  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.
