of thefirst century BCE (probably towards the end of the third century BCE[Burkert 1972a, 53]) and are earlier than most pseudepigrapha. InDiogenes’ life thePythagorean Notes serve as the mainstatement of Pythagoras’ philosophical views. The treatise iswildly eclectic, drawing from Plato’sTimaeus, theearly Academy and Stoicism and the scholarly consensus is that thetreatise is a forgery (Burkert 1961, 26ff., Long 2013, Laks 2014). Itis tempting to suppose that some early material may be preservedamidst later material, but the text is such an amalgam that it is inpractice impossible to identify securely any early material (Burkert1961, 26; Laks 2014, 375). TheNotes are well organized andpresent a complete if compressed philosophy organized around theconcept of purity (Laks 2014). Starting from basic principles (thePlatonic monad and dyad) they give an account of the world, livingbeings, and the soul ending with moral precepts (some of thePythagoreanacusmata). Kahn thought that the treatisereflected a Pythagorean community that was active in the Hellenisticperiod (2001, 83) but Long is more likely to be right that its learnedeclecticism suggests that it is a scholarly creation (Long 2013,158–159). A neglected Pythagorean pseudepigraphon is thetreatise known as theAnonymus arithmologicus, which dates tothe first half of the first century BCE. No actual fragments of theAnonymus survive and it is accordingly not included in Theseff’scollection of the pseudepigrapha. Its existence is deduced fromparallel passages in later sources such as Philo and Theon thatsuggest a common source. It has been recently argued, however, thatthe Anonymus was a crucial influence on the later Neopythagoreantradition (Zhmud 2021). Only a few of the pseudepigrapha survive ascomplete treatises rather than fragments. One of the most interestingcases is the treatise of Bryson on theManagement of theEstate, of which Stobaeus preserved two fragments in Greek butwhich survives entire in an Arabic translation (Swain 2013, Celkyte2023).

4.3 Neopythagorean Metaphysics: Eudorus, Moderatus, Numenius and Hippolytus

“Neopythagorean” is a modern label, which overlaps withtwo other modern labels, “Middle Platonist” and“Neoplatonist,” so that a given figure will be called aNeoplatonist or Middle Platonist by some scholars and a Neopythagoreanby others. It may well be that most of the figures discussed below arebest regarded as part of the Platonic tradition so it has beensuggested that the best description of them is as PythagorisingPlatonists (Bonazzi, 2023, 103). There are several different strandsin Neopythagoreanism. One strand focuses on Pythagoras as a mastermetaphysician. In this guise he is presented as the author of a theoryof principles, which went even beyond the principles of Plato’slater metaphysics, the one and the indefinite dyad, and which showssimilarities to the Neoplatonic system of Plotinus. The firstNeopythagorean in this sense is Eudorus of Alexandria, who was activein the middle and later part of the first century BCE. He evidentlypresented his own innovations as the work of the Pythagoreans (Dillon1977, 119). According to Eudorus, the Pythagoreans posited a singlesupreme principle, known as the one and the supreme god, which is thecause of all things. Below this first principle are a second one,which is also called the monad, and the indefinite dyad. These lattertwo are Plato’s principles in the unwritten doctrines, butEudorus says they are properly speaking elements rather thanprinciples (Simplicius,in Phys.,CAG IX 181.10–30). The system of principles described by Eudorus alsoappears in the pseudo-Pythagorean writings (e.g., pseudo-Archytas,On Principles; Thesleff 1965, 19) and it is hard to becertain in which direction the influence went (Dillon 1977,120–121). On Eudorus’ connection to the pseudo-Pythagoreanwritings see also Bonazzi 2013 and Centrone 2014. Eudorus is a pivotalfigure in the Platonic tradition in that he inaugurates the traditionin which philosophy is identified with exegesis of authoritativetexts, notably theTimaeus, and because he clearly representsthe turn to Pythagoreanism as crucial to understanding Plato incontrast to Hellenistic Platonism, which paid little attention toPythagoras (Bonazzi 2023, 86–90). A generation after Eudorus,another Alexandrian, the Jewish thinker Philo, used a Pythagoreantheory of principles, which is similar to that found in Eudorus, andPythagorean number symbolism in order to give a philosophicalinterpretation of theOld Testament (Kahn 2001, 99–104;Dillon 1977, 139–183). Philo’s goal was to show that Moseswas the first philosopher. For Philo Pythagoras and his travels to theeast evidently played a crucial role in the transmission of philosophyto the Greeks (Dillon 2014). Philo like Eudorus has close connectionsto the Pythagorean pseudepigrapha (Centrone 2014).

Moderatus of Gades (modern Cadiz in Spain), who was active in thefirst century CE, shows similarities to Eudorus in his treatment ofPythagorean principles. Plutarch explicitly labels him a Pythagoreanand presents his follower, Lucius, as living a life in accord with thePythagorean taboos, known assymbola oracusmata(Table Talk 727b). It is thus tempting to assume thatModeratus too lived a Pythagorean life (Dillon 1977, 345). Hisphilosophy is only preserved in reports of other thinkers, and it isoften difficult to distinguish what belongs to Moderatus from whatbelongs to the source.

He wrote a comprehensive eleven volume work entitledLectures onPythagoreanism from which Porphyry quotes in sections 48–53of hisLife of Pythagoras. In this passage, Moderatus arguesthat the Pythagoreans used numbers as a way to provide clear teachingabout bodiless forms and first principles, which cannot be expressedin words. In another excerpt, he describes a Pythagorean system ofprinciples, which appears to be developed from the first twodeductions of the second half of Plato’sParmenides. Inthis system there are three ones: the first one which is above being,a second one which is identified with the forms and which isaccompanied by intelligible matter (i.e. the indefinite dyad) and athird one which is identified with soul. The first two ones showconnections to Eudorus’ account of Pythagorean first principles;the whole system anticipates central ideas of the most importantNeoplatonist, Plotinus (Dillon 1977, 346–351; Kahn 2001,105–110).

Moderatus was a militant Neopythagorean, who explicitly charges thatPlato, Aristotle and members of the early academy claimed as their ownthe most fruitful aspects of Pythagorean philosophy with only smallchanges, leaving for the Pythagoreans only those doctrines that weresuperficial, trivial and such as to bring discredit on the school(Porphyry,VP 53). These trivial doctrines have been thoughtto be the various taboos preserved in thesymbola, but, sincehis follower Lucius is explicitly said to follow thesymbola,it seems unlikely that Moderatus was critical of them. The charge ofplagiarism might suggest that Moderatus was familiar with thepseudo-Pythagorean treatises, which appear to have been forged in partto show that Pythagoras had anticipated the main ideas of Plato andAristotle (see Kahn 2001, 105).

It is with Numenius (see Dillon 1977, 361–379 and Kahn 2001,118–133, and the entry onNumenius, especially section 2), who flourished ca. 150 CE in Apamea innorthern Syria (although he may have taught at Rome), thatNeopythagoreanism has the clearest direct contact with the greatNeoplatonist, 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 ofPlotinus, wrote a treatise entitledConcerning the DifferenceBetween the Doctrines of Plotinus and Numenius (Life ofPlotinus 3 and 17). The third century Platonist, Longinus, to adegree describes Plotinus himself as a Neopythagorean, saying thatPlotinus developed the exegesis of Pythagorean and Platonic firstprinciples more clearly than his predecessors, who are identified asNumenius, his follower Cronius, Moderatus and Thrasyllus, allNeopythagoreans (Porphyry,Life of Plotinus 20). Numeniusalso had considerable influence on Porphyry (Macris 2014, 396),Iamblichus (O’Meara 2014, 404–405) and Calcidius (Hicks2014, 429).

Numenius is regularly described as a Pythagorean by the sources thatcite his fragments such as Eusebius (e.g. Fr. 1, 4b, 5 etc. DesPlaces). He presents himself as returning to the teaching of Plato andthe early Academy. That teaching is in turn presented as deriving fromPythagoras. Plato is described as “not better than the greatPythagoras but perhaps not inferior to him either” (Fr. 24 DesPlaces). Strikingly, Numenius presents Socrates too as a Pythagorean,who worshipped the three Pythagorean gods recognized by Numenius (seebelow). Thus Plato derived his Pythagoreanism both from direct contactwith Pythagoreans and also from Socrates (Karamanolis 2006,129–132). For Numenius a true philosopher adheres to theteaching of his master, and he wrote a polemical treatise, directedparticularly at the skeptical New Academy, with the titleOn theRevolution of the Academics against Plato (Fr. 24 Des Places).Numenius presents the Pythagorean philosophy to which Plato adhered asultimately based on a still earlier philosophy, which can be found inEastern thinkers such as the Magi, Brahmans, Egyptian priests and theHebrews (Fr. 1 Des Places). Thus, Numenius was reported to have asked“What else is Plato than Moses speaking Greek?” (Fr. 8 DesPlaces).

Numenius presents his own doctrine of matter, which is clearlydeveloped out of Plato’sTimaeus, as the work ofPythagoras (Fr. 52 Des Places). Matter in its disorganized state isidentified with the indefinite dyad. Numenius argues that forPythagoras the dyad was a principle independent of the monad; laterthinkers, who tried to derive the dyad from the monad (he does notname names but Eudorus, Moderatus and the Pythagorean system describedby Alexander Polyhistor fit the description), were thus departing fromthe original teaching. In emphasizing that the monad and dyad areindependent principles, Numenius is indeed closer to the Pythagoreantable of opposites described by Aristotle and to Plato’sunwritten doctrines. Since it is in motion, disorganized matter musthave 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 ultimatedominion over matter, thus making the world as good as possible, giventhe existence of the recalcitrant matter.

