Adolf (Abraham) Halevi Fraenkel (February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He is known for his contributions to axiomatic set theory, especially his additions to Ernst Zermelo's axioms, which resulted in the Zermelo–Fraenkel set theory.
The book was written in the 1940s in Hebrew, and was published in five volumes (of varing sizes, holding over 700 pages of actual contents in total). Scans of these volumes are avilable in the following semi-public web-page. As written there, the scans and postings were done with the permission of Fraenkel's family, which reserves the copyrights for the book.
This 49-page text, written in Hebrew, provides an overview of Fraenkel's book. The book is not intended to provide an overview of the contents of mathematics, but rather of it nature, which is captured by its questions and methodology.
The approach of the original text is remarkably conceptual. It starts from the intuitive issues that mathematics wishes to address and explains the methodology employed in order to address them. It views the intuitive issues as driving the process but leaving the stage to the rigorous study, which refers to the actual definitions rather to the underlying intentions. It stresses that definitions do not assert facts nor prove ones, but rather introduce notions that come to exist via them. This introduction is supposedly arbitrary (in principle), but may be justified in retrospect by its fruitfulness (e.g., providing insights about the initial intuitive issues).
The mathematical notions are structures composed of elementary objects and operations on these objects. The representation of the objects is immaterial, and this fact is reflected by an equality that is postulated between different representations (of the same object). The operations applied on these objects are also representation independent, although they may be defined with reference to a specific representation. In such cases, operating on different representations of the same objects yields representations that are equal (i.e., represent the same objects, although they many not be identical).
While the initiation of a mathematical study is pragmatic, its evolution is subjected to general principles that seek clarity, compactness and consistency. These principles have no formal standing in the study, but they do guide it just as non-mathematical considerations may inform the choice of definitions.