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Topic: Skepticism, mysticism, Jewish mathematics
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herbzet@cox.net

Posts: 29
Registered: 7/3/06
Re: Skepticism, mysticism, Jewish mathematics
Posted: Aug 4, 2006 12:26 PM
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"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.

--
hz

"The medium is the message."




















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--
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