how is mathematics done

Mathematics is done through a process of understanding and solving problems related to patterns, structures, and abstract objects by using logic, reasoning, and creativity. Mathematicians study families of objects or problems, learn techniques (often from previous work or papers), and apply these techniques through logical arguments, computations, or even computer programs. Problem-solving in mathematics often involves identifying the components or operations required, then exploring those systematically with curiosity and trial and error. Rigorous proofs are central, ensuring conclusions follow strictly from axioms and previously established results without reliance on intuition or empirical evidence. This approach allows mathematics to model the world, develop theories, and discover truths independent of scientific experimentation.