Where does foundation collide?

In the theory of knowledge, foundationalism is the view that some beliefs are basic, justified on their own without resting on any further belief, and that every other justified belief is supported, directly or through a chain, by this base. René Descartes set out the picture in his 1641 Meditations on First Philosophy, where he cleared away everything doubtful to reach a bedrock of certainty on which the rest of knowledge could be rebuilt. Ernest Sosa's 1980 The Raft and the Pyramid restated the position and set it against coherentism, likening foundational knowledge to a pyramid standing on its base. Here the foundation is a class of justified beliefs, not a substructure that carries a building's weight into soil, and not the axioms and rules that ground a branch of mathematics.[1][2]

Unit

A basic belief, one justified without depending on any other belief

Recovery means

Not applicable

Measured by

Argued for by philosophical analysis of what makes a belief basic and how justification flows from it to the rest

The foundations of mathematics are the systems of axioms, definitions, and rules of inference from which the results of a branch of mathematics are meant to follow with certainty. Gottlob Frege's 1884 The Foundations of Arithmetic opened the modern program by trying to derive arithmetic from logic alone. David Hilbert's 1926 On the Infinite pressed the further goal of showing that such a system is free of contradiction using only finite, checkable steps. Here a foundation is a formal starting point for proof, not a bedrock belief in a theory of knowledge, and not the physical element that carries a building's weight into the ground.[3][4]

Unit

A system of axioms with its rules of inference, judged by what it can derive

Recovery means

Not applicable

Measured by

Studied by proving results within a system and by metamathematical work on its consistency and completeness