My Problem with the Circle

Yus,

I am losing confidence in Euclidean Geometry just like you are losing confidence in Calculus.

This problem has come up for me before in the recent past, but it has once again come to the forefront of my thinking after designing the chocolate mold and hanging out with you talking about math and physics.

Basically the problem I am having is with the basic idea of our math's ability to describe completely and/or accurately something as simple as a circle. As you know, a circle can be created in 3-D either physically or conceptually, the circumference/perimeter of which is just Pi multiplied by the diameter of the circle. My question is this: If the circle can exist perfectly defined by lines and curves in 3-D, why can't its circumference be perfectly defined as a percentage of a line, that line being the diameter? Our math says it can't, since Pi is an irrational number with an infinite number of digits. Yet, the geometry is staring at us, almost laughing at us in the face as it just exists without any problems at all! This points to the fact that either I or we are missing something here. I have a feeling it lies in the very nature of curvature. What is curvature? Can it really be defined mathematically? As you know, calculus attempts to do just this. However, going back to the concept of our number line, how can something analogue in nature (curvature) be defined by something digital (our numbering system)? As you know, calculus is all based on the concept of the limit. And things are not "defined" until you take the limit to infinity. I believe we are forced to go to infinity with it because our very math is faulty or drastically limited. In other words, we are forced to go to infinity because our numbering system is digital in nature and only becomes analogue in the limit as we approach an infinite number of digits of precision.

Ok, so you might say that just the ratio of the diameter to circumference is irrational, but how can that be if it is staring me in the face and can be measured? I can see how the ratio can be irrational even upon finer and finer measurements, but that sort of points to the fact that division itself is flawed, don't you think? In other words, division is attempting to apply something digital in nature to something analogue in nature. It's like trying to describe apples in terms of oranges. Maybe I am going mental...Am I making any sense to you?

The problem of the circle, I think, is that we are trying to describe curvature using a line. And, I have a feeling that that can't be done using the math we currently have. We can only approximate the definition using our math since our math is based on lines. The inherent problem I think lies somewhere in the fact that our math is based on a digital LINEAR numbering system with a digital, discrete number line, which consists of both rational and irrational numbers, the irrational ones described at any point by an infinite number of digits. I think our "discovery" of irrational numbers came after we invented our fundamental math as it currently exists. The discovery of irrational numbers should point to the fact that PERHAPS our math is flawed.

Thoughts?

Steve

posted by Sacred Steve at 8/19/2006 12:06:00 AM