Pythagoras, one of the most famous and controversial ancient Greekphilosophers, lived from ca. 570 to ca. 490 BCE. He spent his earlyyears on the island of Samos, off the coast of modern Turkey. At theage of forty, however, he emigrated to the city of Croton in southernItaly and most of his philosophical activity occurred there.Pythagoras wrote nothing, nor were there any detailed accounts of histhought written by contemporaries. By the first centuries BCE,moreover, it became fashionable to present Pythagoras in a largelyunhistorical fashion as a semi-divine figure, who originated all thatwas true in the Greek philosophical tradition, including many ofPlato’s and Aristotle’s mature ideas. A number oftreatises were forged in the name of Pythagoras and other Pythagoreansin order to support this view. See the entry onPythagoreanism.

The Pythagorean question, then, is how to get behind this falseglorification of Pythagoras in order to determine what the historicalPythagoras actually thought and did. In order to obtain an accurateappreciation of Pythagoras’ achievement, it is important to relyon the earliest evidence before the distortions of the later traditionarose. The popular modern image of Pythagoras is that of a mastermathematician and scientist. The early evidence shows, however, that,while Pythagoras was famous in his own day and even 150 years later inthe time of Plato and Aristotle, it was not mathematics or scienceupon which his fame rested. Pythagoras was famous (1) as an expert onthe fate of the soul after death, who thought that the soul wasimmortal and went through a series of reincarnations; (2) as an experton religious ritual; (3) as a wonder-worker who had a thigh of goldand who could be two places at the same time; (4) as the founder of astrict way of life that emphasized dietary restrictions, religiousritual and rigorous self discipline.

It remains controversial whether he also engaged in the rationalcosmology that is typical of the Presocratic philosopher/scientistsand whether he was in any sense a mathematician. The early evidencesuggests, however, that Pythagoras presented a cosmos that wasstructured according to moral principles and significant numericalrelationships and may have been akin to conceptions of the cosmosfound in Platonic myths, such as those at the end of thePhaedo andRepublic. In such a cosmos, the planetswere seen as instruments of divine vengeance (“the hounds ofPersephone”), the sun and moon are the isles of the blessedwhere we may go, if we live a good life, while thunder functioned tofrighten the souls being punished in Tartarus. The heavenly bodiesalso appear to have moved in accordance with the mathematical ratiosthat govern the concordant musical intervals in order to produce amusic of the heavens, which in the later tradition developed into“the harmony of the spheres.” It is doubtful thatPythagoras himself thought in terms of spheres, and the mathematics ofthe movements of the heavens was not worked out in detail. There isevidence that he valued relationships between numbers such as thoseembodied in the so-called Pythagorean theorem, though it is not likelythat he proved the theorem. In recent scholarship this consensus viewhas received strong challenges, which will be discussed below.

Pythagoras’ cosmos was developed in a more scientific andmathematical direction by his successors in the Pythagorean tradition,Philolaus and Archytas. Pythagoras succeeded in promulgating a newmore optimistic view of the fate of the soul after death and infounding a way of life that was attractive for its rigor anddiscipline and that drew to him numerous devoted followers.

• 1. The Pythagorean Question
• 2. Sources
  • 2.1 Chronological Chart of Sources for Pythagoras
  • 2.2 Post-Aristotelian Sources for Pythagoras
  • 2.3 Plato and Aristotle as Sources for Pythagoras
• 3. Life and Works
• 4. The Philosophy of Pythagoras
  • 4.1 The Fate of the Soul—Metempsychosis
  • 4.2 Pythagoras as a Wonder-worker
  • 4.3 The Pythagorean Way of Life
• 5. Was Pythagoras a Mathematician or Cosmologist?
• Bibliography
  • Primary Sources and Commentaries
  • Secondary Sources
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Pythagorean Question

What were the beliefs and practices of the historical Pythagoras? Thisapparently simple question has become the daunting Pythagoreanquestion for several reasons. First, Pythagoras himself wrote nothing,so our knowledge of Pythagoras’ views is entirely derived fromthe reports of others. Second, there was no extensive or authoritativecontemporary account of Pythagoras. No one did for Pythagoras whatPlato and Xenophon did for Socrates. Third, only fragments of thefirst detailed accounts of Pythagoras, written about 150 years afterhis death, have survived. Fourth, it is clear that these accountsdisagreed with one another on significant points. These four pointswould already make the problem of determining Pythagoras’philosophical beliefs more difficult than determining those of almostany other ancient philosopher, but a fifth factor complicates matterseven more. By the third century CE, when the first detailed accountsof Pythagoras that survive intact were written, Pythagoras had come tobe regarded, in some circles, as the master philosopher, from whom allthat was true in the Greek philosophical tradition derived. By the endof the first century BCE, a large collection of books had been forgedin the name of Pythagoras and other early Pythagoreans, whichpurported to be the original Pythagorean texts from which Plato andAristotle derived their most important ideas. A treatise forged in thename of Timaeus of Locri was the supposed model for Plato’sTimaeus, just as forged treatises assigned to Archytas werethe supposed model for Aristotle’sCategories.Pythagoras himself was widely presented as having anticipatedPlato’s later metaphysics, in which the one and the indefinitedyad are first principles. Thus, not only is the earliest evidence forPythagoras’ views meager and contradictory, it is overshadowedby the hagiographical presentation of Pythagoras, which becamedominant in late antiquity. Given these circumstances, the onlyreliable approach to answering the Pythagorean question is to startwith the earliest evidence, which is independent of the later attemptsto glorify Pythagoras, and to use the picture of Pythagoras whichemerges from this early evidence as the standard against which toevaluate what can be accepted and what must be rejected in the latertradition. Following such an approach, Walter Burkert, in hisepoch-making book (1972a), revolutionized our understanding of thePythagorean question, and all modern scholarship on Pythagoras,including this article, stands on his shoulders. For a detaileddiscussion of the source problems that generate the PythagoreanQuestion see 2. Sources, below.

2. Sources

2.1 Chronological Chart of Sources for Pythagoras

2.2 Post-Aristotelian Sources for Pythagoras

The problems regarding the sources for the life and philosophy ofPythagoras are quite complicated, but it is impossible to understandthe Pythagorean Question without an accurate appreciation of at leastthe general nature of these problems. It is best to start with theextensive but problematic later evidence and work back to the earlierreliable evidence. The most detailed, extended and hence mostinfluential accounts of Pythagoras’ life and thought date to thethird century CE, some 800 years after he died. Diogenes Laertius (ca.200–250 CE) and Porphyry (ca. 234–305 CE) each wrote aLife of Pythagoras, while Iamblichus (ca. 245–325 CE)wroteOn the Pythagorean Life, which includes some biographybut focuses more on the way of life established by Pythagoras for hisfollowers. All of these works were written at a time whenPythagoras’ achievements had become considerably exaggerated.Diogenes may have some claim to objectivity, but both Iamblichus andPorphyry have strong agendas that have little to do with historicalaccuracy. Iamblichus presents Pythagoras as a soul sent from the godsto enlighten mankind (O’Meara 1989, 35–40).Iamblichus’ work was just the first in a ten volume work, whichin effect Pythagoreanized Neoplatonism, but the Pythagoreanisminvolved was Iamblichus’ own conception of Pythagoras asparticularly concerned with mathematics rather than an account ofPythagoreanism based on the earliest evidence. Porphyry alsoemphasizes Pythagoras’ divine aspects and may be setting him upas a rival to Jesus (Iamblichus 1991, 14). These three third-centuryaccounts of Pythagoras were in turn based on earlier sources, whichare now lost. Some of these earlier sources were heavily contaminatedby the Neopythagorean view of Pythagoras as the source of all truephilosophy, whose ideas Plato, Aristotle and all later Greekphilosophers plagiarized. Iamblichus cites biographies of Pythagorasby Nicomachus of Gerasa and a certain Apollonius (VP 251 and254) and appears to have used them extensively even where they are notcited (Burkert 1972a, 98 ff.). Nicomachus (ca. 50–ca.150 CE)assigns Pythagoras a metaphysics that is patently Platonic andAristotelian and that employs distinctive Platonic and Aristotelianterminology (Introduction to Arithmetic I.1). If theApollonius cited by Iamblichus is Apollonius of Tyana (1st CE), hisaccount will be influenced by his veneration of Pythagoras as themodel for his ascetic life, but some scholars argue that Iamblichus isusing an otherwise unknown Apollonius (Flinterman 2014, 357). Porphyry(VP 48–53) explicitly cites Moderatus of Gades as oneof his sources. Moderatus was an “aggressive”Neopythagorean of the first century CE, who reports that Plato,Aristotle, and their pupils Speusippus, Aristoxenus and Xenocratestook for their own everything that was fruitful in Pythagoreanism,leaving only what was superficial and trivial to be ascribed to theschool (Dillon 1977, 346). Diogenes Laertius, who appears to have lesspersonal allegiance to the Pythagorean legend, bases his primaryaccount of Pythagoras’ philosophy (VIII. 24–33) on thePythagorean Memoirs excerpted by Alexander Polyhistor, whichare a forgery dating sometime around 200 BCE and which assign not justPlatonic but also Stoic ideas to Pythagoras (Burkert 1972a, 53; Kahn2001, 79–83).

In thePythagorean Memoirs, Pythagoras is said to haveadopted the Monad and the Indefinite Dyad as incorporeal principles,from which arise first the numbers, then plane and solid figures andfinally the bodies of the sensible world (Diogenes Laertius VIII. 25).This is the philosophical system that is most commonly ascribed toPythagoras in the post-Aristotelian tradition, and it is found inSextus Empiricus’ (2nd century CE) detailed accounts ofPythagoreanism (e.g.,M. X. 261) and most significantly inthe influential handbook of the differing opinions of the Greekphilosophers, which was compiled by Aetius in the first century CE andis based on theTenets of the Natural Philosophers ofAristotle’s pupil Theophrastus (e.g., H. Diels,DoxographiGraeci I. 3.8). The testimony of Aristotle makes completelyclear, however, that this was the philosophical system of Plato in hislater years and not that of Pythagoras or even the later Pythagoreans.Aristotle is explicit that, although Plato’s system hassimilarities to the earlier Pythagorean philosophy of limiters andunlimiteds, the indefinite dyad is unique to Plato(Metaphysics 987b26 ff.) and the Pythagoreans recognized onlythe sensible world and hence did not derive it from immaterialprinciples. In thePhilebus, Plato himself tells a story thatis very much in agreement with Aristotle’s report. Whileacknowledging a debt to the philosophy of limiters and unlimiteds,which is found in Aristotle’s accounts of Pythagoreanism and inthe fragments of the fifth-century Pythagorean Philolaus, Plato makesclear that this is a considerably earlier philosophy, which he iscompletely reworking for his own purposes (16c ff.; see Huffman 2001).How are we to explain the later tradition’s divergence from thistestimony of Aristotle and Plato? The most convincing suggestionpoints to evidence that, for reasons which are not entirely clear,Plato’s successors in the Academy, Speusippus, Xenocrates andHeraclides, chose to present Pythagoreanism not just as a precursor oflate Platonic metaphysics but as having anticipated its centraltheses. Thus the tradition which falsely ascribes Plato’s latemetaphysics to Pythagoras begins not with the Neopythagoreans in thefirst centuries BCE and CE but already in the fourth century BCE amongPlato’s own pupils (Burkert 1972a, 53–83; Dillon 2003,61–62 and 153–154). This view of Pythagoreanism finds itsway into the doxography of Aetius either because Theophrastus followedthe early Academy rather than his teacher Aristotle (Burkert 1972a,66) or because the Theophrastan doxography on Pythagoras was rewrittenin the first century BCE under the influence of Neopythagoreanism(Diels 1958, 181; Zhmud 2012a, 455). Aristotle’s carefuldistinctions between Plato and fifth-century Pythagoreanism, whichmake excellent sense in terms of the general development of Greekphilosophy, are largely ignored in the later tradition in favor of themore sensational ascription of mature Platonism to Pythagoras. Theevidence for the early Academy is, however, very limited and somereject the thesis that its members assigned late Platonic metaphysicsto Pythagoras (Zhmud 2012, 415–432). The key text is found inProclus’Commentary on the Parmenides (pp.38.32–40.7 Klibansky). Proclus quotes a passage in whichSpeusippus assigns to the ancients, who in this context are thePythagoreans, the One and the Indefinite Dyad. Some scholars arguethat this is not a genuine fragment of Speusippus but rather a laterfabrication (see Zhmud 2012a, 424–425 and for a response Dillon2014, 251). If the Academy did not assign the One and the Dyad toPythagoras, however, it becomes less clear how these principles cameto be assigned to him. Theophrastus assigns them to the Pythagoreans(Metaphysics 11a27), but since Aristotle distinguishes thePythagoreans from Plato on this point, Zhmud’s suggestion(2012a, 455) that he is following his teacher and just taking“the next step” does not work. Theophrastus’evidence makes best sense if we accept the traditional view andsuppose that it is on the authority of Plato’s successors in theAcademy that he bases his departure from his teacher’s,Aristotle’s, view.

