N atur non facit saltus? This book is devoted to the fundamental problem
which arises contin- uously in the process of the human investigation of
reality: the role of a mathematical apparatus in a description of
reality. We pay our main attention to the role of number systems which
are used, or may be used, in this process. We shall show that the
picture of reality based on the standard (since the works of Galileo and
Newton) methods of real analysis is not the unique possible way of
presenting reality in a human brain. There exist other pictures of
reality where other num- ber fields are used as basic elements of a
mathematical description. In this book we try to build a p-adic picture
of reality based on the fields of p-adic numbers Qp and corresponding
analysis (a particular case of so called non-Archimedean analysis).
However, this book must not be considered as only a book on p-adic
analysis and its applications. We study a much more extended range of
problems. Our philosophical and physical ideas can be realized in other
mathematical frameworks which are not obliged to be based on p-adic
analysis. We shall show that many problems of the description of reality
with the aid of real numbers are induced by unlimited applications of
the so called Archimedean axiom.