The proof, known to be so hard that a mathematician once offered 10 martinis to whoever could figure it out, connects quantum mechanics to infinitely intricate mathematical structures. Hofstadter was ...

Forbes contributors publish independent expert analyses and insights. The Universe is out there, waiting for you to discover it. You've heard of our greatest scientific theories: the theory of ...

Modal logic, an extension of classical logic, investigates the modes of truth such as necessity and possibility. Its development has been closely intertwined with advances in proof theory, a field ...

All of modern mathematics is built on the foundation of set theory, the study of how to organize abstract collections of objects. But in general, research mathematicians don’t need to think about it ...