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How has modern scholarship approached the relationship between Plato and the Pythagoreans? The unavoidable point of departure is Aristotle. His testimony suggests that Plato did not simply take inspiration from the Pythagoreans. He may even have appropriated their doctrine, if not plagiarised it:

“After the philosophies we have discussed, the doctrine of Plato arose. In many respects it follows that of the Pythagoreans, but it also has features of its own, foreign to the philosophy of the Italians.”

Aristotle therefore speaks of both continuity and difference. But can we draw a clear boundary between the two? That is extremely difficult. The little we know of Pythagoras reached the outside world despite the rule of silence, despite the physical elimination of many Pythagoreans, and despite the later forgeries. And much of what survived was later reconstructed through a Platonic filter.

Heninger writes:

“Pythagoras and his immediate disciples offer us a small and elusive doctrinal nucleus which, in later centuries, was abundantly overlaid with layers of mythical attribution. The Renaissance received this rich tradition with syncretic zeal and reworked it accordingly. Modern historians of Philosophy are more demanding. They distinguish between Pythagoras himself; his earliest school, including Archytas and Philolaus; Plato and the early Academy; and Neo-Pythagoreanism, with its late classical forgeries.”

Zeller, who laid the foundations of modern historiography on ancient philosophy, asked in his Philosophy of the Greeks, published in 1855, whether the original Pythagorean philosophical system could be reconstructed. His conclusion was pessimistic:

“The alleged Pythagorean doctrine, which is unknown to the earliest witnesses, is Neo-Pythagorean.”

Burkert places the birth of Neo-Pythagoreanism within the ancient Academy itself. In his view, it was Speusippus, Xenocrates and Heraclides who identified the doctrine of their master Plato, and therefore their own philosophical positions, with the wisdom of Pythagoras:

“Aristotle views both Academic and Pythagorean philosophy from a certain distance, arriving at a personal position of active polemic. The ‘system of derivation’ is an achievement of Plato and the Academy, a genuinely Platonic transposition of an older Pythagorean philosophy of number, making use of certain improvements introduced within the circle of Archytas. The tradition that attributes the ‘system of derivation’ to the Pythagoreans, or even to Pythagoras himself, must be taken for what it is: testimony for the Ancient Academy, for Speusippus, Xenocrates and Heraclides, but not for historical Pythagoreanism.”

Zhmud is extremely cautious about attributing to Pythagoras ideas which, in the surviving evidence, seem to belong exclusively to the Academy:

“One of the central assumptions of the system described above consists in presenting Plato as the legitimate successor of Pythagoras and the Pythagoreans in an entirely positive sense. Yet the evidence at our disposal does not support the thesis that the early Academics projected Plato’s ‘unwritten doctrine’ onto Pythagoras. Indeed, Plato himself softened his dependence on the Pythagoreans; why, then, should the Platonists underestimate the originality of their master? Plato mentioned Pythagoras only once, and even then only as the founder of the ‘Pythagorean way of life’.”

These are only some of the positions taken by modern scholarship. One might be tempted to adopt a simple criterion when reconstructing Pythagorean doctrine: the older a testimony is, and the closer it appears to the original source, the more “Pythagorean” it should be considered. Yet from a strictly historical point of view, we cannot truly know what was said at the beginning.

We must also remember that Metaphysics is particularly exposed to misunderstanding. And a single misunderstanding, if it occurs early enough, can generate vast traditions built upon an original error. Even so, we may reasonably suppose that the source closest to Pythagoras, namely Philolaus, is the one that preserved the original ideas most faithfully. This is especially plausible if those ideas were bound to an extremely precise diagram, as we shall later argue.

After the violent suppression of the Pythagorean confraternities in Italy, Philolaus survived the fires and the lynchings and reached Thebes, as we know from the Phaedo. There he began to teach in the Greek motherland. Yet we do not know exactly what he taught, nor in what form. He probably confined himself to rather general matters, because when Socrates, that is to say Plato, speaks with two of Philolaus’ pupils, Cebes and Simmias, he seems almost proud to know Pythagoreanism at a much deeper level:

“But how is this, Cebes? Have you and Simmias not heard of such things, you who were pupils of Philolaus?”

For chronological reasons, Philolaus could not have received the doctrine directly from the lips of Pythagoras. We do not know who mediated it to him. Nor do we know what degree of initiation he had reached within the Confraternity of Wisdom. It would therefore be prudent to speak, more narrowly, of the Pythagoreanism of Philolaus.

What we do know is that Plato seems confident in distinguishing this Pythagoreanism of Philolaus, or at least the version he had the opportunity to study, from that of other, more dubious “Pythagoreans”. These figures were heirs to parallel currents strongly contaminated by empiricism, as we can see in a passage from the Republic on harmony, which we shall examine later.

