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oa Litnet Akademies : 'n Joernaal vir die Geesteswetenskappe, Natuurwetenskappe, Regte en Godsdienswetenskappe - Wysgerige perspektiewe op die uniekheid van getal

 

Abstract

Die gegewe dat die "natuurlike" getalle op mekaar volg, dui op die potensiaal om sonder einde voortgesit te word - wat vanself sowel die oneindige as die aritmetiese tydsorde van opeenvolging ten tonele voer. Hierdie fasette is verder ontgin in die transfiniete aritmetika van Cantor. Die resultaat hiervan was onder meer dat die wiskunde as versamelingsleer aangedui is - waarmee die aard van alle vroeëre wiskunde bevraagteken word. Die status van getal word bespreek in die lig van die beperkinge van logiese optelling en die idee van 'n veelheid "eenhede", alvorens oorgegaan word tot 'n ontleding van die aard van die getalsaspek aan die hand van gesigspunte soos getal se funksionele (modus-) aard, die modale universaliteit daarvan, die aard van modale abstrahering, die vraag of getalle God- of mensgemaak is, die gegewe dat "bestaan" nie met "sintuiglike waarneembaarheid" saamval nie, en die vraag of getal gedefinieer kan word. Die analise betrek die klassieke probleem van uniekheid en samehang, sowel as die funderende posisie van getal ten opsigte van ruimte en verskillende struktuurelemente van die ruimte wat implisiet deur getal by ruimte "geleen" word. In hierdie konteks word kernagtig aandag geskenk aan die nuwe lig wat die onderskeiding (en samehang) tussen suksessie en gelyktydigheid op die aard van oneindigheid werp. Dit word veral gemanifesteer in die wiskundige gebruik van oneindige totaliteite (as regulatiewe hipotese) en binne die perspektief van 'n omkering van die ontsluitingsorde tussen getal en ruimte. Wanneer dít wat deur getal by die ruimte "geleen" word nie as inherent aritmetiese eienskappe waardeer word nie, word die sin-kern van getal "bevry" om opnuut en ongestoord kwalifiserend op te tree ten opsigte van alle getalversamelings - hetsy aangedui as "diskreet", "dig" of "kontinu". Binne die sfeer van getal, soos Laugwitz tereg beklemtoon, bly elke (natuurlike, heel-, rasionale en reële) getal onder die "heerskappy" van die diskrete uniek.


"Natural" numbers succeed each other and have the potential to be extended indefinitely, that is to say "without an end". Both the infinite and the arithmetical time order of succession are features found in the transfinite arithmetic of Cantor. However, characterising mathematics as a set theory questions the nature of all earlier mathematics. The status of number is discussed in die light of the limitations of logical addition and the idea of a multiplicity of "units". Before an analysis of the numerical aspect is done, the following points of view are explored, namely that it is a function of reality that displays modal universality, that it can be (theoretically) accessed by means of modal abstraction, that "existence" does not coincide with what "can be observed by the senses", and that it cannot be defined. After discussing the question about who made numbers, the classical problem concerning uniqueness and coherence is addressed, followed by what number "borrowed" from space. In this context a succinct account is given of the new light that the distinctness and coherence between succession and simultaneity shed on the nature of infinity, particularly manifested in the mathematical use of the idea of infinite totalities (as a regulative hypothesis) and within the perspective of a reversal of the order of disclosure between number and space. When what is "borrowed" from space by number is not seen as intrinsic numerical properties, the meaning nucleus of number is "liberated" once more, without any impediment, to serve in a qualifying capacity in respect of all sets of numbers - whether designated as "discrete", "dense" or "continuous". Within the sphere of number (including natural numbers, integers, rational numbers and real numbers) every number is unique in subjection to the "rule" of the discrete, as emphasised by Laugwitz.

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/content/litnet/8/1/EJC62290
2011-01-01
2016-12-04
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