Introduction to Projective Arithmetics

Science and mathematics help people to better understand world, eliminating many inconsistencies, fallacies and misconceptions. One of such misconceptions is related to arithmetic of natural numbers, which is extremely important both for science and everyday life. People think their counting is governed by the rules of the conventional arithmetic and thus other kinds of arithmetics of natural numbers do not exist and cannot exist. However, this popular image of the situation with the natural numbers is wrong. In many situations, people have to utilize and do implicitly utilize rules of counting and operating different from rules and operations in the conventional arithmetic. This is a consequence of the existing diversity in nature and society. To correctly represent this diversity, people have to explicitly employ different arithmetics. To make a distinction, we call the conventional arithmetic by the name Diophantine arithmetic, while other arithmetics are called non-Diophantine. There are two big families of non-Diophantine arithmetics: projective arithmetics and dual arithmetics (Burgin, 1997). In this work, we give an exposition of projective arithmetics, presenting their properties and considering also a more general mathematical structure called a projective prearithmetic. The Diophantine arithmetic is a member of this parametric family: its parameter is equal to the identity function f(x) = x. In conclusion, it is demonstrated how non-Diophantine arithmetics may be utilized beyond mathematics and how they allow one to eliminate inconsistencies and contradictions encountered by other researchers.