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Appendix 2

IS A LINE REALLY COMPOSED
OF POINTS?

THE WHOLE PROBLEM of a continuum of numbers began with the assumption, the concept, that a line—the x-axis—is composed of points, and that to every point there must correspond a number. Does calculus really require that? It is obvious that, in no additive sense, is a line composed of points.

To accept that an infinite number of points of zero length will add up to a positive length, calls for credulity more typical of the demands of religion. What is more, it approves division by 0.

What is a geometrical point anyway? It is simply the word we use for the idea of position. The extremity of a distance from 0 is a point, as is the place where two lines meet. We indicate points and their coördinates one at a time. That is what we do. And having done that, we may then say that that point exists.

Points exist potentially. Calculus has no need for anything more. Points are like pitches on a guitar string. A pitch does not exist until it is sounded; a guitar string is not composed of pitches. And the x-axis is not composed of positions.

Say, however, that a line were inherently composed of points. That is the assumption of the theory of real numbers. Now the most common application of calculus is to motion, where the independent variable is time t. Then if the abstract x-axis is composed of points, its application to time must also be composed of points. That is, the t-axis will consist only of what we call instants. Just as a point is the idea of position only, it is not an interval of length; so an instant is not an interval of time.

At any one instant, then, a body does not move because no time passes; for the same reason it cannot move to another instant, another point on the t-axis. In other words, there could be no motion. That is the arrow paradox of Zeno.  But because time continues and has no inherent components—it is not composed of instants—that paradox is not valid.

Since the t-axis then is not composed of points, then neither can the x-axis—the so-called real line—of which time is but an application.

Thus a continuum of numbers not only does not exist, it is not necessary. If a function is continuous, then it will be continuous at every value of x we name.

End of the lesson

Appendix 1:  Are the real numbers really numbers?

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