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"Cogito, ergo sum."
("I think, therefore I am.")
Anyone who had taken philosophy would have recognized this famous saying by the French philosopher and mathematician Rene Descartes (1596-1650). He united geometry and algebra and founded analytic geometry. He also "proved" the existence of God. Of course, as with any other logical proofs on the existence of God, there was a major flaw in his proof.*
Descartes' idea is that of mind over matter. Mathematics probably best exemplifies this idea. We created a complicated world with just a little thinking and a little insight from each mathematician. (Definitely an understatement for most mathematicians.) In fact, that is what a mathematician does most of the time, deep thinking. I often marvel at the extend the human mind has gone. Let's look at the following two examples.
A person on Earth walks southward a hundred meters, followed by eastward a hundred meters and then northward a hundred meters. Hurrah, he is back to where he starts with. The question is where is this person?
One immediate answer is the north pole. For a long time, people thought this is the only answer for the above question. But as we often say in mathematics, "if you don't know how to solve it, just sit back and watch someone else solve it.", someone did come along and provided an infinite number of solutions.
The answer lies close to the other pole on the earth (figure 1). Somewhere near the south pole, you can find circle that is exactly 100 meters in circumference (a circle with its center at the south pole and a radius of 50/p meters). Then any point on the circle that is 100 meter north of this circle (the circle with a center at the south pole and a radius of (100+50/p) meters) is an answer to the question. Imagine you start at a point on the circle with the bigger radius, you walk south 100 meters to reach the smaller circle, then you walk east 100 meters (i.e., you circle around the south pole exactly once on the smaller circle) and then go back north a hundred meters along the path you come from the bigger circle (figure 2). You are exactly at where you start with. With this said, can you find other points where you could do the same thing?
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Another example involves a 1´1´2 rectangular box (figure 3). A geodesic path on such a box from one point to another is defined as the shortest distance between them. If you start from one corner (A), find a point on the box so that the geodesic path is the longest. For almost a century, people thought the point is the corner (B) on the opposite face, which has a distance of 2Ö2»2.828. A few years ago, Yoshiyuki Kotani discovered that the farthest point is actually 1/4 away from the corner (C) on the opposite face, with a distance of Ö130/4»2.850. The physicist Richard I. Hess heard about the discovery and found the farthest distance on such a geodesic box is about Ö(16-4Ö3)»3.011 but not 3. He proposed this as problem 951 in the Fall 1998 issue of the Pi Mu Epsilon Journal. My solution with generalization to a 1´1´n box appeared in the Fall 1999 issue.
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Such is the power of thinking. Very often people underestimate the power of the mind. I don't think our mind can move something physically. But with a little thinking, we can definitely easily move something physically through other means. Later we will see how our mind destroys our intuition when we deal with infinity.
"Cogito, ergo sum."
(Answer: Any point that is 100 meter north of a circle with radius 50/(ip) meters and center at the south pole is a solution. Here, i = 1, 2, 3, 4, ..., n, ... After you reach the circle, you just have to circle around the south pole i times to complete your 100 meters eastward.)
* Descartes assumed there are three substances in the universe, namely, God, I and the substances in my thinking. Since God is a substance, therefore he exists. Of course, if you are a substance, you exist. It is illogical to say you are a substance, therefore you don't exist. The fundamental problem is how did he know God is a substance? If God had existed physically, there is no need to prove his existence logically. Just as you don't have to prove your great great great great great grandfather had lived at some point and had a son or daughter. And if anyone tries to prove the existence of God logically, he reckons God is a product of our mind.