In contrast to the previous posts about Socrates’s satire of the beautiful city here, here and here, this essay will focus on the second image, the ascent of philosophy and the Pythagoreans. This image plays a central role in the Republic. As part one indicated here, this is the hinted at city in speech for philosophers, closer to the little city of Socrates and his students, Plato’s Academy or the Pythagorean way of life than to Glaucon’s city, the one worked out in the Republic. This is a meaning or image that Plato expected his students to pursue from the Republic, one that conflicts with its surface meaning and is not worked out within it.
If poetry is not exiled, if the Pythagoreans have a way of life with which Plato, in part, identifies and if Plato’s own thoughts, not fully revealed in the dialogues, are in conversation with Pythagoras, then the hints here are very important. I should note at the outset, however, that it is a serious mistake to label Plato a Pythagorean a) because we lack deep knowledge of what the Pythagoreans said,* b) because Plato is a great philosopher and writer in his own right, c) because Plato transformed every story he laid his hands upon, and there is no reason to think that his adaptation of Pythagoras or Pythagoreans is simpler than his ordinary writing (see here on the ring of Gyges) and d) because Socrates’ emphasis on justice and going down to fight tyranny has, from reputation at least, no direct counterpart in Pythagoras or those who followed him.
With regard to c), for example, Pythagoras had an animus against beans for which no evidence survives in Plato. The Pythagorean sect had other putative rules which are eccentric:
Whether this account is accurate or not (much is lost in the mists of time), let us compare the martial way of life in the city of the guardians - Glaucon's city - with that of the Pythagoreans. For the latter is also, broadly speaking, with its focus on inquiry, a metaphor for Socrates's or Plato's way of life.
In the Republic, Plato would have altered such rules in a Glaucon-like direction (the leading male warrior breeds with the most women, the noble lie divides the castes according to supposed metals in their souls, everyone over 10 is expelled, and the like). Thus, in Plato's satiric magic, the "beautiful" city or sect differs markedly from Pythagoras and the Pythagoreans. Imagining that Plato, however, is merely a Pythagorean, Bertrand Russell, for example, neglects such distinctions.
With regard to d - the issue of justice and going down to fight - the Pythagoreans established a way of life different from and critical of ordinary (Crotonian) life, and perhaps took part in politics there. But what they stood for, aside from existing as a contrary way of life, is unclear.
In contrast, even as a private man conversing with those who were thought themselves wise and were not, Socrates's questions, as he suggests in the Apology, stung and made him enemies. At the least, Socrates raised issues of justice (and truth) sharply and more explicitly than Pythagoreans did.
Now, the Crotonians moved against the Pythagoreans, burning their meeting-places, as the Athenians moved against Socrates. But Socrates choose in the Apology to stand up for philosophy, modelling what we call civil disobedience even at the cost of his life. In contrast, Pythagoras left Croton. See here, here and here.
For reasons c) and especially d), it would also be unwise to speak of Pythagoras as a proto-Platonist or a proto-Socrates. But that Plato received something very important, perhaps even in the design of his works, from Pythagoras, now seems, after Jay Kennedy’s work, to be a promising path for investigation – see here; that the Platonic mysteries – where one might eventually get to who fashioned arguments daily, starting with these texts - are, among other matters, fiercely numerical and musical, grows out of Pythagoreanism.
Socrates speaks directly of the Pythagoreans in two central contexts in the Republic, in the passage on the ascent from the cave in book 7 and in the discussion of Homer’s poetry in book 10.
In book 7, Socrates says: ”We may suppose that as we are given eyes to study the stars so we have ears to hear harmonic movements and that consequently a certain kinship exists between the two forms of knowledge. This is the position taken by the Pythagoreans, Glaucon, and we agree with them, don’t we?” Note the leadingness of the rhetoric: whatever Glaucon knows of Pythagoras - though he has interests in music in the text, one might doubt the depth of his knowledge here - he is pushed to make blithely some large assent…
One of the Pythagoreans’ insights was into the relation of music to the ordering of the universe and the soul.** Socrates’s notion here recalls the role of music in the tempering of the warrior athlete in book 2 and more deeply, the notion of the war in the soul of the unjust man, its jangled amusicality, contrasted with the music in the soul of the just man at the end of book 4. Plato does not name the Pythagorean origins of this thought but the wording is plain. Of the just man, Socrates says:
“The reality is that justice is not a matter of external behavior but the way a man privately and truly governs his inner self. The just man does not permit the various parts of his soul to interfere with one another or usurp each other’s functions. He has set his own life in order. He is his own master and his own law. He has become a friend to himself. He will have brought into tune the three parts of his soul: high, middle, and low, like the three major notes of a musical scale, and all the intervals in between. When he has brought all this together in temperance and harmony, he will have made himself one man instead of many." (443c-e)
That the soul is one instead of many or anarchic, that a just man (or woman) is a friend to himself, that the ordering is musical, is here made plain.
