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From: Archimedes Plutonium on 22 Jan 2010 15:51
First I need the actual history events of mathematics, and then I will
proffer my own
speculation as to why Peano went off track, off course with a
Successor Function axiom
rather than to tie into the bulwark of history by giving this axiom a
Series definition. It would
be helpful if Peano left behind alot of notes, personal notes and
whether he gave any
autobiography or notes as to why he constructed a Successor Axiom
rather than a
Series Axiom.

--- quoting by typing from various pages: Cajori's book "History of
Mathematics" 1991 ---

More fortunate in reaching the public was A. L. Cauchy, whose Analyse
Algebrique of
1821 contains a rigorous treatment of series. (page 373)

G. Cantor began his publications in 1870; in 1883 he published his
Grundlagen einer allgemeinen Mannichfaltigkeitslehre. In 1895 and 1897
appeared in Mathematische Annalen his Beitrage zur Begrundung der
transfiniten Mengenlehre.2 (page 400)

In 1891 appeared under G. Peano's editorship, the first volume of the
Rivista di Matematica
which contains articles on mathematical logic and its applications,
but this kind of work was
carried on more fully in the Formulaire de mathematiques of which the
first volume was published in 1895. (page 408)

A new and powerful method of attacking questions on the theory of
algebraic numbers was
advanced by Kurt Hensel of Konigsberg in his Theorie der algebraischen
Zahlen, 1908, and in his Zahlentheorie, 1913. (page 445)

--- end quoting (by my own typing) from Cajori, on various pages the
dates of Cauchy, Cantor, Peano, and Hensel ---

Now the purpose of this outline of math history is that I suspect I
know why Peano and others
veered off course by using a Successor Function for the axioms of the
Natural Numbers
rather than use the more logical choice of the Series, for which the
Series was designed
to be the axiom that creates the Natural Numbers.

My speculation is that, whether we like it or not in science, that
humanity as a whole
is too spiritual, or, over-spiritual, and when it comes to a choice
between two theories
of science as to which is accepted first by the human civilization
that this spiritual or religious component of the general public or
the individual
scientist making the new theory, that the choices selected are those
that have more
"spiritual or religious connotations as the first endorsed theory of
that particular subject."

So for example, the first selected theory of creation was of course
Bible-religion creation
theory. The first selected theory of astronomy was not the
heliocentric theory even though
Ancient Greeks discovered and proved it to be true, but rather the
spiritual-religion theory
of geocentric. The first geology theory of continents was a static
continent theory much like
the static geocentric of astronomy because it is more conforming to
the prevailing social
religion and spiritualism, rather than the alternative of a
Continental drift theory.

And the first cosmology theory to be accepted would of course be the
Big Bang rather than
the Atom Totality which there is plenty of room to fit a religion god
into the Big Bang but in the
Atom Totality, god is the Atom Totality.

That is my hunch as to why mathematics veered off course from Cauchy
to Cantor to Peano
to Hensel. That the first time in math history we examine "infinity"
we of course, laden with
too much spiritualism and religion in the society as a whole would
come down onto acceptance of "infinity" as per Cantor that we lose
sight of truth and reality. That comes
Peano with a choice to use the Series for the Successor and the Series
thus shows us
that the Natural Numbers are a mix of finite-numbers and infinite-
numbers and that would be
rather-- anti-religious or anti-spiritual, for the finite is of
humanity and god is of the infinite.

And really ironic and funny as that the history of mathematics, by the
20th century, had come
to the silly situation that mathematics knew more about "infinity"
with Cantor and then Godel
and Cohen on continuum, that math espoused to have more knowledge and
insights into
infinity than mathematics knew about the lowly "finite", because,
well, anyone wanting a precision definition of finite-number, was out
of luck, because noone was about to offer what
finite meant, yet we have tomes of books waiting at every university
to saturate the student
with infinity.


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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