Royce’s indebtedness to Peirce is principally in the use of modern mathematical material, such as the recent development of the concepts of infinity and continuity, to throw light on fundamental questions of philosophy, such as relation of the individual to God or the Universe. At the end of the nineteenth century mathematics had almost disappeared from the repertory of philosophy (cf. Külpe’s _Introduction to Philosophy_), and Peirce’s essay on the _Law of Mind_ opened a new way which Royce followed in his _World and the Individual_, to the great surprise of his idealistic brethren. In his _Problem of Christianity_ Royce has also indicated his indebtedness to Peirce for his doctrine of social consciousness, the mind of the community, and the process of interpretation. It may be that a great deal of the similarity between the thoughts of these two men is due to common sources, such as the works of Kant and Schelling; but it is well to note that not only in his later writings but also in his lectures and seminars Royce continually referred to Peirce’s views. The ground for the neo-realist movement in American philosophy was largely prepared by the mathematical work of Russell and by the utilization of mathematics to which Royce was led by Peirce. The logic of Mr. Russell is based, as he himself has pointed out, on a combination of the work of Peirce and Peano. In this combination the notation of Peano has proved of greater technical fluency, but all of Peano’s results can also be obtained by Peirce’s method as developed by Schroeder and Mrs. Ladd-Franklin. But philosophically Peirce’s influence is far greater in insisting that logic is not a branch of psychology, that it is not concerned with merely mental processes, but with objective relations. To the view that the laws of logic represent “the necessities of thought,” that propositions are true because “we can not help thinking so,” he answers: “Exact logic will say that _C_’s following logically from _A_ is a state of things which no impotence of thought alone can bring about.”[21] “The question of validity is purely one of fact and not of thinking.... It is not in the least the question whether, when the premises are accepted by the mind, we feel an impulse to accept the conclusion also. The true conclusion would remain true if we had no impulse to accept it, and the false one would remain false though we could not resist the tendency to believe in it.”[22]