The Position of Mathematics in the Hierarchy of Speculative Science

THE POSITION OF MATHEMATICS IN THE HIERARCHY OF SPECULATIVE SCIENCE I. INTRODUCTION I. STATEMENT OF THE PROBLEM Thomistic interest in the problem of the classification of mathematics is not prompted by any desire to minimize the legitimate progress of modern mathematics. Increasing indications of such an interest are, in great measure, due to a belated recognition of the true character of that progress. Faced with overwhelming evidences of major advances in mathematics and the mathematical sciences since the time of St. Thomas, Thomists have been literally forced to a re-examination of the hitherto unchallenged foundations of classical mathematics. It is this renewed study which has served to eliminate many of the apparent conflicts between traditional and modem concepts of the science and to focus attention on the central issue-the position of mathematics among the speculative sciences. From the Thomistic viewpoint of a hierarchy of speculative sciences, all the progress made within the field of pure mathematics is real progress, consonant with the principles of St. Thomas. It is in the extension of mathematics to other fields that difficulties arise. The genuine utility of mathematics in interpreting the data of natural science has led to abuses. Not content with a regulative .role of its own in the physicomathematical sciences, mathematics has refused to concede .a similar role to metaphysics in the philosophy of nature. It has come to deny any higher order of knowledge than its own, to consider itself self-sufficient, independent of all other sciences and capable of interpreting them fully by its superior principles. The conflict here is unequivocal. The traditional conception of a hierarchy of science, while permitting a limited regulative role to mathematics in interpreting data of the lower sciences, 467 468 JOHN F. WHITTAKER demands a certain dependence of mathematics upon metaphysics , the supreme regulator of all science. Any attempt to determine the precise limits of these regulated and regulative functions of mathematics must clearly indicate the mutual relations between mathematics and the other speculative sciences; it must fix the position of mathematics in an adequate division of speculative science. The difficulties inherent in such an endeavor are manifest. Yet, most Thomists agree that the solution to the problem is radically contained in St. Thomas' commentary, In Librum Boetii de Trinitate. An analysis of the pertinent sections in this commentary, supplemented by parallel passages from other works of St. Thomas, has been made the basis for a division of speculative sciences which is proposed as applicable to modern developments in mathematics and compatible with Thomistic principles. The whole project has been animated by the spirit suggested by Maritain: Particularly in relation to the foundations of mathematics much more preliminary work is still required, in my opinion, before Thomist philosophy can propound a systematic interpretation in which all the critical problems offered by modern developments in the mathematical sciences find a solution.1 Admittedly deficient in many respects, the present work is offered as a preliminary step toward the ultimate goal of systematic interpretation set by Maritain. II. TEXTUAL KEY TO THE PROBLEM The key texts of St. Thomas on the division of speculative science are found in the fifth and sixth questions of his opusculum , In Librum Boetii de Trinitate. Each question consists of four articles, distributed as follows: Question V: The Division of Speculative Science. Art. 1: Whether speculative science is conveniently divided into natural, mathematical, and divine science. 1 J. Maritain, The Degrees of Knowledge (New York: 1988), p. xiii. MATHEMATICS IN SPECULATIVE SCIENCE 469 Art. II: Whether natural philosophy is of those things which are in motion and matter. Art. ill: Whether mathematical consideration IS without motion and matter. Art. IV: Whether divine science Is of those things which are without matter and motion. Question VI: The Modes which he attributes to Speculaltive Science. Art. I: Whether it is necessary to proceed in natural science rationabiliter, in mathematics disciplinabiliter, and in divine science intelligibiliter? Art. II: Whether imagination is absolutely relinquished in divine science. Art. III: Whether our intellect can consider the divine form itself. Art. IV: Whether this can be by way of some speculative science. This order of procedure has its origin in the text of Boetius. In...