If we step back for a minute and compare the sources for Pythagoraswith those available for other early Greek philosophers, the extent ofthe difficulties inherent in the Pythagorean Question becomes clear.When trying to reconstruct the philosophy of Heraclitus, for example,modern scholars rely above all on the actual quotations fromHeraclitus’ book preserved in later authors. Since Pythagoraswrote no books, this most fundamental of all sources is denied us. Indealing with Heraclitus, the modern scholar turns with reluctance nextto the doxographical tradition, the tradition represented by Aetius inthe first century CE, which preserves in handbook form a systematicaccount of the beliefs of the Greek philosophers on a series of topicshaving to do with the physical world and its first principles.Aetius’ work was first reconstructed by Hermann Diels (1958) andmore recently by Mansfeld and Runia (2020) on the basis of two laterworks, which were derived from it, theSelections of Stobaeus(5th century CE) and theOpinions of Philosophers bypseudo-Plutarch (2nd century CE). Scholars’ faith in thisevidence is largely based on the assumption that most of it goes backto Aristotle’s school and in particular to Theophrastus’Tenets of the Natural Philosophers. Here again the case ofPythagoras is exceptional. Pythagoras is represented in this traditionbut, as we have seen, Theophrastus in this case either adopted theview that, against all historical plausibility, assigns Plato’slater metaphysics to Pythagoras or Theophrastus’ doxography onthe Pythagoreans was rewritten in the first century BCE. Thus, thesecond standard source for evidence for early Greek philosophy is, inthe case of Pythagoras, corrupted. Whatever views Pythagoras mighthave had are replaced by late Platonic metaphysics in thedoxographical tradition.

A third source of evidence for early Greek philosophy is regarded withgreat skepticism by most scholars and, in the case of most early Greekphilosophers, used only with great caution. This is the biographicaltradition represented by theLives of the Philosopherswritten by Diogenes Laertius. In this case we at first sight appear tobe in luck, at least with regard to the amount of evidence forPythagoras, since, as we have seen, two major accounts of the life ofPythagoras and the Pythagorean way of life survive in addition toDiogenes’ life. Unfortunately, these two additional lives arewritten by authors (Iamblichus and Porphyry) whose goal is explicitlynon-historical, and all three of the lives rely heavily on authors inthe Neopythagorean tradition, whose goal was to show that all laterGreek philosophy, insofar as it was true, had been stolen fromPythagoras. There are, however, some sections in these three livesthat derive from sources that go back beyond the distorting influenceof Neopythagoreanism, to sources in the fourth-century BCE, sourceswhich are also independent of the early Academy’s attempt toassign Platonic metaphysics to the Pythagoreans. The most important ofthese sources are the fragments of Aristotle’s lost treatises onthe Pythagoreans and the fragments of works on Pythagoreanism or ofworks which dealt in passing with Pythagoreanism written byAristotle’s pupils Dicaearchus and Aristoxenus, in the secondhalf of the fourth century BCE. The historian Timaeus of Tauromenium(ca. 350–260 BCE), who wrote a history of Sicily, which includedmaterial on southern Italy where Pythagoras was active, is alsoimportant. In some cases, the fragments of these early works areclearly identified in the later lives, but in other cases we maysuspect that they are the source of a given passage without being ableto be certain. Large problems remain even in the case of thesesources. They were all written 150–250 years after the death ofPythagoras; given the lack of written evidence for Pythagoras, theyare based largely on oral traditions. Aristoxenus, who grew up in thesouthern Italian town of Tarentum, where the Pythagorean Archytas wasthe dominant political figure, and who was himself a Pythagoreanbefore joining Aristotle’s school, undoubtedly had a rich set oforal traditions upon which to draw. It is clear, nonetheless, that 150years after his death conflicting traditions regardingPythagoras’ beliefs had arisen on even the most central issues.Thus, Aristoxenus is emphatic that Pythagoras was not a strictvegetarian and ate a number of types of meat (Diogenes Laertius VIII.20), whereas Aristoxenus’ contemporary, the mathematicianEudoxus, portrays him not only as avoiding all meat but as evenrefusing to associate with butchers (Porphyry,VP 7). Evenamong fourth-century authors that had at least some pretensions tohistorical accuracy and who had access to the best informationavailable, there are widely divergent presentations, simply becausesuch contradictions were endemic to the evidence available in thefourth century. What we can hope to obtain from the evidence presentedby Aristotle, Aristoxenus, Dicaearchus, and Timaeus is thus not apicture of Pythagoras that is consistent in all respects but rather apicture that at least defines the main areas of his achievement. Thispicture can then be tested by the most fundamental evidence of all,the testimony of authors that precede even Aristotle, testimony insome cases that derives from Pythagoras’ own contemporaries.This testimony is extremely limited, about twenty brief references,but this dearth of evidence is not unique to Pythagoras. Thepre-Aristotelian testimony for Pythagoras is more extensive than formost other early Greek philosophers and is thus testimony to hisfame.

2.3 Plato and Aristotle as Sources for Pythagoras

In reconstructing the thought of early Greek philosophers, scholarsoften turn to Aristotle’s and Plato’s accounts of theirpredecessors, although Plato’s accounts are embedded in theliterary structure of his dialogues and thus do not pretend tohistorical accuracy, while Aristotle’s apparently morehistorical presentation masks a considerable amount ofreinterpretation of his predecessors’ views in terms of his ownthought. In the case of Pythagoras, what is striking is the essentialagreement of Plato and Aristotle in their presentation of hissignificance. Aristotle frequently discusses the philosophy ofPythagoreans, whom he dates to the middle and second half of the fifthcentury and who posited limiters and unlimiteds as first principles.He sometimes refers to these Pythagoreans as the “so-calledPythagoreans,” suggesting that he had some reservations aboutthe application of the label “Pythagorean” to them.Aristotle strikingly may never refer to Pythagoras himself in hisextant writings (of the three possible mentionsMetaph.986a29 is probably an interpolation;Rh. 1398b14 is aquotation from Alcidamas;MM 1182a11 may not be by Aristotleand, if it is, may well be a case where “Pythagoreans”have been turned into “Pythagoras” in the transmission).In the fragments of his now lost two-book treatise on thePythagoreans, Aristotle does discuss Pythagoras himself, but thereferences are all to Pythagoras as a founder of a way of life, whoforbade the eating of beans (Fr. 195), and to Pythagoras as awonder-worker, who had a golden thigh and bit a snake to death (Fr.191). Zhmud (2012a, 259–260) argues that in one place Aristotlealso describes Pythagoras as a mathematician (Fr. 191) and in anotheras studying nature (Protrepticus Fr. 20) but in neither caseare the words likely to belong to Aristotle (see paragraph 9 ofsection 5 below and Huffman 2014b, 281, n.7). If Aristotle only foundevidence for Pythagoras as a wonder-worker and founder of a way oflife, it becomes clear why he never mentions Pythagoras in his accountof his philosophical predecessors and why he uses the expression“so-called Pythagoreans” to refer to the Pythagoreanism ofthe fifth-century. For Aristotle Pythagoras did not belong to thesuccession of thinkers starting with Thales, who were attempting toexplain the basic principles of the natural world, and hence he couldnot see what sense it made to call a fifth-century thinker likePhilolaus, who joined that succession by positing limiters andunlimiteds as first principles, a Pythagorean. Álvarez Salas,accepting the three apparent references to Pythagoras inAristotle’s extant writings which are mentioned above asgenuine, suggests that scholars are mistaken to regard Aristotle asslighting Pythagoras, because Aristotle also refers very infrequentlyto other early figures such as Thales. Aristotle’s threereferences to Pythagoras are just his “average treatment of thesages of old” (Álvarez Salas 2021, 256). However, even ifwe accept all three references to Pythagoras as genuine, they are noton a par with what he has to say about the Milesians, since they arenot part of his account of the development of Presocratic philosophyin Book One of theMetaphysics and elsewhere. The Milesiansare mentioned prominently, while Pythagoras is conspicuously missing.Aristotle does mention the Pythagoreans later this account but theyare presented as active around the time of the atomists, so that thereference is clearly to Philolaus and Pythagoreans of his time.Álvarez Salas also argues that Aristotle’s use of theplural Pythagoreans is meant to include Pythagoras, but while this maybe plausible in the discussion of metempsychosis inDe Anima(407b22), it is problematic when Aristotle explicitly dates thePythagoreans to the middle of the fifth-century as he does in theMetaphysics (985b24). Aristotle simply does not includePythagoras in his account of the Presocratic philosopher/scientists.Plato is often thought to be heavily indebted to the Pythagoreans, buthe is almost as parsimonious in his references to Pythagoras asAristotle and mentions him only once in his writings. Plato’sone reference to Pythagoras (R. 600a) treats him as thefounder of a way of life, just as Aristotle does, and, when Platotraces the history of philosophy prior to his time in theSophist, (242c-e), there is no allusion to Pythagoras. In thePhilebus, Plato does describe the philosophy of limiters andunlimiteds, which Aristotle assigns to the so-called Pythagoreans ofthe fifth century and which is found in the fragments of Philolaus,but like Aristotle he does not ascribe this philosophy to Pythagorashimself. Scholars, both ancient and modern, under the influence of thelater glorification of Pythagoras, have supposed that the Prometheus,whom Plato describes as hurling the system down to men, was Pythagoras(e.g., Kahn 2002: 13–14), but careful reading of the passageshows that Prometheus is just Prometheus and that Plato, likeAristotle, assigns the philosophical system to a group of men (Huffman2001). The fragments of Philolaus show that he was the primary figureof this group. When Plato refers to Philolaus in thePhaedo(61d-e), he does not identify him as a Pythagorean, so that once againPlato agrees with Aristotle in distancing the “so-calledPythagoreans” of the fifth century from Pythagoras himself. Forboth Plato and Aristotle, then, Pythagoras is not a part of thecosmological and metaphysical tradition of Presocratic philosophy noris he closely connected to the metaphysical system presented byfifth-century Pythagoreans like Philolaus; he is instead the founderof a way of life.

