Waaldijk F. A. (2005) On the foundations of constructive mathematics: Especially in relation to the theory of continuous functions. Foundations of Science 10(3): 249–324.

Waaldijk F. A.
(

2005)

On the foundations of constructive mathematics: Especially in relation to the theory of continuous functions.
Foundations of Science 10(3): 249–324.
This article describes many foundational issues concerning what is known as constructivism in mathematics. First of all a flaw in the foundations of Bishop-style constructive mathematics, BISH, is discussed. A main theorem shows that the two current BISH definitions of “continuous function” are not equivalent within BISH, and that – together with the natural properties of “continuous function” – they imply the FT (fan theorem) axiom. The theorem sparks an investigation into the realm of topology and the axioms underpinning intuitionism (INT), classical mathematics (CLASS), recursive mathematics (RUSS) and BISH. Some new elegant axioms are introduced to prove theorems showing that CLASS and INT are closer than usually believed (“reuniting the antipodes”). The distance to RUSS is greater, due perhaps to a philosophical difference regarding “real world” phenomena. There is a connection with the old philosophical debate on determinism and perhaps with the debate in modern physics as well. The real-world experiment described in section 7 could cast an alternative mathematical light on this matter. Relevance: The article is entirely concerned with the foundations of constructive mathematics.

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