Plato on the Infinite


Plato (c. 428/7-347 B.C.E)

Pythagoras as we saw, turned the study of cosmology away from matter and towards ideal mathematical structures. Beginning with Pythagoras, physics and mathematics became joined – in fact, they were one and the same. This Pythagorean tactic later led to atomism in Greek science, when the assumption of original units was joined to Ionian materialism. However, and as we’ve seen, this entire approach was rejected by both Heraclitus and Parmenides. Parmenides, for example, made a radical distinction between what can be known and therefore said to exist (the way of truth as grasped by the mind) and what is illusion, mere opinion and therefore non-existent (i.e. the not fully knowable world of non-being and the world of change apprehended by the senses). Distinguishing between truth and opinion Parmenides disparaged the changing world of opinion but affirmed a Monistic unity of being. Heraclitus, by contrast, argued that the world of nature or change was the locus of truth, because the intelligible aspect of the changing world of nature, natural law, was pervaded by the eternal logos. According to Hercalitus, only change is real- the static unity of Parmenidean Being is the illusion.

Enter Plato, the great synthesizer.

Plato will combine aspects from these earlier thinkers in order to present a philosophical system founded on a unique cosmology. Platonism later becomes highly influential with writings outlining a sophisticated system of philosophy that is still taken seriously by many people today.

The theory of the Forms

Plato was strongly influenced by Pythagoreanism but rejected the belief that natural things can be identified with mathematical structures. On the contrary, Plato agreed with Heraclitus that physical objects (or particulars) were constantly changing and impossible to pin down. However, due to the influence of Parmenides, Plato also thought that mathematical structures can represent truths about reality by reference to their connection to a true and unchanging kind of being.

This Parmenidean belief led Plato to establish all of philosophy on a metaphysical foundation. Nonetheless where Parmenides’s One was univocal and unchanging (denying the ontological significance of the information communicated by the senses), Plato's analysis led him to downgrade the senses while still acknowledging that they convey information about the world- albeit about a deficient mode of natural being.

True being for Plato is manifested in ultimate principles that he calls the Forms (in Greek eidos or idea). Plato’s Forms make the natural world intelligible even though they exist beyond it. Form, and not matter, according to Plato is the essence of true being and the Forms are therefore the reason for the changing things that we see and experience. Furthermore, according to Plato, Form and formal truths are not perceptible or able to be seen with our physical eyes. As invisible, intangible principles, Forms are nonetheless said to be supremely intelligible. In Plato's writings, the Forms also seem to have at least two distinct functions:

  1. Forms differentiate things.
  2. Forms make things what they are (i.e. they bestow an essence onto things and, through what Plato sometimes calls 'participation', allow particulars to exist as copies or imitations).

According to Plato's controversial theory, the intelligible world is not a set of concepts abstracted from the world of nature, but instead can be said to exist somehow prior to (and therefore in a more perfect and real sense than) the world of nature. This theory follows the Parmenidean way of truth in accepting that the plenum of being underlies changing phenomena. If, however, like Plato, we choose to follow in the footsteps of Parmenides and hold that only the true world of pure Being (in Plato's case the realm of the Forms) is fully real, then we will be tasked with qualifying how the semi-real world of the senses exists. The full Platonic cosmology responds to this problem by making the visible and changing world of nature (the substance of Thales and his students and the ‘flux’ of Heraclitus) less real than the true world of the Forms. The world of nature, therefore, assumes a different mode of being such that it is dependent on the Forms.

The Forms and the Infinite

No aspect of Plato's doctrine of the Forms is uncontroversial. That said, it seems safe to say that each of Plato's Forms can be described as a fully real and perfectly unchanging kind of being. However, although they later would be associated with 'universals', the Forms are also implied by Plato to exist outside of or beyond space and time. How are we to understand these entities? The Forms of Plato are both the most real and highest modes of being, and, from our natural perspective as individuals living in a spatio-temporal natural world, the least evidently given aspects of reality.

Plato's Forms, like Parmenides’ One, are therefore effectively each complete in themselves (in later philosophical terminology: things in themselves). As such the Forms admit of no imperfection. Moreover, it is because of the Forms that reliable knowledge can be attained, since the invisible Forms are perfectly intelligible.

