To be fully grasped, mathematical ideas have to be rediscovered or reimagined, much like in the translation of poetry.
Mathematical language is becoming more and more pervasive. This phenomenon ranges from the mundane (imprints on T-shirts or mugs) to the more scientific (its use in reporting or in disciplines outside of mathematics) and even includes art in its span. This begs the question, why and how does it work? Or more poignantly: What is the form and function of mathematical language inside and outside its community of speakers?
In the field of mathematics itself, the situation is not as homogenous as one might think. How much truth is contained in a proof by pictures is quite different in algebra versus geometry, and, historically, there is great variation in what is considered a proof—mainly how stylized the language should be. Being too relaxed can lead to foundational crises and questions like those Helmut Hofer is working out in symplectic geometry. An extreme position, which I call Frege’s dream, is also alive today with Vladimir Voevodsky and his colleagues through their endeavors to formalize language as much as possible to maximize verifiability. Some might argue that Bourbaki represented a golden age for striking a balance between the formal, the communal, and the communicable.READ MORE>