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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 nrealdimensional 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 nrealdimensional 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.
https://drive.google.com/file/d/0B46Ee1cnYi6TDRJV3U5VUUzUVk/edit?usp=sh aring
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