How We Count or is it Possible that Two Times Two is Not Equal to Four

People think that their counting is governed by the rules of the conventional arithmetic and that other kinds of arithmetic do not exist and cannot exist. It is demonstrated in this paper that this popular image of counting is incorrect. In many situations, we have to utilize different rules of counting. This is a result of existence of many different arithmetics. To distinct them, we call the conventional arithmetic Diophantine, while all other arithmetics we call non-Diophantine. Theory of non-Diophantine arithmetics is developed in the book of the author "Non-Diophantine arithmetics."