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Ancient Greek Science and the Infinite (Cont.)


Pythagoras of Samos (born circa, 570 B.C.E)

A short while after the Ionians had founded their cosmological school for understanding nature, to the West- in modern day Italy- another group of Greek thinkers decicded to take a more ratioanlist approach for understanding the world. Pythagoras of Samos founded an influential school of advanced thought in Croton (modern day Calabria, Italy) and may have been the first thinker to use the word “philosopher” to describe himself. Pythagoras must have been a charismatic figure since he is said to have started a cult and, along with his many students, developed a community of mystics/mathematicians.

We know very little of the historical Pythagoras, but it is reported that he visualized a universe of geometrical harmony and that he used spheres, circles, and vortical motions as basic forms for describing the design of the universe. According to Diogenes Laertius (a philosopher and historian from the third century), “it was said that Pythagoras was the first thinker to call the heavens ‘cosmos’ and the Earth ‘a sphere’. The Pythagorean community believed that the heavenly bodies were perfect spheres moving in perfect circles around a central cosmic fire that lay beyond the reach of mortal vision. The heavenly bodies emitted melodious notes and their celestial symphony or harmony of the spheres lay beyond the reach of mortal hearing”. Therefore Pythagoras seems to have taught that the Earth was a sphere and, along with the sun, that it circled a great fire in the universe. The great fire may have been a physical expression of Oneness and his anti-geocentric worldview is an idea that did not become popular again until Copernicus and wasn't accepted in the West until the 17th century. More important than astronomy to the Pythagoreans however, were the harmonies in the cosmos and these were of interest because they were said to be governed by mathematical relations.

The Pythagoreans, as followers of Pythagoras were called, were at once both more mystical and more rational than the Milesians. For one thing, the entire cult believed in reincarnation (the Greek term is “metempsychosis” – the transfer of souls) and they also practiced strict meditation. They were vegetarians but were not allowed to eat beans. But more interesting, from a philosophical perspective, they had adopted of numerological belief system holding that the key to understanding ultimate reality could be found in mathematics. To this day, most people still learn the Pythagorean theorem in school, i.e. that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The Pythagoreans were so impressed with the universality and clarity of mathematical knowledge that they started to apply it to the entire universe.

One area where they believed mathematics could shed great light was the movement of the heavenly bodies. Astronomy probably goes back to a time before the first civilizations arose in ancient Mesoptamia. By the time of the Babylonians astronomy was mainly relegated to the domain of priests and prophets. The Greek astronomers, by contrast (as we’ve seen) were philosophers and scientists. Using geometry, part of their knowledge of which they borrowed from the ancient Egyptians, the Greeks made great strides in developing theories for explaining natural processes. Whereas the Babylonians used elementary arithmetic for practical tasks and the Egyptians used geometry in a similar way, mainly as a measuring system to delimit tracts of arable land against the movement and flow (the rise and ebb) of the Nile, the Greeks bagan to practice geometry for its own sake. Subsequently, the Pythagoreans had the brilliant idea that the theoretical and mathematical models they were developing could be applied directly to the world in order to explain its underlying structure. The significance of this idea for later science and philosophy is impossible to underemphasize.

Maintaining that the universe was, in some way, constructed out of numbers and that all things had a numerical essence, the Pythagoreans came to believe that everything had to be clearly limited and well-defined, like numbers in a mathematical equation. In fact, for the Pythagoreans every single aspect of the world could be represented by a finite arrangement of natural numbers (and as Rudy Rucker points out, for Pythagoras “natural number means whole number” Rucker, 1982, 3). To this end, the Pythagoreans maintained that number theory was essential for understanding reality. They thought deeply about the question of what numbers are, and how numbers reflect truths about the nature of the world and came to the following conclusion.

There were two classes of numbers:

  1. the odd,
  2. the even.

