The following is reprinted from the “Q & A Department” of The Objectivist Forum, December 1981. It was inspired by a private discussion with Ayn Rand on this issue. Subsequently, she read this piece and expressed no disagreement with any of it.—Harry Binswanger
Q: Does the law of identity imply that at every instant in time a moving object must be located at a definite point in space?
A: No. The law of identity implies that there are no such things as “instants in time” or “points in space”—not in the sense assumed in the question.
Every unit of length, no matter how small, has some specific extension; every unit of time, no matter how small, has some specific duration. The idea of an infinitely small amount of length or temporal duration has validity only as a mathematical device useful for making certain calculations, not as a description of components of reality. Reality does not contain either points or instants (in the mathematical sense).
By analogy: the average family has 2.2 children, but no actual family has 2.2 children: the “average family” exists only as a mathematical device.
Now consider the manner in which the question ignores the context and meaning of the concepts of “location” and “identity.”
The concept of “location” arises in the context of entities which are at rest relative to each other. A thing’s location is the place where it is situated. But a moving object is not at any one place—it is in motion. One can locate a moving object only in the sense of specifying the location of the larger fixed region through which it is moving during a given period of time.
For instance: “Between 4:00 and 4:05 p.m., the car was moving through New York City.” One can narrow down the time period and, correspondingly, the region: but one cannot narrow down the time to nothing in the contradictory attempt to locate the moving car at a single, fixed position. If it is moving, it is not at a fixed position.
The law of identity does not attempt to freeze reality. Change exists: it is a fact of reality. When a thing is changing, that is what it is doing, that is its identity for that period. What is still is still. What is in process is in process. A is A.