Discuss three major theoretical approaches to the foundations of mathematics: logicism, formalism, and intuitionism. For each approach, explain its fundamental claims about the nature of mathematical objects, the basis of mathematical truth, and the justification of mathematical knowledge. Analyze how each approach addresses foundational challenges, including Russell's paradox, Gödel's incompleteness theorems, and the status of mathematical infinity. Evaluate the implications of these foundational debates for mathematical practice, education, and philosophy.