Since we do not possess Philolaus’ work, we must rely on the accuracy of later quotations, while distinguishing authentic fragments from forgeries. Burkert and Huffman agree on the criteria of authentication. One firm point of the doctrine is the assertion of two opposed principles: the Limit and the Illimit. This confirms that Dialectic was one of the cornerstones of Pythagoreanism. The other firm point was Mathematics. Huffman argues that Philolaus was the first thinker “consciously and thematically to employ mathematical ideas in order to solve philosophical problems”.

One particularly solid example is Fragment 4:

“All that we can know is ‘number’: without it, nothing can be known or comprehended.”

We must remember that in antiquity the term number had a broader meaning than “digit”. It referred, more generally, to a mathematical entity. Philolaus is therefore saying that we are able to know because mathematical entities, numbers and forms, are architectural elements of the mind. We apply them to objects. “To know” means to transform something indefinite into a finite mathematical entity: something measurable, ordered, and capable of being understood.

The crucial distinction here is between “things” and “numbers”. This distinction refutes a naïve interpretation: numbers are not “things”. They are mediators of knowledge. In the relations between mathematical entities, the Formal Relations that structure the Logic of Reality become visible in their clearest form. This is what the fragments of Philolaus suggest, and we shall return to them in the chapter dedicated to Pythagoras.

Now, if our aim were simply to reconstruct the original Pythagorean doctrinal corpus, we might well abandon the project. But this is precisely where our hermeneutic approach changes direction. We are not trying to reconstruct only a doctrine. We are trying to reach what lies beneath doctrine: the logical matrix from which doctrine arises.

Santoro writes:

“Pythagoreanism as a historical category was constructed neither by a precise teaching tradition nor by a clearly defined doctrinal corpus, but rather took shape from a fluid and widespread assemblage of ideas, which were themselves equally fluid and widespread, and which concerned questions ranging from dietary rules to moral prescriptions, from political ideology to views on the nature of life, the universe, and the fundamental constitution of beings in general.”

Burkert offers a highly effective metaphor:

“Just as a city inhabited without interruption over time, while changing its population, presents the investigating archaeologist with far more complicated problems than a site destroyed by catastrophe and then abandoned, the particular difficulty in the study of Pythagoreanism derives from the fact that it never died, as, for example, the system of Anaxagoras, or even that of Parmenides, did.”

This observation seems crucial. The fact that Pythagoreanism never died should make us ask a deeper question. Why, again and again throughout history, have new philosophers brought back to light, with such persistence, principles that appeared to have been buried forever?

Zhmud writes:

“Thanks to the collective efforts of many generations of admirers and interpreters, Pythagoreanism was the only current of Presocratic thought to survive, albeit in a greatly modified form, until the end of antiquity, and Pythagoras rivalled Socrates and Plato, far surpassing their predecessors, in his influence upon thinkers of later ages.”

What is striking about thinkers who call themselves “Pythagorean”, or deliberately place themselves within that tradition, is not mere loyalty to a name. It is the desire to use a method of knowledge: a method capable of giving form to the deepest demands of thought.

For this reason, we fully share Santoro’s approach when he writes:

“What constitutes what I shall call ‘Pythagoreanism’ in this study, is not the presumed criterion of doctrinal and textual preservation, which is the aim and task of the philologist, but the desire for emulation proper to a disciple; or rather, the desire to emulate the master and to simulate his ideas.”

In other words, a “Pythagorean” is not merely someone who repeats, or believes he is repeating, the lesson of Pythagoras in a pedantic way: a Pythagorean is someone who begins from the master’s premises — in our case the study of Formal Relations — and follows a particular method — in our case the dialectical method — in order to reconstruct the Intelligible World.

On this interpretation, Plato may be regarded as an authentic Pythagorean: a true member of the Confraternity of those who aspire to Wisdom. The same may be said of the group of Academics who chose to deepen their study of Pythagoreanism under his guidance, among them Speusippus and Xenocrates. Plato fully embraces the idea that Reality possesses a logic expressible through mathematical language. He follows that path because he believes that only through this logic can one answer Socrates’ challenge concerning the true nature of the Ideas.

No serious discussion of the reconstruction of original Pythagoreanism can avoid quoting, at length, Leonid Zhmud’s severe warning:

“The modern history of the study of Pythagoreanism, which began with August Böckh’s book on Philolaus, now goes back almost two centuries. During this period hundreds, if not thousands, of books and articles have been written on Pythagoras and the Pythagoreans, and yet the corpus of facts on which all scholars would agree is far from extensive, while divergent and often mutually exclusive interpretations are myriad. The Pythagorean question remains one of the most intricate in the history of ancient Greek Science, Philosophy and Religion, and is highly likely to be relegated to the category of insoluble problems.

