Monday, March 29, 2010

The Paradox Of Circles

What is a circle? Essentially an infinitely-sided polygon. But if it has an infinite number of sides, how is it ALL circles are not infinitely long around? These are the kinds of issues Zeno of antiquity brought up. As long as each side of a circle has a length greater than ZERO, should not ALL circles be infinitely long around?

On the other hand, if each side of a circle has a length less than or equal to ZERO, a singularity results thus rendering it something less than a circle. Of course the resolution of this dilemma lies within the realm of recognizing a duality between infinities and limits thereupon! Hence, an infinite entity can be intertwined within something merely finite in scope. Apart from this though, these kind of mind exercises are rather interesting.

This is a perfect example of the "barber paradox". It is not the least bit tempting to believe that a circle, or at least all of them*, are infinite in circumference. But if they are all essentially infinitely-sided polygons with sides > zero length, how could they not all be infinite in circumference? With this then, perhaps it IS tempting to believe all circles are infinitely long around.

But this can't be because I have seen many circles and know them to be finite in circumference! Therefore: if a circle MUST be of infinite circumference, then it CAN'T be! Conversely, if a circle CAN'T be of infinite circumference, then it MUST be!

*A most bizarre implication rears its head. Can there exist a circle infinitely long around outside of an abstract mental construct? What is needed to describe a circle? A definition.....for example, its radius. But does not defining an object or concept systematically render said object or concept to something less than infinite? Perhaps I should now feel compelled to "disbelieve" in circles. They are nothing short of absurd!

My overarching point with this seemingly silly and useless exercise is to demonstrate how much of what we believe or assume about reality is really absurd! Maybe we don't know what we THINK we know? Humility overwhelms me with this realization.....

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