It is a sunny day; you want to take a walk. We here define walking as the taking of repeated steps in any given direction. Our walk will take us into the field of mathematics.
An Unusual Walk
Imagine a stretch of sidewalk that is precisely 6 feet long. An initial 3 foot step is taken. Every step after that must be 1/2 the distance of the previous step. So our second step is 1/2 the length of the first step. The third step is 1/2 the length of the second step, and so on.
Questions:
1. Theoretically, how long would it take you to reach the end of the 6′stretch?
2. How far would you have traveled, after each step?
Answers:
1. You would, again theoretically, never make it to the end.
2. After 1 step, you’ve traveled 3′; after 2 steps, 4.5′. By the third step, you’ve traveled 5.25′, by the fourth, 5.625′. After 5 steps, you’ve traveled 5.8125′, and on, indefinitely.
As you doubtless realize, the more steps you take, the closer you are to the end of the 6-foot sidewalk. However, even if you took steps endlessly, although you would come extremely close to the end of the walk you would never reach it. In the World of Mathematics.
The Real World
Consider once again our 6′ sidewalk and our 3′. This time, every step is set at 3′. I hear you saying, “The traveler reaches the end in two steps.” And you would be correct, even from a mathematician’s perspective. But stop and think… To achieve the second step, the traveler lifts his foot and move it forward. In the process of moving that foot, wouldn’t it pass the 4.5′ mark, the 5.25′ mark, the 5.625′ mark, and the 5.8125′ mark?
How is it you actually succeed in reaching the 6′ mark? Is it because mathematical points do not actually exist in the real world? Mathematics is a supremely useful tool. But is it not also a product of our imagination? In all fairness, I would suggest Mathematics is both. It should not be taken too seriously.
Mathematics Intrinsic Value?
I compare mathematics with the alphabet. Communication is thought transference. The simplest form of thought transference is spoken language. Now spoken language1 consists of combinations of basic sounds called phonemes (watch video).
The alphabet enables words to be put into written form. The alphabet possesses no intrinsic value. Certainly it is a useful tool. It is also artificial. Something similar might be said of mathematics.
1 For obvious reason, we do not include sign-language.
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I first heard of Xeno’s paradox in school and found it most intriguing.