Early Greek Philosophy: Italic Metaphysics — The Eleatics and Pythagoreans

In this lecture, we will trace the historical and logical (theoretical) development of early Greek philosophy, focusing on its second branch: Italic metaphysics. The historical aspect involves the spread of Ionian philosophical teachings westward into the Greek colonies of southern Italy, established by Greeks from the 8th century BCE during the so-called Great Greek Colonization, into an area that came to be known as Magna Graecia. Strictly speaking, we know of only two such philosophical "cultural transmitters" — Xenophanes from the Ionian city of Colophon and Pythagoras from the Ionian island of Samos. These thinkers contributed to the development of the second branch of Greek philosophy — metaphysics — by shifting from a direct sensory representation of nature (phyisic or physiology) to seeking its supersensible principles and foundations. In other words, they transitioned from describing natural phenomena (natural philosophy) to addressing questions about true being (philosophical ontology).

This transition from physics to metaphysics initially occurred within the framework of natural philosophy itself. As a result, the views of early metaphysicians, or ontologists, were heterogeneous: they encompassed elements of both natural philosophical views and metaphysical (ontological) ideas. Let us now detail these general principles by examining the teachings of the founders of metaphysical concepts, Xenophanes and Pythagoras, as well as their successors.

Xenophanes (c. 565—473 BCE) hailed from Colophon, a city north of Ephesus. He spent the latter years of his life in Elea, a Greek colony in Italy (region of Lucania), and is thus considered the founder of the Eleatic philosophical school.

Part of Xenophanes' worldview reflects typical Ionian physics. He regarded the basic elements of the world as dry and moist, that is, earth and water, which also serve as both source and substance. This is evident from his assertions that everything arises from the earth and ultimately returns to it (Fragment 11B27), that we are all born from earth and water (Fragment 11B33), and that we perish whenever, immersed in the sea, the earth turns to mud. This process involves earth being mixed with water but later drying out, a cycle observed in all worlds. Xenophanes cites evidence of this process in the discovery of shells and imprints of marine creatures and plants on stones in the mountains. These beings existed long ago when everything was covered in mud, and their imprints remain after the mud has dried (Fragment 11B33). Here, we see a clear empirical basis for natural philosophical constructs.

These basic elements also serve as substances, as Xenophanes states: "Earth and water are the sources of all that is born and grows" (Fragment 11B29) and "all things consist of earth."

Another aspect of Xenophanes' views is his theology, or rather, theocosmology, which forms the basis and one of the earliest historical forms of metaphysics. According to N. Abbianco, metaphysics throughout its history has appeared in three fundamental forms: theology, ontology, and gnoseology. The characterization of metaphysics as a study of what lies beyond experience primarily pertains to the first of these historical forms — theological metaphysics. The concept of metaphysics as theology involves recognizing the object of metaphysics as a supreme and perfect being upon which all other forms of being and the causes of the world depend.

Having established these general considerations, let us now provide a more detailed historical-philosophical explanation of the connection between theology and metaphysics, returning to one of the fundamental features of Ionian natural philosophy — its natural-theism or pantheism, which we discussed earlier. The identification of nature with divinity, or its deification, led philosophers who emphasized nature to more consistently pursue natural philosophy (phyisic, physiology), while those who focused on the divinity of nature, on god-nature, pursued philosophical theology, creating their distinct conception of gods apart from mythological representations. This was the path to metaphysics, in the classical formulation of the Indian philosopher Sarvepalli Radhakrishnan: "Physics, in alliance with religion, became metaphysics."

This path to metaphysics is demonstrated by Xenophanes' second line of thought. He begins with a critique of mythical representations of gods and develops his philosophical understanding of the divine. Notably, Xenophanes denies that gods are born (Fragment 11B14). As S. Radhakrishnan observes, "A created god is not a god at all."