The monad, which is opposed to the indefinite dyad, is just one ofthree gods for Numenius (Fr. 11 Des Places), who here followsModeratus to a degree. The first god is equated with the good, issimple, at rest and associates only with itself. The second god is thedemiurge, who by organizing matter divides himself so that a third godarises, who is either identified with the organized cosmos or itsanimating principle, the world soul (Dillon 1977, 366–372).Numenius is famous for the striking images by means of which heelucidated his philosophy, such as the comparison of the helmsman, whosteers his ship by looking at the heavens, to the demiurge, who steersmatter by looking to the first god (Fr. 18 Des Places).Numenius’ argument that there is a first god above the demiurgeis paralleled by a passage in another treatise, which showsconnections to Neopythagorean metaphysics,The ChaldaeanOracles (Majercik 1989), which were published by Julian theTheurgist, during the reign of Marcus Aurelius (161–180 CE) andthus at about the same time as Numenius was active. It is hard to knowwhich way the influence went (Dillon 1977, 363).

InThe Refutation of all Heresies, the Christian bishopHippolytus (died ca. 235 CE) adopts the strategy of showing thatChristian heresies are in fact based on the mistaken views of paganphilosophers. Hippolytus spends considerable time describingPythagoreanism, since he regards it as the primary source for gnosticheresy (see Mansfeld 1992 for this and what follows).Hippolytus’ presentation of Pythagoreanism, which groupstogether Pythagoras, Plato, Empedocles and Heraclitus into aPythagorean succession, belongs to a family of Neopythagoreaninterpretations of Pythagoreanism developed in the first century BCEand the first two centuries CE and which also appear in latercommentators such as Syrianus and Philoponus. Hippolytus’interpretation shows similarities to material in Eudorus, PhiloJudaeus, Plutarch and Numenius among others, although he adapts thematerial to fit his own purposes. He regards Platonism andPythagoreanism as the same philosophy, which ultimately derives fromEgypt. Empedocles is regarded as a Pythagorean and is quoted,sometimes without attribution, as evidence for Pythagorean views.According to Hippolytus the Monad and the Dyad are the two Pythagoreanprinciples, although the Dyad is derived from the Monad. ThePythagoreans recognize two worlds, the intelligible, which has theMonad as its principle, and the sensible, whose principle is thetetraktys, the first four numbers, which correspond to thepoint, line, surface and solid. Thetetraktys contains thedecad, since the sum of 1, 2, 3 and 4 is 10, and this is embodied inthe ten Aristotelian categories, which describe the sensible world.The pseudo-Archytan treatise, The Whole System ofCategories, had already claimed this Aristotelian doctrine forthe Pythagoreans (see 4.2 above). Finally, the intelligible world isequated with Empedocles’ sphere controlled by the uniting powerof Love in contrast to the world of sense perception in which thedividing power of Strife plays the role of the demiurge(Refutation of all Heresies 6, 23–25).

4.4 Neopythagorean Mathematical Sciences: Nicomachus, Porphyry and Iamblichus

A second strand of Neopythagoreanism, while maintaining connection tothese metaphysical speculations, emphasizes Pythagoras’ role inthe mathematical sciences. Nicomachus of Gerasa (modern Jerash inJordan) was probably active a little before Numenius, in the firsthalf of the second century CE. Unlike Neopythagoreans such as Eudorus,Moderatus and Numenius, whose works only survive in fragments, twocomplete works of Nicomachus survive,Introduction toArithmetic andHandbook of Music. More than anyone elsein antiquity he was responsible for popularizing supposed Pythagoreanachievements in mathematics and the sciences. TheHandbook ofMusic gives the canonical but scientifically impossible story ofPythagoras’ discovery of the whole number ratios, whichcorrespond to the basic concordant intervals in music: the octave(2:1), fifth (3:2), and fourth (4:3); he supposedly heard the concordsin the sounds produced by hammers of varying weights in ablacksmith’s shop, which he happened to be passing (Chapter 6— translation in Barker 1989, 256 ff.). In the next century,Iamblichus took this chapter over virtually verbatim and withoutacknowledgement in hisOn the Pythagorean Life (Chapter 26)and it was repeated in many later authors. The harmonic theorypresented by Nicomachus in theHandbook is not original andis, in fact, somewhat retrograde. It is tied to the diatonic scaleused by Plato in theTimaeus (35b-36b), which was previouslyused by the Pythagorean Philolaus in the fifth-century (Fr. 6a) andshows no awareness of or interest in the more sophisticated analysisof Archytas in the fourth century BCE. Nicomachus is not concernedwith musical practice but with “what pure reasoning can revealabout the properties of a rationally impeccable and unalterable systemof quantitative relations” (Barker 2007, 447). Nicomachus alsorelies heavily and without acknowledgement on a non-Pythagoreantreatment of music, Aristoxenus’Elementa Harmonica,many of the ideas of which he assigns to the Pythagoreans (e.g., inChapter 2; see Barker 1989, 245 ff.).

TheHandbook was influential because it put forth anaccessible version of Pythagorean harmonics (Barker 2014,200–202). Nicomachus provided a more detailed treatment ofPythagorean harmonics in his lostIntroduction to Music. Mostscholars agree that Books I-III and perhaps Book IV of Boethius’De Institutione Musica are a close paraphrase, which is oftenessentially a translation, of Nicomachus’ lost work (see Bowerin Boethius 1989, xxviii and Barker 2007, 445). Even more influentialthan his work on harmonics was Nicomachus’Introduction toArithmetic. Again Nicomachus was not an original or particularlytalented mathematician, but this popularizing textbook was widelyinfluential. There were a series of commentaries on it by Iamblichus(3rd CE), Asclepius of Tralles (6th CE), and Philoponus (6th CE) andit was translated into Latin already in the second half of the secondcentury by Apuleius. Most importantly, Boethius (5th-6th CE) provideswhat is virtually a translation of it in hisDe InstitutioneArithmetica, which became the standard work on arithmetic in themiddle ages. On Boethius’ use of Nicomachus see Hicks 2014,422–424.

In theIntroduction to Arithmetic, Nicomachus assigns toPythagoras the Platonic division between the intelligible and sensibleworld, quoting theTimaeus as if it were a Pythagorean text(I 2). He also assigns Aristotelian ideas to Pythagoras, in particulara doctrine of immaterial attributes with similarities to theAristotelian categories (I 1). Nicomachus divides reality into twoforms, magnitude and multitude. Wisdom is then knowledge of these twoforms, which are studied by the four sciences, which will later beknown as thequadrivium: arithmetic, music, geometry andastronomy. He quotes a genuine fragment of Archytas (Fr. 1) in supportof the special position of these four sciences. Nicomachus presentsarithmetic as the most important of the four, because it existed inthe mind of the creating god (the demiurge) as the plan which hefollowed in ordering the cosmos (I 4), so that numbers thus appear tohave replaced the Platonic forms as the model of creation (on formsand numbers in Nicomachus see Helmig 2007). It is striking that, alongwith this Platonization of Pythagoreanism, Nicomachus does give anaccurate presentation of Philolaus’ basic metaphysicalprinciples, limiters and unlimiteds, before attempting to equate themwith the Platonic monad and dyad (II 18).

Another work by Nicomachus,The Theology of Arithmetic, whichcan be reconstructed from a summary by Photius and an anonymous worksometimes ascribed to Iamblichus and known as theTheologoumenaArithmeticae (Dillon 1977, 352–353), suggests that helargely returned to the system of principles found in Plato’sunwritten doctrines and did not follow Eudorus and Moderatus inattempts to place a supreme god above the demiurge. Nicomachusapparently presents the monad as the first principle and demiurge,which then generates the dyad, but much is unclear (Dillon 1977,353–358). TheTheology of Arithmetic may have been mostinfluential in its attempt to set up an equivalence between the pagangods and the numbers in the decad, which was picked up later byIamblichus and Proclus (Kahn 2001, 116). Nicomachus also wrote aLife of Pythagoras, which has not survived but which Porphyry(e.g.,VP 59) and Iamblichus used (Rohde 1871–1872;O’Meara 2014, 412–413).

After Plotinus (205–270 CE), Neopythagoreanism becomes absorbedinto Neoplatonism. Although Plotinus was clearly influenced byNeopythagorean speculation on first principles (see above), he was nota Neopythagorean himself, in that he did not assign Pythagoras aprivileged place in the history of Greek philosophy. Plotinus treatsPythagoras as just one among many predecessors, complains of theobscurities of his thought and labels Plato and not Pythagoras asdivine (Enneads IV 8.11 ff.).

The earliest extantLife of Pythagoras is that of DiogenesLaertius, who was active ca. 200 CE. The most recent treatment ofDiogenes’ life is Laks 2014, on which much of what followsdepends. Unlike his successors Porphyry and Iamblichus (see below)Diogenes had no philosophical affiliation and hence no philosophicalaxe to grind in presenting the life of Pythagoras. Indeed, it isstriking that his life shows little influence from the Neopythagoreanauthors discussed above. Diogenes draws on a wide variety of importantsources, some going back to the fourth century and others derivingfrom the Hellenistic period. This material is put together in a veryloose, sometimes undetectable, organizational structure. There is anotable section on Pythagoras’ supposed writings (VIII,6–7). He shows particular interest in the Pythagorean way oflife and quotes a large number of Pythagoreansymbola forsome of which his source was Aristotle (VIII 34–35). The mainsection on Pythagoras’ philosophical doctrines is a longquotation from the first-century polymath Alexander Polyhistor whoclaims to be in turn drawing on a treatise calledPythagoreanNotes (VIII 24–33). For more on this treatise see thesection on Pythagorean pseudepigrapha above (4.2). Diogenes quotes anumber of passages satirizing Pythagoras, including Xenophanes’famous puppy fragment, and presents some of his own epigrams makingfun of the Pythagorean way of life (VIII, 36). However, other parts ofhis life present Pythagoras in a quite postive light so that it ishard to determine precisely what attitude Diogenes took towardsPythagoras (Laks 2014, 377–380).

