We introduce and defend Mathematical Criticism as a field of inquiry that should stand to Mathematics as Literary, Artistic, and Musical Criticism stand to literature, art, and music. Against the frequent reduction of Philosophy of Mathematics to linguistic, logical, and set-theoretical questions, mathematical thought is approached through the creative works, techniques, diagrams, styles, and conceptual forces that form its fabric. The mathematical critic must first enter, know, feel, and live the works themselves, and only then describe, calibrate, evaluate, and explain them. The critical task is threefold: to describe the construction of mathematical thought, to assess the entanglement of such constructions with ideal realms of possibilia, and to contrast those entanglements with the real in a Peircean, inductive sense. The contrast with Analytic Philosophy of Mathematics is explored through Galois and Riemann, whose ambiguity theory, uniformization, algebraic structures, and complex variables disclose dimensions inaccessible to any external reduction of mathematical substance. We also recover neglected pioneers — Shaw, Lautman, and Javier de Lorenzo — and propose RTSK models, combining sheaves, topoi, Kripke models, and Riemann-surface ramifications, to preserve the local-global, archetypal, historical, and cultural stratifications of mathematical creativity.