The marvelous theorem of Pythagoras

The main purpose of this paper is to explore the mode in which the early Pythagorean teachings brought together traditional religious practices and principles of conduct within a framework that accommodated the demand of logos and metron concepts that were used as the cornerstones of the intelligibility of human and cosmic goodness.

By bringing the Orphic tradition under the rule of reason but without abandoning certain fundamental Orphic beliefs, e.g. the immortality of the soul, on the one hand, and by insisting on the general relevance of mathematics for the study of the constant structures of human existence and the processes of nature, on the other, the Pythagoreans offered an original vision of the continuity between moral values and scientific concepts. It was against this broad background that the Pythagoreans advanced a set of distinctions known as the “Tables of  opposites”, to identify and clarify confused and unintelligible practices. But in addition to the “theoretical interests”, and also the espousing of the mystical elements of Orphism, the Pythagoreans gave new impetus to the pursuit of wisdom by formulating a hierarchy of types of life open to human beings: a) the apolaustic life, b) the practical or honor-seeking life, and c) the life of wisdom.

The available fragments of their writings show that the Pythagoreans summarized and clarified the Greek experience that has become known as the philosophical vision in which personal, public and cosmic values converge into a primordial Unity.

I. The Theme

According to Nilsson 1952, Pythagoras occupies a place in the history of philosophy; he fills an equally notable place in the history of religion. In him the sense of formal law was so strongly developed that when he discovered the regulatory of the mathematical laws of number he laid them down as the basis of existence. In religious matters his school was a sect which taught a new doctrine of the greatest importance, the transmigration of souls (201).

The new doctrines, invested with philosophical  significance, draw attention to the role of the divine, how is the divine to be understood, in view of the belief that the soul has properties such as are denied of the mortal bodies of men. The issue that will inevitably come up for examination in this paper reaches beyond the limits of the Pythagorean signification of the divine, although it touches on the ideas of the paligenesia of the psyche, a doctrine no doubt borrowed from the Orphic tradition. As for the Pythagorean scientific cosmology, its author, if it is Pythagoras’s own, according to Burnet (44, 1959) took his point of departure from the Milesians, and Anaximenes in particular.1

Plato informs us that Pythagoras was recognized as a great educator, meaning that he was a great synthesizer of ideas and values who effected a harmonious system that was used as the basis for a way of life. Plato has alerted us to the Pythagorean integration of two distinct fields, religion and science. Perhaps of equal if of a greater importance is the use to which the integration was put. Before the Socratic-Platonic grant contribution to a philosophy of education, aside from the work that the poets and legislators had done in this area of public concern, it may be pointed out that two great movements changed the face of the politics of education and culture in antiquity: the Pythagorean and the Sophistic. With regard to the latter, Werner Jaeger, agreeing with Goperz, has that “what was new in the sophistic movement, and at the same time common to all members, was the educational ideal of rhetoric.”2

Although it cannot be denied that their educational system was founded mainly on a method for developing and transmitting rhetorical skills, their aim was to widen the scope of education, in hope of teaching effective political skills as well. However, the real issue was not whether the sophists satisfied the expectations of their clients by providing them with the skills they themselves had mastered but whether they probed into the foundations of politics. It was in this area of public concern that the Pythagoreans made the presence of their thinking felt.

In discussing the views of the Pythagoreans as a way of life, the emphasis will fall on the exploration of their contribution to human thought and understanding of conduct rather than a reconsideration of the inconsistences and doctrinal difficulties. Such serious difficulties in the teachings of the early Pythagoreans have not escaped the attention of the ancients or the moderns, as for instance, the doctrine that the soul is a harmony as well as a unity. My main concern is to draw attention to the place the Pythagoreans deserve to occupy in the history of philosophy as a result of having revolutionized education by bringing together, science,  religion and morality and by working with a novel set of tools and setting forth the confluence of two traditions, the Orphic-religious and the Ionic scientific. They worked out what Plato eloquently credits them for doing:

“Do we hear that Homer himself in his lifetime became for certain people personally a guide to their education? Are there any who admired him as disciples a master, and handed downto later generations a Homeric way of life, like Pythagoras, who himself was especially admired on this account, and his followers down to this day are conspicuous among the rest of men for the Pythagorean manner of life as they call it?”3

