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 then asks you: 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 slice 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 thus occupies exactly one position; 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 any instant, 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 just clever branding, but doesn’t correspond to what is actually computed. The physicist has, perhaps without quite realizing it, conceded Zeno’s premise just as much: we cannot meaningfully describe the movement of the arrow given just any instant.

Just to clarify what I mean with that – assume you have one of those flipbooks with a series of sequential images that simulate motion if you rapidly flip through the book. The physicist has come up with very clever mathematical machinery to compute the motion between successive pages, no matter how small. But Zeno asks: does the image, on a single page, actually move? Of course not, they’re just static pictures.

The only way the physicist can resolve the paradox is to assert: “the arrow is moving at the instant, and its speed is the instantaneous velocity”. But now, physicist hasn’t proven that the arrow moves, but has simply assumed it! It is no good to assume that the arrow is moving (never mind with what speed) if that was the very thing Zeno was questioning. By assuming the conclusion (or ‘begging the question’), nothing is proven yet.

Reality is a flipbook?

Now, there are two ways out of this. Either, some assumption along the way has gone wrong. Or, we have to assume that the flipbook view is exactly real. This latter view is, in fact, what Bertrand Russell (one of the most prominent public intellectuals, mathematicians and philosophers of the 20th century) held. That is, he fully agreed with Zeno that the arrow is motionless at every instant:

Weierstrass, by strictly banishing from mathematics the use of infinitesimals, has at last shown that we live in an unchanging world, and that the arrow in its flight is truly at rest. […] Motion consists merely in the fact that bodies are sometimes in one place and sometimes in another, and that they are at intermediate places at intermediate times.

This theory is that of the block time universe, which is a four-dimensional view of spacetime where all moments (past, present, and future) exist “simultaneously”. Think of the flipbook. Our experience of “this moment” is simply because we’re on a particular page of the flipbook, rather than another one, but all the other pages exist already. Well, actually, the block universe is a bit more radical than that. There is no single page we are “on” at all, we are present and spread across all pages, simultaneously. This was also, e.g., Einstein’s view:

People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion.

Peculiarly, I think, this view essentially supports Parmenides’ view that change, difference, time, and motion are mere illusions.

Or not? Instants are unreal.

For Zeno’s argument to work, we have to assume 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 at all. 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.

That is – motion disappears if we consider it (or reality in general) is like a flipbook, composed of infinitely many (and in some sense disconnected) instants. But we must draw the opposite conclusion: reality is not like a flipbook, because it is not composed of instants! If reality is not a flipbook, motion is not under any threat of being an illusion at all.

(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 of no issue whatsoever! The arrow moves across an irreducible (or indivisible) temporal interval. That interval cannot be decomposed into instants and then reassembled; the decomposition destroys precisely what it is trying to analyse. Motion is something other than the sum of its parts (positions at instants); and we therefore cannot re-assemble motion out of the considered parts.

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 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. Motion is like that too, you can mark its positions (“instants” and “places”) along it, but those marks are still not what it is made of.

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)