3. Life and Works

References to Pythagoras by Xenophanes (ca. 570–475 BCE) andHeraclitus (fl. ca. 500 BCE) show that he was a famous figure in thelate sixth and early fifth centuries. For the details of his life wehave to rely on fourth-century sources such as Aristoxenus,Dicaearchus and Timaeus of Tauromenium. There is a great deal ofcontroversy about his origin and early life, but there is agreementthat he grew up on the island of Samos, near the birthplace of Greekphilosophy, Miletus, on the coast of Asia Minor. There are a number ofreports that he traveled widely in the Near East while living onSamos, e.g., to Babylonia, Phoenicia and Egypt. To some extent reportsof these trips are an attempt to claim the ancient wisdom of the eastfor Pythagoras and some scholars totally reject them (Zhmud 2012,83–91), but relatively early sources such as Herodotus (II. 81)and Isocrates (Busiris 28) associate Pythagoras with Egyptand one of the oral sayings that may go back to him mentions theEgyptian goddess Isis (Huffman 2019), which at least shows interest inEgypt. Aristoxenus says that he left Samos at the age of forty, whenthe tyranny of Polycrates, who came to power ca. 535 BCE, becameunbearable (Porphyry,VP 9). This chronology would suggestthat he was born ca. 570 BCE. He then emigrated to the Greek city ofCroton in southern Italy ca. 530 BCE; it is in Croton that he firstseems to have attracted a large number of followers to his way oflife. There are a variety of stories about his death, but the mostreliable evidence (Aristoxenus and Dicaearchus) suggests that violencedirected against Pythagoras and his followers in Croton ca. 510 BCE,perhaps because of the exclusive nature of the Pythagorean way oflife, led him to flee to another Greek city in southern Italy,Metapontum, where he died around 490 BCE (Porphyry,VP54–7; Iamblichus,VP 248 ff.; On the chronology, seeMinar 1942, 133–5). There is little else about his life of whichwe can be confident.

The evidence suggests that Pythagoras did not write any books. Nosource contemporaneous with Pythagoras or in the first two hundredyears after his death, including Plato, Aristotle and their immediatesuccessors in the Academy and Lyceum, quotes from a work by Pythagorasor gives any indication that any works written by him were inexistence. Several later sources explicitly assert that Pythagoraswrote nothing (e.g., Lucian [Slip of the Tongue, 5],Josephus, Plutarch and Posidonius in DK 14A18; see Burkert 1972a,218–9). Diogenes Laertius tried to dispute this tradition byquoting Heraclitus’ assertion that “Pythagoras, the son ofMnesarchus, practiced inquiry most of all men and, by selecting thesethings which have been written up, made for himself a wisdom, apolymathy, an evil conspiracy” (Fr. 129). This fragment showsonly that Pythagoras read the writings of others, however, and saysnothing about him writing something of his own. The wisdom and evilconspiracy that Pythagoras constructs from these writings need nothave been in writing, and Heraclitus’ description of it as an“evil conspiracy” rather suggests that it was not (for thetranslation and interpretation of Fr. 129, see Huffman 2008b). In thelater tradition several books came to be ascribed to Pythagoras, butsuch evidence as exists for these books indicates that they wereforged in Pythagoras’ name and belong with the large number ofpseudo-Pythagorean treatises forged in the name of early Pythagoreanssuch as Philolaus and Archytas. In the third century BCE a group ofthree books were circulating in Pythagoras’ name,OnEducation,On Statesmanship, andOn Nature(Diogenes Laertius, VIII. 6). A letter from Plato to Dion asking himto purchase these three books from Philolaus was forged in order to“authenticate” them (Burkert 1972a, 223–225).Heraclides Lembus in the second century BCE gives a list of six booksascribed to Pythagoras (Diogenes Laertius, VIII. 7; Thesleff 1965,155–186 provides a complete collection of the spurious writingsassigned to Pythagoras). The second of these is aSacredDiscourse, which some have wanted to trace back to Pythagorashimself. The idea that Pythagoras wrote such aSacredDiscourse seems to arise from a misreading of the early evidence.Herodotus says that the Pythagoreans agreed with the Egyptians in notallowing the dead to be buried in wool and then asserts that there isa sacred discourse about this (II. 81). Herodotus’ focus here isthe Egyptians and not the Pythagoreans, who are introduced as a Greekparallel, so that theSacred Discourse to which he refers isEgyptian and not Pythagorean, as similar passages elsewhere in Book IIof Herodotus show (e.g., II. 62; see Burkert 1972a, 219).Various linesof hexameter verse were already circulating in Pythagoras’ namein the third century BCE and were later combined into a compilationknown as theGolden Verses, which marks the culmination ofthe tradition of aSacred Discourse assigned to Pythagoras(Burkert 1972a, 219, Thesleff 1965, 158–163; and most recentlyThom 1995, although his dating of the compilation before 300 BCE isquestionable). The lack of any viable written text which could bereasonably ascribed to Pythagoras is shown most clearly by thetendency of later authors to quote either Empedocles or Plato, whenthey needed to quote “Pythagoras” (e.g., Sextus Empiricus,M. IX. 126–30; Nicomachus,Introduction toArithmetic I. 2). For an interesting but ultimately unconvincingattempt to argue that the historical Pythagoras did write books, seeRiedweg 2005, 42–43 and the response by Huffman 2008a,205–207.

4. The Philosophy of Pythagoras

One of the manifestations of the attempt to glorify Pythagoras in thelater tradition is the report that he, in fact, invented the word‘philosophy’. This story goes back to the early Academy,since it is first found in Heraclides of Pontus (Cicero,Tusc. V 3.8; Diogenes Laertius,Proem). Thehistorical accuracy of the story is called into question by itsappearance not in a historical or biographical text but rather in adialogue that recounted Empedocles’ revival of a woman who hadstopped breathing. Moreover, the story depends on a conception of aphilosopher as having no knowledge but being situated betweenignorance and knowledge and striving for knowledge. Such a conceptionis thoroughly Platonic, however (see, e.g.,Symposium 204A),and Burkert demonstrated that it could not belong to the historicalPythagoras (1960). For a recent attempt to defend at least the partialaccuracy of the story, see Riedweg 2005: 90–97 and the responseby Huffman 2008a, 207–208; see also Zhmud 2012a,428–430.

Even if he did not invent the word, what can we say about thephilosophy of Pythagoras? For the reasons given in 1. The PythagoreanQuestion and 2. Sources above, any responsible account ofPythagoras’ philosophy must be based in the first place on theevidence prior to Aristotle and in the second place on evidence thatour sources explicitly identify as deriving from Aristotle’sbooks on the Pythagoreans as well as from the books of his pupils suchas Aristoxenus and Dicaearchus. There is general agreement as to whatthe pre-Aristotelian evidence is, although there are differences ininterpretation of it. There is less agreement as to what should beincluded in Aristotle’s, Dicaearchus’ andAristoxenus’ evidence. What one includes as evidence from theseauthors will have a significant effect on one’s picture ofPythagoras. One particularly pressing question is whether bothchapters 18 and 19 of Porphyry’sLife of Pythagorasshould be regarded as deriving from Dicaearchus, as the most recenteditor proposes (Mirhady Fr. 40), or whether only chapter 18 should beincluded, as in the earlier edition of Wehrli (Fr. 33). It is crucialto decide this question before developing a picture of the philosophyof Pythagoras since chapter 19, if it is by Dicaearchus, is ourearliest summary of Pythagorean philosophy. Porphyry is very reliableabout quoting his sources. He explicitly cites Dicaearchus at thebeginning of Chapter 18 and names Nicomachus as his source at thebeginning of chapter 20. The material in chapter 19 follows seamlesslyon chapter 18: the description of the speeches that Pythagoras gaveupon his arrival in Croton in chapter 18 is followed in chapter 19 byan account of the disciples that he gained as the result of thosespeeches and a discussion of what he taught these disciples. Thus, theonus is on anyone who would claim that Porphyry changes sources beforethe explicit change at the beginning of chapter 20. Chapter 19provides a very restrained account of what can be reliably known aboutPythagoras’ teachings and that very restraint is one of thestrongest supporting arguments for its being based on Dicaearchus,since Porphyry or anyone else in the luxuriant later tradition wouldbe expected to give a much more expanisve presentation of Pythagorasin accordance with the Neopythagorean view of him (Burkert 1972a,122–123). Wehrli gives no reason for not including chapter 19and the great majority of scholars accept it as being based onDicaearchus (see the references in Burkert 1972a, 122, n.7). Zhmud(2012a, 157) following Philip (1966, 139) argues that the passagecannot derive from Dicaearchus, since it presents immortality of thesoul with approval, whereas Dicaearchus did not accept itsimmortality. However, the passage merely reports that Pythagorasintroduced the notion of the immortality of the soul withoutexpressing approval or disapproval. Zhmud lists other features of thechapter that he regards as unparalleled in fourth-century sources(2012a, 157) but, since the evidence is so fragmentary, such argumentsfrom silence can carry little weight. Nothing in the chapter isdemonstrably late or inconsistent with Dicaearchus’ authorshipso we must follow what is suggested by the context in Porphyry andregard it as derived from Dicaearchus.

In the face of the Pythagorean question and the problems that ariseeven regarding the early sources, it is reasonable to wonder if we cansay anything about Pythagoras. A minimalist might argue that the earlyevidence only allows us to conclude that Pythagoras was a historicalfigure who achieved fame for his wisdom but that it is impossible todetermine in what that wisdom consisted. We might say that he wasinterested in the fate of the soul and taught a way of life, but wecan say nothing precise about the nature of that life or what hetaught about the soul (Lloyd 2014). There is some reason to believe,however, that something more than this can be said.

4.1 The Fate of the Soul—Metempsychosis

The earliest evidence makes clear that above all Pythagoras was knownas an expert on the fate of our soul after death. Herodotus tells thestory of the Thracian Zalmoxis, who taught his countrymen that theywould never die but instead go to a place where they would eternallypossess all good things (IV. 95). Among the Greeks the tradition arosethat this Zalmoxis was the slave of Pythagoras. Herodotus himselfthinks that Zalmoxis lived long before Pythagoras, but theGreeks’ willingness to portray Zalmoxis as Pythagoras’slave shows that they thought of Pythagoras as the expert from whomZalmoxis derived his teaching. Ion of Chios (5^th c. BCE)says of Phercydes of Syros that “although dead he has a pleasantlife for his soul, if Pythagoras is truly wise, who knew and learnedwisdom beyond all men.” Here Pythagoras is again the expert onthe life of the soul after death. A famous fragment of Xenophanes,Pythagoras’ contemporary, provides some more specificinformation on what happens to the soul after death. He reports that“once when he [Pythagoras] was present at the beating of apuppy, he pitied it and said ‘stop, don’t keep hittinghim, since it is the soul of a man who is dear to me, which Irecognized, when I heard it yelping’” (Fr. 7). AlthoughXenophanes clearly finds the idea ridiculous, the fragment shows thatPythagoras believed in metempsychosis or reincarnation, according towhich human souls were reborn into other animals after death. Thisearly evidence is emphatically confirmed by Dicaearchus in the fourthcentury, who first comments on the difficulty of determining whatPythagoras taught and then asserts that his most recognized doctrineswere “that the soul is immortal and that it transmigrates intoother kinds of animals” (Porphyry,VP 19).Unfortunately we can say little more about the details ofPythagoras’ conception of metempsychosis. According toHerodotus, the Egyptians believed that the soul was reborn as everysort of animal before returning to human form after 3,000 years.Without naming names, he reports that some Greeks both earlier andlater adopted this doctrine; this seems very likely to be a referenceto Pythagoras (earlier) and perhaps Empedocles (later). Many doubtthat Herodotus is right to assign metempsychosis to the Egyptians,since none of the other evidence we have for Egyptian beliefs supportshis claim, but it is nonetheless clear that we cannot assume thatPythagoras accepted the details of the view Herodotus ascribes tothem. Similarly both Empedocles (see Inwood 2001, 55–68) andPlato (e.g.,Republic X andPhaedrus) provide a moredetailed account of transmigration of souls, but neither of themascribes these details to Pythagoras nor should we. Did he think thatwe ever escape the cycle of reincarnations? We simply do not know. Thefragment of Ion quoted above may suggest that the soul could have apleasant existence after death between reincarnations or even escapethe cycle of reincarnation altogether, but the evidence is too weak tobe confident in such a conclusion. In the fourth century severalauthors report that Pythagoras remembered his previous humanincarnations, but the accounts do not agree on the details.Dicaearchus (Aulus Gellius IV. 11.14) and Heraclides (DiogenesLaertius VIII. 4) agree that he was the Trojan hero Euphorbus andHermes’ son Aethalides in previous lives and Heraclides reportsthat he was also reborn as Pyrrhus, a fisherman from the island ofDelos. Dicaearchus continues the tradition of savage satire begun byXenophanes, when he suggests that Pythagoras was the beautifulprostitute, Alco, in another incarnation (Huffman 2014b,281–285). It has been suggested that one of the most importantfeatures of this wide variety of reincarnations may have been theexperience and hence wisdom that Pythagoras gained from them(Pellò 2018).