TheLife of Pythagoras by Plotinus’ pupil and editor,Porphyry (234–ca. 305) is one of our most important sources forPythagoreanism (For what follows see Macris 2014). It was originallypart of his now lostPhilosophical History. Continuinginterest in Pythagoras in later centuries led theLife ofPythagoras to be preserved separately and it is the only largesection of thePhilosophical History to survive. ThePhilosophical History ended with Plato and clearly regardedPlatonic philosophy as the true philosophy so that Pythagoras seems tohave been highlighted as a key figure in the development ofPlato’s philosophy. Porphyry’sLife of Pythagorasis particularly valuable, because he often clearly identifies hissources. This same penchant for identifying and seeking out importantPythagorean sources can be seen in his commentary on Ptolemy’sHarmonics (2nd CE), in which he preserves several genuinefragments of the early Pythagorean Archytas, along with somepseudo-Pythagorean material. In theLife of PythagorasPorphyry does not structure his information according to anyoverarching theme but instead sets out the information derived fromother sources in a simple and orderly way with the minimum ofeditorial intervention. Although he cites some fifteen sources, somegoing back to the fourth century BCE, it is likely that he did not usemost of these sources but rather found them quoted in the four mainsources, which he used directly: 1) Nicomachus’Life ofPythagoras, 2) Moderatus’Lectures onPythagoreanism, 3) Antonius Diogenes’ novelUnbelievable Things Beyond Thule, and 4) a handbook of somesort. Since these sources come from the first and second centuries CE,Porphyry basically provides us with the picture of Pythagoras commonin Middle Platonism. This Pythagoras is the prototype of the sage ofold who was active as a teacher and tied to religious mystery.However, he is not yet Iamblichus’ priviliged soul sent to savehumanity (Macris, 2014, 390). Porphyry provides little criticism ofhis sources and, although his life has a neutral factual tone, incontrast to Diogenes Laertius in hisLife of Pythagoras, heincludes no negative reports about Pythagoras.

It would appear, however, that Pythagoras was not made the source ofall Greek philosophy, but was rather presented as one of a number ofsages both Greek and non-Greek (e.g., Indians, Egyptians and Hebrews),who promulgated a divinely revealed philosophy. This philosophy is, infact, Platonic in origin as it relies on the Platonic distinctionbetween the intelligible and sensible realms; Porphyry unhistoricallyassigns it back to these earlier thinkers, including Pythagoras.Pythagoras’ philosophy is thus said to aim at freeing the mindfrom the fetters of the body so that it can attain a vision of theintelligible and eternal beings (Life of Pythagoras46–47). O’Meara thus seems correct to conclude thatPorphyry was “…not a Pythagoreanizing Platonist …but rather a universalizing Platonist: he finds his Platonism both inPythagoras and in very many other quarters” (1989, 25–29).Porphyry himself lived an ascetic life that was probably largelyinspired by Pythagoreanism (Macris 2014, 393–394).

Porphyry’s pupil, Iamblichus (ca. 245–ca. 325 CE), fromChalcis in Syria, opposed his teacher on many issues in Neoplatonicphilosophy and was responsible for a systematic Pythagoreanization ofNeoplatonism (see O’ Meara 1989 and 2014), particularly underthe influence of Nicomachus’ earlier treatment of Pythagoreanwork in thequadrivium. Iamblichus wrote a work in ten booksentitledOn Pythagoreanism. The first four books havesurvived intact and excerpts of Books V-VII are preserved by theByzantine scholar Michael Psellus. Book One,On the PythagoreanLife, has biographical aspects but is primarily a detaileddescription of and a protreptic for the Pythagorean way of life. Itmight be that Iamblichus’ Pythagoras is intended in part as apagan rival to Christ and to Christianity, which was gaining strengthat this time. Porphyry, indeed, had written a treatiseAgainst theChristians, now lost. In Iamblichus, Pythagoras’ miraculousdeeds include a meeting at the beginning of his career with fishermenhauling in a catch (VP 36; cf. Matthew 1. 16–20; seeIamblichus,On the Pythagorean Life, Dillon and Hershbell(eds.) 1991, 25–26). O’Meara, on the other hand, doubtsthis connection to Christ (2014, 405 n. 21) and suggests thatIamblichus may have constructed Pythagoras as a rival toPorphyry’s presentation of Plotinus as the model philosopher(1989, 214–215). In the end we cannot be certain whetherIamblichus is responding to Porphyry or Porphyry to Iamblichus, butthey can be seen as battling over Plato’s legacy (O’Meara2014, 403). Porphyry in hisLife of Plotinus and edition ofhis works is promoting Plotinus’ interpretation of Plato.Iamblichus, on the other hand, advocates a return to the philosophythat inspired Plato, Pythagoreanism. Pythagorean philosophy isportrayed by Iamblichus as a gift of the gods, which cannot becomprehended without their aid; Pythagoras himself was sent down tomen to provide that aid (VP 1).

Iamblichus’On the Pythagorean Life is largely acompilation of earlier sources but, unlike Porphyry, he does notusually identify them. Rohde (1871–1872) argued influentiallythat On the Pythagorean Life was largely a compilation fromtwo sources: Nicomachus’Life of Pythagoras and a lifeof Pythagoras by Apollonius of Tyana. O’Meara argues that thisunderestimates both the extent to which Iamblichus reworked hissources for his own philosophical purposes and the variety of sourcesthat he used (O’Meara 2014, 412–415). A particularly clearexample of Iamblichus’ distintive development of ideas found inearlier sources can be seen in his treatment of the doctrine of theharmony of the spheres (O’Meara 2007). It is also true that theremaining books ofOn Pythgoreanism use a variety of sources.Book Two,Protreptic to Philosophy, is an exhortation tophilosophy in general and to Pythagorean philosophy in particular andrelies heavily on Aristotle’s lostProtrepticus. BookThree,On General Mathematical Science, deals with thegeneral value of mathematics in aiding our comprehension of theintelligible realm and is followed by a series of books on thespecific sciences. The treatment of arithmetic in Book IV takes theform of a commentary on Nicomachus’Introduction toArithmetic. Books V-VII then dealt with arithmetic in physics,ethics and theology respectively and were followed by treatments ofthe other three sciences in the quadrivium:On PythagoreanGeometry,On Pythagorean Music andOn PythagoreanAstronomy. Iamblichus was particularly interested in Pythagoreannumerology and his section on arithmetic in theology is probablyreflected in the anonymous treatise which has survived under the titleTheologoumena Arithmeticae and which has sometimes beenascribed to Iamblichus himself. It appears that here again Iamblichusrelied heavily on Nicomachus, this time on hisTheology ofArithmetic.

It is possible that Iamblichus used the ten Books ofOnPythagoreanism as the basic text in his school, but we know thathe went beyond these books to the study of Aristotelian logic and thePlatonic dialogues, particularly theTimaeus andParmenides (Kahn 2001, 136–137). Nonetheless, it wasbecause of Iamblichus that Pythagoreanism in the form of numerologyand mathematics in general was emphasized by later Neoplatonists suchas Syrianus (fl. 430 CE) and Proclus (410/412–485 CE). Proclusis reported to have dreamed that he was the reincarnation ofNicomachus (Marinus,Life of Proclus 28). Proclus did treatPlato’s writings as clearer than the somewhat obscure writingsof the Pythagoreans but his Platonism is still heavily Pythagorean(O’ Meara 2014, 415). The successors of Proclus appear to followhis and Iamblichus’ interpretation of Pythagoras (O’Meara2013).

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

A third strand in Neopythagoreanism emphasizes Pythagoras’practices rather than his supposed metaphysical system. ThisPythagoras is an expert in religious and magical practices and/or asage who lived the ideal moral life, upon whom we should model ourlives. This strand is closely connected to the striking interest inand prominence of Pythagoreanism in Roman literature during the firstcentury BCE and first century CE. Cicero (106–43 BCE) inparticular refers to Pythagoras and other Pythagoreans with somefrequency. InDe Finibus (V 2), he presents himself as theexcited tourist, who, upon his arrival in Metapontum in S. Italy andeven before going to his lodgings, sought out the site wherePythagoras was supposed to have died. At the beginning of Book IV(1–2) of theTusculan Disputations, Cicero notes thatPythagoras gained his fame in southern Italy at just the same timethat L. Brutus freed Rome from the tyranny of the kings and foundedthe Republic; there is a clear implication that Pythagorean ideas,which reached Rome from southern Italy, had an influence on the earlyRoman Republic. Cicero goes on to assert explicitly that many Romanusages were derived from the Pythagoreans, although he does not givespecifics. According to Cicero, it was admiration for Pythagoras thatled Romans to suppose, without noticing the chronologicalimpossibility, that the wisest of the early Roman kings, Numa, who wassupposed to have ruled from 715–673 BCE, had been a pupil ofPythagoras.

In addition to references to Pythagoras himself, Cicero refers to thePythagorean Archytas some eleven times, in particular emphasizing hishigh moral character, as revealed in his refusal to punish in angerand his suspicion of bodily pleasure (Rep. I 38. 59;Sen. XII 39–41). Cicero’s own philosophy is notmuch influenced by the Pythagoreans except inThe Dream ofScipio (Rep. VI 9), which owes even more to Plato.

