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Topic: Skepticism, mysticism, Jewish mathematics
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R. Srinivasan

Posts: 252
Registered: 12/13/04
Re: Skepticism, mysticism, Jewish mathematics
Posted: Aug 4, 2006 12:59 PM
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herbzet@cox.net wrote:
> "R. Srinivasan" wrote:
> > Mike Kelly wrote:
> > > 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?

> >
> > No, Zeno's paradox cannot even be stated in NAFL because it requires
> > open/semi-open intervals of reals to state it. In ZFC the paradox is
> > "resolved" simply by fiat, i.e., we say infinitely many non-zero,
> > non-infinitesimal intervals of reals *can* sum to a finite interval and
> > that is the end of the matter. There are those who may find it
> > counter-intuitive and those who don't.
> >
> > Regards, RS

>
>
>
> (1) This business with the nested intervals is generalized in a theorem
> which is called "Cantor's Theorem" in "Modern Theories of Integration"
> by H. Kestelman [1960]. IIRC, the statement of the theorem (in
> Chapter 1) is (pretty close to) this:
>
> For a [infinite] sequence of non-empty, closed, bounded point-sets {S}
> such that each S_n includes S_(n + 1), the intersection of all the
> sets in the sequence is non-empty. [The inclusions need not be proper.]
>
> I don't recall the proof exactly, but key is to pick a point in each
> set, and then point out that these points converge to a limit.
> That might not be exactly accurate, but that's the basic idea.
>
> For me the surprising part was not that the intersection of
> all the sets was _not_ of non-zero or infinitesmal length (measure?)
> (as your example shows, this is not a necessary consequence) -- the
> surprise for me was that the intersection is necessarily non-empty.
>
> In the book the axiom of choice is introduced, but I forget whether
> it is prior to or subsequent to the proof of this theorem.
>
> In the book "Infinity and the Mind" by Rudy Rucker pages 265-67
> [Bantam edition 1983] he writes:
>
> "Cantor first proved that Aleph_0 < c on December 7, 1873. We
> know this because he communicated his proof to his friend
> Dedekind the next day. Cantor's first proof of the uncountability
> of the reals was a bit different from the diagonal argument now
> used, and this proof will be sketched. The fact that there can
> be no one-to-one correspondence between the sets N and R is the
> first really interesting fact about the transfinite cardinal
> numbers, an it can rightly be said that set theory was born
> on that December day a little more than a century ago.
>
> One must show that there can never be a countable listing r_0,
> r_1, r_2, r_3, ... of real numbers that is _exhaustive_. That
> is, we must show that given any countable set of real numbers
> having the form {r_n: n e N}, there is some real number d that
> is different from all of the r_n's. Cantor's first proof of
> this is simple to present, although a knowledge of the Heine-
> Borel Theorem is necessary fully to grasp why it works. Cantor's
> first proof proceeded as follows: Find a closed interval I_0
> that fails to contain r_0, then find a closed subinterval I_1
> of I_0 such that I_1 misses r_1; continue in this manner,
> obtaining an infinite nested sequence of closed intervals, I_0
> (includes) I_1 (includes) I_2 (includes) ..., that eventually
> excludes every one of the r_n; now, let d be a point lying
> in the intersection of all the I_n's; d is a real number
> different from all of the r_n."
>
> (2) I read a book sometime ago about paradox. It contained
> a lot of discussion of the Zeno paradoxes, and quoted a number
> of contemporary mathmaticians and philosophers. The author's
> opinion was that Zeno's paradoxes have not been finally settled
> by modern analysis. The author was of the opinion that since
> their introduction in ancient times, they have given every
> generation, including ours, a new and refined set of problems
> to deal with.
>
> The book had an amusing picture on the front of the dust jacket,
> of a hammer with a wooden handle bent into a V shape, the striking
> end of the hammerhead facing inward. The handle had a nail driven
> half-way in opposite to the hammerhead, as if the hammer had been
> used to drive the nail into its own handle.
>
> I don't recall the name or author of the book, but I remember it
> was in the low numbers (less than 200) of the Dewey decimal system
> at the main branch of the local library. I'll look for it next
> time I go there.
>
> (3) Sorry to hear about your surly bosses at IBM. You'd think
> with all the research money spent, at IBM and elsewhere, on
> CRAP, they could cough up a little for this kind of research.
>
> (4) People can't absorb too much new stuff at once. It will
> help, I think, if you can make one good, new point, and present
> that free of distraction.
>
> That was part of the problem with, e.g., Marshall Mcluhan.
> He made a big splash for awhile but not much of what he wrote
> has made a lasting impression. There was just too much new
> stuff in everything he wrote. It was too much.
>


Thanks for the information and advice. I will carefully study the
proofs of the theorems you mention and see what I can make of them. I
too have read that Zeno's paradoxes are still very much unresolved even
in modern analysis. I tried my best to convince top IBM people, both
within India and abroad, to let me pursue this work. But to no avail. A
pity, as you mention.

Regards, RS



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