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 xaxis—is composed of points, and that to every point there must correspond a number. But does calculus really require that? Let us begin by remembering that points and lines are mental objects. A line is the idea of length only. A point is the idea of position only. Points and lines exist as ideas. Is that sufficient?
Euclid postulated that a straight line will exist logically—when it has been drawn. We will have brought a representation of that idea into this world. As for a "point," that is the name we give to the extremity of a line. Upon drawing a straight line, its extremitiesthose pointswill logically exist. An instance of that idea will have been brought into this world.
As for the xaxis, we indicate points one at a time. We let a point have coördinate 0. We let the extremity of a distance from 0 have coördinate 1. We say, "Let x take values in the open interval between −4 and 5." Having indicated each point, that is all we need to mean when we say that that point logically exists.
(We may say there are an "infinite" number of points on a line, which is a brief way of saying that there is no limit to the number of points we could indicate.)
Points—like pitches on a violin string—exist potentially. The pitch of a string does not exist until it is sounded: a violin string is not composed of pitches. And the xaxis is not composed of positions.
One could of course completely redefine the meaning of the word composed and the word point.
"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 points. Now the most common and important application of calculus is to motion, where the independent variable is time t. Then if the abstract xaxis is composed of points, its application to time must also be composed of points. That is, time—the taxis—will be composed of points, or, we would say, instants. Is that a valid presumption?
First, like any continuous quantity, time is not inherently composed of intervals; yet we can conveniently decompose time into any intervals,
any units of measure—hours, minutes, seconds—we please, however small.
Time will then be composed of intervals, which will then have common boundaries, to which we give the name "instants."
But time cannot be composed of instants, because they are not intervals—they are not a division of time. (To accept that an infinite number of points of zero time will add up to a positive time, not only calls for credulity more typical of the demands of religion; it approves division by 0.)
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 taxis then is not composed of points, then neither can the xaxis—the socalled 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.
Appendix 1: Are the real numbers really numbers?
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Copyright © 2017 Lawrence Spector
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