Zeno and infinite space

Two and a half millennia ago the Greek sage Zeno said motion is impossible, because to get anywhere you have to get somewhere else first. If you set out to walk a mile, before you can walk a mile you must walk half a mile; but before you can walk half a mile you must walk a quarter of a mile, and before that an eighth, and before that a sixteenth, and so on to infinity. You can never get anywhere, because there is always somewhere else you have to get first.

This famous paradox of Zeno’s applies not only to constructing a line from its smallest parts, but also to analysing it into its smallest parts. Since the line is continuous, it has no smallest parts, but can be divided to infinity.

Zeno’s work survives only in fragments and brief references embedded in the writings of others. So far as we know he did not explicitly address the question of planes. But it is evident the issues which arise in relation to a line arise also in relation to a continuous plane. As a line is made of smaller and smaller lines, so a plane is made of smaller and smaller planes, without it being possible to identify a smallest plane. A plane, like a line, can neither be constructed from its smallest parts nor analysed into them.

The field and faculty of vision

The field of vision is one such continuous plane. The field of vision arrives instantly, as soon as you open your eyes. It is always already complete. You do not witness it being constructed. Like any field, the field of vision is continuous. It has no breaks. It is present as a whole. It is not pixilated. It is not a data stream. Like any continuity, it is divisible in principle to infinity — it is not possible to enumerate all the parts of the field, because any part so picked out is itself a field, with parts of its own to be enumerated in turn. This is not a limitation of any particular eye, but something basic to the nature of vision: whatever part of the field on which one focusses is always itself a field and is itself made of parts. Only a dimensionless point on the field is not itself a field, and it is not possible in principle to see a dimensionless point.

Around the same time as Zeno, the Pythagorean sage Philolaus wrote:

ha physis d’en to kosmo harmochthe ex apeiron te kai perainonton
kai holos ho kosmos kai ta en auto panta.

Life in the cosmos has been harmonised from infinites and limiteds — both the cosmos as a whole and everything in it.

Vision is of an infinity, so we see with an infinite faculty, a faculty which comprehends the infinite plane of vision but which is nonetheless also limited — limited by the size of the field of vision, by the range of visible and invisible colours, by the restricted resolution of the eye.

Infinite depth

So we can see with Zeno that the visual field has infinite depth. It is impossible to analyse the field into its smallest parts, as logically it has none. You plunge into the image right down to the smallest area on which the eye can focus, only to find that this area is still a continuous plane, divisible itself to infinity. For this reason a perfectly sharp resolution is not possible. Whatever is brought into focus can only be an infinite field, of which the component parts will be lost in the depths of infinite divisibility. To see the field sharply would be to see its basic parts cleanly separated from one another. But there are no basic parts, so the image — even for a hawk — must forever lack final sharpness.

Infinite breadth

Being continuous, the field of vision can be broken across an infinite number of parts whether large or small, discrete or overlapping, contiguous or separated. And contrary to the medium of music — in which the notes arrive and depart in a definite sequence — the visual field is present all at once: it has no beginning or end, and so its parts can be read and reread in any order. Just as the field has infinite depth, so we can say it has infinite breadth, meaning that it can be spanned by an infinity of paths or broken into an infinity of relations, which vision can never enumerate or exhaust.

Whole not whole

There is an ancient paradox of the One and the Many, that two cannot properly become one without ceasing to be different from each other, so that unity conceived as the union of many parts in a single whole is always paradoxical. Zeno once said that if anyone could show him what it was for a thing to be one, he would be able to explain how the world could contain many. Vision grasps each of the objects within the field of vision as just such an unstable divided whole, oscillating between one and many. For to pull two things into a unity one must see them as one, yet at the same time one must see them as the separate elements which they must continue to be if they are to fulfil the definition of that very union. So to the field’s infinite depth and infinite breadth we can add this third infinity — the never-resolved paradox of the divided whole.

Recursion

Every part of the image repeats the infinities of the whole. Thus the eye cannot focus on a part, nor draw a relation between two parts, nor construct a whole from a set of parts, without being forced to reread the parts in question endlessly to establish what they actually are.

Elemental vision

In everyday life we swiftly identify in the field of vision the objects of interest in our practical context — a turn in the path, a face in the crowd. Even surveying a landscape, at rest and with no other purpose than contemplation, we are distracted by objects we instinctively recognise and name — trees, hills, houses — so we take them more as objects of thought than as objects of vision. But in gazing at the sky, or over the sea, or across a desert, or into a fire — elemental expanses which contain few or no nameable objects — we are closer to seeing vision itself, as the confrontation between an infinite power and its infinite object.

Glare

What happens then when the infinite eye meets the infinite image? Because the power of vision is infinite the eye is able to grasp the visual field. The field is present, entire, continuous — in its limitless depth and breadth, in all its numberless parts and wholes. Yet because the field is infinite the eye is unable to resolve the field’s three infinities of depth, breadth and divided integrity. We see, and yet we cannot see.

The consequence of an infinite power confronting an infinite image is that the image appears as a kind of glare. There is the blur of imperfect resolution; the discomfort of a constant shift of focus as the eye fails to find a clean resting place; a constant push-back from something independent, other, out-of-reach and out-of-control. Glare reflects the impossibility of subjecting a continuous field to a stable analysis. This seeming weakness of vision, its imprecision and frustration, is not a failing in vision, but the inalienable signature of its infinite power. This is what infinity looks like. It burns the eye.

Grip

The image grips as the eye seeks to dominate. One cannot defeat the image and move past it. One can tear one’s gaze away, but at the cost of leaving the infinity intact, unscathed. The eye gazes at the image — the image glares at the eye. The eye and the image grip each other like two wrestlers with their arms on each other’s shoulders.

Space, light and word

Vision is at the intersection of space, light and word.

We move down an infinite line with an infinite movement — a movement which like the line is divisible forever. And the infinite plane on which we walk is reflected as an infinite image, which it takes an infinite visual power to grasp. And if we further name that plane with words — let us call it the Field of Asphodel — then these words will bear an infinite meaning, which is the infinite plane itself. And the powers of speech and comprehension which grasp and transmit this infinite meaning will be infinite powers.

We see, speak and move infinitely. Only so can plane, image and word — field of movement to field of vision to semantic field — reflect and respond to one another.