In contrast in book 4, Socrates speaks of the cacophony or civil war in the unjust man's soul:
“Our next task is to examine the unjust man. Here the soul’s three parts become contending factions, meddling in one another’s business to the point where civil war [stasis] breaks out. Or one part of the soul rebels against the rest with the purpose of seizing the governing power, usurping the very authority to which it is properly subject. Such anarchy and dissension can only be the product of intemperance, cowardice, ignorance and every kind of vice. All these evils together concoct the essence of injustice." (444b)
Socrates’s discussion in the Republic prefigures Aristotle’s analysis of friendship and tyranny in book 7 of the Nicomachean Ethics. There, too, the parts of the tyrant’s soul are at internal war with one another. A tyrant is no friend to himself.
In book 7, Glaucon answers Socrates’s leading question: “Yes.” Here as is increasingly shown in the book, that there is thought behind his answer beyond trying to please Socrates is unclear. Glaucon is greedy to know everything; leave nothing, he repeats, out. A reader detects how grasping he is, wanting every bit of the philosophical “relishes.” Socrates responds: I’m afraid I must; I’ve said what I can communicate. Of education, Glaucon finally gets it and says ”you outline an enormous undertaking..." (see also line 531c).
In book 7, Socrates responds:
“Then let us consult them [the Pythagoreans] since our task is difficult. Let us see what their opinion is and whether they can add to our knowledge. At the same time, we must guard well our own interests.”
Even without the usual Platonic shifting of characterizations (one would have to know the original), that the dialogue with the Pythagoreans is largely hidden here and that Socrates and probably Glaucon are not Pythagoreans but “guard our own interests” is made clear. “Our own interests” are at least Socrates’s desire to warn Glaucon from tyranny – one core point of the current conversation – and Socrates’ and Plato’s own, largely hidden vision of philosophy, a conversation in which we neither know, more than pointing through the dialogues, where Plato got to, nor more than hints of Pythagoras either. So these are just some initial markers for a long trail for intrepid students, part of which – most of Pythagoras at the least - is probably lost to us…
“What do you mean?”
“We must not permit our students to learn things that are imperfect or that do not lead to results that all studies should attain. We just cited shortcomings in studies of this kind in astronomy. Do you know that the same problems exist in harmony? Those who teach the subject attempt to measure chords within the context of audibility and so repeat the waste of time we discerned in astronomy.”
In astronomy, Socrates recommends plotting the future course and coordinates of stars – seeing the unseen or invisible, future movements, the ideas that govern the seen – not just focusing on the visible as “decorations on a painted ceiling.” (529b) In both astronomy and music, he calls for an ascent to the invisible, to number.
Glaucon suddenly can talk about something he knows about and, to try to impress Socrates, waxes eloquent:
“Yes, by the gods, that’s right [there are many oaths by Glaucon in book 8, a strong, perhaps hasty affirmation beyond what he understands]. And they are so absurd too, with their talk of ‘dense’ notes and the like. They press their ears against the instruments as if they were trying to overhear a voice from next door. Then some claim to detect an extra note between the intervals which should henceforth be accepted as the smallest interval and the basic unit of measurement, others insist that it is no different from the notes already sounded. Both parties prefer their ears to their intelligence.” (531b)
Socrates’s emphasis here is on the theory of numbers in music that go beyond the audible – “they seek numbers in the accords they hear, but they do not make the ascent to the universal problems of number where they might consider which numbers are harmonious and which not, and why.” (531c). The idea here is at least broadly speaking, Pythagorean.
