[New Entry by Benedict Eastaugh on February 2, 2024.]
The question of whether the use of a certain method or axiom is necessary in order to prove a given theorem is widespread in mathematics. Two historical examples are particularly prominent: the parallel postulate in Euclidean geometry, and the axiom of choice in set theory. In both cases, one demonstrates that the axiom is indeed necessary by proving a “reversal” from the theorem to the axiom: that is, by assuming the theorem and deriving the axiom from it, relative to a set of background assumptions known as the base…

Originally appeared on Stanford Encyclopedia of Philosophy Read More