nLab Münchhausen trilemma

Redirected from "Munchhaüsen trilemma".

Context

Philosophy

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

Mathematics

Contents

Idea

The Münchhausen trilemma or the Agrippan trilemma is a thought experiment in epistemology which states that it is theoretically impossible to prove any truth, because one has to resort to one of three unsatisfactory arguments

  • Circular reasoning, in which the proof of a statement relies on said statement itself

  • Infinite regress, in which one needs an infinite sequence of statements, each of which proves the previous statement

  • Foundationalism, in which one selects some statements as unjustified axioms or rules.

 Relevance to the foundations of mathematics

The foundations of mathematics by definition uses foundationalism to resolve the Münchhausen trilemma in mathematics. However, different communities of mathematicians disagree with which unjustified axioms and rules to use for their foundations. For example, classical mathematicians use classical logic, while constructive mathematicians use intuitionistic logic and reject excluded middle.

Subfields of mathematics also use foundationalism by way of synthetic mathematics, in which unjustified axioms and rules are used as a basic foundation for the entire subfield. Examples include synthetic topology, synthetic geometry, synthetic differential geometry, and synthetic (infinity,1)-category theory.

 See also

Created on November 30, 2022 at 19:56:02. See the history of this page for a list of all contributions to it.