Excerpt from Fundamental Conceptions of Modern Mathematics: Variables and Quantities; With a Discussion of the General Conception of Functional Relation

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 promote 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.

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