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The Development of Royce's Later Philosophy a BRUCE KUKLICK ThE STUDY OF THE EvoiArrios of Royce's thought from The World and the Individual 2 to The Problem of Christianity 3 has proceeded independently from an analysis of his work in the foundations of mathematics. This state of affairs is unfortunate since these studies were one of Royce's main interests, if not his main interest, after 1900. He introduced symbolic logic into the Harvard curriculum and, as his son has written, "did a great deal of long and laborious work" on it during the first twelve years of this century.4 In this essay I interpret Royce's doctrines in light of the changes evidenced in his published work in mathematical logic. The bearing of the famous "Supplementary Essay" in the first volume of The World and the Individual on the rest of Royce's metaphysics is so well known that it scarcely needs retelling. Indeed, the essay occupies a place of such prominence in exegeses that it overshadows all else of a mathematical nature. Royce had argued in the body of his first volume for a certain account of the relationship between finite experiencers (the Many) and the Absolute experience (the One). In his "Supplementary Essay" he undertook to defend his position against F. H. Bradley whose Appearance and Reality 5 contended that it was "wholly impossible" for human beings to render "any explicit and detailed reconciliation of the One This paper represents a portion of a chronologically and philosophically more extended work in progress. I am indebted to the chapter on Josiah Royce in M. G. Murphey's and E. Flower's A History o/American Philosophy, soon to be published by Random House. I would also like to thank Michael Dunn for help with the minimal mathematical notions, although he is in no way responsible for what I have done with them. 2 Two volumes (New York, 1899, 1901). These volumes are hereafter referred to as WI, 1 and WI, 2 respectively. s Two volumes (New York, 1913). These volumes are hereafter referred to as PC, 1 and PC, 2 respectively. 4 Daniel Sommer Robinson, Royce and Hocking: American Idealists (Boston, 1968), p. 149. 5 2nd ed. rev. (London, 1925). [349] 350 HISTORY OF PHILOSOPHY and the Many, or any positive theory of how Individuals find their real place in the Absolute." 6 Bradley's magnum opus rests on a dissection of what occurs in any consideration of an object, o, in the world and the relation, R, which links o to the world. To R, as itself an object, there exists R*, R's own relation to the world. We are thus never able to say that o is R-related to the world without becoming involved in an infinite regress for R* has its R**, R** its R***, and so on. As Bradley puts it, "endless fission" unquestionably "breaks out." With this statementof the problem of relational thought Royce agrees. Both Bradley and Royce also agree that the unification of finite experience in the Absolute takes place, but Bradley goes on to say that how it is effected is insoluble for us. We should have to approach this puzzle via an endless succession of relations; we should have to enumerate an "actual infinite," and an "actual infinite multitude" is a self-contradiction for Bradley/ Nevertheless, the infinite regress generated gave Royce the chance to examine new advances in the philosophy of mathematics and, so he thought at the time, to refute Bradley's skepticism. The crux of Royce's reply is his definition of a certain kind of system in which (1) to every element M r in the system there corresponds a unique element called Mr's image or successor that, taken in its order, is the next element, (2) every image is distinct, and (3) at least one element, M, although imaged by another, is itself the image of no other element. The natural numbers best exemplify this type of order: they form an infinite series beginning with zero in which each term is followed by a next tenn. Most importantly, the system is capable of definition "as a single internal...


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