Since odd numbers cannot be divided in half, they related the concept of odd to limit and the concept of the even was related to what is unlimited. The limited, they said, was balanced and good and right and masculine. The unlimited is chaotic or evil and feminine. In addition to odd and even numbers, there were also ‘square’, ‘oblong’ and ‘triangular’ numbers. The Pythagoreans would very likely have used pebbles or stones to visualize these concepts. One way to see square numbers as balanced and limited (opposed to oblong) is to start with one and use a progression called a ‘gnommon’. The one becomes three which expands to a symmetrical five points all balanced and square like. Stating with two, a less symmetrical pattern emerges, 2, 4, 6, in oblong fashion but asymmetrically so manifesting no definite boundary.

gnommon at work

This visualization of number properties is where geometry came in.

For example, if a point was ‘1’, a line = ‘2’, a surface = ‘3’, a solid = ‘4’, and the total of these numbers: 1 + 2 + 3 + 4 = 10 which according to Pythagoras was a sacred ratio. This was expressed in the tetractys.

the Pythagorean tetractys

The natural numbers, therefore, revealed the essence of the universe in the following way. First, points had finite size. Lines and surfaces, built out of points, also had finite thickness (because they were constructed from finite points). It helps that the Greek word for one (hen) can also mean: ‘unit’ (in arithmetic), ‘point’ (in geometry) and ‘indivisible unit’ of matter (in physics). Eventually, the concept of atomism, of a universe constructed from points, would lead to the atomic theory of matter later propounded by thinkers like Leucippus and Democritus. According to Pythagoras and his followers, however, the atoms were not physical, they were numbers. His number philosophy and arithmetical model was joined to geometry in such a way that many problems for understanding the natural world were analyzed in terms of what he called ‘triangular’, ‘square’ and ‘oblong’ numbers.


The Pythagoreans also attached great importance to means. The arithmetic, geometric, and especially the harmonic, mean, were all deemed significant. the Pythagorean worldview maintained that boundaries and finite sums were all important. Musical harmony, for example, was discovered to be expressible in terms of ratios. Two strings tightened, tuned, and stretched out of different lengths, e.g. 2:1 resulted in the shorter string sounding out an octave higher than the longer. If the ratio was 3:2, then a fifth is arrived at and the 4:3 ratio, produces a fourth (5:4 = a major third and 6:5, minor third, etc).

Greek mathematics had no concept of 'zero' so that when the first four numbers were added together (1 + 2 + 3 + 4) the result was the perfect sum of 10 that, as we've seen', represents the tetractys.

The Pythagoreans viewed the tetractys as a reflection of the harmony of the universe (later the term 'music of the spheres' was used to describe these cosmic ratios). Although the cosmos appears chaotic and divided, according to Pythagoreanism, underlying the appearance is a beautiful and harmonious system of finite relations. The cosmos therefore be seen as a musico-mathematical whole. The appearance is one of uneven, changing flux but the reality is unchanging symmetry of perfect but hidden order. This idea seems to surface and resurface periodically in history. Today we have so-called “string theory” an incredibly complex mathematical model of the universe that explains all the laws of physics as resulting from a set of constants, fundamental forces and the collective vibration of quantum mechanical interdimensional strings. Platonism was also heavily influenced by this model of invisible structures establishing the order and intelligibility of visible reality ( see Plato on the Infinite).

Perfect Boundaries and Transfinite Consequences

Since Pythagoreanism was based on mathematical (perfectly logical) principles their dream of reducing the world to the natural numbers and a collection of finite harmonious ratios was undermined by their own mathematical cleverness. One of the Pythagoreans eventually discovered that, in general, the sides and diagonal of a Euclidean figure are incommensurable. Not all geometrical objects can be expressed in whole numbers but instead require integral or fractional terms. This discovery was a disaster to the whole Pythogorean project since it destroyed the harmony of their cosmos and introduced the potential for the mathematically infinite into the foundations of the basic arithmetic that they used to describe the character of simple objects. The finite series of natural numbers therefore had failed to serve the Pythagorean mission and this betrayal led to the suggestion that the infinite suggesting was part of ultimate reality. The conclusion was inevitable since, according to Pythagoreanism, the real is mathematical and the series of rationals never ends…. What is much worse for the Pythagorean agenda, some numbers can only be expressed using apparently infinite decimal expansions: ⅓ = 0.333…. π = 3.14159265…The square root of 2 itself is an example, (√2=1.41421356…). And some of the above (such as both (√2 and π) are ‘irrational’ numbers whose decimal expansion never ends and from which no recurring pattern emerges. Today we say that both the rational and the irrational numbers count as ‘real’ numbers. The spirit of Pythagoras would certainly be horrified by this development, but the failure of Pythagorean metaphysics was not enough to stop the ancient Greek discomfort in dealing with the infinite.

