On the depth of Wiles's proof of Fermat's Last Theorem

What constitutes depth in a mathematical proof? This paper addresses this foundational question through a close case study of Andrew Wiles's proof of Fermat's Last Theorem. We propose a five-criteria framework: Difficulty, Originality, Fruitfulness, Unity, and Explanatory Power. By critically examining the limitations of each criterion and the intricate interrelations among them, we show that they do not merely coexist but capture genuinely complementary aspects of proof depth. Applying this framework to Wiles's proof, we then demonstrate that its celebrated depth does not reside in any single virtue, but rather emerges from a synthesis of all five criteria. Our analysis thus offers not only a novel lens for understanding Wiles's achievement, but also a flexible framework for evaluating depth across mathematical proofs more broadly.