De Noetica Geometriae, Origine Theoriae Cognitionis by Auctore P. Hoenen, S. J. (review)

BOOK REVIEWS De Noetica Geometriae, Origine Theoriae Cognitionis. AucTORE P. HoENEN, S. J., Romae: Apud Aedes Universitatis Gregorianae, 1954. Pp. ~98 cum indice analytico. The latest work by Fr. Hoenen sums up his studies on the foundations of geometry and the relevance of such studies to the formation of a general theory of scientific knowledge. Parts of the text have appeared in various articles in the Gregorianum from 1938 to 1943, but more than half of the work is original and the remainder has been emended and edited to give support to the author's main thesis. Fr. Hoencn's principal purpose is to examine the axiomatic foundations of Euclidean geometry with regard to major points that have come under modern criticism, and on the basis of this to propose an incomplete system of primitive (indemonstrable) propositions from which geometry can be deduced. His study is a noetic one in the sense that it is the function of noesis to make explicit the principles that are intuitively grasped by the mind in actu exercito, and then to clarify these by reflection (so that they can be understood) in actu signata. The author is at pains to show as a major conclusion of his work that this contribution is not only a possible beginning of noetic science, or a theory of knowledge, but is in fact the necessary beginning of all scientific methodology. This main thesis of Fr. Hoenen is not without its controversial aspect. Yet the fundamental work that underlies it is quite sound and of considerable interest to those attempting to evaluate contemporary mathematics and logic from an Aristotelian viewpoint. In order then to recapitulate this contribution of Fr. Hoenen for the benefit of American readers, the first part of this review will be a synopsis of the main points in the development of his thesis. This will then form the basis for a brief analysis of the aspects most open to criticism in the light of traditional Aristotelian doctrine on scientific methodology. * The first chapter of De Noetica Geometriae is devoted to a statement of the problem that is central to all scientific thought, but which is normally encountered first in the mathematical sciences. Actually it is a twofold problem, and it may be stated in this way: Whence arises the necessity of mathematical judgments respecting first principles and indemonstrable propositions, and whence arises the exactitude which is found in mathe381 38~ BOOK HEVIEWS matical concepts and has no counterpart in our sensible experience? These are the fundamental difficulties, but related to them are other problems that are much discussed by contemporary thinkers, such as the arithmetization of the continuum, the status of Euclid's fifth postulate, and the limitations of axiomatics. The author is of the opinion that a good start was made by Aristotle in delineating the general approach to all these problems, but he bemoans the fact that no one in the scholastic tradition has done anything to develop Aristotle's doctrine and apply it to current difficulties. Thus he is aware that he is moving into a virgin field in which he cannot hope to give all the answers, but he is interested more in making a good beginning than in expounding a complete systematic development. The problem of necessity is approached in the second chapter and the solution is given, as might be expected, in terms of the Aristotelian theory of formal abstraction. Fr. Hoenen begins dialectically, furnishing many examples of judgments that are regarded as necessary by ancient and modern mathematicians. Then he seeks the common feature associated with this necessity, and expounds it in terms of the notions of implicit judgment and virtual judgment which he has explained at length in his Reality and Judgment According to St. Thomas (Chicago: Regnery, 195Q). His conclusion is that necessary judgments are arrived at by formal abstraction through which it is seen (intued) that there is a formal connection which necessitates whatever is predicated of the subject. Experience is needed only insofar as these truths are abstracted from sense data, and compared to the other sciences, mathematics requires little experiential knowledge. In casibus mathematicis ... habemus abstractionern forrnalem, et ideo intuernur nexurn formalem, ideo necessarium...