Mathematical Platonism: from objects to patterns

The aim of this paper is to reveal the tacit assumptions of the logicist and structuralist theories on the nature of cardinal numbers. The privileged background theory is the representational theory of language with its main conjecture about necessary correspondence between metaphysical structure of reality and logical structure of language. In such a theory of language we meet a number of thesis on relations between syntax and semantics. Special attention is paid to the so called ‘syntactic priority thesis’: if an expression has the role of a singular term in a true sentence then there must be an object denoted by that term. On these assumptions logicisit develops the view that numerals refer to numbers and that numbers must be a kind of abstract objects. That position we call ‘Platonism of objects’. The inherent weakness of ‘Platonism of objects’ gave rise to the structuralist theory of cardinal numbers. Although the structuralist position in philosophy of mathematics is more congenial to the nature of mathematics as a deductive science, its proponents were reluctant to abandon assumptions on the relations between syntax and semantics as conceived in the representational theory of language. After examining some philosophical implications of measurement theory and intensional logic, we argue for a non-representational theory of language. In such a theory, logical and mathematical structures are not conceived as common form of reality and language, but rather as structures describing a multitude of ‘semantical spaces’ inherent in the language.