1887

n Koers : Bulletin for Christian Scholarship = Koers : Bulletin vir Christelike Wetenskap - Systematic perspectives on diverging mathematical orientations : research article

Volume 70, Issue 4
  • ISSN : 0023-270X
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Abstract


Die populêre opvatting dat die wiskunde "objektief" en "neutraal" is in die sin dat dit geen standpuntverskille impliseer nie, word weerspreek deur die feitelike stand van sake in die moderne wiskunde. In die lig van die dominante eensydige strominge in die geskiedenis van die wiskunde wat gefluktueer het tussen die aritmetisisme en 'n geometrisering van hierdie dissipline, belig hierdie artikel enkele aanknopingspunte vir 'n derde opsie. Hierdie derde opsie word aangepak deur sommige eienskappe van die erkenning van die uniekheid van getal en ruimte te ondersoek, sonder om die intermodale skakels tussen hierdie twee aspekte te verwaarloos. 'n Beredeneringslyn word ontwikkel ten opsigte van die onvermydelike gebruik van analogiese (of elementêre) grondbegrippe en hierdie perspektief word dan geartikuleer in terme van die teorie van modale aspekte. Getals- en ruimteterme word bespreek en uiteindelik toegespits op 'n verdiepte verstaan van die sin van die oneindige. Bykomend tot 'n oorsigtelike weergawe van die redes waarom 'n onhoudbare sirkelredenasie opgesluit lê in die vermeende aritmetisering van die wiskunde (Grünbaum), word ook aandag gevra vir die ooreenkoms en verskil tussen Aristoteles and Cantor aangaande die aard van kontinuïteit - beoordeel in terme van die onherleidbaarheid van die getals- en ruimteaspekte van die werklikheid. Ten slotte word die intuïsionistiese wiskunde en die aksiomatiese versamelingsleer getipeer aan die hand van hul onderskeie ontologiese vooronderstellings.

The popular view that mathematics is "objective" and "neutral" in the sense that it does not know different standpoints is contradicted by the factual state of modern mathematics. In the light of the dominant one-sided trends in the history of mathematics, fluctuating between arithmeticism and a geometrisation of this discipline, this article explores some provisional starting points for a different view. This third option is explored by investigating some features of an acknowledgement of the uniqueness of number and space without neglecting the interaspectual connections between these two modal functions. An argument is advanced regarding the inevitability of employing analogical (or elementary) basic concepts, and this perspective is articulated in terms of the theory of modal aspects. Numerical and spatial terms are discussed and eventually focused on a deepened understanding of the meaning of infinity. In addition to a brief look at the circularity present in the arithmeticist claim that mathematics could be fully arithmetised (Grünbaum), attention is also asked for the agreement between Aristotle and Cantor regarding the nature of continuity - assessed in terms of the irreducibility of the numerical and spatial aspects of reality. Finally a characterisation is given of the ontological assumptions of intuitionism and axiomatic formalism.

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/content/koers/70/4/EJC59230
2005-01-01
2017-01-21

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