It was this way of life that really mattered to Plato, rather than the complex personality of a legendary figure named Pythagoras. Notwithstanding the fact that Pythagoras became a great number of things to many thinkers at different times and different places, depending on the prevailing winds of the doctrine, as we witness in the biographies of the Neoplatonists Porphyry and Iamblichus, among others, Pythagoras effected a synthesis of viewpoints and practices that to others, like Heraclitus, seemed nothing more than polymatheia: the charge being that it was not nous. I will return to this criticism later in this paper, when the Pythagorean contribution to education as preparation for establishing a way of life comes up for discussion. Suffice it to say now that what Heraclitus called polymatheia, Plato found highly relevant to the construction of his own curriculum for the education of the guardians.

II. The assimilation of Orphism

I am inclined to support the view that the Pythagoreans has good reasons for adopting the Orphic belief in the immortality of the soul and its series of transmigrations.4 The material was religious, the concern was ethical, but the evidence on which a follower would have to depend for demonstration was anything but scientific. The belief was speculative, or rather required for the completion of a way of life. Probably, their line of thinking went something like this:

  1. Ultimately, the Universe is Harmony
  2. Human life on earth, as we know it, starts under unfavorable conditions, as a mixture of apeiron and peras, a mingling of portions from entities listed in the table of opposites.
  3. Human existence must develop in appropriate ways to attain the level of teleios the Universe principally has.
  4. Given the fact that a long process is required to bring about this agreeableness between man and the Universe, identity in harmony, and given the initial conditions of human life, the duration it is granted by nature is not long enough to complete the assigned task. Favorable cultural conditions are usually absent and even when present they are not universally available. Hence if death is taken to signify the cessation of individual existence, i.e. the literal extinction of a human being, the teleios can rarely, if ever, be attained and then only by extremely charismatic individuals. If so, the attrition of the parts of the Universe that enter the process of teleios but do not attain it entails the eventual depletion of the whole, and at best, its steady deterioration.
  5. Hence, the belief in immortality is justified for the completion of the process that leads to eukosmia and for the preservation of universal harmony on the parts through their re-integration with the Whole.
  6. Since the duration of the life of each individual on earth is short, not enough time is made available to test the quality of attainment required for the re-integration of the parts with the Whole. Therefore, the soul must return and take another body to complete its unfinished task. In fact a series of bodies may be needed for a given soul to reach the cathartic level that harmony requires.

According to the legend, even Pythagoras himself or rather his soul underwent a number of reincarnations before he was born as Pythagoras. If so, strict abstention from animal flesh becomes mandatory, presumably on the ground that it is a religious and ethical transgression to prevent another soul from occupying its present body or to disrupt its cathartic course. This, and not the usual humorous story that one may be eating one’s own grandparent or friend, lies in the back of the command. Interfering with the right of other souls is a crime against the harmony of the universe itself. The belief presupposes the kinship of all types of living things and life in general with the ultimate principle of the Universe.

III. The Cosmic End of Human Ethics

The Universe itself as such is a living being, zoon. On the other hand, not one body is immortal, though their breath is, which is the soul of the living things, and like the Universe, it is divine, for the breath’s origin is the one and the eternal Universe. The spirit in each human being belongs to the Whole from which it was detached and hence tainted with fragmentary existence. Rejoining the Universe, the Whole, is the goal, and this task may require a number of transmigrations. So long as these are required, alienation persists and individuality is retained, therefore is lasts as the taint of the soul. But once the union is accomplished at the completion of the process of purification, individuality is no longer preserved, let alone desired ⎯ a conviction that today may shock the followers of Western forms of salvational forms of worship who have absolutized the individual as cornerstone of political, economic, religious and artistic life. The Pythagoreans evidently thought differently. Their ideal demanded that the eternal separateness of the self be dropped in favor of the union with the Divine.

