Cristóbal Gallardo Alba

In contrast to the dominant conception, which considers that Pythagoreanism represented a break with respect to the philosophy of nature developed by the school of Miletus, this essay aims to defend the thesis that its development can only be understood as a deepening of the trend already initiated by the Ionian philosophers. Such a change of coordinates would be largely determined by two factors: first, by the replacement of the concept of δῐ́κη as a regulative principle by the concept of harmony. Secondly, by the replacement of ontological monism, represented by water, the άπειρον and the πνεϋμα, by a dualistic ontology, represented by the opposition ἄπειρον-πέρας . This would have largely determined the progressive displacement of the organic-materialistic conception of the φύσις by an increasingly mechanistic-formal vision.

In order to develop the above points, I will first proceed to introduce the defining elements of Ionian and Pythagorean natural philosophy, which will require their historical contextualization, as well as the description of the cultural prejudices in which they developed. I consider it necessary to point out that in the present essay I reject the dogmatic postmodern conception of history, according to which historical events are presented as a series of discontinuous fragments; as opposed to that view, the birth and development of philosophy is considered, in line with Guthrie, a product of temperament, experience and earlier philosophers .¹

Development

Before beginning, it is necessary to define a central concept in the present text, the concept of philosophy (φιλοςσοφια). Etymologically, the word philosophy only expresses curiosity, fondness or desire to know, without referring to any particular object. The oldest source in which it appears corresponds to a fragment of Heraclitus, although according to Cicero and Diogenes, it was Pythagoras himself who coined the term. The term σοφια has a very broad meaning in the Greek literary tradition, being applied indistinctly to all kinds of knowledge, both speculative and practical. In Homer, σοφός means a man skilled in mechanical arts. For Aristotle, philosophy is identical with science , embracing the whole body of human knowledge, distributed in conformity with the diverse variety of its objects.²

Another backbone concept of the present text is the concept of φύσις (physis or nature), which is considered to be the most representative of the new understanding of the world that began in the 6th century BC in the city of Miletus³ . The meaning of this term, already present in Homer's Odyssey, was widely discussed throughout the 20th century⁴ . Some scholars have argued that its basic etymological meaning is related to growth, generation, as well as a force that brings things into being, as an inexhaustible source of "beings". Other authors consider, however, that this term expresses, in the case of the pre-Socratic philosophers, the idea of first and permanent substance, of immutable substratum of being, below all the mutations of things.⁵ Nowadays the idea of the double character, dynamic and static, of the concept of φύσις is widely accepted.

A third fundamental concept to understand the origin of philosophical thought is the concept of ἀρχή (arché), which refers to the originating cause or primary substance from which all things come, from which they are formed, and from which all things are resolved. Although this term appears in literature for the first time in a work by Aristotle, it is very likely that it was already present in the lexicon of the Milesian philosophers. In the case of the pre-Socratic philosophers searching for the first principle, the ἀρχή of things, would itself amount to searching for an ontological reality, since for them physical being and Being were equivalent concepts ⁶⁷ .

A final concept that needs to be defined is that of κόσμος (cosmos), a term untranslatable into English that unites the notion of order, structural perfection with that of beauty, which occupies a central position both in the school of Miletus and among the Pythagoreans.

According to Farrington, Greek philosophy was largely nourished by the substrate provided by the unconscious presuppositions that dominated the cultural context⁸ . It is for this reason that, before entering into a discussion of the paradigm shift implied by the development of the Pythagorean school with respect to the Ionian conception with regard to the philosophy of nature, I will proceed to describe, succinctly, the historical and cultural environment of the period in question.

