The Philosophers and Mathematics pp 201-248 | Cite as

# The Axiom of Choice as Interaction Brief Remarks on the Principle of Dependent Choices in a Dialogical Setting

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## Abstract

The work of Roshdi Rashed has set a landmark in many senses, but perhaps the most striking one is his inexhaustible thrive to open new paths for the study of conceptual links between science and philosophy deeply rooted in the interaction of historic with systematic perspectives. In the present talk I will focus on how a framework that has its source in philosophy of logic, interacts with some new results on the foundations of mathematics. More precisely, the main objective of my brief remarks is to discuss some claims of the late Hintikka (1996, 2001) who brought forward the idea that a game-theoretical interpretation of the Axiom of Choice yields its meaning “evident”. More precisely I will show that if we develop Per Martin-Löf’s (1984) demonstration of the axiom within a dialogical setting, the claim of Hintikka can be upheld. However, the dialogical demonstration, shows that, contrary to the expectations of Hintikka, the meaning that the game-theoretical setting provides to the Axiom is compatible with constructivist rather than with classical tenets.

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