Esta monografía ofrece una revisión enteramente novedosa del campo usualmente denominando «filosofía (de la) matemática». Allende análisis y síntesis, se introducen nuevos modelos (RTHK) para captar toda la complejidad de la matemática, entendida como forma de pensamiento general, donde se incorpora de manera potente un estrato de ideas, imágenes y métodos que entra en diálogo con otro estrato de técnicas, definiciones y pruebas. Mediante modelos de Kripke (K), se vislumbran consideraciones

históricas que permiten manejar simultáneamente perspectivas externalistas e internalistas en la historia de la ciencia. Con los haces (H), se estudian interpretaciones

fenomenológica sobre la variabilidad y la permanencia local de los fenómenos matemáticos. Usando topos (T), se plantean investigaciones

metafísicas sobre la existencia de arquetipos matemáticos que emergen de diversos tipos subyacentes y que, a su vez, se proyectan sobre ellos. Finalmente, gracias a las superficies de Riemann (R), se exploran diversas ramificaciones

culturales del modelo (THK) hacia la literatura, el arte, la música, el cine. El resultado plantea el inicio de una plena

crítica matemática , que debería empezar a surgir paralelamente a la crítica literaria, la crítica de arte, la crítica musical o la crítica cinematográfica. Considerando las obras matemáticas como grandes formas de expresión creativa, un

back-and-forth entre lo concreto y lo abstracto, entre lo particular y lo universal, entre el detalle técnico y el fondo filosófico, recorre multitud de ejemplos de las matemáticas avanzadas, de Galois a Grothendieck, que se reflejan y se entrelazan con muy diversas manifestaciones culturales. La «filosofía (de la) matemática» tiende así a abrirse hacia una «crítica (de la) matemática», que parece ser mucho más afín para captar el hacer propio y específico de la disciplina.

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