Furthermore, Xenophanes was possibly the first to criticize polytheism, not only among the Greeks but also among other peoples known to him. From his observation that different peoples depict gods in their own image, leading to inconsistencies among them, he concludes that polytheism undermines the concept of god itself: if we see numerous differing depictions of gods, it becomes unclear what god truly is or what its true form is. This situation is akin to the saying "not seeing the forest for the trees." Consequently, Xenophanes posits the existence of a supreme god: "There is one god, greatest among gods and men, not similar to mortals in either appearance or thought" (Fragment 11B23). This statement reflects a move away from anthropomorphic views of the divine and towards monotheism, though it represents an early stage, as Xenophanes' view likely aligns with what is termed henotheism (one god + theos), where the single god does not necessarily negate the existence of other gods.

A key aspect of Xenophanes' henotheism was identifying the "one god" with the entire world or the universe. This pantheistic position, expressed both in content and in his words, is reflected in his statement: "God is blended with everything." In Greek, "everything" is πᾶν, and "god" is θεός, suggesting that this statement embodies the term "pantheism." This pantheistic formula of the divinity of the cosmos forms the basis of metaphysical views, as it extends the universe — god beyond specific sensory phenomena and human experience. This is achieved by attributing to the world the properties of the "one god," which are opposed to those of things and phenomena within the natural world. As a result, metaphysical (supersensory and transcendent) traits of the world as a whole emerge: like the divine, the world is ungenerated, eternal, and indestructible (Fragment 11A37); infinite, unique, homogeneous everywhere, limited, and spherical (Fragment 11A3); all is one and unchanging (Fragment 11A34). Xenophanes does not recognize creation or destruction; he asserts that the universe is always like itself (Fragment 11A32). God — the world remains in one place, not moving (Fragment 11B26).

Prominent Soviet philosopher A. S. Bogomolov believed that the essence of Xenophanes' view on the world lay in his inquiry into how to understand the "world as a whole" — the one and only world beyond which there is nothing, which is, therefore, the only and unique being.

In conclusion, Xenophanes' pantheism became a foundation for a new Eleatic (metaphysical) line in philosophy, distinct from Ionian thought, as he began to discuss an abstract "world in general" as the "one god," alongside the concrete-sensory nature.

The epistemological roots of Xenophanes' metaphysical positions were rooted in rationalism, as noted by Aristotle. He wrote that philosophers, including Xenophanes, rejected sensory perceptions and representations, relying solely on logos (reason). Thus, they believed that being is one, that there is nothing else, and that nothing is created, destroyed, or moved (Fragment 11A49). This indicates that rationalism is a significant characteristic of metaphysical concepts, a connection between metaphysics and rationalism that is evident in Xenophanes' followers.

Xenophanes' metaphysical ideas were further developed by Parmenides. Born between 544 and 515 BCE in Elea, he was a prominent statesman of the city and studied under Xenophanes (Fragment 28B1). In his philosophical poem "On Nature," as with all early metaphysicians, Parmenides presents natural philosophical positions typically categorized under "The Way of Opinion" or simply "Opinion" (δόξα); he himself refers to them as "the opinions of mortals." Since opinions reflect knowledge of the sensibly perceived world, whose objects and phenomena are subject to change and vary in perception among different people, the portrayal of the world in these opinions cannot be considered entirely reliable. Parmenides believes that "one must speak plausibly about apparent things" (Fragment 28B1,32). Here are additional lines from his poem:

"I declare to you a plausible account of the cosmos To prevent any mortal view from surpassing you" (Fragment 28B8,60).

I announce to you this likely construction of the world, Nor shall any mortal opinion surpass you. According to the ancient commentator Simplicius, the world-building principles of Parmenides were "two forms"—opposites: On one side, the heavenly fire of flame [ether], Soft, highly rarefied [light], Uniform throughout, identical with itself, And on the other—the opposite of fire, "night, dense and heavy visage" (28B8, 55-59). Ancient commentators interpreted "night" in a material sense as the earth. The principles of light (fire) and night (earth) play the role of the sources of things. Christian writer Clement of Alexandria (2nd century CE) notes: Parmenides "wrote both a cosmogony and by mixing elements of light and dark, produces all phenomena from them and through them" (28B10). The assertion about the substantial nature of light and night can be supported by Parmenides' own words: The weight [of things] is named "Light" and "Night"... All is filled together with Light and impenetrable Night, Both equally... (28B9). Among Parmenides' natural philosophical views, the introduction of the concept of cosmic force—necessity, anankê (Greek avayxr])—is also noteworthy. This concept later became embedded in the teachings of some philosophers. In his poem, it is stated thus: You shall know the all-encompassing Heaven— From whence it arose and how the compelling Anankê bound it To guard the boundaries of the stars (28B10). G. Vlastos believes that Parmenides merges the goddess Dike (whom we mentioned in connection with Heraclitus) with Anankê, thereby moving away from the mythological notion of an external agent monitoring nature. For Parmenides, coercion becomes rational and immanent, a logical-physical necessity. The order of nature is deduced from the rational properties of nature itself.

