III. On our Knowledge of the External World 63 IV. The World of Physics and the World of Sense 101 V. The Theory of Continuity 129 VI. The Problem of Infinity considered Historically 155 VII. The Positive Theory of Infinity 185 VIII. On the Notion of Cause, with Applications to the Free-will Problem 211 Index 243 LECTURE I CURRENT TENDENCIES Philosophy, from the earliest times, has made greater claims, and achieved fewer results, than any other branch of learning. Ever since Thales said that all is water, philosophers have been ready with glib assertions about the sum-total of things; and equally glib denials have come from other philosophers ever since Thales was contradicted by Anaximander. I believe that the time has now arrived when this unsatisfactory state of things can be brought to an end. In the following course of lectures I shall try, chiefly by taking certain special problems as examples, to indicate wherein the claims of philosophers have been excessive, and why their achievements have not been greater. The problems and the method of philosophy have, I believe, been misconceived by all schools, many of its traditional problems being insoluble with our means of knowledge, while other more neglected but not less important problems can, by a more patient and more adequate method, be solved with all the precision and certainty to which the most advanced sciences have attained.