In the opinion of many laymen, mathematics is already today a dead science: after having reached an unusually high degree of development, it has become petrified in rigid perfection. This is an entirely erroneous view of the situation; there are but a few domains of scientific research which are passing through a phase of such intensive development at present as mathematics. Moreover, this development is extraordinarily manifold: mathematics is expanding its realm in all possible directions, it is growing in height, in width, and in depth. It is growing in height, since, on the soil of its old theories which look back upon hundreds if not thousands of years of existence, new problems appear again and again, and ever stronger and more refined results are being achieved. It is growing in width, since its methods permeate other branches of learning, its domain of investigation embraces ever wider ranges of phenomena, and new theories continue to enter into the large circle of mathematical disciplines. And finally, it is growing in depth, since its foundations become more and more firmly established, its methods perfected, and its principles stabilized.It has been my intention in this book to give those readers who are interested in contemporary mathematics, but who are not actively concerned with it, at least a very general idea of that third line of mathematical development, i.e. of its growth in depth.