So far this is religious conviction. However, the prospect of its conciliation with what was taking place as scientific work in Ionia was on the whole tied to the search of means of purification. The means, to be sure, were taken out of the strict supervision of religious rites, i.e. the ritual of Orphism. Pythagoras is credited to have taken the radical step that led to wedding religious values to scientific visions of the universe. According to the new way of thinking, the kinship of humanity to nature is discovered through the simple principles, ultimately the principle of the limit, of form, and by the way of the collection of what belongs to it, represented in the right column of the Table of Opposites.5 The aim is to know that sort of action these principles stand for in the pursuit of purification.

The Universe is a Kosmos, it has the features of order, fitness and beauty. The human bodies as organisms display organization of parts in obedience to a principle of unity that keeps them alive. The real issue is not the unity of the body, understood as its health but the grant theme of the superiority of the soul. It is here that disconnectedness, plurality, and disunity create serious problems for the individual who has to effect order where order is not given ab initio.It is the natural duty of each person to turn the soul into a Kosmos and by becoming Kosmios, aspire to the godlike status of homoiosis theo. To do this one has to realize that neither wishful thinking nor prayers, not even potent rituals and rites have the power to bring the soul closer to its destiny. Thus: 

i) Human beings need to grasp with their nous, what the divine order is so they become fully aware, by attaining theoria, of the unalterable structure of the Divine.

ii) Human beings need to discipline the energies and actions of the soul, all its actions as they originate in each of its parts, from desires to nous, to regulate the actions in accordance with fitting habits and thus make of the soul an ordered micro-universe, indeed, a microcosmos. Here, the form of the body hardly suffices to serve as a guide. The order to be effected must be appropriate to the form of the soul, with the aid of a guide yet to be discovered. Whereas the former is the work of ethics, the latter falls within the domain of science. The close cooperation of both is needed to transform the human being into a kosmios anthropos

Given that the means are both theoretical and practical, the former are what became known as the formalized disciplines, the mathemata, to which Pythagoras himself contributed original discoveries.The latter held an interest of their own. To be sure, the soul must employ both and learn how to effect the peperasmenon. This can be done by imposing the peras (limit) on the apeiron (unlimited) to make the peperasmenon (limited). The method pertains to all activities of the soul, especially the desires.

The Pythagoreans were confident in holding that to be ethical meant to agree with the ways of the Universe, to be in tune with the harmony of the One. Hence, genuine principles that pertain to ethical acts and ethical development are neither subjective nor capricious, neither arbitrary nor relative, and definitely not temporary and subject to change as is the case with the mores and habits of people in different places and at different times. Correct moral principles are derived from the theoria of Being itself, not from the study of becoming parts. Thus where matter and soul display resistance, the agent must show determination to alter the apeiron by means of the peras and make something new and stable, the peperasmenos. To do this is to effect a cosmos or rather a microcosmos in accord with the macrocosmos.

Among the doctrines the Pythagoreans left unexplained is the following: why is it that the real objects and beings as we sense and perceive them, including our own existence, are not perfect as the Universe itself, although they derive from it and are part of its divine harmony. To ask the same question in another way, why is it that the parts which belong to the whole are not exactly as the whole is in its principal state? Furthermore, why the deterioration that follows the emergence of parts or rather the coming to be of parts out of the whole? And much to our puzzlement, why should the parts be tainted? One supposes that the answer could come only from a defensible rational cosmology, but apparently it did not. As tradition has it, the Pythagoreans first posed the problem albeit empirically with the aid of concepts formulated through religious feeling, in response to attitudes regarding values projected as grades of being. Since each part is less than the whole the blemish is in the parts as encountered here and now. Speculation may provide some help but no complete theoretical account is forthcoming. The best one can do is accept the religious teaching of incompleteness of the parts, due to grades of being, view it as ontic deficiency, and complement the view with whatever rational insights speculation can make available in the medium of logos.

IV. The Ethical Foundation of Politics

A brief comment on the political side of Pythagoras’ teachings is perhaps in order at this point. The moral and the political values actually blend in the course of the pursuit of eukosmia. In fact, they blend in a way comparable to that in which moral and cosmic values merge. The polis itself must be understood as a candidate for transformation into a cosmos, and like what takes place in the corresponding moral process, the political catharsis must be initiated pursued and brought about through planning, deliberate choice and action through social effort. Herein lies the idea of the intimate brotherhood as group affiliations and in the form of community whose aims is to promote qualities that sustain unanimity and promote interpersonal harmony.

