The arithmetic foundations of mathematics: constructing new mathematics with negative numbers beyond infinity

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dc.contributor.authorBagdasaryan, Armen G.
dc.date.accessioned2024-07-12T20:50:25Z
dc.date.available2024-07-12T20:50:25Z
dc.date.issued2009en_US
dc.departmentFakülteler, İnsan ve Toplum Bilimleri Fakültesi, Matematik Bölümüen_US
dc.description.abstractThe 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) [1]. We present theoretical statements of a new mathematical conception underlying the construction of a new theory [2] based on: (1) a new method for ordering the integers (first introduced, but in other form, in [3]): 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  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.en_US
dc.identifier.citationBagdasaryan, A. G. (2009). The arithmetic foundations of mathematics: constructing new mathematics with negative numbers beyond infinity. s. 112.en_US
dc.identifier.endpage113en_US
dc.identifier.isbn9.78605E+12
dc.identifier.startpage112en_US
dc.identifier.urihttps://www.maltepe.edu.tr/Content/Media/CkEditor/03012019014112056-AbstractBookICMS2009Istanbul.pdf#page=331
dc.identifier.urihttps://hdl.handle.net/20.500.12415/2339
dc.institutionauthorBagdasaryan, Armen G.
dc.language.isoenen_US
dc.publisherMaltepe Üniversitesien_US
dc.relation.ispartofInternational Conference of Mathematical Sciencesen_US
dc.relation.publicationcategoryUluslararası Konferans Öğesi - Başka Kurum Yazarıen_US
dc.rightsCC0 1.0 Universal*
dc.rightsinfo:eu-repo/semantics/openAccessen_US
dc.rights.urihttp://creativecommons.org/publicdomain/zero/1.0/*
dc.snmzKY07704
dc.titleThe arithmetic foundations of mathematics: constructing new mathematics with negative numbers beyond infinityen_US
dc.typeConference Object
dspace.entity.typePublication

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