Historical and cultural context

It was in the region of Ionia, during the 6th century B.C. that the first fully rationalistic attempts to describe the nature of the world began. It is there that European philosophy was born, in the sense of trying to solve the problems of the Universe solely by means of reason, as opposed to the assumption of purely mythological or theological explanations⁹ . Among the various city-states that made up the region, Miletus stood out above the rest, to which undoubtedly contributed its advantageous geographical position, with its access to the Aegean Sea and its natural harbors, which facilitated maritime trade and cultural exchange with other civilizations, such as Egypt and the cities of ancient Mesopotamia. There, material prosperity and special opportunities for contact with other cultures were allied, at least for a time, with a strong cultural and literary tradition, represented by epic poetry¹⁰

One aspect of Greek thought that would have been of vital importance for the development of natural philosophy was the dominant religious consciousness in Ionia during that period, represented by Hesiod's Theogony, which showed an evolutionary character, or spontaneous development, lacking a theory of creation; unlike Jahwe, the Greek gods had not created the world¹¹ . In this work the omnipotent cosmic force remains Eros, the power of sexual generation, which constitutes the only means of generation for all parts of the universe¹² . Thus, under the prism of the nascent new conception of reality, oriented towards the substitution of mythological causes for natural ones, as a result of the stripping of anthropomorphic imagery, the role of creative force would have been displaced from that of external compulsion, represented by Eros, towards that of internal development, represented by the concept of φύσις.¹³

According to Benjamin Farrington, it is not possible to conceive the birth of natural philosophy without the enormous influence of Homeric religion as a leaven to transform the conception of the world of Greek thought. This author considers that the differential element of Homer is the inseparable link between historical events and human character, which makes man, in a sense, the author of his own destiny and not a mere toy in the hands of the whim of the Fate¹⁴ . It was a natural creed for the man of a heroic age, who equated the goodness of life with bodily prowess in battle, for whom death meant separation from the body, and with it, from life. Indeed, to speak of the human soul as immortal was blasphemy.¹⁵

Another aspect on which Ionian poetry would have exerted a great influence, resulting from the boldness of the poets in freely proclaiming their feelings and their ideas about human life, would have been the development of individuality, and with it in the emergence of the rational self. According to Jaeger, this would have led to a degradation of the old authority figures, as a result of the emergence of a new concept of universal validity, for which only that which can be explained by conclusive reasons, that for which individual thought can give a reason, is true¹⁶ . This would have favored the substitution of religious faith for the faith that forms the basis of scientific knowledge: that autonomous human reason is a sufficient instrument for understanding the causes that govern the natural world.

The origin of Greek mathematics can also be traced back to the philosophers of Miletus, an event closely linked to the cultural exchange with Eastern civilizations. Herodotus himself states that geometry was probably invented in Egypt. For its part, it would have been in Babylon where arithmetic would have emerged for the first time, which would have employed them for the prediction of celestial phenomena with a great degree of accuracy, around 1500 B.C. However, a differential aspect between the natural philosophers of Miletus and the first mathematicians would have been their interest in questioning the "why", instead of focusing only on "how". That is, as opposed to the eminently practical knowledge of the Egyptian and Babylonian thinkers, centered on the plane of the concrete and the material, in the minds of the Milesian thinkers a curiosity for the formal aspects would have begun to develop. The result would be a shift of interest from the individual perceptions of immediate experience to the universal notions conceived by the mind, which required to be able to treat concepts in the abstract, as a unit with its own nature .¹⁷

The social and economic development of the region was slowed down during the 6th century B.C., as a result of the expansion of Persian power over Asia Minor, which resulted in a progressive loss of political autonomy of the Ionian cities, as well as an increase in economic and social inequality. In this new context the Homeric religion, with its capricious deities, whom one had to try to please with material gifts offered in a bargaining spirit, but without any certainty that they would make the required return, began to seem less satisfactory¹⁸ . Such a context would have favored the spread of religious ideas such as Orphism, which offered an ultra-earthly salvation to the victims of injustice in this world¹⁹ . Apparently, these were the religious ideas characteristic of the islands and coastal cities of the Greek fringe of Asia Minor when Pythagoras (570-490 BC) decided to leave the island of Samos to escape the tyranny of Polycrates, taking refuge in Magna Graecia. ²⁰

