Re: mr Normans perpetual flying machine.


"r norman" <r_s_norman@xxxxxxxxxxx> wrote in message
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On Wed, 10 Aug 2011 20:08:02 -0700, "Glenn"
<glennsheldon@xxxxxxxxxxxxxxx> wrote:


"r norman" <r_s_norman@xxxxxxxxxxx> wrote in message
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On Wed, 10 Aug 2011 21:25:06 -0500, Free Lunch <lunch@xxxxxxxxxxxxxx>
wrote:

On Wed, 10 Aug 2011 20:08:41 -0400, r norman <r_s_norman@xxxxxxxxxxx>
wrote in talk.origins:

On Wed, 10 Aug 2011 18:49:53 -0500, Free Lunch <lunch@xxxxxxxxxxxxxx>
wrote:

On Wed, 10 Aug 2011 16:42:00 -0700, "Glenn"
<glennsheldon@xxxxxxxxxxxxxxx> wrote in talk.origins:


"Free Lunch" <lunch@xxxxxxxxxxxxxx> wrote in message
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On Wed, 10 Aug 2011 13:25:21 -0700, "Glenn"
<glennsheldon@xxxxxxxxxxxxxxx> wrote in talk.origins:


"J. J. Lodder" <nospam@xxxxxxxxxxxxxxxx> wrote in message
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r norman <r_s_norman@xxxxxxxxxxx> wrote:

On Wed, 10 Aug 2011 12:42:32 +0200, nospam@xxxxxxxxxxxxxxxx
(J.
J.
Lodder) wrote:

Paul J Gans <gansno@xxxxxxxxx> wrote:

J. J. Lodder <nospam@xxxxxxxxxxxxxxxx> wrote:
r norman <r_s_norman@xxxxxxxxxxx> wrote:

On Mon, 8 Aug 2011 23:04:59 +0200,
nospam@xxxxxxxxxxxxxxxx (J.
J.
Lodder) wrote:

Conan the bacterium
<deinococcus0radiodurans@xxxxxxxxx>
wrote:

This thread is STILL going?

It got started in the early sixteen-hundreds, I
guess,
and it will outlast us.

Geez, it's like Bolzano, Cauchy, and Weierstrass
never
existed...

They don't matter.
0.999... was 1 long before they existed,


But not demonstated satisfactorily.

Archimedes, incidentally, made assumptions about the
infinite
divisibility and the completeness of the geometric
notion
of
line
in
order to do his geometric demonstrations. In other
words, he
assumed
some of the properties of the real numbers. That is
why
geometry
is
not arithmetic; you need analysis.

Incorrect, I'm afraid.
Let's go through it all one more time.
For an example the classic summability of te geometric
series
(according to Archimedes) please refer to the figure top
right in
<http://en.wikipedia.org/wiki/Geometric_series>
We find the area of all purple squares combined
by noting the self-similarity.
Scaled by half the half-size figure is -congruent-
wih the upper left quadrant of itself.
Hence by subtraction and dividing out we see
that the area of all purple squares combined
equals 1/3 of the unit square. [1]

The argument is as rigorous, and uses the same means,
as Euclid's proof of the Pythagorean theorem.
Likewise one easily sees that 0.999... = 1
in the same way.

Now Newton, Leibniz, and followers understood
that this was a degree of rigour and exactness
that their meddling with fluxions or infinitesimals
could not possibly match.

The task for Bolzano, Weierstrass etc.
was -not- to justify the geometric series.
On the contrary, the exact summing of the geometric
series
stood as a landmark for what was to be achieved
for all the new developments of the scientfic
revolution.
Whatever they did had to reproduce what was already
known.

In a technical sense they could re-derive
the convergence of the geometrical series
and you may see that as a 'proof'.
Conceptually this is not a proof at all,
for this whole framework has been built from the ground
up
in such a way that this result must come out of it.

So, summing up, by putting on modern blinders
you can insist that only the latest means of proof will
do.
That means however that you lose all sight of the
development,
and of the conceptual structure of mathematics.

Jan

[1] If you are still there quantum dotty,
please notice that there is no 'last square'
at the top right corner of the diagram, and no remainder
term.
There is exact conguenece, and nothing remains after
subtraction.
Perhaps you -can- see it with shrinking squares
instead of a repeating string of digits?


