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From: Archimedes Plutonium on 21 Jan 2010 15:22

Now Wikipedia is good for an overall perspective of Series in
mathematics, but not good for
tracking down some dates and those mathematicians who contributed to
the progress of
Series as a subject of mathematics.

--- quoting Wikipedia on some pitfalls of Series ---

Potential confusion

When talking about series, one can refer either to the sequence { SN }
of the partial sums, or to the sum of the series,

i.e., the limit of the sequence of partial sums (see the formal
definition in the next section) – it is clear which one is meant from
context. To make a distinction between these two completely different
objects (sequence vs. summed value), one sometimes omits the limits
(atop and below the sum's symbol), as in

∑
an

n

in order to refer to the formal series, that may or may not have a
definite sum.

--- end quoting Wikipedia on Series ----

Now it is not surprizing that Series history goes back to Ancient
Greece especially
with Archimedes of Syracuse with his method of exhaustion which is the
preCalculus.

And not surprizing that Series is a big part of mathematics since it
is the trigonometry
and Calculus. Series is a part of what is known in mathematics as
Analysis.

So it is rather surprizing and rather silly and ridiculous that when
Dedekind and Peano
set about to formalize the Natural Numbers that they would not use
Series as the
foundation of their axiom system but rather invent a newer concept of
Successor, for
which mathematics never had a "successor concept" before, and
beguiling in the fact that
"successor" has already incorporated the concept of counting numbers,
when it is the job and
duty of the axioms to create the counting numbers and not have them
incorporated within
the axioms themselves. So that the Peano Successor axiom is a circular
axiom and not a
legitimate axiom that creates the Natural Numbers. If Peano had used a
Series Axiom, which
does not have the preconceived idea of counting already incorporated,
then Peano would have
a far cleaner and logical axiom set. And by using a Series Axiom
rather than a Successor
Axiom, there would have been no hiding or escape from the fact that
some of the Natural Numbers are finite-numbers whilst others are
infinite-numbers. And so, Peano and Dedekind
would have been faced with providing a precision definition of finite-
number.

I contend that it would have been almost impossible for the technology
of mathematics during the time of Peano and Dedekind or even including
the time period of 1900 to 1990 for mathematics to have ever been able
to define finite-number, since Physics was not mature
until far after 1930s.

So as much as I begrudge Peano, Cantor, Dedekind, Cohen, Godel, and
many others
for failing to precisely define "finite-number" one can easily see
that since the times
in which they lived, they were technologically infeeble to define
finite-number. They did not have these items:

(1) Physics is master and math is subset, especially having Quantum
Mechanics
(2) they did not have frontview with backview of a number where
infinity fits into the middle.

But they did have Series and we can easily see that Hensel made use of
Series to create
the p-adics which have infinite-numbers. And we see that the Hensel p-
adics are what the
Natural Numbers should be like.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
 | 
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