Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 R. Srinivasan Posts: 252 Registered: 12/13/04
Re: Skepticism, mysticism, Jewish mathematics
Posted: Aug 4, 2006 5:25 AM
 Plain Text Reply

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.

> >Again you may not find this counter-intuitive, 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?

Regards, RS

Date Subject Author
7/25/06 David Petry
7/25/06 fishfry
7/25/06 Dr. David Kirkby
7/25/06 Dr. David Kirkby
7/25/06 lloyd
7/25/06 Doug Schwarz
7/25/06 Virgil
7/26/06 Mike Kelly
7/26/06 David Petry
7/26/06 Gene Ward Smith
7/26/06 Brian Quincy Hutchings
7/26/06 dkfjdklj@yahoo.com
7/26/06 herbzet
7/26/06 David Petry
7/28/06 herbzet
7/26/06 herbzet
7/26/06 lloyd
7/26/06 herbzet
7/26/06 dkfjdklj@yahoo.com
7/26/06 herbzet
7/26/06 Gene Ward Smith
7/26/06 Gerry Myerson
7/26/06 David Petry
7/26/06 Randy Poe
7/26/06 mensanator
7/27/06 Richard Herring
7/27/06 mensanator
7/27/06 Richard Herring
7/26/06 Gene Ward Smith
7/26/06 David Petry
7/26/06 Patricia Shanahan
7/26/06 Gene Ward Smith
7/28/06 herbzet
7/26/06 Dr. David Kirkby
7/26/06 Randy Poe
7/27/06 Rotwang
7/28/06 herbzet
7/30/06 Han de Bruijn
7/30/06 Barb Knox
7/30/06 zr
7/30/06 Virgil
7/30/06 Gene Ward Smith
7/30/06 zr
8/1/06 Virgil
7/30/06 T.H. Ray
7/31/06 Brian Quincy Hutchings
7/31/06 T.H. Ray
8/1/06 Brian Quincy Hutchings
8/1/06 David R Tribble
7/30/06 Gene Ward Smith
7/30/06 Ioannis
7/30/06 Dr. David Kirkby
7/30/06 zr
7/30/06 Dave Rusin
8/1/06 David Bernier
8/1/06 David R Tribble
8/2/06 Ioannis
7/31/06 Han de Bruijn
8/2/06 R. Srinivasan
8/3/06 Rupert
8/4/06 R. Srinivasan
8/4/06 R. Srinivasan
8/4/06 Rupert
8/4/06 R. Srinivasan
8/4/06 Rupert
8/4/06 R. Srinivasan
8/4/06 Mike Kelly
8/4/06 R. Srinivasan
8/4/06 Mike Kelly
8/4/06 R. Srinivasan
8/4/06 R. Srinivasan
8/4/06 Mike Kelly
8/4/06 R. Srinivasan
8/4/06 herbzet@cox.net
8/4/06 R. Srinivasan
8/6/06 Newberry
8/4/06 Jack Markan
8/4/06 Brian Quincy Hutchings
8/7/06 R. Srinivasan
8/4/06 Mike Kelly
8/5/06 R. Srinivasan
7/30/06 bogus address
7/28/06 herbzet
7/28/06 Gene Ward Smith
7/28/06 herbzet
7/28/06 mensanator
7/28/06 herbzet
7/27/06 T.H. Ray
7/28/06 herbzet
7/29/06 David Petry
7/30/06 herbzet@cox.net
8/4/06 Jack Markan
8/4/06 T.H. Ray
7/26/06 David R Tribble
7/26/06 Gene Ward Smith
7/26/06 T.H. Ray
7/26/06 toni.lassila@gmail.com
7/26/06 Bennett Standeven
7/26/06 Brian Quincy Hutchings
7/27/06 Rotwang
7/27/06 Craig Feinstein
7/27/06 Toni Lassila
7/27/06 Craig Feinstein
7/27/06 Brian Quincy Hutchings
7/27/06 Rupert
7/28/06 zr
7/28/06 herbzet
7/28/06 T.H. Ray
7/29/06 zr
7/29/06 Virgil
7/29/06 zr
7/29/06 Virgil
7/30/06 herbzet@cox.net
7/30/06 T.H. Ray
7/30/06 LauLuna
7/30/06 LauLuna

© The Math Forum at NCTM 1994-2018. All Rights Reserved.