The interest in Pythagoras and Pythagoreans in the first century BCEis not limited to Cicero, however. Both a famous ode of Horace (I 28)and a brief reference in Propertius (IV 1) present Archytas as amaster astronomer. Most striking of all is the speech assigned toPythagoras that constitutes half of Book XV of Ovid’sMetamorphoses (early years of the first century CE) and thatcalls for strict vegetarianism in the context of the doctrine oftransmigration of souls. These latter themes are true to the earliestevidence for Pythagoras, but the rest of Ovid’s presentationassigns to Pythagoras a doctrine that is derived from a number ofearly Greek philosophers and in particular the doctrine of fluxassociated with Heraclitus (Kahn 2001, 146–149).

This flourishing of Pythagoreanism in Roman literature of the goldenage has its roots in one of the earliest Roman literary figures,Ennius (239–169 BCE), who, in his poemAnnales, adoptsthe Pythagorean doctrine of metempsychosis, in presenting himself asthe reincarnation of Homer, although he does not mention Pythagoras byname in the surviving fragments. Roman nationalism also played a rolein the emphasis on Pythagoreanism at Rome. Since Pythagoras did hiswork in Italy and Aristotle even referred to Pythagoreanism in someplaces as the philosophy of the Italians (e.g.,Metaph.987a10), it is not surprising that the Romans wanted to emphasizetheir connections to Pythagoras. This is particularly clear inCicero’s references to Pythagoreanism but once again finds itsroots even earlier. In 343 BCE during the war with the Samnites,Apollo ordered the Romans to erect one statue of the wisest andanother of the bravest of the Greeks; their choice for the former wasPythagoras and for the latter Alcibiades. Pliny, who reports the story(Nat. XXXIV 26), expresses surprise that Socrates was notchosen for the former, given that, according to Plato’sApology, Apollo himself had labeled Socrates the wisest; itis surely the Italian connection that explains the Romans’choice of Pythagoras. Cicero (not Aristoxenus as suggested by Horky2011) connects the great wisdom assigned to the Samnite HerreniusPontius to his contact with the Pythagorean Archytas (On OldAge 41). This Roman attempt to forge a connection with Pythagorascan also be seen in the report of Plutarch (Aem. Paul. 1)that some writers traced the descent of the Aemelii, one ofRome’s leading families, to Pythagoras, by claimingPythagoras’ son Mamercus as the founder of the house.

Although Rome’s special connection to Pythagoras thus hadearlier roots, those roots alone do not explain the efflorescence ofPythagoreanism in golden age Latin literature; some stimulus probablycame from the rebirth of what were seen as Pythagorean practices inthe way certain people lived. The two most learned figures in Rome ofthe first century BCE, Nigidius Figulus and Varro, both haveconnections to Pythagorean ritual practices. Thus we are told thatVarro (116–27 BCE) was buried according to the Pythagoreanfashion in myrtle, olive and black poplar leaves (Pliny,Nat.XXXV 160). Amongst Varro’s voluminous works was theHebdomadês (“Sevens”), acollection of 700 portraits of famous men, in the introduction towhich Varro engaged in praise for the number 7, which is similar tothe numerology of later Neopythagorean works such as Nicomachus’Theology of Arithmetic; in another work Varro presents atheory of gestation, which has Pythagorean connections, in that it isbased on the whole number ratios that correspond to the concordantintervals in music (Rawson 1985, 161).

It is Nigidius Figulus, praetor in 58, who died in exile in 45,however, who is usually identified as the figure who was responsiblefor reviving Pythagorean practices. In the preface to his translationof Plato’sTimaeus, which is often treated as virtuallya Pythagorean treatise by the Neopythagoreans, Cicero asserts ofNigidius that “following on those noble Pythagoreans, whoseschool of philosophy had to a certain degree died out, … thisman arose to revive it.” Some scholars are dubious about thisclaim of Cicero. They point to the evidence cited above for theimportance of Pythagoreanism in Rome in the two centuries beforeNigidius and suggest that Cicero may be illegitimately followingAristoxenus’ claim that Pythagoreanism died out in the firsthalf of the fourth century (Riedweg 2005, 123–124). While theremay be some evidence that there were practicing Pythagoreans in thesecond half of the fourth century (see above section 3.5), it is hardto find anyone to whom to apply that label in the third and secondcenturies, so that, from the perspective of the evidence available tous at present, Cicero may well be right that Nigidius was the firstperson in several centuries to claim to follow Pythagorean practices.However, the sources for Nigidius are meager and there is no evidencethat he was the leader of a large and powerful group. If there was anorganized group at all, it is more likely to have been a smallercircle (Flinterman 2014, 344).

It is difficult to be sure in what Nigidius’ Pythagoreanismconsisted. There is no mention of Pythagoras or Pythagoreans in thesurviving fragments of his work nor do they show him engaging inPythagorean style numerology as Varro did (Rawson 1985, 291 ff.). InJerome’s chronicle, Nigidius is labeled as Pythagorean andmagus; the most likely suggestion, thus, is that hisPythagoreanism consisted in occult and magical practices. Pliny treatsNigidius alongside theMagi and also presents Pythagoras andDemocritus as having learned magical practices from theMagi.Cicero describes Nigidius as investgating matters that nature hadhidden and this may be a reference to such magical lore (Flinterman2014, 345). Nigidius’ expertise as an astrologer (he is reportedto have used astrology to predict Augustus’ future greatness onthe day of his birth [Suetonius,Aug. 94.5]) may be anotherPythagorean connection; Propertius’ reference (IV 1) to Archytasshows that Pythagorean work in astronomy was typically connected toastrology in first century Rome.

What led Nigidius and Varro to resurrect purported Pythagorean cultpractices? One important influence may have been the Greek scholarAlexander Polyhistor, who was born in Miletus but was captured by theRomans during the Mithridatic wars and brought to Rome as a slave andfreed by Sulla in 80 BCE. He taught in Rome in the 70s. It is anintriguing suggestion that Nigidius learned his Pythagoreanism fromAlexander (Dillon 1977, 117; For critiques of this suggestion seeFlinterman 2014, 349–350 and Long 2013, 145). There is noevidence that Alexander himself followed Pythagorean practices, but hewrote a bookOn Pythagorean Symbols, which was presumably anaccount of the Pythagoreanacusmata (orsymbola),which set out the taboos that governed many aspects of the Pythagoreanway of life. In addition, in hisSuccessions of thePhilosophers, he gave a summary of Pythagorean philosophy, whichhe supposedly found in thePythagorean Notes (See section 4.2above) and which has been preserved by Diogenes Laertius (VIII25–35). The basic principles assigned to Pythagoras are those ofthe Neopythagorean tradition that begins in the early Academy, i.e.,the monad and the indefinite dyad. Since Alexander also assigns to thePythagoreans the doctrine that the elements change into one another,we might suppose that Ovid also used Alexander directly or indirectly,since he assigns a similar doctrine to Pythagoras in theMetamorphoses (XV 75 ff., Rawson 1985, 294).

It is necessary to look in a slightly different direction, in order tosee how magical practices came to be particularly associated withPythagoras and thus why Nigidius was called Pythagorean andmagus. In the first century, it was widely believed thatPythagoras had studied with the Magi (Cicero,Fin. V 87),i.e. Persian priests/wise men. What Pythagoras was thought to havelearned from the Magi most of all were the magical properties ofplants. Pliny the elder (23–79 CE) identifies Pythagoras andDemocritus as the experts on such magic and the Magi as their teachers(Nat. XXIV 156–160). Pliny goes on to give a number ofspecific examples from a book on plants ascribed to Pythagoras. Thisbook is universally regarded as spurious by modern scholars, and evenPliny, who accepts its authenticity, reports that some people ascribeit to Cleemporus. We can date this treatise on plants to the firsthalf of the second century or earlier, since Cato the elder(234–149 BCE) appears to make use of it in hisOnAgriculture (157), when he discusses the medicinal virtues of akind of cabbage, which was named after Pythagoras (brassicaPythagorea).

A clearer understanding of this pseudo-Pythagorean treatise on plantsand a further indication of its date can be obtained by looking at thework of Bolus of Mendes, an Egyptian educated in Greek (see Dickie2001, 117–122, to whom the following treatment of Bolus isindebted). Bolus composed a work entitledCheiromecta, whichmeans “things worked by hand” and may thus refer topotions made by grinding plants and other substances (Dickie 2001,119). Bolus discussed not just the magical properties of plants butalso those of stones and animals. Pliny regarded theCheiromecta as composed by Democritus on the basis of hisstudies with the Magi (Nat. 24. 160) and normally cites itscontents as what Democritus or the Magi said. Columella, however,tells us what was really going on (On Agriculture VII 5.17).The work was in fact composed by Bolus, who published it under thename of Democritus. Bolus thus appears to have made a collection ofmagical recipes, some of which do seem to have connections to theMagi, since they are similar to recipes found in 8th century cuneiformtexts (Dickie 2001, 121). In order to gain authority for thiscollection, he assigned it to the famous Democritus.

Since Democritus was sometimes regarded as the pupil of Pythagoreans(Diogenes Laertius IX 38), Bolus’ choice of Democritus to giveauthority to his work may suggest that someone else (the Cleemporusmentioned by Pliny?) had already used Pythagoras for this purpose andthat the pseudo-Pythagorean treatise on the magical properties ofplants was thus already in existence when Bolus wrote, in the firsthalf of the second century BCE. An example of the type of recipeinvolved is Pliny’s ascription to Democritus of the idea thatthe tongue of a frog, cut out while the frog was still alive, ifplaced above the heart of a sleeping woman, will cause her to givetrue answers (Nat. XXXII 49). Thus, the picture of Pythagorasthe magician, which may lie behind a number of the supposedPythagorean practices of Nigidius Figulus, is based on little morethan the tradition that Pythagoras had traveled to Egypt and the east,so that he became the authority figure, to whom the real collectors ofmagical recipes in the third and second century BCE ascribed theircollections.