It is not clear how Pythagoras conceived of the nature of thetransmigrating soul but a few tentative conjectures can be made(Huffman 2009). Transmigration does not require that the soul beimmortal; it could go through several incarnations before perishing.Dicaearchus explicitly says that Pythagoras regarded the soul asimmortal, however, and this agrees with Herodotus’ descriptionof Zalmoxis’ view. Horky (2021) rejects Dicaearchus’testimony on the grounds that he is assigning Platonic views back toPythagoras. This is a common practice among the Neopythaogreans andappears frequently in the pseudo-Pythagorean writings of the firstcentury BCE and later. However, there is no evidence that Aristotle orhis school followed this practice and, as Burkert has shown (1972a,15, 28, and 79), Aristotle was in fact careful to distinguish what isPythagorean from what is Platonic. Most importantly the goal of such apractice is to glorify Pythagoras and the Pythagoreans by assigningthem developed Platonic (and Aristotelian) doctrine. If this were thegoal of Dicaerarchus he would hardly have assigned just a generaldoctrine of the immortality of the soul to Pythagoras without alsoassigning to him the tripartite Platonic soul as pseudo-Pythagoreantexts do (Huffman 1993, 310). So Dicaearchus’ report thatPythagoras believed in the immortality of the soul should be taken atface value. It is likely that Pythagoras used the Greek wordpsychê to refer to the transmigrating soul, since thisis the word used by all sources reporting his views, unlikeEmpedocles, who useddaimon. His successor, Philoalus, usespsychê to refer not to a comprehensive soul but ratherto just one psychic faculty, the seat of emotions, which is located inthe heart along with the faculty of sensation (Philolaus, Fr. 13).Thispsychê is explicitly said by Philolaus to beshared with animals. Herodotus usespsychê in a similarway to refer to the seat of emotions. Thus it seems likely thatPythagoras too thought of the transmigratingpsychê inthis way. If so, it is unlikely that Pythagoras thought that humanscould be reincarnated as plants, sincepsychê is notassigned to plants by Philolaus. It has often been assumed that thetransmigrating soul is immaterial, but Philolaus seems to have amaterialistic conception of soul and he may be following Pythagoras.Similarly, it is doubtful that Pythagoras thought of thetransmigrating soul as a comprehensive soul that includes all psychicfaculties. His ability to recognize something distinctive of hisfriend in the puppy (if this is not pushing the evidence of a joke toofar) and to remember his own previous incarnations show that personalidentity was preserved through incarnations. This personal identitycould well be contained in the pattern of emotions, that constitute aperson’s character and that is preserved in thepsychê and need not presuppose all psychic faculties.In Philolaus thispsychê explicitly does not includethenous (intellect), which is not shared with animals. Thus,it would appear that what is shared with animals and which ledPythagoras to suppose that they had special kinship with human beings(Dicaearchus in Porphyry,VP 19) is not intellect, as somehave supposed (Sorabji 1993, 78 and 208) but rather the ability tofeel emotions such as pleasure and pain.

There are significant points of contact between the Greek religiousmovement known as Orphism and Pythagoreanism, but the evidence forOrphism is at least as problematic as that for Pythagoras and oftencomplicates rather than clarifies our understanding of Pythagoras(Betegh 2014; Burkert 1972a, 125 ff.; Kahn 2002, 19–22; Riedweg2002). There is some evidence that the Orphics also believed inmetempsychosis and considerable debate has arisen as to whether theyborrowed the doctrine from Pythagoras (Burkert 1972a, 133; Bremmer2002, 24) or he borrowed it from them (Zhmud 2012a, 221–238).Dicaearchus says that Pythagoras was the first to introducemetempsychosis into Greece (PorphyryVP 19). Moreover, whileOrphism presents a heavily moralized version of metempsychosis inaccordance with which we are born again for punishment in this life sothat our body is the prison of the soul while it undergoes punishment,it is not clear that the same was true in Pythagoreanism. It may bethat rebirths in a series of animals and people were seen as a naturalcycle of the soul (Zhmud 2012a, 232–233). One would expect thatthe Pythagorean way of life was connected to metempsychosis, whichwould in turn suggest that a certain reincarnation is a reward orpunishment for following or not following the principles set out inthat way of life. However, there is no unambiguous evidence connectingthe Pythagorean way of life with metempsychosis.

It is crucial to recognize that most Greeks followed Homer inbelieving that the soul was an insubstantial shade, which lived ashadowy existence in the underworld after death, an existence so bleakthat Achilles famously asserts that he would rather be the lowestmortal on earth than king of the dead (Homer,Odyssey XI.489). Pythagoras’ teachings that the soul was immortal, wouldhave other physical incarnations and might have a good existence afterdeath were striking innovations that must have had considerable appealin comparison to the Homeric view. According to Dicaearchus, inaddition to the immortality of the soul and reincarnation, Pythagorasbelieved that “after certain periods of time the things thathave happened once happen again and nothing is absolutely new”(Porphyry,VP 19). This doctrine of “eternalrecurrence” is also attested by Aristotle’s pupil Eudemus,although he ascribes it to the Pythagoreans rather than to Pythagorashimself. (Fr. 88 Wehrli). The doctrine of transmigration thus seems tohave been extended to include the idea that we and indeed the wholeworld will be reborn into lives that are exactly the same as those weare living and have already lived.

4.2 Pythagoras as a Wonder-worker

Some have wanted to relegate the more miraculous features ofPythagoras’persona to the later tradition, but thesecharacteristics figure prominently in the earliest evidence and arethus central to understanding Pythagoras. Aristotle emphasized hissuperhuman nature in the following ways: there was a story thatPythagoras had a golden thigh (a sign of divinity); the Pythagoreanstaught that “of rational beings, one sort is divine, one ishuman, and another such as Pythagoras” (Iamblichus,VP31); Pythagoras was seen on the same day at the same time in bothMetapontum and Croton; he killed a deadly snake by biting it; as hewas crossing a river it spoke to him (all citations are fromAristotle, Fr. 191, unless otherwise noted). Aristotle reports thatthe people of Croton called Pythagoras the “HyperboreanApollo” and Iamblichus’ report (VP 140) that apriest from the land of the Hyperboreans, Abaris, visited Pythagorasand presented him with his arrow, a token of power, may well also goback to Aristotle (Burkert 1972a, 143). Kingsley argues that the visitof Abaris is the key to understanding the identity and significance ofPythagoras. Abaris was a shaman from Mongolia (part of what the Greekscalled Hyperborea), who recognized Pythagoras as an incarnation ofApollo. The stillness of ecstacy practiced by Abaris and handed on toPythagoras is the foundation of all civilization. Abaris’ visitto Pythagoras thus becomes the central moment when civilizing power ispassed from East to West (Kingsley 2010).

Whether or not one accepts this account of Pythagoras and his relationto Abaris, there is a clear parallel for some of the remarkableabilities of Pythagoras in the later figure of Empedocles, whopromises to teach his pupils to control the winds and bring the deadback to life (Fr. 111). There are recognizable traces of thistradition about Pythagoras even in the pre-Aristotelian evidence, andhis wonder-working clearly evoked diametrically opposed reactions.Heraclitus’ description of Pythagoras as “the chief of thecharlatans” (Fr. 81) and of his wisdom as “fraudulentart” (Fr. 129) is most easily understood as an unsympatheticreference to his miracles. Empedocles, on the other hand, is clearlysympathetic to Pythagoras, when he describes him as “ a man whoknew remarkable things” and who “possessed the greatestwealth of intelligence” and again probably makes reference tohis wonder-working by calling him “accomplished in all sorts ofwisedeeds”(Fr. 129). In Herodotus’ report,Zalmoxis, whom some of the Greeks identified as the slave and pupil ofPythagoras, tried to gain authority for his teachings about the fateof the soul by claiming to have journeyed to the next world (IV. 95).The skeptical tradition represented in Herodotus’ report treatsthis as a ruse on Zalmoxis’ part; he had not journeyed to thenext world but had in reality hidden in an underground dwelling forthree years. Similarly Pythagoras may have claimed authority for histeachings concerning the fate of our soul on the basis of hisremarkable abilities and experiences, and there is some evidence thathe too claimed to have journeyed to the underworld and that thisjourney may have been transferred from Pythagoras to Zalmoxis (Burkert1972a, 154 ff.).

4.3 The Pythagorean Way of Life

The testimony of both Plato (R. 600a) and Isocrates(Busiris 28) shows that Pythagoras was above all famous forhaving left behind him a way of life, which still had adherents in thefourth century over 100 years after his death. It is plausible toassume that many features of this way of life were designed to insurethe best possible future reincarnations, but it is important toremember that nothing in the early evidence connects the way of lifeto reincarnation in any specific fashion.

One of the clearest strands in the early evidence for Pythagoras ishis expertise in religious ritual. Isocrates emphasizes that “hemore conspicuously than others paid attention to sacrifices andrituals in temples” (Busiris 28). Herodotus gives anexample: the Pythagoreans agree with the Egyptians in not allowing thedead to be buried in wool (II. 81). It is not surprising thatPythagoras, as an expert on the fate of the soul after death, shouldalso be an expert on the religious rituals surrounding death. Asignificant part of the Pythagorean way of life thus consisted in theproper observance of religious ritual. One major piece of evidence forthis emphasis on ritual is thesymbola oracusmata(“things heard”), short maxims that were handed downorally. The earliest source to quoteacusmata is Aristotle,in the fragments of his now lost treatise on the Pythagoreans. It isnot always possible to be certain which of theacusmataquoted in the later tradition go back to Aristotle and which of theones that do go back to Pythagoras. Most of Iamblichus’ examplesin sections 82–86 ofOn the Pythagorean Life, however,appear to derive from Aristotle (Burkert 1972a, 166 ff.), and many arein accord with the early evidence we have for Pythagoras’interest in ritual. Thus theacusmata advise Pythagoreans topour libations to the gods from the ear (i.e., the handle) of the cup,to refrain from wearing the images of the gods on their fingers, notto sacrifice a white cock, and to sacrifice and enter the templebarefoot. A number of these practices can be paralleled in Greekmystery religions of the day (Burkert 1972a, 177). Indeed, it isimportant to emphasize that Pythagoreanism was not a religion andthere were no specific Pythagorean rites (Burkert 1985, 302).Pythagoras rather taught a way of life that emphasized certain aspectsof traditional Greek religion.

