Examine the historical development of mathematical foundations from Euclidean geometry to formal axiomatic systems. Discuss the impact of non-Euclidean geometries, Cantor's set theory, and Russell's paradox on mathematical thinking. Analyze Hilbert's program and Gödel's incompleteness theorems. Consider the three main schools in philosophy of mathematics: logicism, formalism, and intuitionism. How have foundational debates influenced the practice and teaching of mathematics? What implications do these foundational issues have for our understanding of mathematical truth and certainty?