Science and mathematics are functioning in a sequence of transformations. Sometimes these transformations change the whole foundation of human knowledge. Thus, the Euclidean geometry ^ was believed for 2200 years to be unique (in the sense as an absolute truth as well as a necessary mode of human perception). But almost unexpectedly people began to understand that it is not unique: three great mathematicians of the XIX century (C.F.Gauss, N.I.Lobatchewsky, and Ja.Bolyai ~) discovered a lot of other geometries. It changed to a great extent understanding of mathematics and improved comprehension of the whole world.

In the XX century a similar situation existed in arithmetic. For thousands of years (much longer than for the Euclidean geometry) only one arithmetic existed. It may be called the Diophantine arithmetic because ancient Greek mathematician Diophantus # was the first to make an essential contribution to arithmetic. In a same way as it was with the Euclidean geometry, the Diophantine arithmetic has been unique and nonchallengable - other arithmetics were unknown to people. Its position in human society has been (and is now) even more stable and firm than the position of the Euclidean geometry before the discovery of the non-Euclidean geometries. Really, all people use the Diophantine arithmetic for counting, while Euclidean geometry is only studied at school while in real life it is used by rather few specialists. At the same time, it is arithmetic (and not geometry) which is considered as a base for the whole mathematics in the intuitionistic approach. Prominent mathematician Leopold Kronecker #(1825-1891) wrote: "God made the integers, all the rest is the work of man".

By H.J.S.Smith #, arithmetics (and namely, the Diophantine arithmetic) is one of the oldest branches, perhaps the very oldest branch, of human knowledge. His older contemporary C.O.Jacobi (1805-1851) said: " God ever arithmetizes".

But in spite of such a high estimation of the Diophantine arithmetic, its uniqueness and indisputable authority has been recently challenged. A family of non-Diophantine arithmetics was discovered by the author. Like geometries of Lobatchewski, these arithmetics depend on a special parameter, although this parameter is not a numerical but a functional one. The Diophantine arithmetic is a member of such a family: its parameter is equal to the function f(x)=x .

In some non-Diophantine arithmetics even the most evident truth (like 2 + 2 = 4) may be discarded. Some of them possess similar properties to those of transfinite numbers arithmetics.

An interesting peculiarity of the new arithmetics is their ability to provide means for mathematical grounding of some intuitive constructions used by physicists. As an example, we can take such relations as "much bigger" (denoted by >> ) and "much lesser" (denoted by << ) which are formalized in non-Diophantine arithmetics.

Many scientists (especially, mathematicians) draw the attention of the scientific community to the foundational problems of natural numbers and the ordinary arithmetic. The most extreme view is that there is only a finite quantity of natural numbers. It is one the central postulates of ultra-intuitionism (Yesenin-Volpin, 1960; van Danzig #, 1956). Other authors are more moderate. They write that not all natural numbers are similar in contrast to the presupposition of the ordinary arithmetic (Kolmogorov ^, 1961; Littlewood #, 1953; Birkhoff # and Barti, 1970; Rashevski, 1973). Different types of natural numbers have been introduced, but only inside the ordinary arithmetic.

Non-Diophantine arithmetics giving a mathematical base for these ideas were constructed only in 1975 and the corresponding results were partially published later (Burgin, 1977; 1980).

It is possible to imagine that non-Diophantine arithmetics are absolutely formal constructions, which are very far from the real world. But let us recollect that a similar skepticism and mistrust met the discovery of non-Euclidean geometries. Even Gauss (in spite of being acknowledged as the greatest mathematician of his time) did not dare to publish his results concerning these geometries because he was not able to find anything that is similar to them in nature. Lobatchewski called his geometry an imaginable one. But afterwards it was discovered that the real physical space fits non-Euclidean geometries, and that the Euclidean geometries do not have such essential applications as the non-Euclidean ones. But in spite of the short time, which has passed after the discovery, real phenomena and processes exist that fit the non-Diophantine arithmetics.

As a matter of fact, in (Littlewood, 1953) an example is considered demonstrating how the rules of non-Diophantine arithmetics (in spite of that they were unknown at that time) can be imposed upon the real world. Several similar examples are exposed in (Davis and Hersh, 1986). Thus, a market sells a can of tuna fish for $1.05 and two cans for $2.00. So, we have a + a ¹ 2a. To make the situation, when ordinary addition is inappropriate, more explicit, an absurd but not unrelated question is formulated: If the Mona Liza painting is valued at $10,000,000, what would be the value of two Mona Liza paintings. Another example: when a cup of milk is added to a cup of popcorn then only one cup of mixture will result because the cup of popcorn will very nearly absorb a whole cup of milk without spillage. So, in the last case we have 1+1=1 .

Moreover, in (Rotman, 1997) other examples are given and the problem of elaboration of new (or, as they are called there, non-Euclidean) arithmetics is made explicit. *

In the book the theory of non-Diophantine arithmetics is exposed and its possible applications are considered.