One way to understand his position is to view Plato as holding that in order to make sense of the world we need to accept that every particular changing thing also retains some kind of permanent identity. The apple that changes from being unripe to ripe is both the same and different. Plato would say that the Form of the apple is one, and due to its participation in this Form, each individual apple can be identified and known for what it is. Nonetheless since each Form is what it is and can be known, each Form must also be described as quantitatively finite. Each Form, that is, is singular. Simultaneously, according to Plato, nature- which copies the forms- is ‘a many’, and a realm that is always flowing and changing and therefore it must also be qualitatively infinite. Particulars are 'particular' precisely because they are unique. This means that no two existing apples or two real people are ever perfectly equal. But the form of the equal, as Plato has 'Socrates' tell his interlocutors in the Phaedo, is "absolutely equal". Plato therefore will locate the infinite in the world of change, but since the world we experience is a dependent and deficient, ‘less real’, world, Plato can be seen to continue in the ancient Greek tradition of rejecting the actual or transcendent and fully real infinite.

Fuller reference to some of Plato's many dialogues might help us to better understand his account of infinity and place it in perspective. In Plato’s philosophy the intelligibility of nature depends on the permanent unchanging reality of the Forms that lie beyond the contrary, changing, [non-essential] world of nature. The cosmology presented in the dialogue the Timaeus, for example, is nothing less than an account of how the world of nature, said to be brought about by the activity of a divine craftsman [the Demiurge], is modeled after (and participates in) the world of Forms. Participation in ancient Greek = methexis and Plato’s theory of being (his ontology) can therefore be described as metaxological. The theme of ‘the middle’ or ‘being in between’ is a persistent one in Platonism. Here it comes about in the description of a higher power (the Craftsman), lower properties (the elements) and how their interaction brings about and explains the world of nature. It can be noted that this cosmology places us, as human beings, between the earthly changing world and the perfect divine world. In Platonism, the soul (as separate substance) is what connects us to the unchanging world, while the body (as physical) ties us to the lower world. In Plato’s psychology and epistemology there is likewise said to be a state of the soul between absolute knowledge (episteme) and absolute ignorance (eikasia), namely: opinion (pistis/doxa). Earlier Greek philosophers had sought the absolute truth about nature by examining this world. Plato tells us that natural science can only ever attain a high degree of opinion or probability. True or essential knowledge, by contrast, is attained through dialectic which works by isolating pure rational principles (i.e. making use of logic and philosophy). In this way, practicing philosophy, a person can establish contact with true Being (the Forms).

In the Timaeus Plato also gives us a creation myth through which he attempts to present his cosmological doctrines. For Plato, the universe is apparently a living organism. It is a whole [the English word “uni-verse” actually means one]. In Platonic cosmology, the universe has a “round shape” given to it by God. Plato also seems to have conjectured that there must be symmetry and innate completeness to nature in the sense that the natural is its own realm or exists in itself, even if it is a degraded and imperfect realm in comparison to the Forms. Thus we see earlier Greek ideas synthesized: Thales, Parmenides and others all come together in Plato’s theory with the Pythagorean influence central to his cosmological speculation.

How does the infinite play a role in the Platonic worldview? If we examine the theories presented in the Timeaus we see that, according to Plato, the cosmos consists of two worlds. There is the unchanging world of Forms and the changing world of nature. Accordingly, the world of true Being is not the natural world of sense reality.

  1. The world of Forms: is a world in which everything “always is,” it “has no becoming,” and “does not change” (Timeaus, 28a). We know this world of Being by reason (i.e. through the rational part of our souls).

  1. The world of nature: is a physical world of becoming. Everything in this world “comes to be and passes away, but never really is” (28a). We know this world through sense-perception and opinion, this world, Plato writes, came into being as a model based on the Forms.

To fill out this cosmological picture, Plato describes the natural world of infinite change in greater detail. The physical world, Plato says, is both tangible and visible therefore it must have magnitude and bodily form (Timeaus 31b). Earth and Fire are described as solids that must be combined to form the basis of the physical world. Since Earth and Fire are two, we require two intermediates to balance them out: here Plato proposes Air and Water. All the elements, we are told, are indirectly created by the Demiurge. But these are not the basis of the cosmos for Plato. There is another level beneath the elements, this is the world of atoms understood geometrically and out of which the elements are composed.

