One Mathematic(s) or Many? Foundations of Mathematics in Twentieth-Century Mathematical Practice

Andrei Rodin ()
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Andrei Rodin: Russian Academy of Sciences, Institute of Philosophy

A chapter in Handbook of the History and Philosophy of Mathematical Practice, 2024, pp 2339-2364 from Springer

Abstract: Abstract The received Hilbert-style axiomatic foundations of mathematics has been designed by Hilbert and his followers as a tool for metatheoretical research. Foundations of mathematics of this type fail to satisfactory perform more basic and more practically oriented functions of theoretical foundations such as verification of mathematical constructions and proofs. Using alternative foundations of mathematics such as the univalent foundations is compatible with using the received set-theoretic foundations for metamathematical purposes provided the two foundations are mutually interpretable. Changes in foundations of mathematics do not, generally, disqualify mathematical theories based on older foundations but allow for reconstruction of these theories on new foundations. Mathematics is one but its foundations are many.

Keywords: Metamathematics; Foundations of mathematics; Univalent foundations (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/978-3-031-40846-5_28

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