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

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R. Srinivasan

Posts: 252
Registered: 12/13/04
Re: Skepticism, mysticism, Jewish mathematics
Posted: Aug 4, 2006 3:07 AM
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Rupert wrote:
> R. Srinivasan wrote:
> > Rupert wrote:
> > > R. Srinivasan wrote:
> > > > Barb Knox wrote:
> > > > >
> > > > > Consider that calculus too started out as a half-baked 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, non-zero, non-inifnitesimal 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, non-zero and non-infinitesimal 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 Banach-Tarski paradox or many
of the other paradoxes of classical measure theory. But these are
paradoxes nevertheless, and highly counter-intuitive.

Regards, RS



Date Subject Author
7/25/06
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David Petry
7/25/06
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fishfry
7/25/06
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7/25/06
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7/25/06
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7/26/06
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Mike Kelly
7/26/06
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David Petry
7/26/06
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Gene Ward Smith
7/26/06
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7/26/06
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dkfjdklj@yahoo.com
7/26/06
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herbzet
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dkfjdklj@yahoo.com
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7/27/06
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8/1/06
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7/31/06
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8/2/06
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8/3/06
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Rupert
8/4/06
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8/4/06
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R. Srinivasan
8/4/06
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Rupert
8/4/06
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R. Srinivasan
8/4/06
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Rupert
8/4/06
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R. Srinivasan
8/4/06
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Mike Kelly
8/4/06
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R. Srinivasan
8/4/06
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R. Srinivasan
8/4/06
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R. Srinivasan
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8/7/06
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8/5/06
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7/30/06
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7/28/06
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