Other Pre-Socratics

camp fire

Mysticism and logic were connected together in the thought of Pythagoras, yet his approach failed to secure the universe as finite. In Pythagoreanism mathematical formulas meet reincarnation and ritualistic practice, but the infinity of numerical series and their failure to fit into a symmetrical Euclidean geometrical space conflicted with the idea of a perfectly symmetrical external universe. Whether Pythagoras, as was rumored, had travelled to ancient Indian or not there was a strain of mysticism in ancient Greece and it began to interact with scientific and philosophical framework introduced in Miletus circa the end of the sixth and beginning of the seventh centuries B.C.E. A later thinker who followed in the footsteps of the Milesians but was critical of the School was also from Asia Minor and but from a city thirty miles or so from Miletus. This was Heraclitus of Ephesus (c. 535-475 B.C.E). His colleagues seem to have found him difficult to get along with; as he was branded a misanthrope and pessimist and viewed as an insufferable snob and egotist. They called him Heraclitus the Obscure

Heraclutus is still known today for his terse aphroistic writings wherein we find thoughts such as the following:

Philosophically, Heraclitus’ thought stepped away from Pythagorean abstraction and returned to Ionian materialism. For example, he maintained that an everlasting fire is the first substance and principle from which all the natural world emerges. This everlasting fire is also said by Heraclitus to be the essence of all things. Heraclitus writes: “It [fire] throws apart and then brings together again; it advances and retires” (catologued as Fragment 31 by Wheelwright) Fire is thus the most fluctuating of all things and links the essential reality (the world of being) directly to the world of appearance (externally also mutable nature). Since he embraced this dynamic picture of reality Heraclitus is still known today as 'the philosopher of flux'. Two of his most famous sayings can be found in the following fragments: “Everything flows and nothing abides; everything gives way and nothing stays fixed”. (listed in Wheelwright as Fragment 20) and “You cannot step twice into the same river, for other waters and yet things go ever flowing on” (Fragment 21) . Later philosophers took this teaching of universal change very seriously. In the words of Aristotle, “according to Heraclitus all things are in motion and nothing steadfastly is” (cf. Aristotle’s De Caelo Book III, 1, 289b30). This is a problem for philosophy, since philosophers aim at finding universal principles unchanging laws and truths. But Heraclitus also had another side to his theoretical musings. The counter-balance to his emphasis on constant change can be found in his doctrine of the logos or reason which complements the notion of flux and completes the Heraclitean worldview.

The logos or thinking, like the flux, was said to be “common to all” (Wheelwright 1966,p. 75) but in grasping the logos and gaining understanding of its nature we are both led to wisdom and somehow (perhaps in a quasi-mystical sense) can transcend the unstable flux of nature. Thus, we can see in Heraclitus a far more direct anticipation of the Plato’s doctrine of the one and the many. That is, Heraclitus presents a metaphysical distinction between the temporal ever-changing flux (the world of the senses) and the one divine (and therefore eternally unchanging) logos. Ideas like this can be found in Eastern texts such as the Tao Te Ching and the Bhagavad Gita, (See Eastern Notions of Infinity) and Heraclitus's philosophical thought delves into the realm of paradox and does not systematically articulate the nature of his ontological duality.

