Hyperoperations Wiki
The Hyperoperations Wiki is intended to complement the Tetrationforum. By creating a cartographic tool for the exploration done at the Tetrationforum, keeping track of of definitions, terms, ideas, and bibliographic references; and by formally re-expressing all modern mathematical concepts from a preparatory viewpoint that is compatible with hyperoperative inquiry.
Idea
The Hyperoperations Wiki serves as an online collaborative encyclopedia with a dual purpose. On one hand, it aims to map the definitions and relationships among the mathematical structures that emerge from and are connected to the abstract concept of hyperoperations, thereby situating them within the landscape of modern and contemporary mathematics. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a hyperoperational perspective (hPOV).
The hPOV and the concept of iteration
Just like the concept of a ring, the concept of hyperoperation serves as a potential avenue for exploring the conceptual nature of arithmetical operations through generalization and abstraction. The approach taken by the hPOV lies on the assumption that some of the relationships between addition and multiplication have a hierarchical nature: some of them are suggestive of an unidirectional increase in complexity and reveal pathways to explore further in those directions, above and beyond, i.e. ὑπέρ (hyper), sums and products. The path suggested is the one of repetition, or doing again, once more, i.e. iterum (iteration) that step that got us from addition to multiplication.
There is a circular relation between the hPOV and the iterative point of view a kind of duality.
- At first sight it may seem that iteration supplies the ambient theory where to investigate hyperoperations. Hyperoperations seems to be reduced to an iteration of the link between addition and multiplication that goes beyond them.
- Yet also the opposite is true. Iteration emerges naturally just as one of the many possible ways to understand the connection between addition and multiplication.
This reveals the possibility of an intersection of the two points of view by definition of a deeper and fundamental concept, an intersection where the iteration of iteration itself happens.
Useful pages
Special:AllPages brings you to a list of all so far described terms.