Parmenides' metaphysical doctrine is represented in the concept of true, authentic being. Thus, he laid the foundations of the second, ontological form of metaphysics, while, it should be noted, the first, theological form of metaphysics was established by his teacher Xenophanes. N. Abbagano defines ontology as the second fundamental concept of metaphysics. Ontology "studies the fundamental characteristics of being, characteristics that all being possesses and cannot [choose to] lack."

The section of Parmenides' poem dealing with being is usually called the "Way of Truth" or simply "Truth." He himself refers to it as the "Way of Conviction." It presents the true (authentic) picture of the world (being) created by thought, as opposed to sensations. As the late antique critic of philosophical doctrines Sextus Empiricus writes, Parmenides condemned the theory based on doxa (opinions), that is, arising from unfounded subjective impressions, and recognized as the criterion of truth the scientific, infallible theory. He (Parmenides) explains that one should trust not sensations but only reason, and rejected them altogether (28B1; B7). In the poem itself, it is stated: You must know all: How the infallible heart of persuasive Truth, So too, mortal opinions, in which there is no Unchangeable certainty (28B1, 29-32). "Let not the rich habit of experience Lead you to this path, To gaze with aimless [unseeing] eyes, to listen With a noisy ear And to taste with your tongue. No, judge with reason (xoyoq)... What I have spoken" (28B7, 2-6). Sextus Empiricus explains that the "Many-punishing Truth," as recounted in the poem, is reason endowed with infallible representations of things.

Developing this rationalist approach, Parmenides advances perhaps his most famous thesis: "Thinking and being are the same" (28B3): The same is thought and what is thought, For without the existent, about which it is expressed, You will not find thinking (28B8, 34-36). As a result of the fusion of thought and being, logic becomes the primary tool for depicting the latter, and thus it is no coincidence that Parmenides is seen as a precursor to formal (traditional) logic with its fundamental law—the principle of contradiction, according to which recognizing something as existing means it cannot not exist, thus non-being does not exist. A. S. Bogomolov asserts that Parmenides provided an ontological formulation of the laws of identity and contradiction and constructed the first deductive system. Indeed, according to Simplicius, Parmenides said that two contradictory judgments cannot both be true simultaneously, he condemned those who identify opposites, and reproached those "who consider being and non-being as the same" (28B6). This reproach clearly refers to Heraclitus, and in this regard, we refer to R. Scherer, who notes that Parmenides' static dualism replaced Heraclitus' dynamic dualism: where all judgments were possible, none are possible here except the judgment of identity: A is A.

This logic of identity leads Parmenides, first of all, to the opposition of being and non-being: "The dispute... is this: IS or IS NOT?" (28B11). He resolves in favor of the "way of thinking, [which says]: 'IS'" (28B8, 1). This means affirming the existence of "IS" and denying the existence of "IS NOT," which Parmenides expresses in such identity judgments: "For being is being, / And nothing is not-being" (28B6, 1-2); "One [way]—that something is and that it is impossible not to be... / The other—that [something] is not and that it must necessarily not be" (28B2, 2-8). Identities are meant to emphasize the strict clarity of terms: the word "is" signifies only the fact of existence itself. Secondly, Parmenides draws the thought of the identity of being and knowledge (thought, as already noted), non-being and ignorance: That which is spoken and thought, Must necessarily be existent ["what is"] (28B6, 1). For what is not, you could neither know... Nor speak of (28B, 7-8).