Nigidius’ revival of supposed Pythagorean practices spread toother figures in first century Rome. Cicero attacked Vatinius, consulin 48 and a supporter of Caesar, for calling himself a Pythagorean andtrying to shield his scandalous practices under the name of Pythagoras(Vat. 6). The scandalous practices involved necromancy,invoking the dead, by murdering young boys. Presumably this method ofnecromancy would not be ascribed to Pythagoras, but the suggestion isthat some methods of consulting the dead were regarded as Pythagorean.Cicero later ended up defending this same Vatinius in a speech whichhas not survived but some of the contents of which we know from theancient scholia on the speech against Vatinius. In this speech Cicerodefended Vatinius’ habit of wearing a black toga, which heattacked in the earlier speech (Vat. 12), as a harmlessaffectation of Pythagoreanism (Dickie 2001, 170). Thus, the title ofPythagorean in first century Rome carried with it associations withmagical practices, not all of which would have been widely approved.

Another example of the connection between Pythagoreanism and magic andits possible negative connotations is Anaxilaus of Larissa (Rawson1985, 293; Dickie 2001, 172–173). In his chronicle, Jeromedescribes him with the same words as he used for Nigidius, Pythagoreanandmagus, and reports that he was exiled from Rome in 28BCE. We know that Anaxilaus wrote a work entitledPaignia(“tricks”), which seems to have consisted of some ratherbizarre conjuring tricks for parties. Pliny reports one ofAnaxilaus’ tricks as calling for burning the discharge from amare in heat in a flame, in order to cause the guests to see images ofhorses’ heads (Nat. XXVIII 181). The passion for thingsPythagorean can also be seen in the figure of king Juba of Mauretania(ca. 46 BCE – 23 CE), a learned and cultured man, educated atRome and author of many books. Olympiodorus describes him as “alover of Pythagorean compositions” and suggests that Pythagoreanbooks were forged to satisfy the passion of collectors such as Juba(Commentaria in Aristotelem Graeca 12.1, p. 13).

The connection between Pythagoreanism and astrology visible inNigidius can perhaps also be seen in Thrasyllus of Alexandria (d. 36CE), the court astrologer and philosopher, whom the Roman emperorTiberius met in Rhodes and brought to Rome. Thrasyllus is famous forhis edition of Plato’s dialogues arranged into tetralogies, buthe was a Platonist with strong Pythagorean leanings. Porphyry in hisLife of Plotinus (20) quotes Longinus as saying thatThrasyllus wrote on Platonic and Pythagorean first principles (Dillon1977, 184–185). Most suggestive of all is the quotation fromThrasyllus preserved by Diogenes Laertius (Diogenes Laertius IX 38),in which Thrasyllus calls Democritus a zealous follower of thePythagoreans and asserts that Democritus drew all his philosophy fromPythagoras and would have been thought to have been his pupil, ifchronology did not prevent it. It is impossible to be sure whatThrasyllus had in mind here, but one very plausible suggestion is thathe is thinking of Democritus as a sage, who practiced magic, theDemocritus created by Bolus, who was the successor to the arch magePythagoras, the supposed author of the treatise on the magical uses ofplants (Dickie 2001, 195). Some have argued that the subterraneanbasilica discovered near the Porta Maggiore and dating to the firstcentury CE was the meeting place of a Pythagorean community but theevidence for this suggestion is very weak (Flinterman 2014).

We cannot be sure whether the Pythagoreanism of Nigidius, Varro andtheir successors was limited to such things as burial ritual, magicalpractices and black togas or whether it extended to less spectacularfeatures of a “Pythagorean” life. Q. Sextius, however,founded a philosophical movement in the time of Augustus, whichprescribed a vegetarian diet and taught the doctrine of transmigrationof souls, although Sextius presented himself as using differentarguments than Pythagoras for vegetarianism (Seneca,Ep. 108.17 ff.). One of these Sextians, as they were known, was Sotion, theteacher of Seneca, and it is Seneca who gives us most of theinformation we have about them. It is also noteworthy that Sextius isalso reported to have asked himself at the end of each day “Whatbad habit have you cured today? What vice have you resisted? In whatway are you better” (Seneca,De Ira III 36). Cicerotells us that it was “the Pythagorean custom” to call tomind in the evening everything said, heard or done during the day(Sen. 38, cf. Iamblichus,VP 164). The practicedescribed by Cicero is directed at training the memory in contrast toSextius’ questions, which call for moral self-examination. OnPythagoreanism in Rome see further Flinterman 2014.

Something similar to the Sextian version of the practice is found inlines 40–44 of theGolden Verses, a treatise consistingof 71 Greek hexameter verses, which was ascribed to Pythagoras or thePythagoreans. The poem is a combination of materials from differentdates, and it is uncertain when it took the form preserved inmanuscripts and called theGolden Verses; dates ranging from350 BCE to 400 CE have been suggested (see Thom 1995). It is notreferred to by name until 200 CE. TheGolden Verses arefrequently quoted in the first centuries CE and thus constitute onemodel of the Pythagorean life in Neopythagoreanism, one that is freefrom magical practices. Much of the advice is common to all of Greekethical thought (e.g., honoring the gods and parents; mastering lustand anger; deliberating before acting, following measure in allthings), but there are also mentions of dietary restrictions typicalof early Pythagoreanism and the promise of leaving the body behind tojoin the aither as an immortal. It is not clear that the treatiseshould be called pseudepigraphal, since it was not usually ascribed toPythagoras himself but rather to unnamed Pythagoreans and may havebeen devised as moral instruction for beginners in a Pythagoreancommunity (Thom 2021), although there is no direct evidence for thiscommunity.

Our most detailed account of a Neopythagorean living a life inspiredby Pythagoras is Philostratus’Life of Apollonius ofTyana. Apollonius was active in the second half of the firstcentury CE and died in 97; Philostratus’ life, which was writtenover a century later at the request of the empress Julia Domna andcompleted after her death in 217 CE, is more novel than soberbiography. According to Philostratus, Apollonius identified his wisdomas that of Pythagoras, who taught him the proper way to worship thegods, to wear linen rather than wool, to wear his hair long, and toeat no animal food (I 32). Some have wondered if Apollonius’Pythagoreanism is largely the creation of Philostratus, but thestandard view has been that Apollonius wrote a life of Pythagoras usedby Iamblichus (VP 254) and Porphyry (Burkert 1972, 100), andthe fragment of his treatiseOn Sacrifices has clearconnections to Neopythagorean philosophy (Kahn 2001, 143–145).Rohde thought that large parts of Apollonius’sLife ofPythagoras could be found in Iamblichus’On thePythagorean Life, but recently more and more doubt has arisen asto whether the Apollonius who wrote theLife of Pythagorasused by Iamblichus is really Apollonius of Tyana (Flinterman 2014,357).

Like Pythagoras, Apollonius journeys to consult the wise men of theeast and learns from the Brahmins in India that the doctrine oftransmigration, which Apollonius inherited from Pythagoras, originatedin India and was handed on to the Egyptians from whom Pythagorasderived it (III 19). Philostratus (I 2) emphasizes that Apollonius wasnot a magician, thus trying to free him from the more disreputableconnotations of Pythagorean practices associated with figures such asAnaxilaus and Vatinius (see above). Nonetheless, Philostratus’life does recount a number of Apollonius’ miracles, such as theraising of a girl from the dead (IV 45). On Apollonius as aPythagorean see further Flinterman 2014.

These miracles made Apollonius into a pagan counterpart to Christ. Theemperor Alexander Severus (222–235 CE) worshipped Apolloniusalongside Christ, Abraham and Orpheus (Hist. Aug., Vita Alex.Sev. 29.2). Hierocles, the Roman governor of Bithynia, who wasrigorous in his persecution of Christians, championed Apollonius atthe expense of Christ, inThe Lover of Truth, and drew as aresponse Eusebius’Against Hierocles. As mentionedabove, there is some probability that Iamblichus intends to elevatePythagoras himself as a pagan counterpart to Christ in hisOn thePythagorean Life (Dillon and Hershbell 1991, 25–26).

The satirist Lucian (2nd CE) provides us with a hostile portrayal ofanother holy man with Pythagorean connections, Alexander ofAbnoteichus in Paphlagonia, who was active in the middle of the secondcentury CE. InAlexander the False Prophet, Lucian reportsthat Alexander compared himself to Pythagoras (4), could remember hisprevious incarnations (34) and had a golden thigh like Pythagoras(40). Lucian shows the not often seen negative side to bothPythagoras’ and Alexander’s reputations when he reportsthat, if one took even the worst things said about Pythagoras,Alexander would far outdo him in wickedness (4). Some have seenAlexander as largely a literary construction by Lucian with littlehistorical basis but other evidence confirms that there were travelingPythagorean wonder-workers in the early imperial period (Flinterman2014, 359).

