The Critical Philosophy and the Royce-Bradley Dialogue

The Critical Philosophy and the Royce-Bradley Dialogue BERTRAND P. HELM k GOOD DEAL OF THE NEGATIVE CRITICISM of Kant's philosophy of mathematics is directed almost entirely agmnst certain doctrines in the transcendental aesthetic. The strategy of such criticism is to show, first, that Kant's treatment of space as an a priori form of intuition presupposes the Euclidean geometry and, second, that since the new geometries give different grounds for interpreting space when some other postulate is substituted for the Euclidean postulate on parallels, Kant's particular view of the ideality of space is untenable, t A corollary is sometimes added. It is that the scope of the entire critical philosophy has been drastically limited because of its narrowly conceived base in the transcendental aesthetic. Especially where the corollary is worked out, the importance of the critical philosophy for contemporary philosophy is called into question. The question might be framed as follows: since Kant's philosophy of mathematics can hold only in the locus of Euclidean geometry as interpreted in the transcendental aesthetic, and since the transcendental logic has the aesthetic as one of its necessary conditions, how can the critical philosophy be relevant to contemporary developments in philosophy? Framed in this way, the question assumes that the fortunes of the critical philosophy turn on developments in mathematics. The assumption implies that the methods of philosophy and the methods of mathematics are quite similar, if not even identical. Those who adopt the corollary, with the implicit judgment about the severely limited role of the Kantian philosophy in present developments, proceed as if Part II of the Critique of Pure Reason had never been written. In all probability, however, it was written before Part 1,2 and provides the larger context in which Part I, including the Transcendental Aesthetic, must be construed. It is in Part II that Kant argues that philosophical method and mathematical method differ in all essentials. It... becomes necessary to cut away the last anchor of these fantastic hopes, that is, to show that the pursuit of the mathematical method cannot be of the least advantage in this kind of knowledge (unless it be in exhibiting more plainly the limitations of A typical form of this line of criticism can be seen in Max Black, The Nature of Mathematics (London, 1933), p. 188. T. D. Weldon, Kant's Critique o/Pure Reason (Oxford, 1958), p. 246. [229] 230 HISTORY OF PHILOSOPHY the method); and that mathematics and philosophy, although in natural science they do, indeed, go hand in hand, are none the less so completely different, that the procedure of the one can never be imitated by the other,a Kant goes on to argue that, since mathematics is based upon definitions, axioms and demonstrations of a particular kind, philosophy cannot model its own procedures of defin fion and deduction upon those mathematical paradigms. There is, in the historical development of idealism, a case which neatly illustrates some of the main points at issue between philosophical method and mathematical method. It is the famous exchange between Josiah Royce and Francis H. Bradley on the nature of the Absolute. Both agreed that reality is a systemic whole of some kind. But while Royce held that the Absolute was an infinite self-representative whole, Bradley denied in general that we could have any knowledge of the relation between the parts and the whole, and he denied in particular that the characterization 'self-representative' was an adequate device for construing the part-whole relation within the Absolute. To prove his contention, Royce used a mathematical procedure drawn from new discussions in the philosophy of mathematics concerning the nature of the infinite, while Bradley countered on grounds which were essentially Kantian. The outcome of their dialogue, I believe, corroborates Kant's view that the methods in the two fields of inquiry are quite different. A CASE STUDY OF THE ROYCE-BRADLEY DIAt.O~UE The substance of Royce's position is given in the Supplementary Essay in Volume I of The World and the Individual. 4 In the main, this Essay is an attempt to apply the method and the results of a certain kind of mathematical argumentation to...