A second characteristic of the Pythagorean way of life was theemphasis on dietary restrictions. There is no direct evidence forthese restrictions in the pre-Aristotelian evidence, but bothAristotle and Aristoxenus discuss them extensively. Unfortunately theevidence is contradictory and it is difficult to establish any pointswith certainty. One might assume that Pythagoras advocatedvegetarianism on the basis of his belief in metempsychosis, as didEmpedocles after him (Fr. 137). Indeed, the fourth-centurymathematician and philosopher Eudoxus says that “he not onlyabstained from animal food but would also not come near butchers andhunters” (Porphyry,VP 7). According to Dicaearchus,one of Pythagoras’ most well-known doctrines was that “allanimate beings are of the same family” (Porphyry,VP19), which suggests that we should be as hesitant about eating otheranimals as other humans. Unfortunately, Aristotle reports that“the Pythagoreans refrain from eating the womb and the heart,the sea anemone and some other such things but use all other animalfood” (Aulus Gellius IV. 11. 11–12). This makes it soundas if Pythagoras forbade the eating of just certain parts of animalsand certain species of animals rather than all animals; such specificprohibitions are easy to parallel elsewhere in Greek ritual (Burkert1972a, 177). Aristoxenus asserts that Pythagoras only refused to eatplough oxen and rams (Diogenes Laertius VIII. 20) and that he was fondof young kids and suckling pigs as food (Aulus Gellius IV. 11. 6).Some have tried to argue that Aristoxenus is refashioningPythagoreanism in order to make it more rational (e.g., Kahn 2001, 70;Zhmud 2012b, 228), but Aristoxenus, in fact, recognizes thenon-rational dimension of Pythagoreanism and Pythagoras’ eatingof kids and suckling pigs may itself have religious motivations(Huffman 2012b). Moreover, even if Aristoxenus’ evidence wereset aside Aristotle’s testimony and many of theacusmata indicate that Pythagoras ate some meat. Certainlyanimal sacrifice was the central act of Greek religious worship and toabolish it completely would be a radical step. Theacusmareported by Aristotle, in response to the question “what is mostjust?” has Pythagoras answer “to sacrifice”(Iamblichus,VP 82). Based on the direct evidence forPythagoras’ practice in Aristotle and Aristoxenus, it seems mostprudent to conclude that he did not forbid the eating of all animalfood. The later tradition proposes a number of ways to reconcilemetempsychosis with the eating of some meat. Pythagoras may haveadopted one of these positions, but no certainty is possible. Forexample, he may have argued that it was legitimate to kill and eatsacrificial animals, on the grounds that the souls of men do not enterinto these animals (Iamblichus,VP 85). Perhaps the mostfamous of the Pythagorean dietary restrictions is the prohibition oneating beans, which is first attested by Aristotle and assigned toPythagoras himself (Diogenes Laertius VIII. 34). Aristotle suggests anumber of explanations including one that connects beans with Hades,hence suggesting a possible connection with the doctrine ofmetempsychosis. A number of later sources suggest that it was believedthat souls returned to earth to be reincarnated through beans (Burkert1972a, 183). There is also a physiological explanation. Beans, whichare difficult to digest, disturb our abilities to concentrate.Moreover, the beans involved are a European vetch (Viciafaba) rather than the beans commonly eaten today. Certain peoplewith an inherited blood abnormality develop a serious disorder calledfavism, if they eat these beans or even inhale their pollen.Aristoxenus interestingly denies that Pythagoras forbade the eating ofbeans and says that “he valued it most of all vegetables, sinceit was digestible and laxative” (Aulus Gellius IV. 11.5). Thediscrepancies between the various fourth-century accounts of thePythagorean way of life suggest that there were disputes amongfourth-century Pythagoreans as to the proper way of life and as to theteachings of Pythagoras himself.

Theacusmata indicate that the Pythagorean way of lifeembodied a strict regimen not just regarding religious ritual and dietbut also in a wide variety of aspects of life. The survivingcollection ofacusmata, however, is not extensive enough tocover all features of human life (Thom 2020, 16). Some of therestrictions appear to be largely arbitrary taboos, e.g., “onemust put the right shoe on first” or “one must not travelthe public roads” (Iamblichus,VP 83, probably fromAristotle). On the other hand, some aspects of the Pythagorean lifeinvolved a moral discipline that was greatly admired, even byoutsiders. Pythagorean silence is an important example. Isocratesreports that even in the fourth century people “marvel more atthe silence of those who profess to be his pupils than at those whohave the greatest reputation for speaking” (Busiris28). The ability to remain silent was seen as important training inself-control, and the later tradition reports that those who wanted tobecome Pythagoreans had to observe a five-year silence (Iamblichus,VP 72). Isocrates is contrasting the marvelous self-controlof Pythagorean silence with the emphasis on public speaking intraditional Greek education. Pythagoreans also displayed great loyaltyto their friends as can be seen in Aristoxenus’ story of Damonwho is willing to stand surety for his friend Phintias, who has beensentenced to death (Iamblichus,VP 233 ff.). In addition tosilence as a moral discipline, there is evidence that secrecy was keptabout certain of the teachings of Pythagoras. Aristoxenus reports thatthe Pythagoreans thought that “not all things were to be spokento all people” (Diogenes Laertius, VIII. 15), but this may onlyapply to teaching and mean that children should not be taught allthings (Zhmud 2012a, 155). Clearer evidence is found inDicaearchus’ complaint that it is not easy to say whatPythagoras taught his pupils because they observed no ordinary silenceabout it (Porphyry,VP 19). Indeed, one would expect that anexclusive society such as that of the Pythagoreans would have secretdoctrines and symbols. Aristotle says that the Pythagoreans“guarded among their very secret doctrines that one type ofrational being is divine, one human, and one such as Pythagoras”(Iamblichus,VP 31). That there should be secret teachingsabout the special nature and authority of the master is notsurprising. This does not mean, however, that all Pythagoreanphilosophy was secret. Aristotle discusses the fifth-centurymetaphysical system of Philolaus in some detail with no hint thatthere was anything secret about it, and Plato’s discussion ofPythagorean harmonic theory in Book VII of theRepublic givesno suggestion of any secrecy. Aristotle singles out theacusma quoted above (Iamblichus,VP 31) as secret,but this statement in itself implies that others were not. The ideathat all of Pythagoras’ teachings were secret was used in thelater tradition to explain the lack of Pythagorean writings and to tryto validate forged documents as recently discovered secret treatises.For a sceptical evaluation of Pythagorean secrecy see Zhmud 2012a,150–158.

There is some controversy as to whether Pythagoras, in fact, taught away of life governed in great detail by theacusmata asdescribed above. Plato praises the Pythagorean way of life in theRepublic (600b), but it is hard to imagine him admiring theset of taboos found in theacusmata (Lloyd 2014, 44; Zhmud2012a). Althoughacusmata were collected already byAnaximander of Miletus the younger (ca. 400 BCE) and by Aristotle inthe fourth century, Zhmud (2012a, 177–178 and 192–205)argues that very few of these embody specifically Pythagorean ideasand that it is difficult to imagine anyone following this bewilderingset of rules literally as Burkert argues (1972a, 191). However, theearly evidence suggests that Pythagoras largely constructed theacusmata out of ideas collected from others (Thom 2013;Huffman 2008b: Gemelli Marciano 2002), so it is no surprise that manyof them are not uniquely Pythagorean. Moreover, Thom suggests a middleground between Zhmud and Burkert whereby, contra Zhmud, most of theacusmata were followed by the Pythagoreans but contraBurkert, they were subject to interpretation from the beginning andnot followed literally, so that it is possible to imagine peopleliving according to them (Thom, 2013). It is true that there is littleif any fifth- and fourth-century evidence for Pythagoreans livingaccording to theacusmata and Zhmud argues that theundeniable political impact of the Pythagoreans would be inexplicableif they lived the heavily ritualized life of theacusmata,which would inevitably isolate them from society (Zhmud 2012a,175–183). He suggests that the Pythagorean way of life differedlittle from standard aristocratic morality (Zhmud 2012a, 175). If,however, the Pythagorean way of life was little out of the ordinary,why do Plato and Isocrates specifically comment on how distinctivethose who followed it were? The silence of fifth-century sources aboutpeople practicingacusmata is not terribly surprising giventhe very meager sources for the Greek cities in southern Italy in theperiod. Why not suppose that the vast majority of names inAristoxenus’ catalogue of Pythagoreans, who are not associatedwith any political, philosophical or scientific accomplishment, whoare just names to us, are precisely those who were Pythagoreansbecause they followed the Pythagorean way of life? We would then havelots of people who followed theacusmata (166 of the 222 namein the catalogue appear nowhere else). This suspicion is confirmed bythe fact that one of the names from Aristoxenus’ catalogue(Hippomedon of Argos) is elsewhere (Iamblichus,On the PythagoreanLife, 87) explicitly said to belong theacusmatici.Moreover, other scholars argue that archaic Greek society in southernItaly was pervaded by religion and the presence of similar precepts inauthors such as Hesiod show that adherence to taboos such as are foundin theacusmata would not have caused a scandal and adherenceto many of them would have gone unobserved by outsiders (GemelliMarciano 2014, 133–134).

Once again a problem of source criticism raises its head. Zhmud arguesthat the split betweenacusmatici who blindly followed theacusmata and themathematici who learned the reasonsfor them (see the fifth paragraph of section 5 below) is a creation ofthe later tradition, appearing first in Clement of Alexandria anddisappearing after Iamblichus (Zhmud 2012a, 169–192). He alsonotes that the termacusmata appears first in Iamblichus(On the Pythagorean Life 82–86) and suggests that it isalso a creation of the later tradition. The Pythagorean maxims didexist earlier, as the testimony of Aristotle shows, but they wereknown assymbola, were originally very few in number and weremainly a literary phenomena rather than being tied to people whoactually practiced them (Zhmud 2012a, 192–205). However, severalscholars have argued that the passages in which the split between theacusmatici andmathematici is described as well asthe passage in which the termacusmata is used, in fact, goback to Aristotle (Burkert 1972a, 196; see Burkert 1998, 315 where hecomments that the Aristotelian provenance of the text is “asobvious as it is unprovable”) and even Zhmud recognizes that alarge part of the material in Iamblichus is derived from Aristotle(2012a, 170). Indeed, the description of the split in what is likelyto be the original version (Iamblichus,On General MathematicalScience 76.16–77.18 Festa) uses language in describing thePythagoreans that is almost an Aristotelian signature, “Thereare two forms of the Italian philosophy which is calledPythagorean” (76.16). Aristotle famously describes thePythagoreans as “those called Pythagoreans” and alsodescribes them as “the Italians” (e.g.,Mete.342b30,Cael. 293a20). So the question of whether Pythagorastaught a way of life tightly governed by theacusmata turnsagain on whether key passages in Iamblichus (On the PythagoreanLife 81–87,On General Mathematical Science76.16–77.18 Festa) go back to Aristotle. If they do, we havevery good reason to believe that Pythagoras taught such a life, ifthey do not the issue is less clear.

