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Appendix 3

IS A LINE REALLY COMPOSED
OF POINTS?

THE WHOLE PROBLEM of a continuum of numbers began with the assumption that a line—the x-axis—is composed of points, and that to every point there corresponds a number. We have seen that calculus does not require that. And is a line really composed of points? 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. Effectively, it approves division by 0.)

What is a geometrical point anyway? It is the word we use to indicate a position. The extremity of a distance from 0 is called a point, as is the boundary of an interval. We indicate points and their coördinates one at a time. ("Let x = 5.") 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.

One could of course completely redefine the meaning of the word composed and the word point. And that is exactly what the creators of the theory of real numbers did. See Lesson 1.

"When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean—neither more nor less."

"The question is," said Alice, "whether you can make words mean so many different things."

"The question is," said Humpty Dumpty, "which is to be master—that's all."

Alice in Wonderland

Say, however, that a line were composed of self-existent 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 x-axis is composed of points, its application to time must also be composed of points. That is, the t-axis will be composed of instants.

First, like any continuous quantity, time is not inherently composed of intervals; yet we can conveniently decompose time into any intervals,

Real numbers?

any units of measure—hours, minutes, seconds—we please, however small.

Time will then be composed of those intervals, which will have common boundaries, to which we give the name "instants."

If the minute were the unit of measure, then "10:24 A.M." is the name of an instant.

But time cannot be composed of instants, they are not a unit of time.

If time did consist solely of instants, then at any one instant a body is at rest. It cannot move to another instant, because no time elapses. 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.

End of the lesson

Appendix 2:  Are the real numbers really numbers?

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