Zeno's Arrow Paradox has haunted physicists and philosophers of physics since its formulation by Zeno around 450 BC:
- How can an arrow be moving in space from one point to another, if at each instant of time the arrow is still and is not moving?
- The third is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b.30)
- Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. (Diogenes Laertius Lives of Famous Philosophers, ix.72)
- This argument against motion explicitly turns on a particular kind of assumption of plurality: that time is composed of moments (or ‘nows’) and nothing else. Consider an arrow, apparently in motion, at any instant. First, Zeno assumes that it travels no distance during that moment—‘it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving.
- Motion consists merely in the occupation of different places at different times, subject to continuity as explained in Part V. There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number which is the limit of a certain set of quotients. The rejection of velocity and acceleration as physical facts (i.e. as properties belonging at each instant to a moving point, and not merely real numbers expressing limits of certain ratios) involves, as we shall see, some difficulties in the statement of the laws of motion; but the reform introduced by Weierstrass in the infinitesimal calculus has rendered this rejection imperative.
- According to the “at-at” theory, it is fallacious to conclude from the fact that the arrow does not travel any distance in an instant that it is at rest. Motion has nothing at all to do with what happens during instants; it has instead to do with what happens between instants. In short, motion is merely being in different locations at different times, and that is that. If an object has the same location at the instants immediately neighboring, then we say it is at rest; otherwise it is in motion. Therefore, since the arrow in flight has different positions at different instants, it is surely moving.
- The “at-at” theory is a static theory of motion. In Henri Bergson’s cynical words, “movement is composed of immobilities.” (Bergson 1911, p.308) Continuous motion is simply the occupation, by an object, of a continuous series of places at a continuous series of times. There are no states of motion at an instant, and no instantaneous properties indicate that an object is moving or not.