The Semantic Function of the Axiomatic Method

Evandro Agazzi ()
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Evandro Agazzi: Universities of Genoa, Center for Bioethics of the Panamerican University of Mexico City, Mexico

Chapter Chapter 4 in Axiomatic Thinking I, 2022, pp 43-61 from Springer

Abstract: Abstract Since ancient Greek mathematics the axiomatic method had essentially a deductive function in the sense of granting the truth of the propositions of a certain discipline by starting from primitive propositions that were “true in themselves” and submitting them to a truth-preserving formal manipulation. A new perspective emerged in the foundational research of Peano’s school and Hilbert’s view of mathematical theories. According to this view, the whole axiomatic system provides a sort of global meaning that offers the possibility of understanding he sense of a theory. and also to find possible models for it. We can call this the semantic function of the axiomatic method because it offers the two basic dimensions of semantics, i.e. sense and reference (according to the Fregean perspective). Intellectual intuition is no longer needed for providing the objects of mathematical theories and the requirement of truth is replaced by the requirement of consistency. This approach has immediate consequence of an ontological nature, i.e. regarding the kind of existence of mathematical entities (Is consistency a sufficient condition for existence in mathematics?).

Date: 2022
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DOI: 10.1007/978-3-030-77657-2_4

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