In his cosmology, working in the Greek tradition of philosophy of nature, Heraclitus can be seen to follow in the footsteps of Anaximander. Basically Heraclitus takes the view that the “strife”, the hypothesized movement of the underlying fire (which characterizes the world of change) is something that brings about opposites. “Cool things become warm, the warm grows cool; the moist dries, the parched becomes moist” (22). For Heraclitus however this strife or flux is also viewed as the source of an apparent stability and harmony in the universe. That the order and harmony of the physical world is only apparent can be inferred by Heraclitus’ claims that nature in reality is an ever-living fire. In fact, Heraclitus criticized Anaximander precisely for implying that there was a “cosmic justice” or “moral necessity” at work in the world of nature. By contrast, for Heraclitus war (Polemos) is the father of all things.

The apparent harmony in the world of nature is really the result of constant, unending, tension and clash (strife) in the ever-changing dynamic world-fire. “There is an exchange of all things for fire and of fire for all things, as there is of wares for gold and of gold for wares” (Wheelwright, F 28).

The clash of opposites, however, has a transcendent unity in the logos (a unity in diversity or One in the Many), but the exact relationship between the logos and the eternal flux is very enigmatic and difficult to conceive. The precise connection between the logos and the flux of cosmic fire is never clearly spelled out. Perhaps, Heraclitus understood the physical universe as composed of a kind of mass-energy that could convert itself into the basic substance or nature and in this way make up the reality of all visible things. In this sense the flux, or everlasting fire, can be viewed as a kind of metaphor for what is a qualitatively different kind of underlying reality (eternal space-time energy) that produces (out of itself) the world of nature.

Heraclitus on Oneness and Eternity

In a sense, and placing his thought in context of the earlier Greek philosophers, Heraclitus seems to accept the unity of the world preached by the Milesians. He also seems to accept a kind of qualitative infinity or eternal process of change abiding in nature. On the oneness and eternity of the universe, Heraclitus writes:

We can end our overview of Heraclitus with a longer fragment and a passage where he tells us about how his method of inquiry would work:

Although the Logos is eternally valid, yet men are unable to understand it—not only before hearing it, but even after they have heard it for the first time. That is to say, although all things come to pass in accordance with this Logos, men seem to be quite without any experience of it—at least if they are judged in the light of such words and deeds as I am here setting forth. My own method is to distinguish each thing according to its nature, and to specify how it behaves; other men, on the contrary, are as neglectful of what they do when awake as they are when asleep” (Wheelwright, F 1)

Heraclitus paradoxical philosophy of change was answered by the logical and metaphysical approach of Parmenides of Elea.

metal orb

Parmenides of Elea (c. 540-470 B.C.E).

A student of by Xenophanes and influenced by Pythagoras, Parmenides has been called “the father of Western metaphysics” and this legacy rests on his attempt to challenge the whole account and approach of cosmology that earlier thinkers (specifically the Ionian naturalists and Heraclitus) had pioneered. For Parmenides all of nature and being itself is one, eternal and unchanging. This sounds like the exact opposite of what Heraclitus had argued, and in many ways it is. This is also a strange claim. Obviously the natural world always changes. Nature, according to the evidence of our senses, is a series of processes that appear to be in a state of constant movement. What Parmenides must have meant by his description of the universe as static is that the underlying reality or substance that allows for the apparently shifting appearance of nature is in reality one space-filling completely connected mass and an expression of what can be called the plenum of being.

Being is. This is the Parmenidean motto- and logically it is difficult to find fault with this approach. If I'm thinking then I'm thinking about something, if anything exists, then it must exist. What is non-being? Can we even conceive of it? Taking this approach seriously, the infinite, if it exists can only be described as the fullness of being.

Since being is everything and non-being doesn’t exist, the infinite is the One. However, Parmenides (like Heraclitus before him) takes issue with Anaximander. This is not to say that his thought has nothing in common with Anaximander. Parmenides radically reinterprets the unbounded and constantly changing infinite substance. Change always involves a transition from what something is into a new state in which the previous manifestation of the substance is no longer. Parmenides, accepting his logical premise that Being always is, denies that change in the above sense is possible. Part of Parmenides argument is also a critique of Pythagoras and his number universe. Pythagoras wanted perfectly harmonious structures, ideal numerical ratios, to determine reality. Parmenides believed that if these structures existed they still could not account for space. Numbers are perfect forms, but if they sit in an empty non-existing void, the universe would not have a boundary. Therefore, numbers cannot be the principle substance of the universe. Only being as a whole can be the primary substance. But this fullness of being is everywhere One.