In a parallel vein, Socrates conjures knowledge of the invisible movement of the stars. Such knowledge ascends to a noetic realm in which there are no contingent assumptions, where one reasons from idea to idea. This is a kind of glorious metaphor though it differs from all science and mathematics, however influenced by or otherwise realizing of Plato’s vision in contrast to all forms of empiricism, for in these fields, one may alter but not escape assumptions or auxiliary statements (see Gilbert, Marx’s Politics, ch. 1, 11 ; Hilary Putnam, "On the Corroboration of Theories")
The other invocation of the Pythagoreans is in book 10. There, Socrates offers the supposed indictment of the poets or painters as imitators of imitations of reality – reality at a third remove, shadows reflected from the reflections of the visible world. There is the idea of a chair (comparable to the idea, or the proof in the mind, of a geometric theorem), the particular realization in a real chair by a worker in wood, and the third remove or representation in a painting of the chair.
On behalf of this reductionist argument, one might say that Socrates in the Republic spends a lot of time cleaning up and debunking poetry. So surely, many have concluded, censorship is a main message of the city in speech.
But book 10 concludes with a fabulous poetic story, the myth of Er. It goes along with the resonant myths of the cave and the wrenching ascent, as well as the ring of Gyges. The Republic is full of metaphors of going down, and beautifully spun images to give one a glimpse or indication of, though not an argument about, where Plato’s vision goes. Are all these poems, particularly the ring of Gyges which is central to the illusions of all politics (that any American President is a nice guy rather than, as the head of the Empire, inevitably a killer), simply noble lies? That seems unlikely.
There are noble lies – lies which allegedly further a common good and advance the political purposes of the legislator – and stories which go beyond this (the cave, for example or the myth of Er). What they go beyond this to is hinting at what the vision is, not the steps of argument, if, indeed, there are such steps, which get one there (on the face of it, Socrates and Plato may be closer to Buddha, though with a higher measure of mathematics, music and argument than the latter – though, of course, Buddhists are famously interested in mathematics and music, too). See here.
Students might notice: the beautiful poetry in the writing satirizes the overt censorship theme of the text.
“Then if a man were able to produce both realities and illusions, do you think he would choose to work with illusions and devote himself to them as the best that life could offer.’ [Socrates]
“No” [Glaucon]. But what if the experience of illusions, the question of opinion in the cave leads, and is the only path, to the ascent to the light? Then one would have to question illusions to turn wrenchingly and ascend toward the light.
“I would suppose, if he actually knew the things he imitates, he would do those things and leave off imitating. Would he not try to leave as his memorial a tangible record of many splendid deeds? Would he not rather be the one who is praised than the one who praises?” To the extent that Socrates knows – that he alone among the dead of the cave can see - he does not leave famous deeds, except, of course, the most famous one: going to his death to preserve philosophy. And of course, Plato’s written accounts we still read and teach 2400 years later (now, there is a fire…).
There is something unpleasant and arrogant in Socrates’s line of questioning here, something which perhaps Plato wanted his students to consider…
“I should think he would. In deeds there is far greater honor and profit.”
“Well, let us not demand an accounting from Homer or any of the other poets whether any of them were really doctors or whether they merely copied off medical language. Nor will we ask whether any poet, past or present, has ever healed men as Asclepius healed them [See here on Asclepius, Epidauros and legislation], nor whether he has left behind heirs to his medical knowledge the way Asclepius did. And we shall forgo questions about any of the other arts. But surely it is fair to inquire of Homer concerning the greatest and most estimable things he undertook to speak of: wars, military skills, the governing of cities, and the education of men.”
“’Beloved Homer’” we would say…” A student might take in the word beloved here.
“if your words about virtue are not at third remove from the truth, if you don’t fit our definition of the imitator as an inventor of dreams – if you are instead next neighbor to truth and can identify the kinds of behavior that make men better or worse in both private and public life – then tell us what city has ever been better governed thanks to you. Sparta owes thanks to Lycurgus. Many other cities, small and great, owe a similar debt of thanks. What city gives you credit for having been a good lawgiver or benefactor? In Italy and Sicily, Charondas is so credited, among us it is Solon. Who credits you? Will there be any he can name?”