Orphism is, above all, a doctrine of salvation, permeated by a pessimistic concept of human nature according to which men carry within themselves an evil, titanic element, and a good, Dionysian one. This dualism finds expression in the distinction between ψυχή (psyche, soul) immortal, of divine origin, and mortal body, the former being enclosed in the latter as in a prison or tomb.²¹ The ψυχή , understood as breath or vital principle that animates the body, acquires in Orphism a role completely different from that represented in the Homeric epics. In Homer the soul, despite surviving death, lacks any transcendental purpose, and its existence is tied to physical life. Its existence after death is bleak and resembles a meaningless existence²² . For Orphism, as well as for the Pythagoreans, the ψυχή is conceived as an immortal entity (άθάνατος) that transcends the body and can undergo a cycle of reincarnations, which implies a purpose and destiny beyond physical life. Life in the material world is seen as a test or a period of purification, from which derives an express normative element, necessary to guarantee its purity in its earthly state of union with the body²³

One of the central figures of the Orphic doctrine is represented by the god Apollo, who will occupy a prominent place in the Pythagorean philosophy, to whose cult they were devoted. Apollo symbolizes the exaltation of the ideas of limit, moderation and order, as well as the triumph of the λογος (logos), conceived as the intelligible, definite and measurable, as well as the proportion of things, both in their own nature and in their relation to the whole . ²⁴

Beyond the influence of epic poetry and Orphism, in order to understand the philosophers of the sixth century, we must first of all get rid of the Cartesian dualism of matter and mind. The only form of existence recognized by the pre-Socratic philosophers was existence in space, i.e., they were not able to understand that something can exist without spatial extension²⁵ . Being and physical being are for the pre-Socratics equivalent concepts. Another element from which it is necessary to get rid of is the atomistic conception of dead matter in mechanical motion. For the early philosophers the world was a living, eternal, divine creature and lived breathing air or breath of the infinite that surrounded it²⁶ . In fact, for the Greeks, the divine character was deduced from the fact of immortality: eternal life is the mark of the divine, and of nothing else.

Natural philosophy in the school of Miletus.

According to tradition, Thales (626/623-548/545 BC) is considered the first European philosopher, although it is not possible to affirm that with him the line separating the pre-rational, mythical or anthropomorphic conceptions and the purely rational and scientific conception was crossed, since it is an entelechy, typical of the positivist conception of history, to consider the existence of a clear demarcation between both conceptions ²⁷²⁸ .

For Thales, the ἀρχή , the original cause, was water. However, this is not where his originality lies, since this was a common conception in Eastern cosmogonies. The originality of his formulations, despite still being strongly affected by the preceding mythological and anthropological worldview, lies in the fact that he was the first to attempt to explain natural phenomena from modifications in nature itself, or what is the same, that the causes that operated in the beginning must have been similar to those that operate now, which earned him the recognition of Aristotle himself .²⁹

It is with Anaximander (610-546 BC), considered to be the most important figure among the Milesian physicists, with whom appears the first description in purely natural terms of the origin of the world and of life, described in terms of organic life in line with the threadzoist conception of reality³⁰ . The latter considers that the first substance, the ἀρχή , must not have been a state of tangible matter, but a kind of basic common denominator of all sensible things arrived at by a process of abstraction, to which he gave the name άπειρον (the indefinite).³¹ Such a concept has both a temporal sense (the arché of all things cannot itself have a beginning), as well as a spatial one, that is a body without internal limits, or what is the same, without distinguishable elements (it is unlikely that Anaximander was able to grasp the notion of strict spatial or quantitative infinity).³² This would represent a great progress towards the abstract understanding of matter, for whose comprehension the senses are not enough, but it requires to be apprehended by the mind³³ . Thus arises the notion of the non-perceptible.

The άπειρον is presented in Anaximander as a fertile matrix, from which the κόσμος germinates³⁴ through a process of separation (εκκκρισις, άπόκρισις), or of differentiation of opposites³⁵ . The organic conception of reality would have determined Anaximander to understand φύσις as intrinsically dynamic, as the driving force of a process of unfolding characterized by eternal motion, thus linking in a single concept, ἄπειρα φύσις, the forces of nature and the material origin of all that exists³⁶ . Such a process would be governed by the δῐ́κη , a regulating principle through which order would be achieved.

