Explanation of the Arrow Paradox, Attributed to Zeno

In the arrow paradox, Zeno uses the example of an arrow in flight. He states that for motion to occur, the arrow must change the position which it occupies, but for every such position we see no motion, thus motion does not exist.

Modern formulation: (From: https://bit.ly/2v30Q8X) “At every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.”

As reported by Aristotle: “If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.”

This is a paradox, as we know that motion exists.

So, what’s wrong with Zeno’s syllogism?

Explanation:

The concept of infinity helps us with mathematical functions “at the limit”, but it is possibly confusing when, in the physical world, we need to evaluate motion (from point A to point B) or speed (an interval of space divided an interval of time).

The term infinite means “not finite”.

A finite interval (of space, or time), can be thought of as the sum of a finite number of very small intervals, as small as we want (infinitesimal), but each with its own dimension.

Conversely, an infinite number, of dimensionless points or instants cannot add up to a finite interval(1).

In the first case, we can calculate the intervals of space and time, thus we can evaluate motion and speed. In the second case, we can’t.

Zeno, in Aristotle’s words, asks us to use the concept of a dimensionless(2) instant to imagine the arrow “at rest”.

Today, with our familiarity with photography, we can easily imagine such a snapshot of the arrow, but we cannot evaluate its state of motion without some other information.

In other words, by considering only that dimensionless instant, without measurements of time and space, we cannot evaluate if, or in which direction, the arrow may be moving.

However, Zeno suggests that we could evaluate the arrow’s locomotion at any such moment and declare it “motionless” (presumably with respect to the ground). Thus, Zeno’s syllogism breaks down.

We could explain this contradiction to Zeno a bit more figuratively: We distinguish between a snapshot of the arrow, taken at one instant in time, and the arrow it depicts.

Zeno transfers the motionlessness we observe on the snapshot of the arrow, to the arrow that is the subject of the snapshot. This is not a valid logic operation, at any time.

Now, Zeno may respond that he knows nothing about photography, and that he is talking about the real arrow, as we imagine it in a dimensionless instant in time.

We can then respond by saying that, if we are given information about an arrow in a dimensionless instant, and no other space/time information, we cannot say that the arrow is motionless, even if we can imagine it motionless.

NOTES:

(1) An infinite series can never complete, and form a finite interval. It can converge, or get arbitrarily close to, but not actually equal a finite sum.

(2) If that instant had measurable dimensions, we could measure the corresponding intervals of space and time, and calculate the speed of the arrow, invalidating the premise.