Parmenides

Zeno of Elea posed a cluster of paradoxes which appear to be mere intellectual puzzles, but they were created with a different aim – an aim of defending the philosophy of Parmenides, whose position was remarkably simple.

Parmenides held that reality is one completely unchanging blob. His position was this: (1) what is, is; (2) what is not, is not (i.e., “nothing” or “non-being” cannot be referred to, thought about, or described). Put differently, his point is that reality must be describable without ever using the word “not”. But change, difference, time, and motion all require negation to be made sense of: to move, is to not be where you were; to change, is to not be what you were. Therefore, change, motion, and time are not real, on Parmenides account. They are mere appearances, “the way of illusion”.

Zeno’s paradoxes were meant to press this conclusion home by showing that motion, examined carefully, collapses into contradiction. One such is the paradox of the arrow.

The paradox of the arrow

Consider an arrow in flight, travelling from point A at time t₀ to point B at time t₂. Zeno asks a simple question: is the arrow moving at any instant t₁ between t₀ and t₂?

An instant, by definition, has no duration. It is a point in time, not a stretch of it. As an instant has no duration, there is no time within that instant for the arrow to travel any distance. At any given instant, the arrow occupies exactly one position, thus it cannot move during the instant.

But this holds for every instant. The flight from A to B is supposedly composed of instants, and at each of them the arrow is not moving. So Zeno poses a follow-up question: when, exactly, does the arrow move? Well… there is no instant you can point to! The arrow is not moving at any moment of its so-called flight. Whatever motion you seemed to see, was merely an illusion, because the arrow never moved! (if you disagree, tell me at which instant it moved?)

Physics agrees with Zeno — and doesn’t notice

The standard modern response is to define motion as the change of position over time and to deploy the calculus of limits. We consider two positions (separated by Δx) at two times (separated by Δt), compute the ratio Δx/Δt, and then let Δt shrink toward (but never reach) zero. The result is the “instantaneous velocity.” Problem solved, the textbooks say.

But look at what has actually been conceded. The physicist agrees that at a true instant (Δt = 0) the ratio Δx/Δt is undefined. This is precisely why the limit is never allowed to reach zero. “Instantaneous velocity” is, on inspection, a limit of ratios over intervals, not a property of a genuine instant at all. The word “instantaneous” is a convenience, not a description.

The physicist has given us a powerful tool for calculating the appearance of motion: for predicting where the arrow will be at any future time. But Zeno’s challenge was not predictive but metaphysical. He asked: in what sense is the arrow actually moving? To this, physics replies: “at every instant, it has a well-defined velocity.” But that velocity is always computed over an interval, never at a true point. The physicist has, perhaps without quite realizing it, conceded Zeno’s premise: the arrow is not moving at any instant.

Bertrand Russell, writing not as a defeat to Zeno but as a statement of his own block-universe metaphysics, made the point plainly:

we live in an unchanging world and … the arrow, at every moment of its flight, is truly at rest.

Even the most sophisticated modern picture of time leaves Zeno’s challenge standing, IMO.

The way out: instants are not real

Where, then, does the argument go wrong? I want to suggest it goes wrong at the very first step.

Zeno assumes that time is composed of instants, in the same way a line might be composed of points. The idea is that if you could collect all the instants together, you would recover the full stretch of time. Movement, on this view, must be reconstructible from what is happening at each instant, and since nothing is happening at any instant, nothing is happening at all.

But this assumption is precisely backwards. An instant (a durationless “now”) is not a building block of time. It is an abstraction extracted from time. We begin with the experience of duration, something with a temporal width. From this temporal continuum we can, by intellectual analysis, extract the concept of an instant. A limiting ideal point that doesn’t endure – something that is, in some sense, not temporal at all! But we cannot reverse this operation. We cannot start with instants and build up time, any more than we can start with points (which have no length) and build up a line by accumulating them.

Why not? Because zero, added to itself any number of times, even infinitely many, remains zero. An instant has zero duration. An infinite collection of zero-duration moments has zero duration. Time cannot be reconstructed from instants, because instants are not parts of time at all; they are just a particular way of thinking about time. Henri Bergson made exactly this argument against Zeno: once you have replaced movement with a series of frozen positions, you have lost movement permanently. No number of positions and instants, however many, will give it back.

(For readers who may be thinking of integration: an integral does sum infinitely many “slices” to give a finite area, but this works precisely because each slice has an infinitesimal width, not zero width. The machinery of the integral is designed to avoid the very problem Zeno poses; it is not evidence that zero-duration instants can sum to a finite duration.)

The conclusion is this: Zeno is right that the arrow does not move at any instant!

But, since instants are not real constituents of time (since they are abstractions from time, not atoms of it) this is no threat to motion whatsoever! The arrow moves across an irreducible temporal interval. That interval cannot be decomposed into instants and then reassembled; the decomposition destroys precisely what it is trying to analyse.

The physicist’s calculus, interpreted carefully, actually supports this conclusion. To compute the velocity at “an instant”, one must situate that instant inside an interval over which a limit is taken first. The interval is prior; the instant is derived. Even in the mathematics, the full time interval comes first.

The strangeness that remains

There is something vertiginous about this conclusion. It means the “now”, the present moment, the knife-edge between past and future, may be a pure abstraction, doesn’t really exist as an instant. No matter how many mindfulness teachers advise us to inhabit it, the instant might simply not be there to inhabit.

The way to think of it is like this: time is something like a river, you can mark positions (“instants”) along it, but the marks are not what it is made of, and removing them does not stop its flow. Time is really very weird.

What then is time? If no one asks me, I know: if I wish to explain it to one who asks, I know not.

(St. Augustine, ~400 AD)