Let us now turn to how Parmenides characterizes being. He names many "marks" of being: it is ungenerated, imperishable, whole, homogeneous, and complete; "it is now—altogether [~ simultaneously], / One, continuous" (28B8, 2-6). Ancient commentators well understood that Parmenides describes the attributes of absolute being. The absoluteness of being, or rather its stability, is safeguarded by the previously mentioned cosmic force, "irresistible Anankê," which here also appears under the names of Truth (Dike) and Moira. It firmly holds being within the bounds of great chains (28B8, 14-15, 26).

The summary integral character of absolute being is represented in such definitions: it is indivisible, continuous, homogeneous, within its boundaries, all together in it, it is the ultimate limit, and as a result—one (28B8). This summation and integrality of being is expressed by Parmenides metaphorically, "substantially, coarsely, visibly" as its monolithic nature: he compares it to a "lump of a sphere," since it encloses everything within itself and as thought is not outside of it but within it itself (28B3). Simplicius reports that the one existent "is like a lump of a perfectly round sphere" (28B8).

This image allows some historians of philosophy to see in being a certain ultimate materiality. For instance, R. Scherer proposes to consider Parmenides' "one" as matter, understanding it as a "homogeneous mass" and writes that in Parmenides' "unity" all structures dissolve, akin to the dynamic flow in Heraclitus. Scherer's proposal is appealing because it aligns closely with the spirit of metaphysics. If being is a "homogeneous mass," that is, formless substance, it means that Parmenides paves the way for a metaphysical understanding of matter as some primal matter, devoid of any sensory attributes and only thinkable, which Aristotle later discussed and which was considered a complete absurdity by natural philosophers.

The diverse content of Parmenides' poem shows that he laid the groundwork for the division of reality into the intelligible (metaphysical) world and the sensory (physical) world. Simplicius wrote that philosophers acknowledged a dual reality: the reality of truly existing, intelligible being, and the reality of becoming, sensory, which they did not consider possible to call "being" in an absolute sense, but "what seems to be" (28B1). What Simplicius said can be considered a classic explanation of Parmenides' worldview and all Eleatics, as contemporary and earlier historians of philosophy discuss the Eleatic dualism, the division between the phenomenal world and pure being, ontological and physical reality.

On the clear distinction introduced by Parmenides between the intelligible truth and opinion based on sensory experience, and accordingly, what truly exists and merely seems to exist, the following Eleatic philosopher—Zeno (~490-430 BCE)—built his arguments.

Aristotle already designated Zeno’s contribution as the invention of dialectics—the art of achieving truth through disputation or interpretation of opposing views. Zeno applied dialectics to prove and justify Parmenides' metaphysical positions concerning being. The content of his dialectical arguments consists of the famous "paradoxes" or aporiae (a7TOptai) which demonstrate the impotence of sensations in grasping truth, and consequently, the impossibility of recognizing multiplicity and movement as truly existing. Hence, through the method of contradiction, it inevitably followed the recognition of the truth of Parmenides' positions about the purely speculative path to the knowledge of being, its unity, and immobility.

Let's turn our attention to some of his arguments (aporia) on the aforementioned issues, starting with the "millet grain" aporia. The ears detect the noise of a substantial quantity of grains falling, measured by a medimnus (a unit of volume equivalent to 52 liters), but they do not perceive the noise made by a single falling grain. Although the noise of a single grain is not audible, we can prove its existence through calculations, establishing a proportion between the noises and volumes of the medimnus of grains and a single grain. Thus, reason, through rational operations, leads us to knowledge and truth despite the senses. This leads to another significant consequence, employed by later philosophers: there exists something that we cannot see, hear, or touch, but only understand.

In the situation of the "millet grain," we dealt with proving the existence of what we do not perceive. Conversely, there arises a situation where we must justify the existence, or the attribution to being, of what is given to us through sensation, since the senses are unreliable, and we need to "verify harmony through algebra."

This line of reasoning by Zeno is found in his exploration of the validity of sensory facts such as the multitude and movement of the things around us. Thus, the attempt to find arguments proving the validity of sensory perception of a multitude of things leads to aporias. Here is the simplest one: those who admit the existence of a multitude verify it based on obviousness: there are a horse, a man, and any other single thing, the combination of which constitutes a multitude. Thus, a multitude, such as that of people, consists of individual people, and these people are units. But a single person, a unit within the multitude—such as Socrates—is not only "Socrates," but also "white," "philosopher," "pot-bellied," and "snub-nosed." It turns out that the same person will be both one and many; therefore, Socrates is not one (a unit), and thus there will be no multitude composed of units. From this, it follows that being must necessarily be one. This aporia was described by the Byzantine philosopher Philoponus (29A21).