In coherence with the organic conception of the existent, what is born and generated, once completed, must degenerate and die. This is the framework from which the existence of infinite worlds, contained in Anaximander's cosmogony, should be interpreted. This interpretation is in contradiction with the one presented in Fraile's work, in which it is stated that the destruction of the worlds would result from the struggle between equal opposites . ³⁷This divergence can be understood on the basis of two highly questionable premises. First, Fraile equates the term δῐ́κη to the ethico-legal concept of the Law of Talion, according to which justice would be equivalent to equality between opposites. Second, Fraile assumes the pair κόσμος limited ( πεπρασμμένοι ) and άπειρον unlimited as a contraposition of equal contraries.

Contrary to the premises assumed by Fraile, the concept of δῐ́κη should be understood not only as a redistributive principle, but as a means by which everything exists in proper relation to the other and by which the good is realized, in accordance to the conception of Homeric poetry, in which the concept of δῐ́κη is opposed both to "injustice" and to uncivilized or savage behavior³⁸ . Second, despite the notion of opposites being a fundamental feature in Greek thought of the time, which probably influenced Anaximander's thought, it is necessary to point out that before the formation of the κόσμος , opposites as such were still non-existent³⁹ . That is to say, to consider that the κόσμμος and the άπειρον are related as a pair of equal contraries does not seem accurate, since it would require assuming an ontological dualism that has no place in Anaximander's thought. It is therefore logical to think that the notion of δῐ́κη would not correspond to that of a law tending to the reestablishment of equality between opposites, but as a principle regulating the development of κόσμος , which once it fulfills its possibilities, would undergo a process of degradation, to return again to the άπειρον, from which everything arises, and to which everything must return.

Finally, according to Jaeger, Anaximander's conception of the earth and the world would represent the triumph of the geometric spirit, which is described as constructed by means of rigorous mathematical proportions, developing the assumption, already present in Homer and Hesiod, that the world is ordered and determinable ⁴⁰⁴¹ .

Anaximenes (585-528/524 BC), a disciple of Anaximander, is considered the last of the great philosophers of the school of Miletus. For him, as for his teacher, the κόσμος was conceived as a living animal, endowed with respiration; likewise, the ἀρχή should be in perpetual self-caused motion, which would make its changes possible. From here probably comes his concept of ἀρχή as ἀήρ (air) , given the relationship existing in that period the living, i.e., that which possesses ψυχή, the πνεῦμα (breath, wind) and the ἀήρ ⁴²⁴³ . However, it is not the above where the originality of his thought lies, but in his attempt to rationally explain the mechanism by which the ἀρχή would have given rise to the various elements. To this end, Anaximenes would have tried to identify processes that could be verified in actuality to explain the transformations of matter, which led him to propose condensation and rarefaction ⁴⁴⁴⁵ . According to Guthrie, by attempting to explain all qualitative differences in matter by the different degrees of condensation and rarefaction of the one basic matter, his proposal would represent the first attempt to explain qualitative changes in matter from quantitative differences.⁴⁶ Finally, according to extant sources, Anaximenes did not think that all natural substances derived directly from the πνεϋμα, but that there were certain basic forms, such as fire, water, and earth, of which other types were compounds. If true he would be the pioneer in considering the existence of simple substances and compound substances .⁴⁷

Natural philosophy in the Pythagorean school

Before introducing the foundations of Pythagorean natural philosophy, it is necessary to clarify that the scarcity of original writings has greatly hindered the understanding of its doctrines, Aristotle being the main source of information through numerous references scattered in various books⁴⁸ . This is not only due to the loss of records, but also to the very nature of the school , whose intention was to make it difficult for anyone who was not a member of the brotherhood itself to understand its teachings⁴⁹ . This has conditioned Pythagoreanism to be considered as a unity in which it is difficult to distinguish significant internal differences despite the multiple personalities that contributed to its development .⁵⁰

