? Goue draad wat deur die geskiedenis van die Wiskunde loop

Abstract

Die aanvanklike sukses wat opgesluit gelê het in the Pythagoreïese oor tuiging dat rasionale verhoudinge alles getalsmatig toeganklik maak, het spoedig gegewens teëgekom wat nie met behulp van rasionale getalsverhoudinge hanteer kon word nie. Die onvermoë om hierdie probleem getalsmatig tot ? oplossing te bring sou daartoe lei dat die Griekse wiskunde die weg van ? geometrisering bewandel het. Daar is nie probeer om hierdie impasse met behulp van die idee van oneindige totaliteite te bowe te kom nie. Nie alleen het dit die aard van getal ondergeskik aan die ruimte gemaak nie, want dit het ook die Middeleeuse spekulatiewe “metafisika van die syn” tot gevolg gehad. Eers via Descartes sou die moderne tyd ? toenemende terugkeer na ? aritmetiese perspektief vergestalt. Die ontdekking van die differensiaal- en integraalrekene deur Newton en Leibniz het egter opnuut probleme geskep, met name in die verantwoording van die aard van limiete. Die geykte benadering om limiete met behulp van konvergerende rye rasionale getalle te bepaal, het ? sirkelredenasie bevat, want om as limiet te kan optree moes elke sodanige limiet rééds ? getal gewees het. Skynbaar het Weierstrass daarin geslaag om in sy statiese teorie ? suksesvolle alternatiewe waardering van ? veranderlike te gee. Limiet-waardes kan willekeurig gekies word uit ? domein van onderskeie getalle wat as ? oneindige totaliteit opeens staties voorhande is. Toe hierdie be nadering egter in die versamelingsteorie van Cantor tot Russell se kontradiksie lei, het twee teëgestelde reaksies die toneel oorheers. Die eerste was ? aksiomatisering van die versamelingsleer en die tweede was ? terugkeer na die oorspronklike Griekse sentimente waar die oneindige bloot as ? onvoltooibare suksessie gesien is. Benewens aksiomatisering en ? terugval op die potensieeloneindige het die kentering by Frege – tot die siening dat die wiskunde wesentlik meetkunde is – en by die Franse kontinuumwiskundiges, aan ons getoon dat die Griekse geometrisering alternatief selfs in die 20ste en 21ste eeu nog wiskundige navolging sou vind. In die algemeen word egter die oortuiging nog steeds gevind dat die versamelingsleer die basis van die hele wiskunde vorm. Die goue draad wat derhalwe klaarblyklik deur die ganse geskiedenisvan die wiskunde loop, is enersyds in die komplekse aard van die oneindige gegee (hetsy in die sin van die potensieël-oneindige of in die sin van oneindige totaliteite of die aktueel-oneindige) en andersyds is dit in die kader van die immer-teenwoordige metgeselle van die oneindige te vinde, naamlik die “aritmetiese” en die “geometriese”. Bernays verkies om hier van diskreetheid en kontinuïteit te praat.

The initial success of the Pythagorean conviction that rational ratios would provide access to everything, soon had to face spatial relation ships exceeding the grip of rational numbers. This led to the geometrization of Greek mathematics. It was not attempted to resolve this impasse by using the idea of infinite totalities. Instead, num ber was subjected to space and in addition this geometrization provided a starting-point for the entire medieval legacy of a metaphysics of space (of “being”). Descartes started a new development heading towards a new process of arithmetization. However, the dis covery of the differential and integral calculus by Newton and Leibniz created new problems, specifically in respect of the account given of limits. The standard approach was to obtain limits by means of converging sequences of rational numbers. Yet this procedure contains a vicious circle, because in order to serve as such a limit it already had to be a number. Apparently Weierstrass successfully advanced an alternative assessment of a variable in terms of a static theory. Limit values can be chosen arbitrarily from distinct numbers which are at once present in an infinite totality. When this approach, in the set theory of Cantor, generated Russell’s contradiction, two opposing reactions dominated the scene. The first pursued an axiomatization of set theory and the second reverted to the original Greek sentiment of viewing infinity merely as an unfinished succession. In addition to axiomatization and acknowledging only the potential infinite, the later development in the thought of Frege, as well as the continuum mathematicians from France, demonstrates that the Greek alternative of geometrization still had adherents in the 20th and 21st centuries. In general the conviction still is that set theory forms the foundation of mathematics. Therefore, the golden thread apparently running through the entire history of mathematics on the one hand is given in the complex nature of the infinite (either in the sense of the potential infinite or as an infinite totality) or, on the other, it is found in the context of two ever-present companions, the “arithmetical” and the “geometrical”. Bernays prefers to speak of discreteness and continuity.