The reality of movement, or the possibility of attributing it to being, is refuted by a series of aporias, of which we will present the simplest. This is the "dichotomy" (division in half): if something moves along a finite line, it must, before it covers the entire line, first cover half of the line, and before covering half, it must cover a quarter, and before a quarter, an eighth, and so on to infinity, as the continuous is divisible to infinity. But the infinite cannot be traversed from start to finish. This aporia is also presented by Philoponus (29A25). This reasoning shows that movement is unprovable, and one must admit the immobility of truly existing things.

In conclusion, it is worth noting that Zeno's aporias had not only specific historical and situational significance for defending Parmenides' metaphysical theses but also a general historical-philosophical significance concerning the fundamental philosophical question of the relation between thought and being, the question of the reflection of being in consciousness. This epistemological aspect of Zeno's aporias is universally noted by historians of philosophy. For example, V. Ya. Komarov speaks of the clash in Zeno's aporias between sensory representation of the finite and the idea of the infinite, the clash between the real fact of movement and its conceptual understanding. A very insightful observation on the logical-epistemological meaning of Zeno's aporias, which is not exhausted by mathematical descriptions of the processes of movement, is made by the author of the article in the "Philosophical Encyclopedic Dictionary."

Parmenides' views on being were developed by his successor Melissus. His peak (akmè) is dated to around 440 BCE and is usually associated with a significant event in his life—the victory over the Athenian fleet. His teachings are mainly known through testimonies. Plutarch refers to him as "Melissus the physicist" (30A3), which corresponds to the title of his attributed work "On Nature, or On Being," indicating that the philosopher considered nature as being (30A4).

Melissus presented this nature—being—in the spirit of Parmenidean metaphysics and termed it using a word inherited from Xenophanes, "to pan"—"all," meaning the universe, or "en kai to pan"—"one-and-all." To denote the physical nature (world), he uses the word "kosmos."

Based on the usual Eleatic skepticism towards sensory knowledge, Melissus did not consider it possible to recognize the cosmos as genuinely existing. According to Melissus' logic, phenomena (sensory data) themselves refute their existence (30A14). In one of his fragments, there is reasoning of this kind: if people say that earth, water, air, fire, iron, gold exist (truly exist), then they must be like being, that is, unchanging; but they change; therefore, we do not know what exists (30B8).

The denial of the reality of sensory phenomena leads Melissus to construct a universe of metaphysical reality, opposite in its characteristics, expressed in the already familiar "Parmenidean" theses: "there cannot be many beings"; "being is not many but one, eternal, infinite, and wholly like itself" (30A5); "All [= universe] is one, immobile, and unlimited" (30A8).

In general, it should be noted that the characteristics of Melissus' universe are listed in many sources (see, for example: 30A1). But in the most detailed exposition, we find them in the work of an unknown commentator, known as Pseudo-Aristotle (30A5). Pseudo-Aristotle presents Melissus' doctrine in the form of a coherent system of propositions, one following from another. Their sequence is as follows: if something exists, it is eternal, as nothing comes from nothing; if it is eternal, it is infinite (apeiron), as it has neither beginning nor end; if it is infinite, it is one, as it has no boundaries; if it is one, it is the same everywhere (homoioun), as it cannot differ from itself; if it is eternal, infinite, and the same everywhere, it must be immobile.

Among these features of the universe, we will linger on two: its similarity to itself and immobility. If being were to change (becoming different), Melissus argues, even by a hair's breadth every ten thousand years, it would be entirely destroyed over the course of time (30B7). As for the immobility of being, it follows, firstly, from its essential properties; secondly, from the absence of emptiness. Thus, being is immobile because it is one and infinite in all directions, and there is simply no place for it to move (30A10). Moreover, in the physical world, everything moves through the change of place due to emptiness. But the fact is that "emptiness does not belong to being," "for emptiness is nothing, and nothing could not be," from which it follows that being is immobile (see: 30A5; 8; B7). In concluding the issue of the characteristics of the universe, we note a notable difference between Melissus and Parmenides, marked in various testimonies about his doctrine, which lies in the fact that Melissus declared the universe (to pan) to be infinite, whereas Parmenides spoke of its finiteness (30A5; 9).

