In discussing the existence of mathematical entities, it is convenient to distinguish two points of view. To entertain a certain community of existence between abstract objects such as relations, qualities, numbers, and so on, and real objects such as chairs and apples, is to adhere to what we shall call Platonism. Often the common type of being which Platonists ascribe to all objects, abstract or not, is distinguished from that of actual things: they then speak of “subsistence” (Ansichbestand) (cf. VON FREYTAG-LÖRINGHOFF, 1951), reserving “existence” for actuals, and claim that subsistence is the ontological essence of real things, which possess in addition other characteristics such as spatio-temporality. We will call Aristotelianism the opposite view, that there is no type of being common to objects of different categories: such objects may perhaps exist in related or analogous ways, but there is no kind of existence which can serve as common denominator for all types of objects. These appellations will be useful in bringing out the flavour of some theories to be examined below.
KeywordsMathematical Object Category Theory Categorial Formulation Mathematical Entity Mathematical Construct
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