Open AccessSystematic perspectives on diverging mathematical orientations
Author(s) -
Danie Strauss
Publication year - 2005
Publication title -
koers
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.166
H-Index - 5
eISSN - 2304-8557
pISSN - 0023-270X
DOI - 10.4102/koers.v70i4.286
Subject(s) - intuitionism , epistemology , mathematics , uniqueness , irreducibility , acknowledgement , modal , closeness , mathematical proof , pure mathematics , computer science , philosophy , geometry , mathematical analysis , chemistry , computer security , polymer chemistry
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 mathe-matics, 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 inter-aspectual 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 assumpt-ions of intuitionism and axiomatic formalism