Overall, in his doctrine of the intelligible universe, Melissus expressed both the theological and ontological components of Eleatic metaphysics, which is confirmed by ancient commentators. He had the right to consider the universe as a deity, being a singular immobile and infinite principle of all that exists (30A13) and had the right to deny it corporeality, since a single being (and such is the universe), if it had the properties of a body (thickness and parts), would no longer be one (30B9). Thus, the universe is a non-corporeal being of metaphysics. Finally, sources suggest the possibility of interpreting Melissus' universe as matter in its metaphysical understanding, that is, as a qualityless and intelligible substance, as we have already discussed in connection with Parmenides' being. Thus, Aristotle has an interesting remark related to the above-mentioned divergence between Melissus and Parmenides regarding the understanding of the universe, noting that Parmenides investigated the formal, that is, corresponding to the concept of the One (finite One), whereas Melissus investigated the material One (infinite One) (30A7). The well-known physician and philosopher Galen calls Melissus' "one-and-all" matter. According to him, Melissus meant that there is a common substance underlying the four elements, which is neither created nor destroyed, later called matter (hyle), and he named it "one-and-all" (30A6).

Regarding Melissus, as with all Eleatic philosophers, it can be said that in his metaphysical constructions, he significantly detached from reality. As Aristotle wrote, in theory, that is, in an abstract sense, their assertions seem true, but to consider this true of real things is akin to madness (30A8).

The second center of the formation of the metaphysical line in emerging Greek philosophy was the Pythagorean philosophical school. Its founder, Pythagoras (~570-500 BCE), originally from Ionia, from the island of Samos, moved to the Italian city of Croton (to which his peak is dated), which became the first major city of the Pythagorean community (society). The description of the Pythagorean society by the term "thiasos"—a religious assembly or festival—corresponds to some features of the life of the Pythagoreans. Plutarch, for instance, describes home festivals in honor of the saved Aeginians after the war against Troy, during which sacrifices are made to Poseidon, referred to as "thiasoi," or banquets, where they feast in complete silence. This description by Plutarch can be compared with the Pythagoreans' seclusion from outsiders, their "domesticity," and their vow of silence.

During political struggles and military conflicts, the Pythagoreans in Croton faced persecution. Polybius reported on the defeat of the Pythagoreans in Italy: "The best men in each city perished." The Pythagoreans relocated to the city of Tarentum (Latin; Greek Taras, Tarentum). The most famous Pythagorean philosophers of this period (after Pythagoras) are Philolaus of Croton (born around 470 BCE) and Archytas of Tarentum (born around 400 BCE).

The period from the 6th to the 4th century BCE, from Pythagoras to Archytas, is called ancient Pythagoreanism. During this time, Pythagorean philosophy evolved from natural philosophy to metaphysics. Sources for studying the views of the Pythagoreans are primarily the testimonies of later authors.

As mentioned earlier, the doctrines of the Pythagorean philosophers, like those of the Eleatics, combined natural philosophical and metaphysical views. An example of this can be found in the characterization of Archytas' work in the ode dedicated to him by Horace (Odes 1.28.1-6):

"O you who measured the countless seas, lands, and sands, Archytas ... you explored the realm of the ether and all the heavens with your mind..."

Pythagorean natural philosophers taught about material principles in the spirit of the Ionian philosophers. For instance, Hippasus posited that the fundamental principle is fire; Hippon, that it is water. Alcmaeon and Anaxagoras spoke of cold and heat; others, of dry and moist. A distinctive feature of Pythagorean natural philosophy was the doctrine of the formation of the world from the interaction of two principles—the limit and the unlimited—traced back to Pythagoras himself. It is believed that the unlimited in Pythagoras' thought was the material principle, specifically the air or breath (pneuma). Pneuma, when confined within certain spatial limits, that is, when influenced by the limit, becomes the world. L. Y. Zhmud, a prominent scholar of Pythagoreanism, suggests that perhaps Xenophanes' critique of Pythagoras' boundless pneuma—that the divine world does not breathe—was directed against it.