The Structure of Matter

Plato’s answer to the Pre-Socratics question, 'What is the world made of?' is given starting in Paragraph 48b of the Timaeus. Plato thinks that the natural world is a combination of space, atoms and elements What is called the receptacle is said to be something that exists in addition to the archetypes or models after which the world is constructed (i.e. the Forms) and the process of imitation of the model (the act of creation bringing forth the world of becoming) requires “the receptacle of all becoming” (49a). The receptacle, therefore, might be viewed as a tangible manifestation of space but it also relates to the Forms since it encompasses the connection between the ideal geometrical patterns and the particulars that emerge in the natural world from these patterns.

According to Plato each kind of matter (earth, air, fire, water) is made up of particles (“primary bodies”). Each particle, in turn, is said to be a regular geometrical solid. There are four kinds of particles, one for each of the four kinds of matter all composed of elementary right triangles. The particles are the molecules of Plato’s theory; the triangles are his atoms. The different shapes account for the properties of natural kinds and for the cosmological structure of nature. Thus, the universe as a whole is a giant dodecahedron, the earth a sphere, but the elements composing nature as a whole can be reduced to Pythagorean interpretations of Milesian elements: thus a tetrahedron represents fire, an octahedron= water, etc.

The four elements are “the most excellent four bodies that can come into being” (53e). And the above can all be broken down into triangles or triangular figures. At Timaeus (54a1) Plato tells us that there is only one kind of isosceles (right angled) triangle (angles = 45°/45°/90°), whereas there are “infinitely many” kinds of scalene triangles. Plato then describes an “ideal” scalene triangle as “one whose hypotenuse is twice the length of its shorter side” (54d). (The angles = 30°/60°/90°.)

From these figures, he derives the faces for his particles. The triangles.   These can be combined and taken apart as needed to construct the natural world. In this vein, the isosceles triangle is described as a “square-builder” and the scalene the building block of equilaterally faced geometrical figures. So fire particles are tetrahedrons (4-sided solid), made of 4 isosceles triangles consisting of 24 scalene’s altogether. Air is modelled on octahedrons (8-sided solids), made of 8 isosceles’s consisting of 48 scalene’s altogether. Regarding the scientific importance of this theory we can say that the numerical ratios are not as important as the principle behind them, namely that the world can be stood using mathematical models.

Finally, the material or natural world, represented by the only uniform solid (the sphere), is complemented by what Plato calls “a world soul”. Recall that the soul has another mode of being for Plato, existing somehow between the Forms and the natural world acting as a kind of intermediary medium between the two. The end of the Timaeus makes it clear that Plato’s cosmological speculation is motivated by more than mere scientific curiosity, there is a deeper ethical agenda at play. Accordingly:

…if a man has seriously devoted himself to the love of learning and to true wisdom, if he has exercised these aspects of himself above all, then there is absolutely no way that his thoughts can fail to be immortal and divine, should truth come within his grasp (Timaeus, 90c).

The didactic lesson presented here seems to be that, the contemplation of the changing world is necessary for achieving knowledge of what is unchanging and divine, i.e. the Forms Since the goal is to arrive at communion with true Being the infinite can be said to be representative of the distraction and lack of definiteness that represents life in the natural world.

The One and the Many

In his dialogue, the Philebus we see more clearly how infinity for Plato also touches on the problems faced by the embodied human soul. The Philebus features a discussion about a very Socratic question: 'What is the human good?'

Socrates is, in fact, the spokesman and protagonist of this dialogue and in it we see Plato’s skill as a dialectician on display here. Very briefly: Socrates has offered the question of what is the highest human good. Philebus has answered that it is pleasure of some sort. Socrates has apparently challenged this answer and his challenge is being met by Protarchus. Philebus’ position can be viewed as follows: the good life is about being happy, partaking of enjoyment and in general having a good time all the time… Socrates, by contrast, thinks that some kind of truth seeking is more important to human beings. Plato has him say:

[K]nowing, understanding, and remembering, and what belongs with them, right opinion and true calculations, are better than pleasure and more agreeable to all who can attain them; those who can, get the maximum benefit possible from having them both now alive and future generations (Philebus 11c)

