Number vs. Magnitude

In light of the various questions about the distinction between number and magnitude, the following is an excerpt from Chapter 2 of Victor J. Katz's book:
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Another of Aristotle's contributions was the introduction into mathematics of the distinction between number and magnitude. The Pythagoreans had insisted that all was number, but Aristotle rejected that idea. Although he placed number and magnitude in a single category, "quantity," he divided this category into two classes, the discrete (number) and the continuous (magnitude). As examples of the latter, he cited lines, surfaces, volumes, and time. The primary distinction between these two classes is that magnitude is "that which is divisible into divisibles that are infinitely divisible," while the basis of number is the indivisble unit. Thus magnitudes cannot be composed of indivisible elements, while numbers inevitably are.

Aristotle further clarified this idea in his definition of "in succession" and "continuous." Things are in succession if there is nothing of their own kind intermediate between them. For example, the numbers 3 and 4 are in succession. Things are continous when they touch and when "the touching limits of each become one and the same." Line segments are therefore continuous if they share an endpoint. Points cannot make up a line, because they would have to be in contact and share a limit. Since points have no parts, this is impossible. It is also impossible for points on a line to be in succession, that is, for there to be a "next point." For between two points on a line is a line segment, and one can always find a point on that segment.

Today, a line segment is considered to be composed of an infinite collection of points, but to Aristotle this would make no sense. He did not concieve of a completed or actual infinity. Although he used the term "infinity," he only considered it as potential. For example, one can bisect a continuous magnitude as often as one wishes, and one can count these bisections. But in niether case does one ever come to an end. Furthermore, mathematicians really do not need infinite quantities such as infinite straight lines. They only need to postulate the existence of, for example, arbitrarily long straight lines.

Victor J. Katz,
A History of Mathematics: An Introduction (2nd Ed.),
1998, Addison-Wesley Educational Publishers, Page 56.