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Re: Ariadne's thread
Posted:
Jan 2, 2012 3:26 AM
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Let me return to the paradox encountered on the advanced
level of RMP 41 and formulate it more simply.
Imagine a cylinder. Halve the height and double the
diameter. Halve the height and double the diameter.
Halve the height and double the diameter. Go on like
this forever. The cylinder turns into an ever thinner
and larger disc whose volume, doubled with every step,
increases beyond any limit. In the end you get a circle
of an infinite diameter and an infinite volume!
Back to the original cylinder. Double the height and
halve the diameter. Double the height and halve the
diameter. Double the height and halve the diameter.
Go on like this forever. The cylinder turns into an
ever thinner and longer rod whose volume, halved with
each step, gets smaller and smaller. In the end you
have a straight line of infinite length and zero volume.
A line has no volume, but how can a circle have one?
The above circle has an infinite diameter and an
infinite volume.
The lesson to be learned may be that the infinite is full
of surprises. Our mathematical insights, powerful as they
may be, are but glimpses of a divine knowledge.
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