Intuition and rigor are two inevitable components in mathematical problem-solving. Traditionally, features of education in mathematics show that those two themes are seen to be mutually exclusive rather than complementary. The famous foundational debate between L.E.J Brouwer and David Hilbert shows that intuition and rigor are often seen as being at odds. The two characteristics of mathematics play a pivotal and dynamic role in education, reasoning problem-solving, and different applications of maths. This essay delves into the intricate relationship between these seemingly contrasting approaches, underscoring their significance in the evolution of mathematical development. The research employs a blend of historical analysis and philosophical inquiry to explore the roles of intuition and rigor and how they interact. The essay argues that while intuition often sparks the initial insights in mathematical discovery, it is the rigor that ensures the robustness and validity of these insights, leading to their acceptance within the mathematical community. Through case studies of renowned mathematicians and well-known problems, this research uncovers the symbiotic nature of intuition and rigor and is demonstrating how they collectively contribute to the richness of mathematical problem-solving. Those findings suggest that a balanced integration of both features is crucial for the integrity including consistency and completeness of mathematical knowledge.
