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Topic: A new number zero
Replies: 0

 Jonathan Cender Posts: 75 Registered: 12/13/04
A new number zero
Posted: May 28, 2014 11:21 AM

Here's an alternative number zero to compare and contrast with the
number 0. A link to a paper on the subject, Replacing 0: A
"NonEuclidean" Arithmetic, follows the paper's abstract and
introduction.

Abstract.
The idea of nothing as conditional absence exemplified by placeholders
provides a way to replace the number 0 with a new number not based
on the empty set. Notation coincides with, and expands upon, Wheeler's
and
Penrose's array notation for n-real-dimensional space.
The replacement differs from the number 0 in its definition as a
divisor with
unique quotients and in its arithmetization of nothing. Some
consequences of
it are a nonPeano or "nonEuclidean" arithmetic, an extension of the
number
system, and building n-real-dimensional space operationally.

1. Plan of the investigation

The concept "nothing" or "nothingness" is represented in mathematics in
several
ways. Representations consist of the number zero and the placeholder
zero as well as the lack of one or more dimensions of the point, line,
and plane. The aim of this paper is to expand the mathematics of
nothing by introducing a novel representation of nothing; a new number
zero that replaces the number 0.
For many, nothing and 0 are equivalent; 0 is the number of nothing. A
different
number of nothing makes no sense because there is simply no other
nothing possible than that represented by 0. I therefore devote the
next section to pointing out that indeed another notion of nothing is
extant in mathematics, and from it, develop a hypothesis suitable for
an alternative to 0.
In the third section terms are defined for one such alternative zero
followed by
an effort to work out a "nonPeano", or "nonEuclidean", arithmetic.[14]
Switching
from a zero defined in terms of subtraction to a zero defined in terms
of division with unique quotients does not change any existing, defined
arithmetic. However, it does address what Patrick Suppes refers to as
"the vexing problem of defining the operation of division in the
elementary theory of arithmetic".[28] So addressing the "vexing
problem" extends the number system, and, together with a new definition
for a point, provides an interesting insight into the basis for the
Dirac delta function.
Lastly, a suggestion for an alternative to the empty set based on an
amendment
to predicate logic is given along with some comments on the
arithmetization of
nothingness and on calculus.