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Ancient Greek Science and the Infinite

parthenon The Parthenon of Athens

The ancient Greeks were one of the first peoples to think about nature systematically. What we call science today arose in the speculation of the ancient Greek cosmologists or natural philosophers. These thinkers focused their attention on the natural world and then asked ‘What is nature made up of?’

Their questioning approach to nature quickly led them to ask a follow up question, namely: ‘What is the essence of the natural world?’ This is the first time in recorded history that a distinction was applied between how things appear and how they actually are. As philosophers, moreover, which is what these ancient Greek thinkers were (science grows out of philosophy and these thinkers are also known as the pre-Socratics since they flourished before the great ancient Greek philosopher Socrates [died 399 BCE]) the earliest Greek philosophers also made use of external observations about the order and complexity of nature to ponder the enigma of how we too, even as we observe it, are somehow part of the natural structure and its laws.

For example, it was said by Anaximander of Miltetus that the universe as a whole was governed by ‘moral necessity’.


Pictured above: The Didyma Temple of Apollo

The Ionian school and the Infinite in nature

Amongst the important thinkers of the so-called Ionian school (as the early Greek natural philosophers were also known, since they flourished on Western coast of Ionia -modern day Turkey) we can here highlight the school of Thales. This school included, its founder Thales of Miletus (c. 630/20- 546/45 B.C.E) and his two pupils Anaximander (c. 610/11- 547/46 B.C.E), and Anaximenes (c. 546/5- 528/5 B.C.E).

This is the first important group of thinkers we can examine for scientific insights regarding the infinite. The Ionians flourished from between the 7th and 6th century B.C.E. Aristotle, who was himself scientifically minded, called the Ionians ‘theorists of nature’ (i.e. Phusiologoi from the Greek: Phusis= nature, and logos = an account or theory of something). Furthermore, according to Aristotle, this title was held to be especially apt because the central unifying theme of the Ionians was the asking of, and attempts to give an answer to, the question: ‘What is the world made of?’

Insofar as their answers went, we can here add, the Ionian systems were usually reductionist. By calling their thought reductionist what is meant is that, for the most part, they sought an explanation by reducing the many facts at hand to a limited series of causes or reasons. In fact almost all the early Ionians attempted to explain the plurality of natural processes as determined by one underlying stuff or category (i.e. one fundamental substance). The technical term substance (which we will make a great deal of use of throughout this and subsequent sections) is an important one philosophically and in ancient Greek nature philosophy as well. In general a substance can be viewed as more or less synonymous with an independently existing thing. The most popular candidates for the primal category or primal substance, within Ionian thought, was usually one or another combination of the four Greek elements: Earth, Water, Air, and Fire (as they were later definitively described and categorized by Empedocles).

In general, the pre-Socratics make use of a speculative and metaphorical approach to describing and understanding the world. All the Ionians, however, seem to have accepted the fact that the existence of the external world or cosmos should be taken as a given. From this immediately perceived ‘external’ nature, they also noticed that an ‘internal dimension’ was manifested: some natural things (including human beings) are animate and some are not. Some natural things are self-moving and some are not. But the question that unifies their research is always the following: “What is common to all natural things?” In other words, what is the underlying property of all of existence and how can it be understood? Here is where the concept of infinity arises, because if there is one unified underlying nature, we must, in order to understand it, try to establish its characteristics and definite attributes. The Ionians, for the most part, took a qualitative rather than explicitly quantitative approach. The Ionian answer will be that all natural things are made of an underlying substance. A primal substance as most basic ‘kind of being’ or ‘mode of [independent] existence’ is what the Ionians sought to find. More importantly, that substance (whatever it was) had to be able to determine the world, according to these first scientists, and therefore had to be such that it could be understood and named.


Loving both symmetry and order, a reluctance is manifested in the ancient Greeks for accepting an actually infinite natural world. This reluctance to explore infinity is evident in Thales. Thales viewed the natural world as ultimately made up solely of water. He is also said to have conceived of the earth as floating on a giant cosmic ocean. Although he is said to have written entire works, what surive of Thales thought has been passed down through collected fragmants (these have been collected together and translated by Philip Wheelwright in his book the PreSocratics ). Both the entirety of the natural world, and the earth as part of it, can be inferred to be finite. If the boundary of the universe is a container holding a universal ocean, then it would appear that the persisting coherence of this ‘water world’ model requires a boundary. According to Thales moreover, all moving things can be said to be both filled with water and to have souls. If the gods create, or are responsible for, the soul then perhaps Thales is implying that the natural cosmos can be viewed as a self-moving ‘god-filled’ thing. Another possible interpretation of Thales’ view of the cosmos might be the following. Perhaps he believed that the natural world was a divine organism or cosmic animal that grew out of water. In any case, both interpretations of Thales’ surviving work lead us to the conclusion that the natural world is finite. No one today takes seriously Thales model of the cosmos, nonetheless we should take seriously the deep philosophical insights that Thales had and which were the following: (1) the entire natural world must be made of some fundament substance, what is this substance? And (2) the visible world of independent changing things is not the ‘real’ world, (3) Thales answers that the unifying substance or essential property connecting the world of appearance is one fundamental reality (on his model, water). To ask ‘What is the world really like?’ is another way to ask: ‘What is the essence of the world?’ Thales’ answer was that essentially everything is a manifestation of water.