The testimony of fourth-century authors such as Aristoxenus andDicaearchus indicates that the Pythagoreans also had an importantimpact on the politics and society of the Greek cities in southernItaly. Dicaearchus reports that, upon his arrival in Croton,Pythagoras gave a speech to the elders and that the leaders of thecity then asked him to speak to the young men of the town, the boysand the women (Porphyry,VP 18). Women, indeed, may haveplayed an unusually large role in Pythagoreanism (see Rowett 2014,122–123), since both Timaeus and Dicaearchus report on the fameof Pythagorean women including Pythagoras’ daughter (Porphyry,VP 4 and 19). Theacusmata teach men to honor theirwives and to beget children in order to insure worship for the gods(Iamblichus,VP 84–6). Dicaearchus reports that theteaching of Pythagoras was largely unknown, so that Dicaearchus cannothave known of the content of the speech to the women or of any of theother speeches; the speeches presented in Iamblichus (VP37–57) are thus likely to be later forgeries (Burkert 1972a,115), but there is early evidence that he gave different speeches todifferent groups (Antisthenes V A 187). The attacks on thePythagoreans both in Pythagoras’ own day and in the middle ofthe fifth century are presented by Dicaearchus and Aristoxenus ashaving a wide-reaching impact on Greek society in southern Italy; thehistorian Polybius (II. 39) reports that the deaths of thePythagoreans meant that “the leading citizens of each city weredestroyed,” which clearly indicates that many Pythagoreans hadpositions of political authority. On the other hand, it is noteworthythat Plato explicitly presents Pythagoras as a private rather than apublic figure (R. 600a). It seems most likely that thePythagorean societies were in essence private associations but thatthey also could function as political clubs (see Zhmud 2012a,141–148), while not being a political party in the modern sense;their political impact should perhaps be better compared to modernfraternal organizations such as the Masons. Thus, the Pythagoreans didnot rule as a group but had political impact through individualmembers who gained positions of authority in the Greek city-states insouthern Italy. See further Burkert 1972a, 115 ff., von Fritz 1940,Minar 1942 and Rowett 2014.

5. Was Pythagoras a Mathematician or Cosmologist?

In the modern world Pythagoras is most of all famous as amathematician, because of the theorem named after him, and secondarilyas a cosmologist, because of the striking view of a universe ascribedto him in the later tradition, in which the heavenly bodies produce“the music of the spheres” by their movements. It shouldbe clear from the discussion above that, while the early evidenceshows that Pythagoras was indeed one of the most famous early Greekthinkers, there is no indication in that evidence that his fame wasprimarily based on mathematics or cosmology. Neither Plato norAristotle treats Pythagoras as having contributed to the developmentof Presocratic cosmology, although Aristotle in particular discussesthe topic in some detail in the first book of theMetaphysicsand elsewhere. Aristotle evidently knows of no cosmology of Pythagorasthat antedates the cosmological system of the “so-calledPythagoreans,” which he dates to the middle of the fifthcentury, and which is found in the fragments of Philolaus. There isalso no mention of Pythagoras’ work in geometry or of thePythagorean theorem in the early evidence. Dicaearchus comments that“what he said to his associates no one can say reliably,”but then identifies four doctrines that became well known: (1) thatthe soul is immortal; (2) that it transmigrates into other kinds ofanimals; (3) that after certain intervals the things that havehappened once happen again, so that nothing is completely new; (4)that all animate beings belong to the same family (Porphyry,VP 19). Thus, for Dicaearchus too, it is not as amathematician or Presocratic writer on nature that Pythagoras isfamous. It might not be too surprising that Plato, Aristotle andDicaearchus do not mention Pythagoras’ work in mathematics,because they are not primarily dealing with the history ofmathematics. On the other hand, Aristotle’s pupil Eudemus didwrite a history of geometry in the fourth century and what we find inEudemus is very significant. A substantial part of Eudemus’overview of the early history of Greek geometry is preserved in theprologue to Proclus’ commentary on Book One of Euclid’sElements (p. 65, 12 ff.), which was written much later, inthe fifth century CE. At first sight, it appears that Eudemus didassign Pythagoras a significant place in the history of geometry.Eudemus is reported as beginning with Thales and an obscure figurenamed Mamercus, but the third person mentioned by Proclus in thisreport is Pythagoras, immediately before Anaxagoras. There is nomention of the Pythagorean theorem, but Pythagoras is said to havetransformed the philosophy of geometry into a form of liberaleducation, to have investigated its theorems in an immaterial andintellectual way and specifically to have discovered the study ofirrational magnitudes and the construction of the five regular solids.Unfortunately close examination of the section on Pythagoras inProclus’ prologue reveals numerous difficulties and shows thatit comes not from Eudemus but from Iamblichus with some additions byProclus himself (Burkert 1972a, 409 ff.). The first clause is takenword for word from Iamblichus’On Common MathematicalScience (p. 70.1 Festa). Proclus elsewhere quotes long passagesfrom Iamblichus and is doing the same here. As Burkert points out,however, as soon as we recognize that Proclus has inserted a passagefrom Iamblichus into Eudemus’ history, we must also recognizethat Proclus was driven to do so by the lack of any mention ofPythagoras in Eudemus. Even those who want to assign Pythagoras alarger role in early Greek mathematics recognize that most of whatProclus says here cannot go back to Eudemus (Zhmud 2012a,263–266). Thus, not only is Pythagoras not commonly known as ageometer in the time of Plato and Aristotle, but also the mostauthoritative history of early Greek geometry assigns him no role inthe history of geometry in the overview preserved in Proclus.According to Proclus, Eudemus did report that two propositions, whichare later found in Euclid’sElements, were discoveriesof the Pythagoreans (Proclus 379 and 419). Eudemus does not assign thediscoveries to any specific Pythagorean, and they are hard to date.The discoveries might be as early as Hippasus in the middle of thefifth century, who is associated with a group of Pythagoreans known asthemathematici, who arose after Pythagoras’ death (seebelow). The crucial point to note is that Eudemus does not assignthese discoveries to Pythagoras himself. The first Pythagorean whom wecan confidently identify as an accomplished mathematician is Archytasin the late fifth and the first half of the fourth century.

Are we to conclude, then, that Pythagoras had nothing to do withmathematics or cosmology? The evidence is not quite that simple. Thetradition regarding Pythagoras’ connection to the Pythagoreantheorem reveals the complexity of the problem. None of the earlysources, including Plato, Aristotle and their pupils shows anyknowledge of Pythagoras’ connection to the theorem. Almost athousand years later, in the fifth century CE, Proclus, in hiscommentary on Euclid’s proof of the theorem (ElementsI. 47), gives the following report: “If we listen to those whowish to investigate ancient history, it is possible to find themreferring this theorem back to Pythagoras and saying that hesacrificed an ox upon its discovery” (426.6). Proclus gives noindication of his source, but a number of other late reports (DiogenesLaertius VIII. 12; Athenaeus 418f; Plutarch,Moralia 1094b)show that it ultimately relied on two lines of verse whose context isunknown: “When Pythagoras found that famous diagram, in honor ofwhich he offered a glorious sacrifice of oxen...” The author ofthese verses is variously identified as Apollodorus the calculator orApollodorus the arithmetician. This Apollodorus probably dates beforeCicero, who alludes to the story (On the Nature of the GodsIII. 88), and, if he can be identified with Apollodorus of Cyzicus,the follower of Democritus, the story would go back to the fourthcentury BCE (Burkert 1972a, 428). Two lines of poetry of indeterminatedate are obviously a very slender support upon which to basePythagoras’ reputation as a geometer, but they cannot be simplyignored. Several things need to be noted about this tradition,however, in order to understand its true significance. First, Proclusdoes not ascribe a proof of the theorem to Pythagoras but rather goeson to contrast Pythagoras as one of those “knowing the truth ofthe theorem” with Euclid who not only gave the proof found inElements I.47 but also a more general proof in VI. 31.Although a number of modern scholars have speculated on what sort ofproof Pythagoras might have used (e.g., Heath 1956, 352 ff.), it isimportant to note that there is not a jot of evidence for a proof byPythagoras; what we know of the history of Greek geometry makes such aproof by Pythagoras improbable, since the first work on the elementsof geometry, upon which a rigorous proof would be based, is notattested until Hippocrates of Chios, who was active after Pythagorasin the latter part of the fifth century (Proclus,A Commentary onthe First Book of Euclid’s Elements, 66). All that thistradition ascribes to Pythagoras, then, is discovery of the truthcontained in the theorem. The truth may not have been in general formbut rather focused on the simplest such triangle (with sides 3, 4 and5), pointing out that such a triangle and all others like it will havea right angle. Modern scholarship has shown, moreover, that longbefore Pythagoras the Babylonians were aware of the basic Pythagoreanrule and could generate Pythagorean triples (integers that satisfy thePythagorean rule such as 3, 4 and 5), although they never formulatedthe theorem in explicit form or proved it (Høyrup 1999,401–2, 405; cf. Robson 2001). Thus, it is likely that Pythagorasand other Greeks first encountered the truth of the theorem as aBabylonian arithmetical technique (Høyrup 1999, 402; Burkert1972a,429). It is possible, then, that Pythagoras just passed on tothe Greeks a truth that he learned from the East. The emphasis in thetwo lines of verse is not just on Pythagoras’ discovery of thetruth of the theorem, it is as much or more on his sacrifice of oxenin honor of the discovery. We are probably supposed to imagine thatthe sacrifice was not of a single ox; Apollodorus describes it as“a famous sacrifice of oxen” and Diogenes Laertiusparaphrases this as ahecatomb, which need not be, as itliterally says, a hundred oxen, but still suggests a large number.Some have wanted to doubt the whole story, including the discovery ofthe theorem, because it conflicts with Pythagoras’ supposedvegetarianism, but it is far from clear to what extent he was avegetarian (see above). If the story is to have any force and if itdates to the fourth century, it shows that Pythagoras was famous for aconnection to a certain piece of geometrical knowledge, but it alsoshows that he was famous for his enthusiastic response to thatknowledge, as evidenced in his sacrifice of oxen, not for anygeometric proof. What emerges from this evidence, then, is notPythagoras as the master geometer, who provides rigorous proofs, butrather Pythagoras as someone who recognizes and celebrates with aritual act thanking the gods certain geometrical relationships as ofhigh importance.