The soul, or some “possession or state of the soul” is said to be necessary to render life happy for all human beings. Since happiness is connected to consciousness by most thinking people, this seems like a valid enough point; for example: can you be truly happy if you are unaware of anything? This highlights the tensions emphasized in the Platonic worldview between the mind and the body, here they are connected to the distinction between pleasure and knowledge. Plato has Socrates raise the following point: “What if it should turn out that there is another possession [of the soul] better than either of them? Would the result not be that, if it turns out to be more closely related to pleasure, we will both lose out against a life that firmly possesses that, but the life of pleasure will defeat the life of knowledge?” (11e)

Both pleasure and knowledge, Plato thinks, are complex, not simple states. Knowledge, for example, is not completely without its own mode of pleasure, i.e. the pleasure of discovery and knowing, and even a sober man takes a kind of pleasure in his sobriety….(12d). What seems to be at work here is an acknowledgement that opposites exist in reality although Plato refuses to say that opposite phenomena are really the same. There are different kinds of shapes and different kinds of pleasures, just as there are different kinds of knowledge.

Plato has Socrates introduce the principle of the One and the Many at paragraph 14 c. “It has an amazing nature…that the many are one and the one many are amazing statements…” There are two ways to view this principle. First what can be called the childish view, I have many moods but I’m one person and second, the serious, “the one is many and indefinitely many, and again that the many are only one thing” (14e). This, once again is an expression of the problem of the Forms or the Platonic theory of universals. The universal or general nature of things is a fact of reality. We experience individual concrete things and we also discern patterns and qualities that are general and universal. The white swan shares its shape and color with many others, yet each is an individual. The One somehow participates in the many and the problem remains how the many particulars can have repeatable and similar characteristics or qualities. For example, we say that there are humans but also that 'humanity' exists as a species. We say there are instances of falling things but that gravity exists as a universal force. To make sense of the world, we need an account of universals. Infinity too, falls under this principle. We say that the infinite is one quality or thing, i.e. the whole that encompasses everything, but we can also explain the infinite as what is unbounded or continuous such that it is never completed.

Plato summarizes the problem of the One and the Many starting at paragraph 15e:

….it is through discourse that the same thing flits around, becoming one and many in all sorts of ways in whatever it may be that is said at any time, both long ago and now. And this will never come to an end, nor has it just begun, but it seems to me that this is an “immortal and ageless” condition that comes to us with discourse (logos). Whoever among the young first gets a taste of it is as pleased as if he had found a treasure of wisdom. He is quite beside himself with pleasure and revels in moving every statement, now turning it to one side and rolling it all up into one, then again unrolling it and dividing it up. He thereby involves first and foremost himself in confusion, but then also whatever others happen to be nearby, be they younger or older or of the same age, sparing neither his father nor his mother nor anyone else who might listen to him. He would try it on other creatures, not only on human beings, since he would certainly not spare any foreigner if only he could find an interpreter somewhere (Philebus 15e-16a).

The above, spoken by the Platonic character of “Socrates”, reminds us of the dilemma that faced ancient Greek philosophy and science. Some thinkers wanted the world to be One (Parmenides), others mainttained that it was many (Pythagoras), both concepts seem necessary but do we cogently unite them?

Socrates will explain Plato's point as follows:

…whatever is said to be consists of one and many, having in its nature limit and unlimitedness. Since this is the structure of things, we have to assume that there is in each case always one form for every one of them, and we must search for it, as we will indeed find it there. And once we have grasped it, we must look for two, as the case would have it, or if not, for three or some other number. And we must treat every one of those further unities in the same way, until it is not only established of the original unit that it is one, many and unlimited, but also how many kinds it is. For we must not grant the form of the unlimited to the plurality before we know the exact number of every plurality that lies between the unlimited and the one. Only then is it permitted to release each kind of unity unto the unlimited and let it go (Philebus 16d-17)

The letters of the alphabet are limited, their combination and the possible words that can be derived appears unlimited. The Platonic theory of the One and the many is therefore central for understanding an important characteristic of infinity. More to the point, the relevance of these concepts becomes clear if we ask two questions:

  1. How many kinds of things are there?


  1. How many things are there?

Especially in science, we often try to reduce individual things to more basic natural or logical kinds, to subsume the many to the one. Philosophically we can take a top down approach “All is One” (Parmenides) and arrive at Monism or take a bottom up approach holding that “units compose the whole of reality” (Pythagoras) and arrive at Pluralism. But the problem that remains is the following: Can everything- the collection of all things in the world- be considered one thing or many?