“The Human Worth of Rigorous Thinking.” Essays and Addresses. Columbia University Press, 1916.
Selection from contents: The human worth of rigorous thinking. The human significance of mathematics. The walls of the world; or concerning the figure and the dimensions of the Universe of space. The universe and beyond. The existence of the hypercosmic. The axiom of infinity: A new presupposition of thought. Research in American Universities. Mathematical productivity in the United States.
“Mathematical Philosophy, the Study of Fate and Freedom. Lectures for Educated Laymen.” Forthcoming Book.
Selection from contents of general interest: The mathematical obligations of philosophy. Humanistic and industrial education. Logic the muse of thought. Radiant aspects of an over-world.—Verifiers and falsifiers. Significance and nonsense.—Distinction of logical and psychological. A diamond test of harmony.—Distinction of doctrine and method.—Theoretical and practical doubt.—Mathematical philosophy in the rôle of critic. A world uncriticised—the garden of the devil. “Supersimian” Wisdom. Autonomous truth and autonomous falsehood. Other Varieties of truth and untruth. Mathematics as the study of fate and freedom. The prototype of reasoned discourse often disguised as in the Declaration of Independence, the Constitution of the United [pg 223] States, the Origin of Species, the Sermon on the Mount.—Nature of mathematical transformation. No transformation, no thinking. Transformation law essentially psychological, Relation function and transformation as three aspects of one thing. Its study, the common enterprise of science. The static and the dynamic worlds. The problem of time and kindred problems. Importation of time and suppression of time as the classic devices of sciences.—The nature of invariance. The ages-old problem of permanence and change. The quest of what abides in a fluctuant world as the binding thread of human history. The tie of comradeship among the enterprises of human spirit.—The concept of a group. The notion simply exemplified in many fields, is “Mind” a group. The philosophy of the cosmic year.—Limits and limit processes omnipresent as ideals and idealization, in all thought and human aspiration. Ideals the flint of reality.—Mathematical infinity, its dynamic and static aspects. Need of history of the Imperious concept. The rôle of infinity in a mighty poem.—Meaning of dimensionality. Distinction of imagination and conception. Logical existence and sensuous existence. Open avenues to unimaginable worlds.—The theory of logical types. A supreme application of it to definition of man, and the science of human welfare.—The psychology of mathematics and the mathematics of psychology. Both of them in their infancy. Consequent retardation of science. The symmetry of thought. The asymmetry of imagination.—Science and engineering. Science as engineering in preparation. Engineering as science in action. Mathematics the guide of the engineer. Engineering the guide of humanity. Humanity the civilizing or Time-Binding class of life. Qualities essential to engineering leadership. The ethics of the art. The engineer as educator, as scientist, as philosopher, as psychologist, as economist, as statesman, as mathematical thinker—as a Man.
Appendix II. Biology And Time-Binding
The life of one man is short, and to very few is it given to achieve much in their lifetime. Extensive achievements are made almost entirely by many men taking up the work done by a discoverer. In such a case, we arrive at a complete “truth,” not by the production of one man but by a chain of men, but the initial discovery not only has to be produced but correctly defined before it can be used and that is the important point to be made. What we do not realize is the tremendous amount of mental work that is lost by an incorrect use of words.
Human thought—that unique, subtle and yet most energetic phenomenon of nature—is in the main wantonly wasted, because we do not use, or take pains to use, suitable language; at the same time, false definitions lead to consequences not merely wasteful but positively harmful. When ideas and facts are falsely defined, they tend to bring us to false conclusions, and false conclusions lead us in wrong directions, and life and knowledge greatly suffer in consequence. Our progress is not a well ordered pursuit after truth, as pure chance plays too large a part in it.
Until lately, logic was supposed to be the science of correct thinking, but modern thought has progressed so far that the old logic is not able to handle the great accumulated volume—the great complicated mass of existing ideas and facts—and so we are forced to look for another instrument much more expedient and powerful. There is no need to establish a new science to replace logic; we simply have to look closer [pg 225] into the sciences at hand and realize the fact, which was with us all the time, namely, that mathematics and mathematical reasoning is nothing else than the true logic of nature—nature's universal tongue—the one means of expression that is the same for all peoples. This is not a play on words, it is a fact which, after investigation, everybody must admit. Everybody who wants to think logically must think mathematically or give up any pretense of correct thinking—there is no escape and all who refuse to investigate the justice of this statement put themselves outside the pale of logically thinking people. The application of rigorous thinking to life will even revolutionize scientific methods by the introduction of right definitions, correct classifications, just language, and so will lead to trustworthy results. Very probably all our doctrines and creeds will have to be revised; some rejected, some rectified, some broadened; bringing about unanimity of all sciences and thus greatly increasing their effectiveness in the pursuit of truth. This application of mathematics to life will even revolutionize mathematics itself. In App. I it is suggested tentatively how this may be accomplished.