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Zeno's Paradoxes
Many readers of my paper on the calculus have assumed that I am opposed to the idea of infinities or limits. They imagine that I wrote my paper exposing the faults in the derivation of the calculus because I had philosophical problems with infinite progressions or with infinitely small quantities. But the fact is that I have no problem with the infinite as a concept, either mathematical or physical. Indeed, I find it much easier to believe that the universe is infinite in space and time. To believe the opposite is to beg so many questions. I also find it easier to believe in a real infinite progession in size, both large and small. That is (to give a concrete example) I find it more likely than not that particles smaller than photons will eventually be discovered or postulated. Everything I have said so far is pretty standard. But I suspect that Zeno's point was even more subtle than that, and that we might look to an even more clever solution than the ones that have been offered historically. Summing the series certainly solves these problems, and Zeno probably meant to give the clue in that direction. But there is an even simpler answer, one that requires no limit and no summing. It does not even require looking at the series. This solution concerns noticing that Zeno never disallows us from assuming that Atalanta actually reaches the halfway mark. In fact it is a postulate of the problem. Zeno creates the paradox by
Another paradox of Zeno concerns an arrow flying through the air. Zeno states that at each instant the arrow must be imagined to be immobile—frozen in one spot. If it is frozen at each instant it must be frozen at all instants. If it is frozen at all instants it must not be moving. Therefore motion is an illusion.
I would like to make one final point concerning the paradox in mathematics. The paradox is now seen as a sign of distinction. From transfinite math to Special Relativity to QED, the paradox is seen not as a fault but as an esoteric badge, a hint that the transcendent is being approached. This belief could not be more false or deluded. A paradox is not a good sign. It is a very bad sign. It is a sign of disease. It is a very loud noise warning of a coming train. It is usually a sign that the question has been badly posed, as with Zeno. It is the outcome of a contradiction. If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many |