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Hi Yanniru,
I am sorry, but Paul is absolutely correct. I read that essay a number of years ago and Paul and I discussed the funny result that you could fill the figure with paint but could not paint its interior. That result is true, mathematically, because area and volume are significantly different concepts. When one tries to convert the result into the concept of painting and filling the interior, it is very natural to jump to some very unwarranted intuitive conclusions: i.e., if you can fill it, you can paint it.
The amount of paint required to paint the figure is given by the area times the thickness of the paint coat. No such conceptual constraint exists on the amount of paint needed to fill the figure. The problem in the comparison occurs as soon as the diameter of the figure becomes less than the thickness of the paint which will always occur if you go to sufficiently large x. The only paint thickness which avoids this problem is a thickness of zero. In that case, the amount of paint is given by zero times the area and one can conclude that the volume of paint required may not be infinite.
In fact, if one uses the radius of the figure as the thickness of the paint (we use a thinner coat as we proceed to large x) then the quantity of paint required to paint the interior is exactly equal to the quantity of paint required to fill the figure.
These are just a sampling of the problems which occur when one allows things to go to infinity! Paul has a very strong position when he complains about the problems one discovers when one believes in "infinity".
There is another very significant issue arising from the use of infinity which may turn out to be a significant discovery. Take a look at http://www.thinhtran.com . Thin Tran has come up with some serious complaints about probability theory. He apparently has spent a big portion of his life calculating extremely improbable events (related to electronic analysis of noise) and found that "extremely improbable events" turn out to be experimentally more improbable than the math predicts. He thinks the solution is the failure of probability theory to take into account that any "real" physical device is actually a finite state machine and that, given enough time it must repeat itself. Now I don't know what he is doing but I asked him to send me a preprint of his book which he says will be published sometime in July. If his results conform better to the experimental results, he may have discovered a fundamental concept.
Yanniru, I don't want to make you feel bad but I think your education was somewhat lacking in the fundamental concepts beneath physics. You strike me as someone who has learned to plug equations and what equations to plug. The problem with that approach is it is awfully hard to remember which equations one is suppose to be using (there are so many).
Have fun -- Dick
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