Another aspect to highlight before going deeper into the study of the Pythagorean school is the fact that for Pythagoreanism the religious aspect, largely inspired by Orphism, and the philosophical doctrine, were not two separate dimensions, but two inseparable factors of a single way of life. In fact, prominent members of the school were named in antiquity as referents of the Orphic tradition .⁵¹

With Pythagoreanism the motive of philosophy ceased to be primarily what it had been for the Ionian philosophers, namely intellectual curiosity or technical improvement, and became the search for a way of life by which a right relationship with the universe could be established⁵² . For them a full human life ultimately required a total detachment from the body, an escape from the wheel of rebirth and the ultimate bliss of losing oneself in the universal, eternal and divine soul, for which philosophy, and more specifically music and mathematical research, proved to be the most convenient way ⁵³⁵⁴ .

ἁφύσις δ᾿ ἐν τῷ κόσμῳ ἁρμόχθη ἁρμόχθη ἐξ ἀπείρων τε καὶ περαινόντων καὶ ὅλος ὁ κόσμος καὶ τὰ ἐν αὐτῷ πάντα⁵⁵ . Such an expression, attributed to Philolaus of Tarentum, would represent the change that the concept of φύσις underwent under the influence of the Pythagorean school. The φύσις would cease to be eternal, becoming the product of the harmonic union of two principles, ἀπείρων (the unlimited) and the περαινόντων (the limited), which are themselves eternal and pre-exist the κόσμος. With this, the φύσις would lose its character of organic development, adopting a passive character .⁵⁶

Harmony as the organizing element of the cosmos

The term ἁρμονία (harmony) appears already in Homer's Odyssey, in the sense of union or interlocking of things, as well as the material support with which they were joined. From the 5th century onwards it was associated especially with the stringing of a stringed instrument of different tensions, and thus also with the creation of the musical scale⁵⁷ . Its mathematical dimension, understood as the relationship and balance between different elements through proportions and numerical patterns, is, however, due to Pythagoras himself, who is considered the discoverer that the basic intervals of Greek music could be represented by the proportions 1:2, 3:2 and 4:3⁵⁸ . According to Jaeger, the discovery that music responded to a mathematical order represented a turning point in Pythagoras' thinking, since the generalization of the numerical legislation of music to the whole cosmos would have led him to try to explain all reality in mathematical terms, and thus to the discovery of the "mathematics of nature⁵⁹⁶⁰ .

The discovery of harmony would also lead to a new conception of the divine, which was linked not only to the eternal, as in the case of the Milesian philosophers, but above all to the fact of being constituted by a combination of elements in harmonic order, that is, according to the rules of mathematical proportions⁶¹ . Undoubtedly, this determined to a great extent that numbers possessed for the Pythagoreans a mystical meaning and an independent reality, to which phenomena were secondary. We must not forget that the ultimate goal of Pythagoreanism was assimilation to the divine, and since the divine possessed a numerical character, what better than the study of the mathematical structure of the κόσμος to achieve assimilation, since they considered that the philosopher, by associating himself with what is divine manages to divinize himself⁶² . This would explain, at least in part, the shift of interest in Pythagorean natural philosophy from the basic matter of the universe and the physical changes by which it had come into existence, to an understanding of order.

Ontological dualism

Although the general doctrine of opposites in nature was shared both by the thinkers of the school of Miletus and by the Pythagoreans, it is with the latter that it would have acquired a leading character. According to Guthrie, the monistic interpretation of the Pythagorean school, which grants a central role to the ideal One and unity as the origin of all that exists, is undoubtedly Platonic in character .⁶³

A key element that distinguishes the Pythagorean doctrine from Ionian philosophy is the fact that it is rooted in the moral concepts of good and evil, another aspect it shares with Orphism. This determined that in Pythagorean thought, despite lacking full internal coherence, is dominated by a pair of opposites, which manifest themselves in many different ways. Thus, unity, limit, even, clear, etc. are associated with goodness, while plurality, unlimited, odd, dark, etc. are considered evil⁶⁴ . This duality is also reflected in the anthropological aspect, according to which man consists of two distinct parts: the body, composed of material elements, and the soul, of celestial origin⁶⁵ . And not only that, but the ψυχή itself would possess for the Pythagoreans a dualistic character, distinguishing a soul of material character, which would "vanish like smoke" after death, and the δαίμων (demon), of immortal and divine character, which underwent transmission through many bodies .⁶⁶