The introduction of such principles as the limit and the unlimited gave Pythagorean natural philosophy a dualistic character. The dual universe of the Pythagoreans is represented by a list of ten pairs of opposites that form the structure of the world. This list was compiled by Aristotle. It begins with the pair limit and unlimited, followed by pairs such as one and many, rest and motion, light and darkness, and others.

Philolaus viewed the interaction and integration of opposites within existing entities as a means of introducing order, proportion, and harmony into the world. According to his perspective, harmony emerges from opposites, forming a network of diverse mixtures and the reconciliation of discord.

The metaphysical aspect of Pythagorean doctrine was grounded in the same rationalist foundation as that of the Eleatics. The Pythagoreans rejected sensory perception as imprecise and believed that physical theory could not be established through it. The strength of rationalism in knowledge lies precisely in its ability to create theories that explain phenomena in the physical world. However, unlike the Eleatics, the Pythagoreans developed their metaphysical conceptions not on a general logical-epistemological basis or mere analysis of philosophical categories but on a scientific-theoretical basis, specifically mathematics. The Pythagoreans are credited with founding the mathematical science, which is underscored by the ancient view that Pythagoras coined the term "philosophy." It is also reasonable to consider him the originator of the term "mathematics," as the Greek word (μαθηματικός) was given a specific meaning related to the knowledge of numbers.

In a general philosophical or epistemological sense, the role of Pythagorean mathematics was that number was seen as a limit that constrains the boundless, thus enabling the understanding of the infinite, which itself is unknowable. As Philolaus explained, the limitless in number and magnitude cannot be known. We can only comprehend what we can measure and calculate, which means expressing it in numbers. To know is to find the number or measure of an object (parameter; σύγκριση - comparison). Thus, without the aid of numbers, nothing can be known.

The methodological role of mathematics in the Pythagoreans' scientific pursuits was evident. They applied it to explain sensory phenomena, including astronomical and acoustic facts, and developed mathematical expressions for musical tones. Their colleague in this endeavor was Zeno of Elea, who logically "calculated" the sound of a falling grain of millet. But the Pythagoreans approached this mathematically, showing that musical tones are related to each other as whole numbers. The development of mathematical theory of music posed a significant philosophical and aesthetic question: Can mathematics provide a complete understanding of music? Can the theory of musical harmony replace or exhaust sensory perception of sounds? These questions were debated among the Pythagoreans, with one of them, Aristoxenus, advocating for greater trust in auditory perception over mathematical interpretation. A similar debate is explored in A. S. Pushkin's tragedy "Mozart and Salieri."

Describing natural objects and phenomena using mathematics led to the establishment of numerical theory as the core of Pythagorean metaphysics (ontology): number replaced the Eleatic notion of the true being. Thus, the metaphysical part of Pythagorean philosophy is the doctrine of intelligible, immutable entities—numbers. This doctrine developed over time through several stages. At the first stage, Pythagoras equated everything with numbers but did not speak of numbers existing independently of the physical world.

The actual development of numerical philosophy began with Philolaus, in the second half of the 5th century BCE—this marked the second stage. Philolaus introduced a spatial understanding of numbers, associating them with geometric objects: 1 (monad) - point, 2 (diad) - line, 3 (triad) - plane, 4 (tetrad) - body (three dimensions). A. F. Losev referred to such numbers as sculptures. In Philolaus' views, numbers began to transform into independent entities as he introduced the concept of the movement of numbers: the motion of the monad forms a line, the movement of the diad forms a plane, and the movement of the triad forms a body.

The recognition of numbers as distinct entities was also facilitated by the early Pythagoreans' method of "counting" or "calculating" by forming geometric shapes and outlines of objects from counting pebbles (psephoi). A pebble was equal to one. Diogenes Laertius quotes the verses of Epicharmus, a student of Pythagoras:

• If we add or subtract a single pebble to an even or odd number, Will the number remain the same?
• Clearly, it will change (III, 11).

