INFINITY
What is really meant when the mathematician uses the concept of infinity in his operations? Suppose that we take a line of finite length and divide it into halves, and then divide each half into halves, and so on ad infinitum. We make cuts in the line, and these cuts have no magnitude, so that the sum of the lengths into which we divide the line is equal to the length of the undivided line. We can divide the line into as many parts as we choose, that is, into an “infinite” number of parts.
Suppose that we are making a thing which is to match another thing, and suppose that we can make the thing as great as we choose. If, then, no matter how great we make the thing, it is still too small, the thing that we are trying to match is infinitely great.
Substitute “small” for “great,” and this is also a definition of the infinitely small.
Clearly the idea of infinity does not reside in the results of an operation, but in its tendency. It inheres in our intuition of striving towards something, but not in the results of our striving.