Despite these attacks on figures such as Apollonius and Alexander whomodeled themselves on Pythagoras, the Pythagorean way of life was ingeneral praised; the Neopythagorean tradition which portraysPythagoras as living the ideal life on which we should model our ownreaches its culmination in Iamblichus’On the PythagoreanLife and Porphyry’sLife of Pythagoras

5. Pythagoreanism in the Middle Ages and Renaissance

The influence of Pythagoreanism in the Middle Ages and Renaissance wasextensive and was found in most disciplines, in literature and art aswell as in philosophy and science. Here only the highlights of thatinfluence can be given (see further Heninger 1974, Celenza 1999,Celenza 2001, Kahn 2001, Riedweg 2005, Hicks 2014, Allen 2014, and theessays in Caiazzo, Macris and Robert (eds.) 2022 to all of whom thefollowing account is indebted). It is crucial to recognize from thebeginning that the Pythagoras of the Middle Ages and Renaissance isthe Pythagoras of the Neopythagorean tradition, in which he isregarded as either the most important or one of the most importantphilosophers in the Greek philosophical tradition. Thus, RalphCudworth, inThe True Intellectual System of the Universeasserted that “Pythagoras was the most eminent of all theancient Philosophers” (1845, II 4). This is a far cry from thePythagoras that can be reconstructed by responsible scholarship.Riedweg has put it well: “Had Pythagoras and his teachings notbeen since the early Academy overwritten with Plato’sphilosophy, and had this ‘palimpsest’ not in the course ofthe Roman empire achieved unchallenged authority among Platonists, itwould be scarcely conceivable that scholars from the Middle Ages andmodernity down to the present would have found the pre-Socraticcharismatic from Samos so fascinating” (2005, 128).

5.1 Boethius/Nicomachus, Calcidius, Macrobius and the Middle Ages

In the Middle Ages Pythagoras and Pythagorean philosophy were regardedas the height of Greek philosophical achievement, although, somewhatparadoxically Pythagoreanism was not still an active philosophy aswere Platonism and Aristotelianism but instead belonged to an“imagined history” of philosophy (Hicks 2014, 420). Theview of Pythagoreanism in the Middle Ages was heavily determined bythree late ancient Latin writers: Calcidius, Macrobius and Boethius.It was in particular the mathematical Pythagoreanism of Nicomachus astransmitted by Boethius that determined the medieval picture ofPythagoras. In ethics, Christians were able to embrace somePythagorean maxims such as the principle labeled Pythagorean byBoethius: “Follow God” (Consolation of Philosophy1.4). Some attention was also paid to other Pythagoreansymbola oracousmata as is shown later in thissection. On the other hand the doctrine of metempsychosis wasrepugnant to Christian doctrine. John of Salisbury(Policraticus 7.10) says “When the Pythagoreans teachus about innocence, frugality and contempt for the world, we shouldlisten to them; when they force souls that have ascended into theheavens back into the bodies of beasts, even Plato must be reftued,for on this point he followed Pythagoras too closely” (tr.Hicks, 2014, 419–20). When it comes to Pythagoras’ life itis crucial to recognize that Iamblichus’ and Porphyry’slives of Pythagoras were not known in the Middle Ages so thatPythagoras’ activities were mostly known through passages fromclassical authors and church fathers (Hicks 2014, 421). Pythagoras wasincluded in medieval encyclopedic works and was given particularlythorough treatment by Vincent of Beauvais (before 1200–1264) inhisSpeculum historiale (3.24–26), by John of Wales(fl. 1260–1283) inCompendiloquium (3.6.2) and inThe Lives and Habits of the Philosophers ascribed to, butprobably not actually composed by, Walter Burley (1275–1344; seeRiedweg 2005, 129; Heninger 1974, 47; Hicks 2014, 421).

The most influential texts for the conception of Pythagoras in theLatin Middle Ages and early Renaissance were Boethius’(480–524 CE)De Institutione Arithmetica andDeInstitutione Musica, which are virtually translations of theNeopythagorean Nicomachus’ (second century CE)Introductionto Arithmetic andIntroduction to Music (this largerwork is now lost, but a smallerHandbook of Harmonicssurvives). Boethius followed Nicomachus’ classification of fourmathematical sciences depending on the nature of their objects(arithmetic deals with multitude in itself, music with relativemultitude, geometry with unmoving magnitudes and astronomy withmagnitude in motion). Boethius introduced the termquadrivium, “fourfold road” to understanding, torefer to these four sciences and following Nicomachus made Pythagorasthe father of thequadrivium, a depiction which laststhroughout the Middle Ages (Panti 2022). In music theory, Boethiuspresents the Pythagoreans as taking a middle position, which gives arole in harmonics to both reason and perception. His presentation ofthe Pythagorean position was central to music theory for over athousand years (Hicks 2014, 424 and 2022, 98–104). Boethiusrecounts the apocryphal story of Pythagoras’ discovery in ablacksmith’s shop of the ratios that govern the concordantintervals (Mus. I 10).

Pythagoreanism as found in Boethius’InstitutioArithmetica was developed into the Medieval ChristianNeopythagoren theology that is found particularly in the writings ofThierry of Chartres (1100–1150) and Nicholas of Cusa(1401–1464). In this mathematical theology God is the source ofall numbers and contains the arithmetical blueprints of the world(Albertson, 2022, 390). On the other hand, Thomas Aquinas(1225–1274) primarily dervied his knowledge of Pythagoras andPythagoreanism from his study of Aristotelian texts. He findsphilosophical interest in three Pythagorean doctrines which he, likeAristotle, ultimately rejects: the transmigration of souls (which wasalmost universally rejected in the Middle ages – See Caiazzo2022), number as a substantail principle of sensible things, the tableof opposites as providing the basic principles of all reality. He alsocriticizes the Pythagorean doctrine of the harmony of the spheres(Borgo and Costa 2022).

The medieval picture of Pythagoras as a natural philosopher and themedieval understanding of his theory of the nature of the soul wereheavily influenced by the Latin commentary on Plato’sTimaeus by Calcidius (4th century CE) and theCommentaryon the Dream of Scipio by Macrobius (5th century CE). Calcidiusregarded Plato’sTimaeus as a heavily Pythagoreandocument. Under the influence of the Neopythagorean Numenius,Calcidius assigned to Pythagoras the view that god was unity andmatter duality (Hicks 2014, 429). Calcidius describes Plato’sWorld-Soul in a way that highlights its harmonic structure andMacrobius explicitly ascribes to Pythagoras the view that the soul isa harmony (Commentary on the Dream of Scipio 1.14.19). Thedoctrine of the harmony of the spheres, which portrays the cosmos as aharmony that is expressed in the music made by the revolutions of theplanets, follows from the numerical structure of the World-Soul andwas also assigned to Pythagoras by Calcidius. Most medievalNeoplatonic cosmoligies adopted the doctrine, but the reintroductionof Aristotle’s criticism of it in the thirteenth century causedmany to abandon the theory until it was revived in the Renaissance byFicino (Hicks 2014, 434). Later, Shakespeare refers to the doctrinememorably inThe Merchant of Venice (V i. 54–65).Cicero’s presentation of it in theDream of Scipio wasalso influential in the Renaissance (Heninger 1974, 3).

Pythagoras was also known for moral precepts in the Middle Ages (seeRobert 2022) and one of the most important sources for these was St.Jerome’sApology against Rufinus (402 CE). Jeromereported precepts such as “Avoid excess … in all thingalike” and the famous “Friends have all things incommon.” In addition Jerome quoted several of the Pythagoreanacousmata which he calledaenigmata, e.g.“Never jump over the scale” and “Never stir the firewith the sword.” Theseaenigmata came to circulateseparately from Jerome’s text and were known as theAenigmata Aristotelis. The oldest evidence for them dates tothe 9th century and they circulated widely in the 12th to 15thcenturies. In the 14th century they came to be accompanied by a moraland theological commentary calledAenigmata moralizata. Theywere also incorporated into theGesta Romanorum, which wasone of the most widely circulated collections of moral examples in theMiddle Ages. The first chapter of this work portrayed Aristotle asteaching the Pythagoreanacousmata to Alexander the Great.The author then provides commentary on each of theacousmata,often appealing to Christian scripture. Moral maxims of Pythagorasalso circulated inOn the Foolishness of the Philosophersascribed to a fictional character named Caecilius Balbus. OtherPythagorean sayings reached the Latin West through translations ofArabic gnomologies such as that by Al-Mubashshir (see below).Helinandus of Froidmont’sChronicon (compiled between1211 and 1223) was the basis for the medieval tradtion about the lifeof Pythagoras. It consisted of quotations from classical literatureand the church fathers and provided a favorable portrait ofPythagoras, which stressed his moral virtue. Helinandus was closelyfollowed, with some additional material, by Vincent of Beauvais (d.1264) inThe Mirror of History, John of Wales in hisCompendiloquium de vita e dictis illustrium philosophorum andtheLiber de vita et moribus philosophorum illustrium, whichwas usually ascribed to Walter Burley (b. 1275). “The collectionof Pythagoras’exempla anddicta served notonly to provide material for scholarly works, but also providedclerics with a pagan mirror in which to contemplate, with shame, theirown shortcomings” (Robert, 2022, 265).

Pythagorean influence also appeared at less elevated levels ofmedieval culture. A fourteenth-century manual for preachers, whichcontained lore about the natural world and is known asThe Lightof the Soul, ascribes a series of odd observations about natureto Archita Tharentinus, who is presumably intended to be the fourthcentury BCE Pythagorean, Archytas of Tarentum. These are mostly citedfrom a book, which was evidently forged in Archytas’ name andknown asOn Events in Nature. Some of the observations areplausible enough, e.g., that a person at the bottom of a well seesstars in the middle of the day, others more puzzling, e.g., that adying man emits fiery rays from his eyes at death, while still othersmay have connections to magic, e.g., “if someone looks at amirror, before which a white flower has been placed, he cries.”Some magical lore ascribed to an Architas is also found in thethirteenth-centuryMarvels of the World (ps.-AlbertusMagnus), e.g., “if the wax of the left ear of a dog be taken andhung on people with periodic fever, it is beneficial…”These texts seem to continue the connection between Pythagoreanism andmagic, which developed in the third and second centuries BCE, and isprominent in Rome during the first-century BCE (see above section4.5).

