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Topic:
Skepticism, mysticism, Jewish mathematics
Replies:
115
Last Post:
Aug 7, 2006 1:30 AM




Re: Skepticism, mysticism, Jewish mathematics
Posted:
Aug 4, 2006 3:07 AM


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



