Re: mr Normans perpetual flying machine.

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

Jan

.

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 ... We find the area of all purple squares combined ... Here you argue that the ancients proved the convergence of the ...
    (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 ... We find the area of all purple squares combined ...
    (talk.origins)
  • Re: mr Normans perpetual flying machine.
    ... Lodder) wrote: ... Archimedes, incidentally, made assumptions about the infinite ... For an example the classic summability of te geometric series ... We find the area of all purple squares combined ...
    (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 ... We find the area of all purple squares combined ...
    (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 ... We find the area of all purple squares combined ...
    (talk.origins)