Medieval Arabic biographical accounts of Pythagoras such as those ofAbū al-Ḥasan Muḥammad ibn Yūsufal-ʿĀmirī (d. 992) in hisOn the Afterlife andAbū l-Fatḥ Muḥammad al-Shahrastānī(11th-12th c.) in hisBook of Religions and Sects presentedPythagoras as transmitting the Eastern wisdom of Egypt and Solomon tothe West and as a sage who had direct experience of the celestialrealms and heard the harmony of the spheres. One of the most importantArabic sources for Pythagoras is Abū al-Wafāʾal-Mubashshir ibn Fātik’s (11th c.)Book of theChoicest Maxims and Best Sayings. It combines a biography ofPythagoras (a shortened and altered version of Porphyry’sLife of Pythagoras) with a collection of Pythagorean maxims.Al-Mubashshir regarded this gnomology as of more than historicalinterest and as genuinely helpful in religious and practical matters.Most of these maxims were derived from thePythagoreanSentences but another important source isThe GoldenVerses, which had already been translated into Arabic in the 8thcentury.The Golden Verses were regarded by many in theArabic world as the main source for the teaching of Pythagoras.Another important collection of anecdotes and sentences about Greekand Arabic philosophers was The Cabinet of Wisdom, which wasput together around 1000 AD. Many of the sayings ascribed toPythagoras are assigned to other thinkers in the Greek tradition.Pythagoras was presented as the first philosopher and as an ascetic.Some of the material in this collection is derived from thepseudepigraphal letter of Pythagoras to Hieron I (Thesleff 1965, 185),which was knows asThe Letter of Pythagoras to the Tyrant ofSicily. Another set of maxims attributed to Pythagoras is foundin The Spiritual Contents of Greek Maxims collected by IbnHindū (d. 1019 or 1029). The section on Pythagoras includes 14sentences, all of which are not found in other Arabic gnomologies. Thefifth one starts out “And he said to his son, I recommend tenthings and you should learn them: do not appear to be harsh, do notdrink with the one who is too eager, do not live with a jealous one…” (tr. Izdebska 2022). These gnomological collections donot include the Pythagorean symbola, which were however translatedinto both Syriac and Arabic and circulated in collections even moreextensive than than those preserved in Greek. In the gnomologicaltradtion Pythagoras is especially presented as a teacher and moralguide for a community of followers. The Arabic doxographies such asthose of Pseudo-Ammonius (second half of 9th century), who relied onHippolytus’Refutation of all Heresies (3rd centuryCE), and al-Shahrastānī (d. 1153) portrayed Pythagoras as aNeoplatonist, whose insights into the unity of god, whose essence isbeyond human comprehension, and who transcends all other unities,could serve as a guide to crucial Islamic tenets such as God’sunity and oneness (De Smet, 2022). For more on Pythagoras in theArabic tradition see Izdebska 2022. Nicomachus’Introductionto Arithmetic was translated into Arabic twice. One translationin 822 CE was based on a previous Syriac translation and is lost andonly now known through a Hebrew translation completed in 1317 CE(Freudenthal, 2022). The other was completed in the second half of the9th century from the Greek and survives in one copy. Thesetranslations insured that Nichomachus exerted in important influenceon Arabic mathematical treatises, teaching textbooks and encyclopedias(Brentjes, 2022).

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

In the Renaissance, Pythagoreanism played an important role in thethought of fifteenth- and sixteenth century Italian and Germanhumanists. The Florentine Marsilio Ficino (1433–1499) is mostproperly described as a Neoplatonist. He made the philosophy of Platoavailable to the Latin-speaking west through his translation of all ofPlato into Latin. In addition he translated important works of writersin the Neoplatonic and Neopythagorean tradition, such as Plotinus,Porphyry, Iamblichus and Proclus. From that tradition he accepted anddeveloped the view that Plato was heir to an ancienttheology/philosophy (prisca theologia) that was derived fromearlier sages including Pythagoras, who immediately preceded Plato inthe succession (Allen 2014, 435–436). Ficino like theNeopythagoreans had no conception of an early and a latePythagoreanism, for him Pythagoreanism was a unity as indeed was theentire tradition of ancient theology (Celenza, 1999, 675–681).Ficino regarded works ascribed to the Chaldaean Zoroaster, theEgyptian Hermes Trismegistus, Orpheus and Pythagoras, which modernscholarship has shown to be forgeries of late antiquity, as genuineworks on which Plato drew (Kristeller 1979, 131). Ficino provided acomplete translation of the writings ascribed to Hermes Trismegistusinto Latin as well as translations of 39 of the short Pythagoreansayings known assymbola, many of which are ancient, andHierocles’ commentary on the pseudo-PythagoreanGoldenVerses (Heninger 1974, 63 and 66). TheGolden Verses(see Thom 1995) were, in fact, one of the most popular Greek texts inthe Renaissance and were commonly used in textbooks for learningGreek; other pseudo-Pythagorean texts, such as the treatises ascribedto Timaeus of Locri and Ocellus, were translated early and regarded asgenuine texts on which Plato drew (Heninger 1974, 49, 55–56).Indeed, Ficino regarded the Pythagorean pseudepigrapha as a whole asgenuine and thought that Plato relied on these texts as well as directinfluence from Pythagorean teachers such as Philolaus in composingTimaeus, Phaedo, Gorgias, Philebus, Sophist andParmenides. He followed Iamblichus in regarding theRepublic, and in particular the divided line passage, ascomposed under the influence of pseudepigrapha by Brontinus andArchytas (Robichaud 2018, 149–186). Ficino thought, moreover,that this whole pagan tradition could be reconciled with Christian andJewish religion and accepted the view that Pythagoras was born of aJewish father (Heninger 1974, 201). For Ficino and the Renaissance asa whole, Pythagoras was the most important of the Presocraticphilosophers but he never overshadowed Plato, who was the highestauthority, in part because there was no extensive body of texts byPythagoras himself to compete with the Platonic dialogues (Allen 2014,453).

Ficino translated Iamblichus’ four works on Pythagoreanism forhis own use and Iamblichus’On the Pythagorean Life hadparticular influence on him. Ficino felt that in his time there was aneed for a divinely inspired guide on earth and fashioned himself assuch a prophet under the influence of Iamblichus’ presentationof Pythagoras as a divine guide sent by the gods to save mankind(Celenza 1999, 667–674). The Pythagorean musical practice thathe found in Iamblichus’On the Pythagorean Life, withits emphasis on the impact of music on the soul, shaped his own musicmaking and his presentation of himself as a Pythagorean and Orphicholy man (Allen 2014, 436–440). Ficino and other Renaissancethinkers grappled with the challenge that the Pythagorean notion ofmetempsychosis presented to Christiantiy and how it might bereconciled with Christian views (Allen 2014, 440–446). Ficinowas eager to absolve Plato from such a heresy. He does this in part bytreating metempsychosis metaphorically as referring to thesoul’s ability to remake itself, but he also emphasized thatmetempsychosis was not present in Plato’s latest work,Laws, and made the Pythagoreans scapegoats by suggesting thatother passages in Plato refer not to Plato’s own doctrines butthe Pythagoreans (Celenza 1999, 681–691). Ficino saw his ownarithmology as Pythgorean and study of Neopythagorean mathematicaltreatises by Nicomachus and Theon led Ficino to conclude thatPlato’s nuptial number in Book 8 of theRepublic was 12(Allen 2014, 446–450. For a full account of Pythagorean numbermysticism in the Rennaissance see Brandt 2022). Ficino also mistakenlyand paradoxically followed the Neopythagoreans in thinking that thePythagoreans occupied the crucial position in the history ofphilosophy of the first philosophers to distinguish between thecorporeal and incorporeal and to assert the superiority of the latter,an achievement that is more reasonably assigned to Ficino’s heroPlato (Celenza 1999, 699–706).

The Pythagoreansymbola were important to Ficino and theRenaissance. They had already been interpreted as moral maxims by theearly church fathers (e.g., Clement, Origen and Ambrose). Ambrose, forexample, interpreted the Pythagorean “do not take the publicpath” to mean that priests should live lives of exceptionalpurity (Ep. 81). Jerome discussed 13 symbola in hisEpistle Against Rufinus (see 5.1 above) and this list becamethe basis for medieval discussions of thesymbola in textssuch as theSpeculum historiale of Vincent of Beauvais andtheLives and Habits of the Philosophers of Walter Burley(Celenza 2001, 11–12). Ficino particularly encountered them inIamblichus’On the Pythagorean Life andProtrepticus. For Ficino, their brevity was appropriate torevealing the supreme reality, since he argued that the closer themind approaches to the One the fewer words it needs (Allen 2014,450–451). In addition, he found them relevant to the preparationand purification of the soul (Celenza, 1999, 693). They were widelydiscussed by Ficino’s contemporaries and successors (Celenza2001, 52–81). Some figures wrote treatises devoted to theirinterpretation (Ficino’s mentor Antonio degli Agli, his followerGiovanni Nesi [for an edition of Nesi’s work see Celenza 2001],Filippo Beroaldo the Elder and Lilio Gregorio Giraldi), while othersdiscussed them as part of larger works (Erasmus and Reuchlin). Noteveryone took thesymbola seriously; Angelo Poliziano, thegreat Florentine philologist and professor, presents a satire on themin the fashion of Lucian, joking about Pythagoras’ ability totalk to animals and ridiculing the prohibition on beans (Celenza 2001,33).

