Sabinet | Accounting for primitive terms in mathematics : research article
n Koers : Bulletin for Christian Scholarship = Koers : Bulletin vir Christelike Wetenskap - Accounting for primitive terms in mathematics : research article
Volume 70, Issue 3,
p. 515 - 534
Keyword(s) : Arithmeticism, Aritmetisisme, Geheel-dele-verhouding, Geometricism, Geometrisisme, Mutual coherence, Primitiewe / basiese terme, Primitive / basic terms, Uniekheid, Uniqueness, Wederkerige samehang and Whole-parts relation
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Die wysgerige probleem van eenheid en verskeidenheid bevat &apos;n uitdaging vir die rasionalistiese strewe om alles te definieer. Hierdie tipe definisies kom in verskillende akademiese dissiplines na vore in formulerings soos uniekheid, onherleibaarheid, en wat bekend staan as "primitiewe terme". Selfs nie die mees "eksakte" dissiplines, soos die wiskunde, kan daarin slaag om te ontkom aan die implikasies wat opgesluit lê in die rekenskapgewing of verantwoording van sulke primitiewe terme nie. Deur te let op die historiese ontwikkeling van die wiskunde word die feit belig dat alternatiewe perspektiewe deurlopend bestaan het - vanaf die aritmetisisme van die Pytagoreërs, die eventuele geometrisering daarvan ná die ontdekking van "inkommensurabiliteit", tot by die herlewing van die aritmetisisme in the wiskunde van Cauchy, Weierstrass, Dedekind en Cantor (met die latere oriëntasie van Frege teen die einde van sy lewe toe hy die sirkel voltooi het met sy oortuiging dat die wiskunde wesenlik geometrie is). Met verwysing na die sienings van Hilbert, Weyl en Bernays word die artikel afgesluit deur die voorstel dat daar teenoor dieerfenis van aritmetisisme en geometrisering &apos;n alternatiewe opsie ontgin behoort te word - een waarin sowel die uniekheid as die onverbreeklike wederkerige samehang tussen die getalsaspek en die ruimte-aspek erken word.
The philosophical problem of unity and diversity entails a challenge to the rationalist aim to define everything. Definitions of this kind surface in various academic disciplines in formulations like uniqueness, irreducibility, and what has acquired the designation "primitive terms". Not even the most "exact" disciplines, such as mathematics, can avoid the implications entailed in giving an account of such primitive terms. A mere look at the historical development of mathematics highlights the fact that alternative perspectives prevailed - from the arithmeticism of Pythagoreanism, the eventual geometrisation of mathematics after the discovery of incommensurability up to the revival of arithmeticism in the mathematics of Cauchy, Weierstrass, Dedekind and Cantor (with the later orientation of Frege, who completed the circle by returning to the view that mathematics essentially is geometry). An assessment of logicism and axiomatic formalism is followed by looking at the primitive meaning of wholeness (and the whole-parts relation). With reference to the views of Hilbert, Weyl and Bernays the article concludes by suggesting that in opposition to arithmeticism and geometricism an alternative option ought to be pursued - one in which both the uniqueness and mutual coherence between the aspects of number and space are acknowledged.
Persistent Link : http://hdl.handle.net/10520/EJC59202