The only surviving work wherein Parmenides worldview is described (outside of the suriviving paradoxes of his student Zeno see Zeno's Arrow) are in the long poem On Nature

We can analytically break down the doctrines spelled out in Parmenides’ poem On Nature as follows:

  1. Either something exists or it doesn't (Principle of Non-Contradiction).
  2. Being exists. It is one, eternal and unchanging.
  3. If being is equivalent to nature (viewed as all space-filling mass), then non-being = empty space.
  4. But being always is.
  5. Therefore, being cannot have a beginning or an end and it cannot come to be from non-being nor cease to be once it is.
  6. In that case, being is really unchanging and non-moving: everywhere the same.
  7. Conclusion: Non-being cannot exist.

Parmenides concludes that, since “to be” (existence) is equivalent to “always existing,” movement and plurality as well as empty space, are not real. Reality is conceived as a completely full eternal space. Non-being, on the other hand, is logically impossible and, subsequently, truth is completely foreign to the appearance of nature as known to us through the senses. Therefore, only unchanging being exists (in the only way Parmenides allows the meaning of that word) and we can grasp this fact only with the mind and logic. Appearance and movement as perceived through the senses are therefore the cause of all intellectual or epistemic error. In effect, Parmenides speculations lead us to a purely metaphysical doctrine that claims that the only way we can know reality is through the structure of thought. This position, however, is the strongest possible condemnation of Ionian cosmology and naturalism (i.e. empirical science) that can be conceived. That is: if movement is not real, and if the senses do not lead us to any kind of reliable knowledge about reality, then philosophy of nature is condemned to failure before we can even begin. Empirical science starts with both theories and observations. It is partly logical but it needs experience or what philosophers call a posteriori data (information obtained after experience takes place). Parmenides wants to restrict us to an a priori understanding of science and philosophy. He implies that the only way to understand and explain the universe is through logic and independent of sense experience. On the positive side, as mentioned above, this tendency to move from cosmology to metaphysics is one that will repeat itself in subsequent history of western philosophy. Parmenides would be a highly influential source on the metaphysical thought of Plato and subsequently on a great deal of later thought up to the present day.


Melissus of Samos (born circa 441-0 B.C.E.)

Parmenides students’ took sides on how to interpret his fundamental idea that ‘being is’ Zeno of Elea, adhered to the orthodox interpretation supported in Parmenides’ own cosmology that the universe was finite, but another student of his from Samos, by the name of Melissus, broke ranks with his teacher and affirmed the infinity of being. Melissus’ argument follows the logic of Parmenides’ own logical approach but comes to a different conclusion. If, as Parmenides asserts, the world cannot come into being or pass out of existence, and if nothing comes from nothing, the world, which exists, not only cannot not have existed, but must also be eternal and unlimited, i.e. infinite. This position is difficult to escape if, like Parmenides, we deny non-being. What therefore can exist beyond the finite sphere of the universe? The Parmenidean might argue that the universe bounds itself, but given the materialist understanding of nature maintained by Parmenides- this response is not so convincing. If the universe is extended and physical, we can ask, then what , precisely, is it extended into? Parmenides denies anything like a void, vacuum or empty space (which would resemble non-being), the only conclusion we can draw is that the finite unity of matter is all that exists. But no proof can be given that there is not a dimension beyond the physical or that what is extended in space is not also somehow bounded. In Melissus of Samos we can therefore find the second affirmation of an actual infinity in nature (Anaximander of Miletus was the first). Although his argument should have put the idea of an actual infinity into play as a living option for ancient Greek philosophers, the hesitation about accepting an actually infinite nature would win out and no metaphysical infinity is seriously entertained by Greek philosophy until the Neoplatonic thinkers begin to reform Plato and Aristotle.