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Re: Skepticism, mysticism, Jewish mathematics
Posted:
Aug 4, 2006 4:46 AM


R. Srinivasan wrote: > Rupert wrote: > > R. Srinivasan wrote: > > > Rupert wrote: > > > > R. Srinivasan wrote: > > > > > Rupert wrote: > > > > > > R. Srinivasan wrote: > > > > > > > Barb Knox wrote: > > > > > > > > > > > > > > > > Consider that calculus too started out as a halfbaked theory laden with > > > > > > > > paradoxes, and that one of the great mathematical achievements was to > > > > > > > > put it on a rigourous footing. And indeed, set theory was one of the > > > > > > > > important tools in that enterprise. > > > > > > > > > > > > > > The presently accepted foundations of the calculus still does not > > > > > > > resolve Zeno's paradoxes. > > > > > > > > > > > > Why not? What's the paradox? > > > > > > > > > > The paradox can be stated in two ways. Consider the example of Achilles > > > > > chasing the tortoise along a straight line at a constant velocity > > > > > higher than that of the tortoise's (constant velocity). The paradox is > > > > > that Achilles has to "complete" infinitely many opertaions to catch up > > > > > with the torotoise  from the starting point he sees the tortoise at a > > > > > particular location ahead, and he first has to reach that location. > > > > > When he reaches, he sees the toroise ahead at another location, and > > > > > Achilles has to reach there. And ad infinitum. The Greeks thought that > > > > > this was a paradox presumably because they viewed infinity as > > > > > "potential", i.e., the infinitely many operations required to reach the > > > > > torotoise cannot never be completed in finite time. > > > > > > > > > > >From the modern point of view the paradox can be stated as follows. > > > > > How can infinitely many finite, nonzero, noninifnitesimal intervals > > > > > of reals sum to a finite interval (why isn't the sum infinite)? I..e, > > > > > suppose starting from location zero, Achilles first reaches 1/2, then > > > > > 3/4, then 7/8,...... Then Achilles covers the distance > > > > > 1/2+1/4+1/8.....=1, where he catches up with the tortoise. The paradox > > > > > is  why isn't the sum infinite, given that there are infinitely many > > > > > finite, nonzero and noninfinitesimal intervals being summed (assume > > > > > we are using some standard version of real analysis). > > > > > > > > > > > > > Well, it just isn't. I don't see any reason why it should be. You > > > > haven't shown a contradiction in standard real analysis. > > > > > > > > > That is true. The same can be said of the BanachTarski paradox or many > > > of the other paradoxes of classical measure theory. But these are > > > paradoxes nevertheless, and highly counterintuitive. > > > > > > Regards, RS > > > > Well, they may be counterintuitive for some people. I agree the > > BanachTarski paradox is a rather surprising result. But the fact that > > the sum of an infinite series of positive numbers can be finite I don't > > find counterintuitive at all, myself. Do you really claim to have a > > formulation of analysis where this result is avoided? Can you tell me > > what it is? > > Another way to state the paradox is as follows. Consider the infinite > series of nested real intervals [1,1], [1/2, 1/2], [1/4,1/4],... The > intersection of these intervals contains the single point 0, but *each* > of these infinitely many intervals is of nonzero and noninfinitesimal > length. So why doesn't their intersection contain infinitely many > points?
Because they keep getting smaller, but never "reach" 0????
>Again you may not find this counterintuitive, but many do.
How many?
 mike.



