Mathematics in Eternity
This chapter addresses ontological questions as formulated in ancient Greece. Mathematics is about something, but it is unclear what kind of objects mathematics is dealing with. The chapter examines different suggestions. According to Plato, mathematics is about immutable entities that constitute a world of ideas. This world is real—although not palpable to our senses. We cannot sense mathematical objects, but we can grasp them by means of our rationality. This rationality, then, is a unique system of perception through which we reach beyond the capabilities of our senses and “see” the objects in the world of ideas, including the real mathematical objects.
The location of mathematical objects in a world of ideas is not merely an expression of Plato’s personal way of thinking. Platonism in the philosophy of mathematics reoccurs time and time again. For example, great mathematicians and logicians such as Frege and Gödel voiced Platonist notions. Thus the chapter addresses Platonism after Plato, Platonism before Plato, as well as Plato’s Platonism. Furthermore, the chapter examines the idea of axiomatisation, and how this structures Euclid’s Elements. This work got a paradigmatic significance, not only with respect to the formulation of mathematical knowledge, but with respect to knowledge in general.
KeywordsAxiomatisation Euclid’s Elements Ontology Plato’s Platonism Platonism after Plato Platonism before Plato
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