Mathematical Philosophy: A Studyof Fate and Freedom

In this collection of twenty-one lectures, Dr. Keyser has "endeavored to present in the language current among educated men and women some of the maturer fruits" of the study of "the nature of mathematics, upon its significance in thought, and upon its bearing on human life."
These lectures are designed primarily for students of thought, but the author hopes that they "may not be ungrateful to a much wider circle of readers and scholars" among whom he instances "all readers who desire to acquire a fair understanding of such genuinely great mathematical ideas as are accessible to all educated laymen and to come into touch with the universal spirit of the science which Plato called divine."
What mathematical training is necessary for a proper appreciation of these lectures? "So much knowledge of algebra, geometry and trigonometry as a capable student can acquire in one collegiate year."
In propounding the question, "how much mathematical discipline is essential to the appropriate education of men and women as human beings?" The author clearly and forcibly states the "evident tokens and the cardinal constituents of that which in human beings is human," and which they possess in some recognizable measure: a sense for language, for expression in speech-the literary faculty; a sense for the past, for the value of experience-the historical faculty; a sense for the future, for prediction, for natural law-the scientific faculty; a sense for fellowship, cooperation and justice-the political faculty; a sense for the beautiful-the artistic faculty; a sense for logic, for vigorous thinking-the mathematical faculty; a sense for wisdom, for world harmony, for cosmic understanding -the philosophical faculty; and a sense for the mystery of divinity-the religious faculty.
The aim of mathematics, says the author, is "to think vigorously whatever is vigorously thinkable, or whatever may become vigorously thinkable, in course of the upward striving and refining evolution of ideas." And thinking he defines as the "handling of ideas as ideas-the formation of concepts, the combination of concepts into higher and higher ones, discernment of the relations subsisting among them, embodiment of these relations in the forms of judgments or propositions, the ordering and use of these in the construction of doctrine regarding life and the world."