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From the Preface. THE present work is essentially one of constructive criticism. It is, we believe, the first attempt made on any extensive scale to examine critically the fundamental conceptions of Mathematics as embodied in the current definitions. The purpose of our examination is not solely or even chiefly to show the presence of error, but to pro mote the development of a more scientific doctrine. In expounding our own views we have often been obliged to find fault with those of others; but we have not gone out of our way for the sake of mere criticism; we have merely cleared away false doctrine preparatory to replacing it with true. Our work, though in a sense dealing with definitions, does not have as its essential scope questions as to the words to be used in expressing something about which there is universal agreement; it really deals with the conceptions underlying the definitions where there is, as will be shown, a great diversity of view. Further than a discussion of definitions (in this sense) we do not go, and though we have at times occasion to enunciate axioms and theorems we never set down a demonstration. It is indeed undeniable that a discipline consisting of definitions alone would be perfectly futile, but this is no argument against deeming the definitions of a science worthy of a separate exposition. How far from being systematic is the treatment of the definitions of Mathematics in most mathematical writings will be appreciated by all who have given their attention to the matter. Definitions are laid down only as they are needed for the work in hand, and in their formulation attention is given, not to the needs of mathematical science as a whole, but to those of a single book too often a book whose sole purpose is to enable more or less stupid youths to pose as graduates of a course in Mathematics. As to the articles of original research published in mathematical journals, definitions are hardly to be found in them at all. This state of affairs has reacted upon the demonstrations of Mathematics. When a systematic nomenclature and its concomitant, a clear and connected view of matters, are lacking, precision in statement cannot be expected. Nor is it to be found, and by far the most difficult task to the reader of a work on advanced Mathematics is not appreciating the cogency of the reasoning employed or depreciating it, as one is sometimes compelled to do but ascertaining what the author really means. This in no small number of cases is something very different from what he has said. Such a state of affairs does not rule in elementary Geometry; due in large measure to the Euclidean custom of beginning a demonstration with a precise statement of the fact about to be proven; this in turn necessitating more attention to matters of definition than modern mathematicians have thought fit to give....

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