Logicism, formalism, intuitionism, structuralism, and modern perspectives on the foundations of mathematics.
Foundations programs study different ways of explaining the nature, meaning, and justification of mathematics, and they ask what mathematics is about, what counts as a proof, and how mathematical knowledge should be organized.
The chapter begins with logicism, which attempts to reduce mathematics to logic by showing that mathematical concepts and theorems can be derived from purely logical principles.
Formalism is then introduced as the view that mathematics can be studied through formal systems, where symbols, axioms, and rules of inference become the primary objects of analysis.
Intuitionism is studied as a constructive approach to mathematics, where existence claims require explicit constructions and where proof has a central role in determining mathematical truth.
Structuralism is then discussed as the view that mathematics studies structures and positions within structures, rather than isolated objects considered independently of their relations.
The final part of the chapter presents modern perspectives, where foundational questions are studied using tools from logic, set theory, category theory, type theory, computation, and the practice of formalized mathematics.