It is striking that a very similar picture of Pythagoras emerges fromthe evidence for his cosmology. A famous discovery is attributed toPythagoras in the later tradition, i.e., that the central musicalconcords (the octave, fifth and fourth) correspond to the whole numberratios 2 : 1, 3 : 2 and 4 : 3 respectively (e.g., Nicomachus,Handbook 6 = Iamblichus,On the Pythagorean Life115). The only early source to associate Pythagoras with the wholenumber ratios that govern the concords is Xenocrates (Fr. 9) in theearly Academy, but the early Academy is precisely one source of thelater exaggerated tradition about Pythagoras (see above). One storyhas it that Pythagoras passed by a blacksmith’s shop and heardthe concords in the sounds of the hammers striking the anvil and thendiscovered that the sounds made by hammers whose weights are in theratio 2 : 1 will be an octave apart, etc. Unfortunately, the storiesof Pythagoras’ discovery of these relationships are clearlyfalse, since none of the techniques for the discovery ascribed to himwould, in fact, work (e.g., the pitch of sounds produced by hammers isnot directly proportional to their weight: see Burkert 1972a, 375). Anexperiment ascribed to Hippasus, who was active in the first half ofthe fifth century, after Pythagoras’ death, would have worked,and thus we can trace the scientific verification of the discovery atleast to Hippasus; knowledge of the relation between whole numberratios and the concords is clearly found in the fragments of Philolaus(Fr. 6a, Huffman), in the second half of the fifth century. There issome evidence that the truth of the relationship was already known toPythagoras’ contemporary, Lasus, who was not a Pythagorean(Burkert 1972a, 377). It may be once again that Pythagoras knew of therelationship without either having discovered it or havingdemonstrated it scientifically. The relationship was probably firstdiscovered by instrument makers, and specifically makers of windinstruments rather than stringed instruments (Barker 2014, 202). Theacusmata reported by Aristotle, which may go back toPythagoras, report the following question and answer “What isthe oracle at Delphi? Thetetraktys, which is the harmony inwhich the Sirens sing” (Iamblichus,On the PythagoreanLife, 82, probably derived from Aristotle). Thetetraktys, literally “the four,” refers to thefirst four numbers, which when added together equal the number ten,which was regarded as the perfect number in fifth-centuryPythagoreanism. Here in theacusmata, these four numbers areidentified with one of the primary sources of wisdom in the Greekworld, the Delphic oracle. In the later tradition thetetraktys is treated as the summary of all Pythagoreanwisdom, since the Pythagoreans swore oaths by Pythagoras as “theone who handed down thetetraktys to our generation.”Thetetraktys can be connected to the music which the Sirenssing in that all of the ratios that correspond to the basic concordsin music (octave, fifth and fourth) can be expressed as whole numberratios of the first four numbers. Thisacusma thus seems tobe based on the knowledge of the relationship between the concords andthe whole number ratios. The picture of Pythagoras that emerges fromthe evidence is thus not of a mathematician, who offered rigorousproofs, or of a scientist, who carried out experiments to discover thenature of the natural world, but rather of someone who sees specialsignificance in and assigns special prominence to mathematicalrelationships that were in general circulation. This is the context inwhich to understand Aristoxenus’ remark that “Pythagorasmost of all seems to have honored and advanced the study concernedwith numbers, having taken it away from the use of merchants andlikening all things to numbers” (Fr. 23, Wehrli). Some mightsuppose that this is a reference to a rigorous treatment ofarithmetic, such as that hypothesized by Becker (1936), who arguedthat Euclid IX. 21–34 was a self-contained unit that representeda deductive theory of odd and even numbers developed by thePythagoreans (see Mueller 1997, 296 ff. and Burkert 1972a, 434 ff.).It is crucial to recognize, however, that Becker’sreconstruction is rejected in some recent scholarship (e.g., Netz2014, 179) and no ancient source assigns it even to the Pythagoreans,let alone to Pythagoras himself. There is, moreover, no talk ofmathematical proof or a deductive system in the passage fromAristoxenus just quoted. Pythagoras is known for thehonor hegives to number and for removing it from the practical realm of tradeand instead pointing to correspondences between the behavior of numberand the behavior of things. Such correspondences were highlighted inAristotle’s book on the Pythagoreans, e.g., the female islikened to the number two and the male to the number three and theirsum, five, is likened to marriage (Aristotle, Fr. 203).

What then was the nature of Pythagoras’ cosmos? Thedoxographical tradition reports that Pythagoras discovered thesphericity of the earth, the five celestial zones and the identity ofthe evening and morning star (Diogenes Laertius VIII. 48, AetiusIII.14.1, Diogenes Laertius IX. 23). In each case, however, Burkerthas shown that these reports seem to be false and the result of theglorification of Pythagoras in the later tradition, since the earliestand most reliable evidence assigns these same discoveries to someoneelse (1972a, 303 ff.). Thus, Theophrastus, who is the primary basis ofthe doxographical tradition, says that it was Parmenides whodiscovered the sphericity of the earth (Diogenes Laertius VIII. 48).Parmenides is also identified as the discoverer of the identity of themorning and evening star (Diogenes Laertius IX. 23), andPythagoras’ claim appears to be based on a poem forged in hisname, which was rejected already by Callimachus in the third centuryBCE (Burkert 1972a, 307). The identification of the five celestialzones depends on the discovery of the obliquity of the ecliptic, andsome of the doxography duly assigns this discovery to Pythagoras aswell and claims that Oenopides stole it from Pythagoras (AetiusII.12.2); the history of astronomy by Aristotle’s pupil Eudemus,our most reliable source, seems to attribute the discovery toOenopides (there are problems with the text), however (Eudemus, Fr.145 Wehrli). It thus appears that the later tradition, finding noevidence for Pythagoras’ cosmology in the early evidence,assigned the discoveries of Parmenides back to Pythagoras, encouragedby traditions which made Parmenides the pupil of Pythagoras. In theend, there is no evidence for Pythagoras’ cosmology in the earlyevidence, beyond what can be reconstructed fromacusmata. Aswas shown above, Pythagoras saw the cosmos as structured according tonumber insofar as thetetraktys is the source of all wisdom.His cosmos was also imbued with a moral significance, which is inaccordance with his beliefs about reincarnation and the fate of thesoul (West 1971, 215–216; Huffman 2013, 60–68). Thus, inanswer to the question “What are the Isles of the Blest?”(where we might hope to go, if we lived a good life), the answer is“the sun and the moon.” Again “the planets are thehounds of Persephone,” i.e., the planets are agents of vengeancefor wrong done (Aristotle in PorphyryVP 41). Aristotlesimilarly reports that for the Pythagoreans thunder “is a threatto those in Tartarus, so that they will be afraid”(Posterior Analytics 94b) and anotheracusma saysthat “an earthquake is nothing other than a meeting of thedead” (Aelian,Historical Miscellany, IV. 17). Zhmudcalls these cosmologicalacusmata into question (2012a,329–330), noting that some only appear in Porphyry, but Porphyryexplicitly identifies Aristotle as his source and we have no reason todoubt him (VP 41). Pythagoras’ cosmos embodiedmathematical relationships that had a basis in fact and combined themwith moral ideas tied to the fate of the soul. The best analogy forthe type of account of the cosmos which Pythagoras gave might be someof the myths which appear at the end of Platonic dialogues such as thePhaedo,Gorgias orRepublic, wherecosmology has a primarily moral purpose. Should the doctrine of theharmony of the spheres be assigned to Pythagoras? Certainly theacusma which talks of the sirens singing in the harmonyrepresented by thetetraktys suggests that there might havebeen a cosmic music and that Pythagoras may well have thought that theheavenly bodies, which we see move across the sky at night, made musicby their motions. On the other hand, there is no evidence for“the spheres,” if we mean by that a cosmic model accordingto which each of the heavenly bodies is associated with a series ofconcentric circular orbits, a model which is at least in part designedto explain celestical phenomena. The first such cosmic model in thePythagorean tradition is that of Philolaus in the second half of thefifth century, a model which still shows traces of the connection tothe moral cosmos of Pythagoras in its account of the counter-earth andthe central fire (seePhilolaus).

If Pythagoras was primarily a figure of religious and ethicalsignificance, who left behind an influential way of life and for whomnumber and cosmology primarily had significance in this religious andmoral context, how are we to explain the prominence of rigorousmathematics and mathematical cosmology in later Pythagoreans such asPhilolaus and Archytas? It is important to note that this is not justa question asked by modern scholars but was already a central questionin the fourth century BCE. What is the connection between Pythagorasand fifth-century Pythagoreans? The question is implicit inAristotle’s description of the fifth-century Pythagoreans suchas Philolaus as “the so-called Pythagoreans.” Thisexpression is most easily understood as expressing Aristotle’srecognition that these people were called Pythagoreans and at the sametime his puzzlement as to what connection there could be between thewonder-worker who promulgated theacusmata, which hisresearches show Pythagoras to have been, and the philosophy oflimiters and unlimiteds put forth in fifth-century Pythagoreanism. Thetradition of a split between two groups of Pythagoreans in the fifthcentury, themathematici and theacusmatici, pointsto the same puzzlement. The evidence for this split is quite confusedin the later tradition, but Burkert (1972a, 192 ff.) has shown thatthe original and most objective account of the split is found in apassage of Aristotle’s book on the Pythagoreans, which ispreserved in Iamblichus (On Common Mathematical Science,76.19 ff). Theacusmatici, who are clearly connected by theirname to theacusmata, are recognized by the other group, themathematici, as genuine Pythagoreans, but theacusmatici do not regard the philosophy of themathematici as deriving from Pythagoras but rather fromHippasus. Themathematici appear to have argued that, whiletheacusmatici were indeed Pythagoreans, it was themathematici who were the true Pythagoreans; Pythagoras gavetheacusmata to those who did not have the time to study themathematical sciences, so that they would at least have moralguidance, while to those who had the time to fully devote themselvesto Pythagoreanism he gave training in the mathematical sciences, whichexplained the reasons for this guidance. This tradition thus showsthat all agreed that theacusmata represented the teaching ofPythagoras, but that some regarded the mathematical work associatedwith themathematici as not deriving from Pythagoras himself,but rather from Hippasus (on the controversy about the evidence forthis split into two groups of Pythagoreans see the fifth paragraph ofsection 4.3 above). For fourth-century Greeks as for modern scholars,the question is whether the mathematical and scientific side of laterPythagoreanism derived from Pythagoras or not. If there were nointelligible way to understand how later Pythagoreanism could havearisen out of the Pythagoreanism of theacusmata, the puzzleof Pythagoras’ relation to the later tradition would beinsoluble. The cosmos of theacusmata, however, clearly showsa belief in a world structured according to mathematics, and some ofthe evidence for this belief may have been drawn from genuinemathematical truths such as those embodied in the“Pythagorean” theorem and the relation of whole numberratios to musical concords. Even if Pythagoras’ cosmos was ofprimarily moral and symbolic significance, these strands ofmathematical truth, which were woven into it, would provide the seedsfrom which later Pythagoreanism grew. Philolaus’ cosmos and hismetaphysical system, in which all things arise from limiters andunlimiteds and are known through numbers, are not stolen fromPythagoras. They embody a conception of mathematics, which owes muchto the more rigorous mathematics of Hippocrates of Chios in the middleof the fifth century; the contrast between limiter and unlimited makesmost sense after Parmenides’ emphasis on the role of limit inthe first part of the fifth century. Philolaus’ system isnonetheless an intelligible development of the reverence formathematical truth found in Pythagoras’ own cosmological scheme,which is embodied in theacusmata.

The picture of Pythagoras presented above is inevitably based oncrucial decisions about sources and has been recently challenged in asearching critique (Zhmud 2012a). Zhmud argues that the consensus viewof Pythagoras’ cosmos as presented above is based on the faultyassumption that there was a progression from myth and religion toreason and science in Pythagoreanism. In many cases, he argues, theevidence suggests that early Pythagoreanism was more scientific andthat religious and mythic elements only gained in importance in thelater tradition. The consensus picture of Pythagoras’ cosmosassigns number symbolism a central role and treats thetetraktys, the first four numbers, which total to the perfectnumber ten, as a central concept. Zhmud argues that thetetraktys and the importance of the number ten do not go backto Pythagoras but flourish in the Neopythagorean tradition, whilehaving roots in number speculation in the Academy associated with suchfigures as Plato’s successor Speusippus. One of the centralpieces of evidence for this view is that thetetraktys doesnot first appear until late in the tradition, in Aetius in the firstcentury CE (DK 1.3.8). However, thetetraktys does appear inone of theacusmata in a section (82) of Iamblichus’On the Pythagorean Life that is commonly regarded as derivingfrom Aristotle. Zhmud himself agrees that sections 82–86 ofOn the Pythagorean Life as a whole go back to Aristotle butsuggests that theacusma about the tetraktys was apost-Aristotelian addition (2012a, 300–303). Once again sourcecriticism is crucial. If theacusma in question goes back toAristotle then there is good evidence for thetetraktys inearly Pythagoreanism. If we regard it as a later insertion intoAristotelian material, the early Pythagorean credentials of thetetraktys are less clear.