The dualistic conception of reality appears also reflected in their understanding φύσις. For the Pythagoreans the basic notions that would allow to explain development of the world are those of πέρας (limit) and ἄπειρον (unlimited)⁶⁷ . While it is true that, as various sources state, they revered the One (τὸ ἓν) as the divine, this would not have existed eternally, but would be assimilated to a kind of seed, the seed of the world, the limited ( περασμένον ), resulting from the intersection of πέρας and ἄπειρον⁶⁸ . Such a conception is closely linked to the concept of harmony. Thus, in analogy with music, just as melodic order derives from the ordered delimitation, through the notes, of the unlimited element of blank intervals, the order of the cosmos would derive from the imposition of the πέρας on the ἄπειρον⁶⁹ . In this scheme, the unit was considered as the starting point of the numerical series, but not as belonging to it, since every real number must be either even or odd, and the unit, curiously enough, was conceived as a combination in itself of even and odd. That is, it is both the one-dimensional unit of construction and the dimensionless point of contact between two sections .⁷⁰

Before continuing, I consider it necessary to make three points about the limitations of the conceptual apparatus available to the Pythagorean philosophers, which will allow us to better understand in role that numbers in relation to the φύσις . First, the fact that they conceived of everything in existence as extensive largely prevented them from being able to distinguish between concrete and abstract numbers ⁷¹⁷² . Second, the Pythagoreans would have experienced difficulty in distinguishing clearly between the concepts of "similarity" and "identity," a difficulty deeply rooted in the Greek consciousness of the time, given that there was only one word to designate both terms: όμοιος⁷³ . Finally, the distinction between form and matter had not yet received a clear formulation, so that in describing the structural scheme of things they believed they were also describing their material nature⁷⁴ , which would have largely determined that Pythagorean mathematics was inseparable from their physics .⁷⁵

Returning to the generation of the One, for the Pythagoreans the Unity-point, generated from the imposition of the πέρας on the, would also constitute the basis of physical matter, something like a primitive form of atom⁷⁶ . The emergence of the Unity-point would thus represent the first manifestation of the harmony of the κόσμος, and with it the birth of the φύσις, as reflected in the work of Philolaus of Tarentum. With all this, despite the manifestly mechanical character of such a conception, the Pythagoreans would not have abandoned the idea of the infinite breath of the Universe, through which the void would be introduced, defined as "that which separates and divides the things that are next to each other"⁷⁷ . In this we also perceive the loss of the organic character of the world, since the concept of πνεῦμα ceases to be linked to the concept of ψυχή, common in the Milesian philosophers. Thus, as a result of the introduction of the void into the bosom of the Point-Unit, and its consequent division, the rest of the elements of the numerical series would arise (the number two, for example, would result from two Point-Units, separated by the void). In a following stage, from the spatial order of the dot-units, grouped in sets, would emerge the geometric figures, each of which, in turn, was associated with the various elements ⁷⁸⁷⁹ . Thus, for example, the tetrahedron would correspond to the properties of fire, and the cube to those of the earth. E n synthesis, ὁ ἀριθμός ἐστιν ἀρχὴ πάντων τῶν ὄντων .⁸⁰

From the above it does not seem adventurous to affirm that, if we had to choose a term to describe the changes that the notion φύσις underwent in its journey from the Ionian school to the Pythagorean school, this would be μεταβολή (transformation), understood as a change in the nature of something. And the fact is that, although the Pythagorean conception of number as the principle of things is preformed both in the rigorous geometric symmetry of the cosmos of Anaximander ⁸¹⁸²and in the intention of Anaximenes to explain qualitative changes from quantitative changes⁸³ , with the Pythagorean school, by generalizing the concept of harmony of opposites as the backbone of its conception of reality, the very concept of φύσις would have undergone a mutation.

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