Ficino’s friend and younger contemporary, Giovanni Pico dellaMirandola (1463–1494), advanced an even more radical doctrine ofuniversal truth, according to which all philosophies had a share oftruth and could be reconciled in a comprehensive philosophy(Kristeller 1979, 205). HisOration on the Dignity of Manshows the variety of ways in which he was influenced by thePythagorean tradition. He equates the friendship that the Pythagoreanssaw as the goal of philosophy (see, e.g., Iamblichus,VP 229)with the peace that the angels announced to men of good will (1965,11–12); the Pythagoreansymbola forbidding urinatingtowards the sun or cutting the nails during sacrifice are interpretedallegorically as calling on us to relieve ourselves of excessiveappetite for sensual pleasures and to trim the pricks of anger (1965,15); the practice of philosophizing through numbers is assigned toPythagoras along with Philolaus, Plato and the early Platonists (1965,25–26); Pythagoras is said to have modeled his philosophy on theOrphic theology (1965, 33). Finally, on the basis of thepseudo-Pythagorean letter of Lysis to Hipparchus, Pythagoras is saidto have kept silent about his doctrine and left just a few things inwriting to his daughter at his death. In observing such silence,Pythagoras is portrayed as following an earlier practice symbolized bythe sphinx in Egypt and most of all by Moses, who indeed published thelaw to men but supposedly kept the interpretation of that law asecret. Pico equates this secret interpretation of the law with theCabala, an esoteric doctrine in which the words and numbers of Hebrewscripture are interpreted according to a mystical system (1965, 30;see alsoHeptaplus 1965, 68).

Pico’s interest in reconciling the Cabala with Christianity andthe pagan philosophical tradition, including Pythagoreanism, wasfurther developed by the German humanist, Johannes Reuchlin(1445–1522). In the dedicatory letter for hisThreeBooksOn the Art of the Cabala (1517), which wasaddressed to Pope Leo X, Reuchlin says that as Ficino has restoredPlato for Italy so he will “offer to the Germans Pythagorasreborn,” although he cannot “do this without the cabala ofthe Hebrews, because the philosophy of Pythagoras took its beginningfrom the precepts of the cabalists” (tr. Heninger 1974, 245).Thus, in an earlier work (De verbo mirifico) he had equatedthe four consonants in the Hebrew name for God, JHVH, with thePythagoreantetraktys, and gave to each of the letters, whichare equated with numbers as in Greek practice, a mystical meaning. Thefirst H, which also stands for the number five that the Pythagoreansequated with marriage, is thus taken to symbolize the marriage of thetrinity with material nature, which was equated with the dyad by theNeopythagoreans (Riedweg 2005, 130). In the 18th century Johann JakobBrucker (1696–1770) in hisCritical History ofPhilosophy looked back on Pico as spreading a disease thatcorrupted Reuchlin. Under the influence of Richard Bentley’s Dissertation upon the Epistles of Phalaris (1699) Bruckercame to regard Porphyry and Iamblichus not only as wretched historiansbut also as having deliberately constructed their accounts ofPythagoras “in order to fabricate Pythagoras into ananti-Christian thaumaturge to rival Jesus” (Robichaud, 2022,433).

At the level of popular culture, several fortune-telling devices weretied to Pythagoras, the most famous of which went under the name ofthe Wheel of Pythagoras (Heninger 1974, 237). Pythagoras was probablymost widely known, however, through Ovid’s presentation of himat the beginning of Book XV of theMetamorphoses, which wasimmensely popular in the Renaissance (Heninger 1974, 50). Ovidrecounts the story, which had already been recognized as apocryphal byCicero (Tusc. IV 1), that the second Roman king, Numa,studied with Pythagoras. Pythagoras is presented inaccurately by Ovidas a great natural philosopher, who discovered the secrets of theuniverse and who believed in a doctrine of the flux of four elements.On the other hand, Ovid’s emphasis on the prohibition on eatinganimal flesh and on the immortality of the soul have some connectionto the historical Pythagoras. In the Renaissance, Pythagoras was notprimarily known for the “Pythagorean Theorem,” as he istoday. Better known was the doubtful anecdote (Burkert 1960, Riedweg2005, 90–97), going back ultimately to Heraclides of Pontus butknown to the Renaissance mainly through Cicero (Tusc. V3–4), that he was the first to coin the word“philosopher” (Heninger 1974, 29).

In the sixteenth century, Pythagorean influence was particularlyimportant in the development of astronomy. The Polish astronomerCopernicus (1473–1543), in thePreface and Dedication toPope Paul III attached to his epoch making work, On theRevolution of the Heavenly Spheres, reports that, in hisdissatisfaction with the commonly accepted geocentric astronomicalsystem of Ptolemy (2nd century CE), he laboriously reread the works ofall the philosophers to see if any had ever proposed a differentsystem. This labor led him to find inspiration not from Pythagorashimself but rather from later Pythagoreans and in particular fromPhilolaus. Copernicus found in Cicero (Ac. II 39. 123) thatthe Pythagorean Hicetas (4th century BCE — Copernicus mistakenlycalls him Nicetas) had proposed that the earth revolved around itsaxis at the center of the universe and in pseudo-Plutarch (Diels 1958,378) that another Pythagorean, Ecphantus, and Heraclides of Pontus(both 4th century BCE), whom Copernicus regarded as a Pythagorean, hadproposed a similar view. More importantly, he also found inpseudo-Plutarch that the Pythagorean, Philolaus of Croton (5th centuryBCE), “held that the earth moved in a circle … and wasone of the planets” (On the Revolutions of the HeavenlySpheres 1. 5, tr. Wallis).

Copernicus reports to the Pope that he was led by these earlierthinkers “to meditate on the mobility of the earth.”Pythagorean influence on Copernicus was not limited to the notion of amoving earth. In the same preface he explains his hesitation topublish his book in light of the pseudo-Pythagorean letter of Lysis toHipparchus, which recounts the supposed reluctance of the Pythagoreansto divulge their views to the common run of people, who had notdevoted themselves to study (for further Pythagorean influences onCopernicus see Kahn 2001, 159–161). A number of the followers ofCopernicus saw him as primarily reviving the ancient Pythagoreansystem rather than presenting anything new (Heninger 1974, 130 and144, n. 131); Edward Sherburne reflects the common view of the late17th century in referring to the heliocentric system as “thesystem of Philolaus and Copernicus” (Heninger 1974,129–130), although in the Philolaic system it is, in fact, acentral fire and not the sun that is at the center of theuniverse.

The last great Pythagorean was Johannes Kepler (1571–1630— see Kahn 2001, 161–172 for a good brief account ofKepler’s Pythagoreanism). Kepler began by developing theCopernican system in light of the five regular solids (tetrahedron,cube, octahedron, dodecahedron and icosahedron), to which Platoappealed in his construction of matter in theTimaeus (seeespecially 53B-55C). He followed the Renaissance practice illustratedabove of regarding Greek philosophy as closely connected to the wisdomof the Near East, when he asserted that theTimaeus was acommentary on the first chapter ofGenesis (Kahn 2001, 162).In the preface to his early work,Mysterium Cosmographicum(1596), Kepler says that his purpose is to show that God used the fiveregular bodies, “which have been most celebrated from the timeof Pythagoras and Plato,” as his model in constructing theuniverse and that “he accommodated the number of heavenlyspheres, their proportions, and the system of their motions” tothese five regular solids (tr. Heninger 1974, 110–111).

In ascribing geometrical knowledge of the five regular solids toPythagoras, Kepler is following an erroneous Neopythagorean tradition,although the dodecahedron may have served as an early Pythagoreansymbol (see on Hippasus in section 3.4 above and Burkert 1972,70–71, 404, 460). Thus, this aspect of Kepler’s work ismore Platonic than Pythagorean. The five solids were conceived of ascircumscribing and inscribed in the spheres of the orbits of theplanets, so that the five solids corresponded to the six planets knownto Kepler (Saturn, Jupiter, Mars, Earth, Venus, Mercury). There weresix planets, because there were precisely five regular bodies to beused in constructing the universe, corresponding to the five intervalsbetween the planets. This view was overthrown by the later discoveryof Uranus as a seventh planet. Kepler’s cosmology was, however,far from a purelya priori exercise. Whereas hiscontemporary, Robert Fludd, developed a cosmology structured bymusical numbers, which could in no way be confirmed by observation,Kepler strove to make his system consistent with precise observations.Kahn suggests that we here see again the split “between arational and an obscurantist version of Pythagorean thought,”which is similar to the ancient split in the school betweenmathematici andacusmatici (2001, 163).

Close work with observational data collected by Tycho Brahe led Keplerto abandon the universal ancient view that the orbits of the planetswere circular and to recognize their elliptical nature. More clearlyPythagorean is Kepler’s consistent belief that the data showthat the motions of the planets correspond in various ways to theratios governing the musical concords (see Dreyer 1953,405–410), so that there is a heavenly music, a doctrine attestedfor Philolaus and Archytas, which probably goes back to Pythagoras aswell. For Kepler, however, the music produced by the heavenly motionswas “perceived by reason, and not expressed in sound”(Harmonice Mundi V 7). In his attempt to make the numbers ofthe heavenly music work, he joked that he would appeal to the shade ofPythagoras for aid, “unless the soul of Pythagoras has migratedinto mine” (Koestler 1959, 277).

Kepler has been described “as the last exponent of a form ofmathematical cosmology that can be traced back to the shadowy figureof Pythagoras” (Field 1988, 170). It is true that Kepler’swork led the way to Newton’s mechanics, which cannot bedescribed in terms of ancient geometry and number theory but relies onthe calculus and which relies on a theory of physical forces that isalien to ancient thought. On the other hand, many modern scientistsaccept the basic tenet that knowledge of the natural world is to beexpressed in mathematical formulae, which is rightly regarded as acentral Pythagorean thesis, since it was first rigorously formulatedby the Pythagoreans Philolaus ( Fr. 4) and Archytas and may, in arudimentary form, go back to Pythagoras himself.

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