Mathematics seems certain and absolute, yet its foundations have faced several crises throughout history. Explore the foundational debates in mathematics from the discovery of irrational numbers through set theory paradoxes to Gödel's incompleteness theorems. Analyze the competing foundational programs of formalism (Hilbert), logicism (Frege, Russell), and intuitionism (Brouwer). Discuss how these approaches attempted to resolve mathematical uncertainty and what limitations remain. How does this philosophical uncertainty about mathematics' foundations affect our understanding of mathematical truth and knowledge?