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