This is a grander version of tyranny - great legislation that shapes a people, a Socratic or Platonic version - than the one that possesses Glaucon (he merely aims for great personal pleasure: the “relishes”). It moves Glaucon who is also interested in virtue, in great accomplishments, in appearing good, to take in a wider picture, to think…
But a cleverer student of Plato’s might ask: despite the thinking about lawgiving in the Republic and the Laws, have “Socrates” and the Stranger shaped laws? In the Seventh Letter, Plato criticizes Dionysius for writing a bad book on legislation and says that he, Plato, who knows so much, will never write down what he knows about law-giving – what he means is not write so that it can be discerned without study. But note: on the account of the Seventh Letter, neither the Republic nor the Laws nor the Statesman, inter alia, is a book about laws…
Do Socrates or Plato pass through this seemingly Socratic criterion for Homer or does the great-contribution-to-war-and-legislation filter, in the most obvious sense, exclude not only Homer but these philosophers as well?
Socrates was but a soldier, though a formidable one (see Alcibiades’ account at the end of the Symposium and here).
“I don’t suppose so. Even Homer’s followers have never mentioned any.”
“Can we recall any war in Homer’s time won under his command or as a result of his advice?”
“What about the inventions and practical devices that we expect from men skilled in crafts? Thales the Milesian and Anarcharsis from Scythia are credited with many. [Thales predicted an eclipse on May 28, 585 BC. Anacharsis invented the anchor and the potter’s wheel.] How about Homer?”
“Well, if Homer has no record of public service, what report do we have of him as a private educator around whom men gathered because they cherished his teachings? Did he and they pass on to posterity a kind of Homeric way of life after the manner of Pythagoras? Such a legacy is counted among Pythagoras’s greatest achievements. Even now his successors use the term Pythagorean to denote a certain way of life that many of our contemporaries look upon with respect.”
“Once again, Socrates, there is no report. To look on this companion Creophylus [his name means member of the beef tribe, perhaps a poet] as representative of Homeric culture might turn out to be a more promising subject for ridicule than the name Creophylus itself. For if the things said about Homer are true, he was neglected by that child of the flesh during most of his lifetime.”
Glaucon goes quickly off the deep end once again here, making fun of Homer as of the member of the beef tribe. (Athletes like Polydamas who eat “great amounts of beef” are a theme in the Republic, as Socrates uses this example to harm Thrasymachus’s argument about the advantage of the stronger in book 1, line 339c-d****). The Pythagorean way of life probably included vegetarianism, so that is the source of the joke (Glaucon knew that much about Pythagoreans…).*****
Pythagoras is known, though he left no writings, through Pythagoreans, for thoughts on the transmigration of souls, human and animal. This is, of course, a feature of the Myth of Er in book 10 (Plato’s tale probably also reflects on the Athenian mystery religions). But here we have a seventh powerful memory of Pythagoras in the text of the Republic: 1) the divided line which obeys the golden mean, 2) the importance of numbers, 3) music (and numbers), 4) the stars (and numbers), 5) the way of life, 6) the musical ordering of the soul, and 7) the transmigration of souls.
Even without much knowledge of Pythagoras, the conversation with Pythagoras seems powerful, inspiring.
For Pythagoreans, consuming the bodies of animals was as objectionable as consuming people.
One might ask two questions of Plato in book 10. First, why is the Pythagorean way of life good? It is not said directly here (though recall book 7), but it is a life of philosophy, mathematics and music. The philosophy that a student slowly ascends toward from here, however, is perhaps more Plato’s.
If one takes the 7 Pythagorean aspects above together, they go far beyond the supposed single mention of the Pythagorean way of life in the Republic in the Stanford Encyclopedia of Philosophy. Yes, this connection is somewhat difficult to decipher, but the path of thought in the dialogues points beyond itself in ways which are sometimes Pythagorean or more aptly, in conversation with Pythagoreans (again, since no writing of Pythagoras survives and there are but tales of the way of life and writings by later Pythagoreans, the conversation of Plato and Pythagoras is not quite audible).
In addition, Jay Kennedy has just cast new light on the very organization of the dialogues. The divided line in book 6 is a ratio between the perceptible world and the noetic world which illustrates the golden mean (1.618: discovered by Theano, Pythagoras’s wife). See here and here. It occurs , Kennedy has suggested, .618 through the Republic. This ratio is powerful in architecture and in the human body, among other things, so Plato might indeed have adapted this beautiful number to the organization of the Republic.
