Examine competing theories about the nature of mathematics including Platonism, formalism, intuitionism, and fictionalism. Analyze Hilbert's program and Gödel's incompleteness theorems. Discuss the implications of set-theoretic independence results such as the continuum hypothesis. Consider whether mathematics is discovered or invented, and what this means for mathematical truth. Explore the relationship between mathematical formalism and mathematical practice, including the role of intuition and visualization in mathematical reasoning.