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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 6:02 AM


R. Srinivasan wrote: > Mike Kelly wrote: > > 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???? > > > > None of the endpoints of the nested intervals ever get "infinitely > close" to 0 either. This seems to *suggest* that the intersection of > these intervals should contain uncountably many points.
But the sequence of nested intervals does get "as close as you like" to zero, if you wait long enough. Any point other than zero will be "passed" eventually.
> > >Again you may not find this counterintuitive, but many do. > > > > How many? > > I have seen this paradox crop up from time to time. Probably mainstream > mathematicians are no longer bothered by these kinds of issues. Maybe > some philosophers and philosophically inclined logicians?
I don't understand why this is a paradox. Not trying to be patronising or belittling, I just don't understand what the root of the conceptual difficulty is here.
 mike.