Moreover, I have previously emphasized, that in book 9, the just man’s happiness is 729 times the “pleasure” of the tyrant (587e); correspondingly, “the tyrant’s pain may be measured by the same number.” 729 is 9 to the power of 3 or 3 to the power of 6 and this - otherwise inexplicable - calculation seems startlingly Pythagorean (or Plato inventing in a Pythagorean vein). As I have recently discovered, Nietzsche, a great classicist, has a striking essay which talks about the mysticism of numbers for Pythagoreans - 9 being the number for justice - and so this seems an eighth aspect.
"We mention, in characterizing their method of equations, that justice consists of like times like - in other words, of quadratic number; for this reason [the number] 4, or especially 9 (the first uneven quadratic number), was called justice." (see "the Pythagoreans" in Nietzsche, The Pre-Platonic Philosophers, trans. Greg Whitlock, p. 136.)
One wonders about the series of multiples and sums and squares which Pythagoreans (or Plato) may have worked on (multiples of 9 for example, are in sum also such multiples i.e 9 x 9=81 whose sum is 9, 9 x 67=603 whose sum is 9, 9 x 505 is 4545 whose sum is 18, 9 x 1041= 9369 whose sum is 27...)
Thus, some of the philosopher’s journey that Plato encourages is Pythagorean, and the suggestions go far deeper than the surface. If Jay Kennedy is right, Pythagorean principles are employed by Plato in the very writing of the dialogues and where ideas occur, how they are to be taken. If my way of reading the dialogues is right, however, these are no more than hints, since the bearing of the ideas would have to be assessed in the interpretation of the dialogue and not externally according to a code. How would one know, for example, that the satires of censorship of poetry in book 2, 3 and 10 are that, rather than to be taken, along with the supposed city in speech, seriously? That all the stories or myths in the Republic somehow aren’t poetry also to be censored as opposed to resonant clues, eventually, perhaps, to be figured out by argument (does the ring of Gyges’ story have a negative number in the text as a whole?)?
Leo Strauss, too, is into numerical clues (see Thoughts on Machiavelli), but the idea that one infers truth mechanically from the placement rather than a student’s thinking about the idea or metaphor is, prima facie, implausible. When I say such divisions are but a hint to provoke thought, I mean they are not simply a code, not a secret and final hidden meaning to be discovered solely by cryptography.
But second, is there no Homeric way of life? Plato takes Homer as a single, though wandering author. But like all epic poems, we now know that Homer was constructed by many singers, who travelled around, and told their tales in rhythm and rhyme, both poetic devices helping them remember. And are not rhythms, following the Pythagoreans, musical…?
In contrast to Homer, Socrates mentions Protagoras and Prodicus who took pay for teaching which Socrates did not.
“After all, Protagoras of Abdera and Prodicus of Ceos and others are able to attract many of their contemporaries to their private lessons, where they impress upon their listeners that without instruction from them, they will be able to govern neither their cities nor their own homes. This is the wisdom that makes them so beloved that their own students all but carry them about on their shoulders. (600c-d)
Socrates protested against sophists like Thrasymachus, thinking them too tied to the rich and to contemporary illusions (the cave). Perhaps he had more respect for his own teachers, however, and one should be careful of this judgment, too, think out the relationship many-sidedly. For many teachers, though perhaps not Thrasymachus, are partly outside, though partly still within the cave.
Now supposedly, by this argument, Homer and his fellow singers, are a contrast:
“If Homer had helped men to achieve excellence, surely his contemporaries would not have let him and Hesiod live as itinerant reciters of poetry. Would they not sooner have parted with gold than with their poets? [recall again that Socrates does not teach for pay] Would they not have insisted that they stay with them and dwell in their homes? Failing that, would they not have followed them and attended them until they were able to receive from them an adequate education.” (600d-e)
But was not Socrates lifted up on the “broad shoulders’ of Plato (what the name Plato means)? Was he not fed, in his self-chosen poverty, by his students and friends?
Were not the poets named Homer fed when they recited the stories of Greece as the night fell, as the lights of the fire and the stars flickered in the darkness (remember the points of light in the torch-race on horses of book 1)?
Do not the Greeks honor Homer, “beloved” Homer, above others? Is there not a different and dialectical relationship between poetry and philosophy than book 10 and the authoritarian and censorious passages suggest?