The principle behind the institution of brotherhood is that all individuals are parts of the universe and that they differ quantitatively and qualitatively from each other as distinct parts, although in the case of the human species the soul of each being shares in the same structure and form. However, the peperasmenon as action remains still elusive. Its attainment, which is the sine qua non for securing harmony within the self, and by extension with the Universe, from the political point of view is the same for all individuals. Given that human beings share the same ultimate Form, the completion of the cathartic process is the same for all, and once the completed vision is communicated, presumably through the revelations of the master teacher, it serves both as standard and the universal end. Being therefore the ultimate common good, it is not open to dispute or doubt. The problem human beings must solve does not lie in the essence of the standard but in charting the paths the cathartic process demands to reach the end.

Evidently, the decisions that comprise the task we may call “ethical development” differ with each individual, not only because the taint differs from part to part, but mainly because the differences in each case of apeiron present in each soul is a dark and unfathomable entity. One simply does not possess self-knowledge when the journey begins. Hence, the aim of politics is to secure the conditions that will facilitate the way of life Pythagoras taught and make it available to all individuals so that they may proceed to build the kind of character that will accord with the harmony of the Universe, and by the same token stay in harmony with the ethos of other individuals who have elected to cultivate the life of social unanimity. The process therefore is one that leads from difference to similarity and from diversity to community. To the liberal imagination of the modern this translates as the assimilation of the sacred principle of individuality into the lifeless and objective structures of the state.

However, that was not the Pythagorean thought of the relation of the self to the universe and to a brotherhood conceived as the highest and best embodiment of politics. What is sacred and divine is the Universe, not the individual.

In practice, it is almost impossible to realize this ideal in a world in which process and diversity persist as stubborn facts as much as harmony and unity are ultimate. Being and Becoming are both principles in nature, and so long as tensions are generated between them, and they will so long as there is change in nature, the process of converting diversity into similarity remains and on-going task, and ultimate unity can be envisioned only as an ideal, though a reality in its own terms, which is what the divine is. This ideal, as the object of rational vision, and not simply the feared divinity demanding ritual sacrifices to grant favors to the hierophants, is humanity’s best guide to enlightened conduct, the avoidance of evil, and in the long run, further reincarnations. As a science, therefore, politics derives its authority from the study of the universe and the principles of Nature, but as a practical art it is the still the art of the possible, of what is ephikton.

As human beings beget new human beings, they in turn inherit the taint of being parts, the imperfection of the apeiron. Furthermore, not all citizens, polites, are members of the Pythagorean brotherhood, and as such have no access to the truths of harmony; nor are all city-states, including those that have only partial knowledge of what the master means, are fully aware of Pythagoras’ doctrines. As a result, they lack the wisdom that grants control over their actions. If individual harmony is so difficult to achieve, social harmony verges on the impossible.

Hence we must not confuse the vision of harmony in the ethical and political dimensions of the Pythagorean way of life with the real events as they occur in the daily affairs of human beings. Current realities aside, the Pythagorean teaching are offered more as reflective recommendations to follow, and not as legal ordinances demanding total obedience, although as guidelines they draw support from theoria. Logos and freedom, in order to move the human being toward intelligent action, depend on whether logos is given the requisite priority that makes action the result of deliberate choice, and on the whole, turns the way of life into an enlightened commitment, what the Greeks call hairesis biou. Unless this condition is met, individuals fall easy prey to the quest for power and passion to recruit disciples. We have no evidence that Pythagoras indulged either of these forms of oppression. The silence of the followers need not to be turned against the bonds of friendship that the idea of brotherhood sought to promote.

V. Toward a Theory of Education 

The process toward harmony is actually a fundamental phase of education, a leading out of the pupil onto a life of light in metron and the implementation of deliberate choice. The way is paved through disciplined study leading to the rediscovery of harmony, and this can be done only through continuous practice and further reinforcement by means of additional knowledge until the way of life is firmly established as a rational habit. Neither the performance of ritual nor the forced imposition of rules provide the kind of education one needs as preparation for the Pythagorean way of life. A favorable environment is need as well. One begins to wonder at this point whether this was the reason Pythagoras left Samos in 530, B.C., and went to Croton, via Delphi, in hope of finding there the right environment for the great education experiment in cosmic politics.

