Re: *** T. Winter says: Definition: sum{i in N} i = 0

On 25 Jun., 12:54, Klaus Cammin <netzkl...@xxxxxxxx> wrote:
WM schrieb:

As far as we can sum natural numbers, the sum will
increase.

Well, we don't just sum up numbers when investigating sequences.

That's exactly what we do. We sum a much as we like. The sum converges
closer and closer to some limit (or it does not) but does never reach
it.

That's what so nice about the contemporary notion of a limit: in comprises
all kinds of sequences, monotonous, alternating, you name it, under no more
but *one* notion. So you need to show that your notion is as strong as the
usual one. Well, it is not.

Try to store as much numbers as possible on
your hard disk.

Adding to hagman:
1. So the limit depends on the size of the hard disk?
Your limit is different from mine, if our harddisks differ?
2. Hence there is not one limit, but possibly many of them?

I only pointed you to the kindergarden in order to prepare you for the
real life.

3. If |a_n - g| is small enough, |a_(n+1) - g| is not?
n+1 is clearly different from a_n, so it exists.

If you have an old computer with only 256 bits, then you can
accelerate the procedure to store as many numbers as you can. When I
was a young student, I had a wonderful programmable pocket calculator,
a TI 59 from Texas Instruments. In order to do numerical calculations
for my vacuum polarization experiments, I let it run for several days
sometimes. This calculator could store 60 numbers. Nevertheless I
managed to do sums over several 10^4 data points. Of course some
tricky programming was necessary. There it was for the first time that
I had to delete all numbers which were not immediately required for
the next steps. This was an apt account of the mathematics you can do.

But we do agree, that the whole dispute is just about ideas, don't we?

Correct. About things which do not exists unless someone thinks or
notes them.

Because if that is not true, we'd have to actually perform the hard disk
test. Since I'm not dumb enough to do that, I'm sorry that I must leave
that important physical experiment to you ...

So because it's all about ideas, and no idea has more right than another,
there must be criterias to tell which notion should be preferred.
Usefulness would be a nice suggestion. And unique limits are more useful
than ambigous ones. And I don't want to have a hard disk in my mind when
calculating limits. It's too blocky ...

Your memory is comparable to a hard disk with less than 10^11 bits.

BTW: I looked up "eindeutig" and LEO came up with "clear","well-
defined","definite", and many other nice terms mathematicians prefer to use

You should say: mathematician to talk about that, while in fact they
prefer cloudy sets of undefined numbers.
...


The main issue was to show you some roots of the idea of actual
infinity. I hope you appreciate them.

Sure I do. However religion was by no means the promininent idea, although
you tried to badmouth it as usual. BTW: did you apologize to Mr. Sponsel by
misleading him so badly with the Poincaré source?

There is no reason at all. I did not write a book on history but a
book on mathematics. My responsibility is that the mathematical
contents is ok. For biographical notes I took biographical and
personal data from reliable experts, as is usual. And Poincaré's quote
it spread all over the literature. Already Skolem in 1929 reports it.
Dauben is a distinguished expert concerning Cantor. And, in adition,
from other quotes of Poincaré and Cantor's reactions we see that
Poincaré's opinion is correctly reproduced.

Poincaré: Es gibt kein aktual Unendliches, das haben die Cantorianer
vergessen und haben sich in Widersprüche verwickelt. [H. Poincaré, Les
mathématiques et la logique III, Rev. métaphys. morale 14, p. 316,
(1906).]

Cantor an Hilbert, 24.6.1908: Wie bisher rühre ich keinen Finger zu
meiner Vertheidigung wider die seit Jahren fortgesetzten boshaften
Angriffe Poincaré's.
Die letzte Attaque desselben hat meine Augen nur noch sehender gemacht
in Bezug auf die Dürftigkeit, Oberflächlichkeit, das Schaumschlagen,
die zu Grunde liegende gemeine Gesinnung dieses Gelichters, das sich
einbildet, an der Spitze der Wissenschaft zu stehen und ihr
Vorschriften machen zu können, wie sie sich in Zukunft zu gestalten
habe.
Seit Jahren weiss ich zwar, daß die beiden Akademien, die Berliner und
die Pariser in seniler Decadence sind; daß ihr letztes Stündchen bald
schlagen wird, auf diese Meinung bin ich erst in diesem Frühjahr,
durch die Poincarésche Glanzleistung gekommen.

Cantor to Russell, 19.9.1911

I am Baconian in the Bacon-Shakespeare question and I am quite an
adversary of Old Kant, who, in my eyes has done much harm and mischief
to philosophy, even to mankind; as you easily see by the most
perverted development of metaphysics in Germany in all that followed
him, as in Fichte, Schelling, Hegel, Herbart, Schopenhauer, Hartmann,
Nietzsche, etc. etc. on to this very day. I never could understand
that and why such reasonable and enobled peoples as the Italians, the
English and the French are, could follow yonder sophistical
philistine, who was so bad a mathematician.
And now it is that in just this abominable mummy, as Kant is,
Monsieur Poincaré felt quite enarmoured, if he is not bewitched by
him. So I understand quite well the opposition of Mons. Poincaré, by
which I felt myself honoured, so he never had in his mind to honour
me, as I am sure. If he perhaps expect, that I will answer him for
defending myself, he is certainly in great a mistake.

Therefore, whether Poincaré did mention the illness or not: The quote
shows his real opinion.

BTW,I for my person agree with the "abominable mummy" Kant who pleads
in favour of the freedom "von seiner Vernuft in allen Stücken
öffentlichen Gebrauch zu machen" while I pity the Cantorists, who,
with respect to the binary tree, seem not even dare to do so in
private.

The method seems
comparable: Take the most casual phrase and misdisplay it as the most
important. The same seems to have taken place here, and so I won't believe
a bit that Cantor was religious.

Georg Cantor Briefe H. Meschkowski, W. Nilson (Herausgeber), Springer,
Berlin (1991), p 15: Cantor ist wohl der letzte große Vertreter der
Newtonschen Geisteshaltung in Sachen Religion.

Regards, WM

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