ABSTRACT
This chapter shows how, from the indispensable apparatus of general logical notions, the theory of finite integers and of rational numbers without sign could be developed. It examines the problems raised by endless series and by compact series—problems which, under the names of infinity and continuity, have defied philosophers ever since the dawn of abstract thought. One of the generalizations of number, namely complex numbers, has been excluded completely, and no mention has been made of the imaginary. The theory of imaginaries was formerly considered a very important branch of mathematical philosophy, but it has lost its philosophical importance by ceasing to be controversial. The examination of imaginaries led, on the Continent, to the Theory of Functions—a subject which, in spite of its overwhelming mathematical importance, appears to have little interest for the philosopher. Complex numbers first appeared in mathematics through the algebraical generalization of number.
