(Co)Arithmetic: Can numbers help with the theory of quanta

We start with the arithmetic of natural numbers (eventually completed to form the integers) and introduce the concept of coaddition, comultiplication, counit, antipode, etc. of arithmetic. A careful study of graded duality unveils relations which are fruitfully used on index sets in the theory of symmetric functions, quantum mechanics, and quantum field theory (QFT). The amazing occurance of number theoretic functions in QFT becomes natural. A pathway is opened to transport number theoretic methods into QFT and vice versa. We give arguments, that renormalization has to do with multiplicativity versus complete multiplicativity of (arithmetic number theoretical) functions, and that hard number theoretic problems and hard QFT problems emerge from the same structural source. Our main conjecture is, that the arithmetic structure developed is functorial and can be established in a large class of categories.