Thales of Miletus


Thales’ student, Anaximander, is said to have left a whole book behind with the title On Nature. Unfortunately no extant copy if this work exists. Like Thales before him however, Anaximander was interested in cosmology. Philosophically, Anaximander builds on Thales’ insights but also expands and develops his teacher's ideas. This is especially evident as regards the infinite. Anaximander held that the first substance or the true reality of nature was undifferentiated matter (He might have meant earth but there is no consensus on this). Anaximander called this undifferentiated matter ‘the unlimited’ or ‘unbounded’. Anaximander’s theories also differed in other ways from those of his teacher. For example, where Thales believed that the earth rested on a fundamental watery ground, Anaximander maintained that it is not a flat raft or plate-like structure but a thing like a body, possibly cylindrical in shape, and floating free in a surrounding medium. Also where Thales identified all natural things as really ‘water’, Anaximander said, no, the medium or fundamental substance is not water but must be some kind of undifferentiated stuff. This stuff was precisely what Anaximander identified as the primary substance of reality and described as ‘unbounded' (in Greek: ‘to aperion’) but an equally valid translation would be the infinite.

Actually, this idea is a great improvement on Thales since in making water the underlying substance Thales was giving priority to an already existing natural kind of thing and saying that this one natural element or thing is actually more substantial and essential to the make up of the world. But why and how the plurality of natural kinds arises from one already existing natural substance is a problem that Thales never clarifies. Anaximander, in effect, seems to be implying that Thales' answer to the question what primarily exists was not sufficient. The boundless or unlimited, by contrast, can-accoerding to Anaximander- account for all the elements because it is:

  1. unlike the already bounded and individuated elements and natural kinds, somehow infinitely mutable
  2. and,
  3. indeterminate, at least in quality, i.e. essentially lacking all special characteristics and attributes—but also, somehow, able of acquiring or taking them on.

The unlimited thus seems to have been viewed by Anaximander in this way, in the words of a later Greek thinker (Simplicius of Cilicia):‘indefinite in both kind and extent’.

Anaximander's concept of the unbounded is the first explicit mention of the infinite in ancient Greek philosophy and science.


The last of the Ionians is Anaximenes of Miletus who is often viewed today as a less significant thinker than Anaximander. Anaximenes’ went back to Thales’ assumption that the one underlying substance could be identified with an already existing natural element. For Anaximenes that element was air and his writings imply that ‘air’ is the underlying reality of the natural world. Anaximenes furthermore disagreed with Thales and rejected the claim that the earth floated on water. Although Anaximenes viewed the earth as flat, like his teacher, it was said by him to rest on a cushion of air pressing down on it. The fundamental substance or ‘air’ although determinate of quality is infinitely changing and indeterminant in essence. Anaximenes’ most important contribution to the philosophy of nature and later science, moreover, may well have been his insistence that, although the natural elements are continuous and connected, the differentiations they manifest can be accounted for by a study of an analysis of structural movement. Thus, air (as the fundamental substance) can become water when it is condensed and it can become earth or fire when it is rarefied.

Condensation and rarefaction are scientifically important notions because the earlier Ionians (specifically Anaximander) had asked: ‘if the various natural kinds are all made up of one substance or original substratum why do they behave in noticeably different ways?’ Anaximander’s answer was along the lines that opposites are somehow differentiated and segregated by the rotary motion and movement of the unbounded. But this answer still leaves open the question of why rotary movement in an undifferentiated substance (of itself) should lead to the generation of what the Greeks called contraries or opposites (hot-cold, black-white, wet-dry, etc.). Anaximenes provided an answer by establishing a rational explanation based on changing states of matter. He thus provided a key for explaining the phenomenon of change. Nonetheless his explanation was filtered through his air-centered worldview. Anaximenes thought that low pressure or condensed air becomes heated and then warm like fire while coolness was caused by the rarefied movement of air (high pressure change) and led to the creation of earth and water. Thus, air assumes opposite qualities under the influence of motion. However, although changes of state might be potentially infinite for Anaximenes, the cosmos – constructed of air - is finite and bounded like Thales’ watery universe.