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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 9:29 AM
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R. Srinivasan wrote:
> R. Srinivasan wrote:
> > Mike Kelly wrote:
> > > 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 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????
> > > > >

> > > >
> > > > None of the end-points 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.

> >
> > Note carefully the difference between your approach and mine. I am
> > considering the *totality* of nested intervals (which classical
> > standard real analysis permits) and saying that not one interval in
> > this totality has end-points "infinitely close" to zero.
> >
> > You are considering a process in time, where you are at a particular
> > interval, and then pass on to the next smaller nested interval, and so
> > on. If you asked the Greeks, they would have probably told you that
> > this infinite process of sub-diviision can never be "completed" and
> > you will "always" be an interval's length away from zero. Whereas you
> > have assumed that you can somehow complete this process, and after
> > completion, only the point zero is never passed. That precisely is the
> > issue in Zeno's paradox.

>
> Note also that your argument essentially postulates the existence of
> totalities of real numbers like (0,1] or [-1,1]\{0}, i.e.,
> open/semi-open intervals and other non-closed sets of reals. This is
> permitted in classical logic, but not in my proposed logic NAFL. So my
> way of looking at Zeno's paradox is necessarily different from yours.


So is Zeno's paradox a paradox in NAFL? What about, say, ZFC?

What is the crux of the paradox?

--
mike.



Date Subject Author
7/25/06
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David Petry
7/25/06
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7/26/06
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Mike Kelly
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David Petry
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Gene Ward Smith
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herbzet
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lloyd
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herbzet
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7/30/06
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8/4/06
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