Essays On The Theory Of Numbers

The mathematical reading public unacquainted with German is under considerable obligation to Professor Beman for the present faithful rendering of these two celebrated essays of Dedekind. Modern logical views of continuity and arithmetic are largely based on the results which Dedekind and his contemporary, G. Cantor, furnished (the first of their essays was published in 1872), and it is good that these investigations should be made accessible to all readers in their original form. Furthermore, the German of these essays is not the easiest imaginable reading, and the interpretation of the forms of expression which Professor Beman has given and which has involved considerable study, will also be welcome to readers of German who are in the habit of purchasing but not perusing German books. “In science," remarks Dedekind, “nothing capable of proof ought to be accepted without proof." But in laying the foundations of the simplest of the sciences, “viz., that part of logic which deals with the theory of numbers,” this postulate has not been complied with. In saying that arithmetic (algebra and analysis) is a branch of logic, Dedekind by implication asserts that the number-concept “is entirely independent of the notions or intuitions of space and time . . . an immediate result of the laws of thought." His answer to the question of the nature and meaning of numbers is that “numbers are free creations of the human mind and serve as a means of apprehending more easily and more sharply the difference of things.” He continues in the preface to his essay of 1887: “It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind. If we scrutinise closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation, . . . must in my judgment, the whole science of numbers be established.” Such is the fundamental idea of Dedekind's thought. He succeeded by means of it in building up a continuous number-system which included not only rational numbers but also, and necessarily, irrational numbers. These memoirs can, as their own author remarks, “be understood by any one possessing what is usually called good common sense; no technical philosophic, or mathematical, knowledge is in the least degree required." –The Monist, Vol. 13