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

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Rupert

Posts: 3,810
Registered: 12/6/04
Re: Skepticism, mysticism, Jewish mathematics
Posted: Aug 4, 2006 2:54 AM
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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.

> The NAFL resolution of these paradoxes is given in Sec. 4 of
> <http://arxiv.org/abs/math.LO/0506475> (see Remarks 14-16). Basically
> open/semi-open intervals of reals do not exist in the NAFL version of
> real analysis -- so the proposition that Achilles is confined to the
> interval [0,1) fails and cannot even be stated. Secondly it is not
> legal in NAFL to ask *how many* intervals (or reals) are present in the
> super-class of intervals ([0,1/2], [1/2,1/4] ....[1,1]), because direct
> quantifiication over reals (or intervals of reals), which are infinite
> classes/super-classes, is banned. But there is a way to quantify
> indirectly, as explained in my paper -- and represent intervals, etc.
> as "super-classes" mentioned above.
>
> Regards, RS




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7/25/06
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David Petry
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Mike Kelly
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herbzet
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8/3/06
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Rupert
8/4/06
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R. Srinivasan
8/4/06
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R. Srinivasan
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Rupert
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