Aren’t these words, on the lips of all, notably Plato’s Socrates who describes the relation of philosophy to the cave by summoning Achilles’s cry: better to be the meanest slave on earth, to convey the dazzlingness of the ascent? Is not this the setting on fire of Greek and later Arab and Western imagination? If Plato inspires Wilde and Tolkien and this is brilliant and enduring, are we to ignore the light shining from Homer inspiring Plato on the meaning of the cave and the sun-like idea of the good? See here.
Let us now consider a potential quasi- or one-sidedly Pythagorean objection on behalf of the ostensible seriousness of the city in speech. The way to philosophy, this Pythagorean might say, is mathematics and this is shown in the Republic in book 7. One works with numbers and the stars (and music) to discover the invisible movement of the heavens, the ideas, mathematical and physical, which govern where the planets will be, when an eclipse, as Thales predicted, will occur.
In fact, this Pythagorean might argue, teaching mathematics in the city in speech is sufficient to engender philosophers. Consider the education of Dionysius in the Seventh Letter; when Plato tries to teach him geometry in their first and only session together, Dionysius finds this too trying and turns away. Mathematics is a clue and paradigm, the entry point of philosophy, and the way to wherever philosophy gets.
One might acknowledge the aspect of truth in the point (perhaps it is the way for some). Yet, in contrast, one might insist: Socrates is profoundly into justice. Plato’s idea of the good fuses these two things (a concern about justice, a concern about the invisible order of the cosmos). While a possibility, this Pythagorean’s overemphasis on mathematics and a cocoon until one reaches the age of 50 is not right (the Republic's age for starting to do philosophy, echoed by a proposal in the Laws). Even about mathematics, one needs, as the dialogue with the slave in the Meno shows, questioning…
And if one can ask questions about the diagonals of squares, why not about justice? Why not about poetry? Why must strict censorship occur except to encourage the advantage of the stronger – the philosophical tyrant – and what happens, as Socrates asks Thrasymachus in book 1 of the Republic, if the stronger mistakes his advantage?
The approach of books 2-5 to philosophy is a betrayal of the Socrates who neither knows nor thinks that he knows.****** This Pythagorean objection suggests the impossible: questioning about mathematics, but keeping one’s mouth firmly closed about justice. To draw a picture of this quasi-Pythagorean universe, these books deny Socrates, force the neophyte philosopher to hop around on one leg, and breed stupidity and holding one’s tongue about the most important matters. Note that the Pythagoreans lived differently from other Greeks and no sign survives in their way of life of the warrior authoritarianism in kallipolis...
Perhaps one might say of their silence about justice in the larger community somewhat anachronistically: there's was a utopian view, embodied in their way of life (we do not know how or whether they participated in the politics of Croton). In contrast, Socrates ultimately advocated a reformation of political life (for instance, in suggesting longer trials in cases with a potential death sentence about the gods, protecting philosophical inquiry...).
At the least, Socrates and Plato differed from such (quasi-)-Pythagoreans******* in this central respect.
*Bertrand Russell's History of Philosophy, 1945, ch. 3 is the source for this list, and one might be skeptical of this, too. His account later in the History of "Plato's" views is literal, as if reading a tract. In ch. 14, Russell makes the good point that cultivation of a noble lie - for instance, the myth of the metals - is incompatible with philosophy:
But Plato is not Socrates or a character in the dialogues.
Russell fails, at real intellectual cost, to ask himself the questions: what is the point of a dialogue? and what did Plato intend or hope for philosophical students - as opposed to ordinary readers - to learn?
**Nietzsche claims Pythagoras was not a philosopher or mathematician but transmitter of a way of life and secret doctrines about the transmigration of souls
***This image also occurs in the Athenian Stranger’s account in book one of the Laws.
****"You are not going to argue that we non-athletes have a like interest in such a diet or that it would be good for us to follow it?”
*****Again, Pythagoras famously rules out beans. And one must not eat parts of even those animals (perhaps non-sacrificial) permitted, i.e. hearts. So as recorded in legend, the Pythagorean stance, as (semi-)vegetarians, is, seemingly, mixed...
******This point is explored in the Apology, and in a more worked out way, in Alcibiades 1, lines 110e-118c.
*******And the Pythagoreans, with the notion of transmigration of souls from human to animal to human, were also spiritually innovative in a way which challenged the Greek status quo and particularly the gods of Athens (the charge against Socrates). Thus, even for Pythagoreans, mathematics is important but insufficient. Even for Pythagoreans, there was a broad tapestry of thought about spirituality and social/political connection...