Pure mathematics focuses on the study of mathematical concepts for their inherent intellectual challenge and aesthetic beauty, rather than for direct real-world applications. This approach has roots in ancient Greece, where figures like Plato and Euclid advocated for "mathematics for its own sake." The concept was significantly elaborated around 1900, driven by the emergence of counter-intuitive theories such as non-Euclidean geometries and Cantor's theory of infinite sets, which necessitated a renewed emphasis on logical rigor and axiomatic methods.
Despite this focus, many purely mathematical theories have unexpectedly found crucial practical applications. For instance, Isaac Newton utilized ancient Greek studies of conic sections to describe planetary orbits, and the problem of factoring large integers, once an abstract concept, now forms the basis of the modern RSA cryptosystem for secure internet communications. Therefore, the distinction between pure and applied mathematics today is often seen more as a philosophical viewpoint or a mathematician's preference, rather than a rigid separation.