The arithmetic foundations of mathematics: constructing new mathematics with negative numbers beyond infinity
AuthorBagdasaryan, Armen G.
MetadataShow full item record
CitationBagdasaryan, A. G. (2009). The arithmetic foundations of mathematics: constructing new mathematics with negative numbers beyond infinity. s. 112.
The basic and fundamental concept underlying the foundations of mathematics is the notion of natural number. Negative numbers had been introduced to extend natural numbers to the set of all integers. Some properties of negative numbers had long been remaining unclear, in particular, the order relation between positive and negative numbers. There existed at least two approaches: (1) negative numbers are less than ”nothing” (zero), −1 < 0 (Descartes, Girard, Stifel), (2) negative numbers are ”greater” than infinity, −1 > ∞ (Wallis, Euler, and probably Pascal) . We present theoretical statements of a new mathematical conception underlying the construction of a new theory  based on: (1) a new method for ordering the integers (first introduced, but in other form, in ): let a, b ∈ Z, then a ≺ b ⇔ −1 a < −1 b , thus getting Z = [0, 1, 2, ..., −2, −1]; the set Z can be geometrically represented as cyclically closed; (2) a new class of real regular functions f(·) and the definition of Pb a f(·) that extends the classical definition to the case b < a: let Za,b = [a, b] if a ¹ b and Za,b = Z \(b, a) if a Â b, Z \(b, a) = [a, −1]∪[0, b], then ∀a, b ∈ Z,Pb k=a f(k) = P k∈Za,b f(k); (3) a set of conditions imposed on regular functions. From these we define a new regular method for infinite series summation and find a unified approach to summation of divergent series, and to determination of limits of unbounded and oscillating functions. In this new setting we recently elementarily evaluate the zeta function and the zeta alternating function at integer points [4-5]. We discover various surprising phenomena and unexpected results concerning some areas of mathematics, obtained within the framework of this new theoretical background, which is being futher developed. We also discuss some aspects of future research which will be based on the theory to be formulated as a paradigm.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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