Some Aspects of Understanding Mathematical Reality: Existence, Platonism, Discovery

Axiomathes 25 (3):313-333 (2015)
The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving aside its application, are completely virtual and yet clearly organized. This organization can be recognized in the creation of axioms or in the arrival at new theorems and definitions. It results either from a formalization of an intuitive idea or from a combinatorics that has not been guided by intuition. In all four possible cases—therefore, also in the two in which there is no intuitive “lead”, we can plausibly talk about a discovery of mathematical facts and thus support the Platonist view
Keywords Reality  Existence  Discovery  Proof  Platonism
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DOI 10.1007/s10516-014-9253-8
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Realism in Mathematics.Penelope Maddy - 1990 - Oxford University Prress.
What is Mathematics, Really?Reuben Hersh - 1997 - Oxford University Press.

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