For the educational process to succeed, as the brotherhood understood it, must take place within a properly controlled environment and only after the rites of initiation have been duly performed to secure the confidence of the candidates. The latter appears to have been the adaptation of certain religious practices. One can only guess what was the justification for the rule to maintain control over a set of secret truths called the aporreta? Aristoxenus, Aristotle’s pupil and a friend of the Pythagoreans, reports (fragment 4) on the rules of education that “not everything was to be divulged to all men”. It seems that the Pythagoreans preferred to have interested people go to them as candidates and ask to receive the doctrines rather than engage in a policy of recruitment. We may recall Plato’s advice at this point, who also recommended that the teacher had better wait for the person in need of enlightenment go to him rather than recruit pupils as the sophists did. Knowledge is like medicine. The sick seek it because they want their health restored; to do otherwise is to waste the fruits of wisdom on those who are not ready to receive them.

There is something peculiar about popularizing advanced knowledge, especially knowledge that is fundamental to cathartic conduct that alleges to secure perfection. Knowledge, once it is diluted, loses both its potency and its original goal; it is easily misunderstood and worse becomes easy prey by being available for evil applications and indiscriminate misuses. Hence the Pythagorean confidentiality regarding the secret and sacred character of the esoteric knowledge needed to guard the treasures of the spirit. To become a celebrant of wisdom one had first to provide of his worth and sincerity o intentions. This rather priestly control of the power of knowledge, seems to have been the main reason why the Pythagoreans made use of the practical side of religion. They understood the need to protect the special quality of salvational knowledge within a scientific community. To conclude this part of my paper, I suspect that the Pythagoreans stood for a real advancement of the effort to liberalize the demanding curriculum of higher education by introducing new criteria as well as restrictions to judge the qualifications of candidates who wished to participate in research and discovery.

The Pythagorean tradition of education was continued and broadened in the curriculum Plato designed for the academy. In Republic, BK VII, Socrates recommends a curriculum, which in essence incorporates and extends the cycles of studies Pythagoras had established for the education in the brotherhoods of South Italy: there is no reason to doubt that it included cosmology, as Philolaus and Archytas report. We have it on Plato’s word that Timaeus of Locri was a Pythagorean cosmologist, and his teachings, if not also Plato’s own amplifications, are those of a Pythagorean. Socrates, building on the principles of the new logic, the higher dialectic, demands that the guardians be educated in arithmetic, geometry, stereometry, astronomy, and music before they would qualify for the responsibility to rule as philosophers-kings. Pythagorean mathematics soon became prized possession of higher education.6 In Plato’s works the pursuit of knowledge and the training in higher dialectic preserved something of the sacred character of wisdom the Pythagoreans essayed to add to the advancements that rational inquiry had already established as a fundamental to the critique of public ideals. Plato, though not an orthodox Pythagorean himself, put the Pythagorean tradition to its most creative use. After Plato it was down the hill. The original way of life Pythagoras envisaged gradually lost its initial vitality. How that happened and why is another story.


1 Burnet, op.cit. makes no effort to answer this aspect issue of the divine.
2 Paideia, I, 293.
3 Republic 600 b; translation in Guthrie 1962 I, 148.
4 Herodotus IV, 95; II, 81. Guthrie’s Orpheus and Greek Religion (London 1935) is still one of the best studies of this theme.
5 According to Aristotle (Meta. I. 896a22) they are as follows: limit/unlimited, odd/even, one/many, right/left, male/female, rest/movement, straight/curved, light/dark, good/evil, square/oblong.
6 As W. Jaeger notes, thanks to the Pythagoreans, the mathematical science brought “a new element in Greek culture.” Paedeia I, 163.


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The research paper was presented at the international conference on Pythagorean philosophy at the Pythagorion of Samos 22-27 August 1991.The conference was organized by the International Association of Greek Philosophy & the International Centre for Greek Philosophy and Culture.


John Anton

John Anton


was professor of Greek Philosophy and Culture at the University of South Florida, USA. He specialized in Classical Greek Philosophy, History of Philosophy, Philosophy of Art and Metaphysics.


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