👉 Formal mathematics is the rigorous, symbolic language used to express mathematical concepts and proofs with precision and clarity. It involves the use of axioms, definitions, theorems, and logical deductions to build a structured framework for mathematical reasoning. At its core, formal mathematics relies on set theory and logic to define mathematical structures, such as numbers, functions, and spaces, using symbols and rules. This framework allows mathematicians to derive theorems through deductive reasoning, ensuring that conclusions are logically and necessarily true given the initial axioms. Formal systems often include formal languages with syntax (rules for forming valid expressions) and semantics (interpretations that assign meaning to these expressions). This approach not only enhances the rigor and consistency of mathematical arguments but also facilitates communication and collaboration across the mathematical community, enabling the verification and discovery of new mathematical truths.
