Mathematical Philosophy

This work examines whether mathematics can bridge the abstract ideas and the real world. Mathematical formalism cannot fully explain everything on its own. Mathematics depends on basic ideas like identity and difference, coherence, and comparability, which must already exist before any formal system can be built. These ideas are not derived from mathematics; rather, they provide the foundation for mathematical thinking. The main task of mathematical philosophy is to ask what must be in place for formal systems to function, rather than merely to reflect on mathematics or to formalize philosophy. Mathematics does not create meaning by itself. It helps organize and stabilize meaning, but only after the basic groundwork is set. Although mathematics has made significant progress, these achievements do not mean that its core ideas are always secure. Scientific theories, even if they seem strong and well-structured, can still have shaky foundations and unclear links to reality.