This is not because every generation adopts a new view of the personality and teaching of Pythagoras. That is a phenomenon common to all Greek thinkers who, like Socrates, Plato and Aristotle, retain their intellectual fascination in the modern world. Nor is it because of the existence of almost insurmountable differences of opinion within each generation of scholars, all divided into their respective disciplines and national schools of thought, each based primarily upon its own tradition.

The problem, as I see it, lies in the fact that within the academic community no consensus has yet been reached on the most fundamental facts, nor on the separation between problems that are soluble and those that are fundamentally insoluble. Although no ‘definitive’ interpretation of Plato’s philosophy is possible, there is general agreement that Plato was a pupil of Socrates, the teacher of Aristotle, and the author of philosophical dialogues. But whether Pythagoras was instructed by Egyptian priests or by Pherecydes of Syros, whether he studied Philosophy and Science, whether there were texts that he himself wrote, whether among his students there were mathematicians and philosophers, all this remains a matter of debate.

To hope for a solution to the Pythagorean question under these circumstances would be an unforgivable illusion. Having spent many years studying Pythagoreanism, I have no such illusions. If I return to this problem, it is only because I remain convinced that, like any other complex scientific problem, it can be broken down into a series of smaller and more particular problems, which may prove soluble. There are many facts on which agreement can be reached. There is also an undoubted hierarchy of interpretations, ranging from those recognised as impossible or unverifiable to those that are more probable and internally coherent. The fact that the situation is not hopeless, provided, of course, that one is willing to accept the facts and to take account of the successes and errors of our predecessors, is indicated by the resolution of one particular question of great importance: the authenticity of the fragments of the Pythagorean Philolaus.”

After this premise, Zhmud reaches the crucial question:

“Is Pythagoreanism possible without Pythagoras, as Orphism is possible without Orpheus, whose personality we feel no need for? Or is Pythagoreanism possible despite Pythagoras, like certain intellectual and spiritual movements that have evolved in directions opposed to the designs of their founders? I do not believe that much can be achieved by following this path.”

There is little to add to these words. Our interpretative wager is this: Pythagoreanism is not, at its core, a doctrine. It is a logical system. And a logic can indeed be reconstructed even when the logician is no longer available to us.

As we have already said, Pythagoras interests us only secondarily as a historical phenomenon. For us, the reconstruction of Synthetic Logic will be the result of Anamnesis and Speculation. In this sense, it falls within the range of what Zhmud himself considers potentially reconstructable. But our hypotheses about historical events, whenever they are not supported by overwhelming documentary evidence, remain conjectures. The reader must be warned of this.

Our account, we must repeat, fills documentary gaps through deductions that arise from the complete reconstruction of the logic. What we shall try to demonstrate is that Pythagoreanism is not a dogmatic doctrinal corpus, but a potential logical system rooted in the human psyche. For this reason, we shall not concentrate primarily on the philological and historical authenticity of what was actually written, by whom, and at what time, although it would of course be fascinating to reconstruct its origins and developments exactly.

Our real task is to formalise Pythagorean logic. For that logic remains, even today and at a universal level, far more than a historiographical curiosity. It is one of the deepest impulses of Philosophy, because it represents the first philosophical and scientific Theory of Everything.

We shall therefore reconstruct the Synthetic Logic that presumably guided Pythagoras on his path of knowledge. This logic is the human, still-evolving expression of the sub-logic of the Part as it strives to resolve itself, gradually, into the super-logic of the Whole. Such a discipline requires, first of all, a detachment from empirical thought.

The final goal of Synthetic Logic is Archilogic: the complete and transcendent representation of the Logic of Reality. Synthetic Logic tends towards Archilogic, whose archetypal Form is Unity of Multiplicity. Analytic Logic, by contrast, tends towards Ontology, whose archetypal Form is Multiplicity of Unity.

On these premises, “Pythagoreanism” need not be confined to the tradition founded by Pythagoras, nor to the transmission of dogmatic contents supposedly elaborated by him. It may be extended to all philosophers who reason according to the same logic. The term “Pythagorean” could therefore be replaced by three other terms: the first is Dialectical, one who reasons according to Relations of Opposition; the second is Synthetic, one who seeks formal Identities while placing empirical Differences in the background; the third is Mathetic, one who uses the instruments of Ideal Mathematics to express and represent this logic.

With these necessary premises in place, the distinction between Pythagoras and Plato becomes clearer. We cannot compare the Dialogues with a supposedly “historical, pure and original Pythagoreanism”, because we do not possess such a thing. What we can do is make the only comparison available to us: a comparison between three symbols: the Tetractys of Pythagoras, the Harmonic Octave of Philolaus, and Plato’s Psyché.

Why these three? Because, according to the criteria of Synthetic Logic, symbolic constructions in diagrammatic form are the only way to represent the Intelligible World. And the Intelligible World is the true object of Metaphysics. “To the Principle!”