I'm sorry, but I cannot agree with you. What is lacking
is
the
notion of an infinite series, the very thing that dotty
is
hung
up about.

For the geometric proof to work, one has to have the
notion
of
infinity. And one has to have the notion of convergence.
It
is not at all axiomatic that these infinite series
converge.

If it is not perfectly obvious to you that the sum of the
areas
of the purple square in
<http://en.wikipedia.org/wiki/Geometric_series>
must be smaller than that of the unit square they are
contained in
I'm afraid there is nothing I can do to make that more
clear.


If it is not perfectly obvious to you that the sum of the
areas
of
the
purple squares involves an infinite (unbounded) number of
squares,
I'm
afraid there is nothing I can do to make that more clear.
The
problem
absolutely reeks of infinity and infinitely small and
infinite
divisibility. The ancients already understood that there
are
problems
with dealing with infinities. Archimedes saw that for some
reason
those problems were absent in the particular case of the
geometric
series. We now understand why and easily justify doing
infinite
series and sums.

See my other reply.
The problems are equally absent (and for the same reasons)
in the case of 9/10 + 9/100 + 9/1000 + ... = 1

Never, ever, ever will that addition equal 1. That is evident
from
the 0
in
front of the decimal and the first 9 behind the decimal in
0.999...

So you claim.

No claim, just fact.

You are clearly wrong. Your ignorance of decimal expansion is not an
argument against how it is defined.

Please explain why 1/3 + 2/3 does not equal 1.

They do, I have not said otherwise, and fractions are real numbers,
not
infinitely repeating decimals. A square peg just will not fit in a
round
hole, and neither will thirds fit into a decimal hole.


Just as
_
.3 is defined as the decimal expansion of 1/3, and
_
.6 is defined as the decimal expansion of 2/3, so
_
.9 is defined as the decimal expansion of the sum of those. It's 1.
Deal.

What you have just done is define away any connection to decimal
place
notation.

Not really. I understand your point, but disagree to some extent
within
the context. The fact that we can do a decimal expansion only works if
we recognize that there is no boundary on that expansion.

That has been my point all along. Problems with that unbounded
property have to be addressed to prove that 0.999... meaning an
unending series of 9's equals 1. That is what some people have
difficulty with. There is also the question of why, 0.333... * 3 =
0.999... if 0.3bar simply is a way of writing "1/3". The only way it
makes sense is to multiply each '3' by 3.

Get to it then, and let me know when you finish. Take the turtle with you
if
you wish.


No problem. For each place in the sequence, multiply the digit by n.
Each digit in the resulting sequence is well defined. No matter which
one you point to, I can tell you exactly how to calculate its value.
The infinite sequence forms a decimal expansion that we have already
proved represents a real number whose value we calculate as the
infinite sum of a geometric series. We have already demonstrated how
to calculate such a sum without going through all the steps. The
result is exactly 1.0. Not "going to". Not "approaching". The sum
IS 1.0.

There is no "exactly" in infinity, Richard.

Now, you tell me whether Homer reaches his destination or whether
Achilles passes the tortoise.

And did Mary finally marry her childhood sweetheart Sven?

The preSocratics understood that Homer and Achilles have no difficulty
doing their things. Why, more than two thousand years later, do you
have difficulty understanding that?

I'd rather not even consider how they "did their things".


.

Relevant Pages

  • Re: mr Normans perpetual flying machine.
    ... Archimedes, incidentally, made assumptions about the infinite ... For an example the classic summability of te geometric series ... equals 1/3 of the unit square. ... Your ignorance of decimal expansion is not an ...
    (talk.origins)
  • 0.999...
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  • Re: mr Normans perpetual flying machine.
    ... Archimedes, incidentally, made assumptions about the infinite ... For an example the classic summability of te geometric series ... equals 1/3 of the unit square. ... Your ignorance of decimal expansion is not an ...
    (talk.origins)
  • Re: mr Normans perpetual flying machine.
    ... Archimedes, incidentally, made assumptions about the infinite ... For an example the classic summability of te geometric series ... equals 1/3 of the unit square. ... Your ignorance of decimal expansion is not an ...
    (talk.origins)
  • Re: mr Normans perpetual flying machine.
    ... Archimedes, incidentally, made assumptions about the infinite ... For an example the classic summability of te geometric series ... equals 1/3 of the unit square. ... Your ignorance of decimal expansion is not an ...
    (talk.origins)