Zhmud supports Pythagoras’ position as genuine mathematicianrather than someone interested only in number symbolism by pointing togaps in the development of early Greek mathematics. Although there isno explicit evidence, Pythagoras is the most likely candidate to fillthese gaps. Thus between Thales, whom Eudemus identifies as the firstgeometer, and Hippocrates of Chios, who produced the firstElements, someone turned geometry into a deductive science(Zhmud 2012a, 256). Similarly, Hippasus’ experiment with bronzedisks to show that the concordant intervals of the octave, fifth andfourth were governed by whole number ratios is too complex to be afirst attempt so that once again someone must have discovered theratios in a simpler way earlier (Zhmud 2012a, 291). In each case Zhmudsuggests that Pythagoras is that someone. Finally, the study ofproportion ties together arithmetic, geometry and harmonics and Zhmudargues that, although there is no explicit fourth-century evidence,later reports which assign Pythagoras the discovery of the first threeproportions (Iamblichus,Commentary on Nicomachus’Introduction to Arithmetic 100.19–101.11) are likely to goback to Eudemus (2012a, 265–266). Such speculations have someplausibility but they highlight even more the puzzle as to why, ifPythagoras played this central role in early Greek mathematics, noearly source explicitly ascribes it to him. Of course, some scholarsargue that the majority have overlooked key passages that do assignmathematical achievements to Pythagoras. In order to gain a roundedview of the Pythagorean question it is thus appropriate to look at themost controversial of these passages.

Some scholars who regard Pythagoras as a mathematician and rationalcosmologist, such as Guthrie, admit that the earliest evidence doesnot support this view (Lloyd 2014, 25), but maintain that theprominence of Pythagoras the mathematician in the late tradition mustbe based on something early. Others maintain that there is evidence inthe sixth- and fifth-century BCE for Pythagoras as a mathematician andcosmologist. They argue that Herodotus’ reference to Pythagorasas a wise man (sophistês) and Heraclitus’description of him as pursuing inquiry (historiê), showthat he was regarded as practicing rational cosmology (Kahn 2002,16–17; Zhmud 2012a, 33–43). The concept of a wise man inHerodotus’ time was very broad, however, and includes poets andsages as well as Ionian cosmologists; the same is true of the conceptof inquiry.Historiê peri physeos (inquiry concerningnature) is later used to refer specifically to the inquiry into naturepracticed by the Presocratic cosmologists, but Herodotus’ usageshows that at Heraclitus’ timehistoriê referredto inquiry in a quite general sense and has no specific reference tothe cosmological inquiry of the Presocratics (Huffman 2008b). In oneinstance in Herodotus it refers to inquiry into the stories ofMenelaus’ and Helen’s adventures in Egypt (II. 118).Heraclitus may be thinking of Pythagoras’ inquiry into andcollection of the mythical and religious lore that is found in theacusmata (Huffman 2008b; see also Gemelli Marciano 2002,96–103). Thus the description of Pythagoras as a wise man whopracticed inquiry is simply too general to aid in deciding what sortof figure Herodotus and Heraclitus saw him as being. It is certainlytrue that Empedocles shows that the roles of rational cosmologist andwonder-working religious teachercould be combined in onefigure, but this does not prove these roles were combined inPythagoras’ case. The only thing that could prove this inPythagoras’ case is reliable early evidence for a rationalcosmology and that is precisely what is lacking.

There is more controversy about the fourth-century evidence. Zhmudargues that Isocrates regards Pythagoras as a philosopher andmathematician (2012a, 50). However, it is hard to see how the passagein question (Busiris 28–29) supports this view. Nowherein it does Isocrates ascribe mathematical work or a rational cosmologyto Pythagoras. He reports in general terms that Pythagoras brought“ other philosophy” to Greece from Egypt but what heemphasizes is that Pythagoras was “more clearly interested thanothers in sacrificial rites and temple rituals.” It is true thatearlier, in a passage that does not mention Pythagoras(Busiris 22–23), Isocrates had said that some of theEgyptian priests studied mathematics but if Isocrates thoughtPythagoras also brought mathematical learning from Egypt he has chosennot to say so explicitly. What Isocrates emphasizes about Pythagorasis what the rest of the early tradition emphasizes, his interest inreligious rites. Fr. 191 from Aristotle’s lost work on thePythagoreans reports that Pythagoras “dedicated himself to thestudy of mathematical sciences, especially numbers” and Fragment20 from Aristotle’sProptrepticus says that Pythagorassaid that human beings were born to contemplate the heavens anddescribed himself as an observer of nature (Zhmud 2012a, 56 and259–260). Unfortunately, in neither case are the words inquestion likely to be Aristotle’s. Fr. 191 comes from a book onmarvels by Apollonius (2nd BCE?). The words in question come beforeApollonius mentions Aristotle and, as Burkert pointed out (1972a,412), are overwhelming likely to be by Apollonius himself, since theyserve as the transition sentence between Apollonius’ account ofPherecydes and his account of Pythagoras. In the face of the hugeextant corpus of Aristotle’s works in which he never ascribesany mathematical work to Pythagoras, a single sentence that is notascribed directly to Aristotle and that, in terms of function, appearsto be the work of Apollonius and not Aristotle cannot with anyconfidence be used as evidence that Aristotle regarded Pythagoras as amathematician. The same situation arises with Fr. 20 of theProtrepticus. If the words in question were by Aristotle theywould be his sole statement that Pythagoras was a natural philosopher.The case of Fr. 20 is even more tenuous than that of Fr. 191. Fr. 20comes from Iamblichus’Protrepticus, large parts ofwhich are likely to derive from Aristotle’s lostProtrepticus but, as is his practice, Iamblichus does notmake any explicit reference to Aristotle. The further problem with Fr.20, as Burkert noted (1960, 166–168), is that the same story istold first about Pythagoras and then immediately afterwards aboutAnaxagoras: both are asked why human beings were born and both answer“to contemplate the heavens” (Iamblichus,Protrepticus 51.8–15). This awkward repetition of thesame story about two different people immediately suggests that onlyone story was in the original and the other was added in the latertradition. This suggestion is strikingly confirmed by the fact thatAristotle does tell this story about Anaxagoras in his extant works(Eudemian Ethics 1216a11–16) but not about Pythagoras.Thus, if the passage in Iamblichus’Protrepticus is, infact, from Aristotle, it is very likely that only Anaxagoras appearedin Aristotle’s version and that Pythagoras was added in thelater tradition, perhaps by Iamblichus himself. Since these twopassages are unlikely to be from Aristotle, there are no references toPythagoras as a mathematician or as a natural philosopher either inAristotle’s extant works or in the fragments of his works.Aristotle only knows Pythagoras as a wonder working sage and teacherof a way of life (Fr. 191). Aristotle’s attitude is similar tohis predecessors in the earlier fourth century: Plato’s solereference to Pythagoras is as the founder of a way of life andIsocrates emphasizes both the way of life and the interest inreligious ritual.

What about the pupils of Plato and Aristotle? As discussed in thesecond paragraph of section 5 above, Eudemus, who wrote a series ofhistories of mathematics never mentions Pythagoras by name. Argumentsfrom silence are perilous but, when the most well-informed source ofthe fourth-century fails to mention Pythagoras in works explicitlydirected towards the history of mathematics, the silence meanssomething. There are only two passages in which Pythagoras isexplicitly associated with anything mathematical or scientific bypupils of Plato and Aristotle. First, Aristotle’s pupilAristoxenus reports that Pythagoras “most of all valued thepursuit (pragmateia) of number and brought it forward, takingit away from the use of traders, by likening all things tonumbers” (Fr. 23). Zhmud translates pragmateia as“science” (2012a, 216) so that he has Aristoxenusattributing the invention of the science of number to Pythagoras but,while Aristoxenus does usepragmateia to mean science in somecontexts, it more commonly simply means “pursuit” (Huffman2014b, 292). Here surely it must mean “pursuit,” becausePythagoras is presented as taking it away from the traders and we canhardly suppose that the traders were engaged in the theoreticalscience of arithmetic. Moreover, Aristoxenus explains what he means inthe final participial phrase. He is not ascribing rigorous mathematicswith proofs to Pythagoras but rather says that Pythagoras was“likening all things to numbers”. This is consistent withthe moralized cosmos of Pythagoras sketched above in which numbershave symbolic significance. The second important passage isPlato’s pupil Xenocrates’ assertion that Pythagoras“discovered that the intervals in music, too, do not arise inseparation from number” (Fr. 9). Xenocrates is being quoted herein a fragment of a work by a Heraclides (Barker 1989, 235–236),perhaps Heraclides of Pontus. There is controversy whether thequotation of Xenocrates is limited just to what has been quoted in theprevious sentence or whether the whole fragment of Heraclides is aquotation of Xenocrates. Burkert (1972a, 381) and Barker (1989, 235)argue that it is probably just the first sentence that Heraclidesascribes to Xenocrates, while Zhmud would include at least a secondsentence in which Heraclides presents Pythagoras as pursuing a programof research into “the conditions under which concordant anddiscordant intervals arise” (Zhmud 2012a, 258). If the secondsentence is accepted then Xenocrates clearly presents Pythagoras as anacoustic scientist. It seems most reasonable, however, to accept onlythe first sentence as belonging to Xenocrates. If the quotation fromXenocrates does not break off at that point, there is no other obviousbreaking point in the fragment and the whole two pages of text must beascribed to Xenocrates. The problem with ascribing it all toXenocrates is that Porphyry introduces the passage as a quotation fromHeraclides, which would be strange if everything quoted, in fact,belongs to Xenocrates. If just the first sentence comes fromXenocrates, then all he is ascribing to Pythagoras is the recognitionthat the concordant intervals are connected to numbers. It is easy toassume, as Zhmud does, that Xenocrates is saying that Pythagoras wasthefirst to discover that the concordant intervals aregoverned by whole number ratios but Xenocrates’ remarks need notmean this. Xenocrates’ comments might well come from a contextlike that in the fragment of Aristoxenus, above, i.e., a context inwhich Pythagoras is presented as likening all things to numbers andarguing that numbers in some sense explain or control things. In sucha context Xenocrates would not be making the point that Pythagorasdiscovered the whole number ratios but rather that he found out thatconcords arose in accordance with whole number ratios, perhaps frommusicians (who discovered them first not being the issue), and usedthis fact as another illustration of how things are like numbers.Thus, the fragments of Aristoxenus and Xenocrates show that Pythagoraslikened things to numbers and took the concordant musical intervals asa central example, but do not suggest that he founded arithmetic as arigorous mathematical discipline or carried out a program ofscientific research in harmonics.

Controversy concerning Pythagoras’ role as a scientist andmathematician will continue. Indeed, Hahn has recently endorsed manyof Zhmud’s arguments and argues that Pythagoras was a rationalcosmologist, who was further developing a project begun by Thales toconstruct the cosmos out of right triangles. Hahn admits, however,that his thesis is “speculative” and “acircumstantial case at best” (2017, xi). It should now be clearthat decisions about sources are crucial in addressing the question ofwhether Pythagoras was a mathematician and scientist. The view ofPythagoras’ cosmos sketched in the first five paragraphs of thissection, according to which he was neither a mathematician nor ascientist, remains the consensus.

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