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

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Mike Kelly

Posts: 344
Registered: 3/30/06
Re: Skepticism, mysticism, Jewish mathematics
Posted: Aug 4, 2006 4:46 AM
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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 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

> >
> > Well, they may be counter-intuitive for some people. I agree the
> > Banach-Tarski 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 non-zero and non-infinitesimal
> 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 counter-intuitive, but many do.

How many?

--
mike.



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7/25/06
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8/4/06
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8/4/06
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8/4/06
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8/4/06
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8/4/06
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Mike Kelly
8/4/06
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8/4/06
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8/4/06
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8/6/06
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