Using this method, Euripides, a student of Philolaus, would find the "number" of an object by forming its shape from pebbles. The numbers themselves thus acquired a spatial structure: they had "top" and "bottom," "right" and "left," order (sequence) of parts. Numbers began to express the essence not only of geometric objects but also of physical things. However, as L. Ya. Zhmud notes, Euripides did not yet equate things with the number of pebbles and did not state that they consist of numbers.

The decisive step in turning numbers into independent substances was made by Ecphantus, another student of Philolaus. He was the first to consider monads (units) as corporeal. Monads became the smallest bodies out of which the world is composed—this is "numerical atomism." Aristotle summarized this concept with the statement that for the Pythagoreans, number was both the principle (ἀρχή) and the matter (ὕλη) from which the world was created, and that "everything is number" according to the Pythagoreans. Aristotle's conclusions about the essence of Pythagorean doctrine were based on the views of Euripides and Ecphantus.

1. Ya. Zhmud distills Pythagorean numerical philosophy into two propositions: firstly, that sensibly perceived things are composed of units; and secondly, that everything in the world is arranged in accordance with numerical principles. This means that numbers exist beyond the cosmos as its substance, while things are viewed as imitations or reflections of numbers.

To illustrate how this numerical theory operated, consider the general worldview created by the Pythagorean philosophers. According to A. F. Losev, the archetypal essence of the entire cosmos, its form or eidos, was the decade (ten), as they considered ten to be a perfect number. Philolaus provided numerous arguments supporting this view. For example, the number ten contains an equal number of even and odd numbers; it includes a point (monad), a line (diad), a plane (triad), and volume (tetrad), among other features.

In accordance with the decade, Philolaus depicted the cosmos as follows: around the central fire, which he called the Hearth (ἑστία) and Counter-Earth (ἀντίχθον), orbit the Moon, the Sun, and five planets (Venus, Mars, Mercury, Jupiter, Saturn) along with the sky in the form of the Milky Way. The outer boundary of the cosmos is the peripheral fire.

It should be noted that the Pythagorean cosmos was not merely numerical but also musically numerical. This was expressed in the famous Pythagorean idea of "celestial harmony" and the music of the spheres. Celestial bodies produce sounds as they move. As Cicero ironically observed, Pythagoras believed the world sang in notes. Of course, this was a special kind of music. Kepler rightly described it as "music perceived not by hearing but by reason." In other words, it was "mathematical" music, in which numbers serve as the tones.

Philolaus' concept of a central primordial fire as the "inner" boundary of the cosmos significantly influenced the development of the heliocentric model of our universe. Additionally, the Pythagoreans stimulated discussions about the nature and status of mathematical objects and their place in the cosmos. This problem was actively discussed in both ancient and medieval philosophy, often positioning mathematical objects between the world of incorporeal and corporeal entities.

In conclusion, let us compare metaphysical and natural philosophical concepts. Natural philosophy deals with the surrounding world and nature, remaining within the bounds of sensory experience and can be equated with science or theoretical natural history. Its content consists of natural-scientific hypotheses, thus aligning it with the history of science (physics, cosmology). If the emergence of natural philosophy is described as a movement from mythology towards science, then the emergence of metaphysics represents a transition from science to philosophy. In this sense, natural philosophy can be seen as pre-philosophy, while metaphysics is the core of philosophy. This interpretation is justified by the fact that natural philosophy, like science, has a ready-made object of study—nature. Metaphysics, on the other hand, creates its own domain of inquiry. It deals with concepts and categories, analyzing ideal objects and the intelligible world. Hence, metaphysical teachings exhibit the unique philosophical issue of the relationship between thought and being, as the concepts created within it serve to make sense of reality and the world of experience. It is thus natural that philosophy was long referred to as metaphysics.

Metaphysics was not homogeneous. The Eleatic and Pythagorean schools represent two distinct branches of metaphysical speculation. Eleatic metaphysics can be defined as the doctrine of the supra-sensory material, as their being was conceived. This idea of supra-sensory materiality later influenced natural philosophers and laid the foundation for materialist philosophy. Pythagorean metaphysics, on the other hand, is the doctrine of the intelligible ideal, represented by numbers and other mathematical objects. Pythagorean principles were also employed in natural philosophical constructions but within an idealistic framework.