This is a re-print of a book which scholars of philosophy, mathematics, and natural science consider as one of those "classics" every one in those fields must have read and mastered. It is also of interest to social scientists, like economists, because it delves into fundamental issues with the depth of philosophical thought and the rigor of a mathematician. The book was initially published by the Princeton University Press in 1949 (thus, 60 years ago); then it was, by and large, a translation and an expansion of a previous essay : the article "Philosophie der Mathematik und Naturwissenschaft" published in 1926 as a part of R. Oldenbourg's Handbuch der Philosophie (hence, nearly 80 years ago). As Nobel Prize Frank Wilczek recalls in the Introduction, Weyl was, with Einstein and von Neumann, part of the "trinity of refugee stars" that place Princeton's Institute for Advanced Studies at the forefront of research in many a field as they embodied the grand German literary and pan-European cultural tradition and rocketed into American pragmatism. – The book is divided into two parts, dealing respectively with Mathematics and Natural Science. Even though it includes several theorems and algorithms completely new and original at the time when they were formulated (and both the book and the previous essay published), it is not a technical text written only or mostly for specialists of the two main fields (mathematics and natural science). It is for a much broader readership with its roots in the European (not solely German) literature and thought. Throughout the book , we walk into the concepts of number and continuum, of the infinite, of geometry, of space, of time, of formation of theories, of the physical picture of the world. However, Hermann Weyl reminds us that he is not holding our hands in this voyage through mathematics and science. On the contrary, we and he have as guides Decartes, Leibnitz, Hume, Kant and all the other founding fathers of pan-European culture. Also, the voyage is not toward the discovery of new math and science technicalities : the final chapter deals with law, chance and especially freedom and the last appendix with how to reconcile basic questions on the origin of life and evolution and deep beliefs, like religious beliefs. M.-M. V.