the principle of relativity: from ungar’s gyrolanguage for physics to weaving computation in mathematics

this paper extends the scope of algebraic computation based on a non standard × to the more basic case of a non standard +, where standard means associative and commutative. two physically meaningful examples of a non standard + are provided by the observation of motion in special relativity, from either outside (3d) or inside (2d or more), we revisit the “gyro”-theory of ungar to present the multifaceted information processing which is created by a metric cloth w, a relating computational construct framed in a normed vector space v , and based on a non standard addition denoted ◦+ whose commutativity and associativity are ruled (woven) by a relator, that is a map which assigns to each pair of admissible vectors in v an automorphism in aut w. special attention is given to the case where the relator is directional.