What do we mean by "the foundations of mathematics"?

The phrase “foundations of mathematics” is bandied about frequently these days, but it’s clear that there is widespread confusion about what it means. Some say that a proof assistant must be based on a foundation of mathematics, and therefore that the foundations of mathematics refers to some sort of formal system. And yet, while set theory is frequently regarded as the foundation of mathematics, none of the mainstream proof assistants are based on set theory. These days we see everything from category theory to homotopy type theory described as a possible foundation of mathematics. There is a lot of wrongness here. N. G. de Bruijn made the remarkable claim We do not possess a workable definition of the word “mathematics”. (AUT001, p. 4) He seemed to be referring primarily to the difficulty of defining mathematical reasoning, but the dictionary definition – “the abstract science of number, quantity, and space” – does not begin to scratch the surface of the topics studied by mathematici

What do we mean by "the foundations of mathematics"? Machine Logic At the junction of computation, logic and mathematics What do we mean by "the foundations of mathematics"? 01 Nov 2023 [ philosophy logic type theory Principia Mathematica AUTOMATH ] The phrase “foundations of mathematics” is bandied about frequently these days, but it’s clear that there is widespread confusion about what it means. Some say that a proof assistant must be based on a foundation of mathematics, and therefore that the foundations of mathematics refers to some sort of formal system. And yet, while set theory is freq

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