This work deals with the “noetics of geometry,” that is, with that part of philosophical noetic science whose object is the cognitive nature of the human mind insofar as it constructs geometric science.
The first part of the things expounded here (chapters I-IV) is a second edition of a series of articles which were published several years ago in the journal Gregorianum 1, with certain adaptations which seemed necessary so that they might be composed into the unity of the present work. The other chapters (V-VII) contain those things which were already foreseen and promised in the last article (of the year 1943, pag. 234), but whose elaboration had been impeded up to now by wartime and postwar circumstances.
In these chapters we noetically examine the axioms which are required for the construction of geometry — the so-called “foundations of geometry” — only insofar as they relate to the principal points of modern criticism; these, however, are examined deeply, for they are of greater philosophical importance in all epistemological investigations. Hence certain other axioms, e.g., those which concern the comparative measurement of lengths of lines of different curvature, are not examined. In this respect, the axioms which are examined constitute an incomplete system of indemonstrable or primitive propositions for the deduction of geometry, even for classical deduction.
But we also have a deeper and more decisive reason why we are content, and indeed must be content, with an incomplete system of primitive, indemonstrable propositions. For, as we shall show in chapter V, it is impossible to establish such a complete system. For in the very deduction — not only of geometry, but also of other sciences which can be continuously evolved — it seems necessary for the human mind always to invoke new primitive propositions again and again, indemonstrable and yet known as necessary; this seems not only necessary, but at the same time, in those sciences, possible; to infinity. The noetic method of inquiring into these propositions seems to be of the greatest importance, for it manifests the noetic nature of the human mind, not only in inquiring into the beginnings, the foundations, of sciences, but also in building up a science deductively. Such repeated noetic exercise, also because of the “habit” which it begets, seems to be, while not strictly necessary, nevertheless very useful for general noetics, in which it is usually neglected, although it is its task to carry out such investigations. Because of such negligence it can happen, and does happen, that certain noetic systems fall into error. Yet this does not necessarily happen; for the things which are discovered in such a noetic analysis are implicitly present in the cultivated human mind and naturally direct its investigations intellectively; but they are present only implicitly, or “in exercised act” (in actu exercito); the task of noetics is to establish such principles explicitly, to lead the exercised act to a “signified act” (actum signatum).
We will often encounter this in what follows.
But in the title of this work we say even more. We call the noetics of geometry the “origin” of the theory of knowledge. And we intend to say that this discipline can not only begin with the noetics of geometry (at least with its principles) but that it must necessarily begin thus. This we shall attempt to establish in the last paragraph of chapter VII, from the nature of the thing, but also from the doctrine of St. Thomas, whose doctrine concerning the judgment of the human mind we have expounded at length in another work; and there we already frequently referred back to the noetics of geometry 2.
It is historically established that scholastic cosmology cannot be built up, and fully saved from antinomies, without such preliminary or simultaneous investigations. But also for other
philosophical disciplines this continuous practical exercise, to which the human mind applies itself in this noetic analysis, seems to bring the greatest utility, including that part which examines the notion of actual “being” (esse) itself. So that this may be illustrated, we add to this book as an appendix our article Concerning the Necessary Connections Between Existential Acts (De connexionibus necessariis inter actus existentiales) from the journal Gregorianum 1953 pag. 603-639.
Let one thing be added here. There are philosophers, perhaps not a few, who, when they encounter geometric or generally mathematical pronouncements in a philosophical elaboration, even if it concerns the simplest things, are immediately deterred from reading and studying. Let them not fear; this small labor, unless I am mistaken, will generate the most abundant fruits. No more is demanded than that which Plato is said to demand from every philosopher.
Rome, in the commemorative days of the completion of the fourth century since the constitution of the Gregorian University, in the year [^1953].
P. HOENEN S. J.
1938, 498-514, De philosophia scholastica cognitionis geometricae; 1939, 19-54, De problemate necessitatis geometricae; 1939, 321-350, De problemate exactitudinis geometricae I; 1943, 171-234, De problemate exactitudinis geometricae II. Cf. also Gregorianum, 1951, 434-452, De noetica geometriae, reply to the observations of the very distinguished H. Freudenthal. ↩
La théorie du jugement d’après St. Thomas d’Aquin ed. 1, 1946, ed. 2 enlarged and amended, 1953, English version Reality and Judgment according to St. Thomas, Chicago 1952. We will refer to it under the siglum Th. d. J. or R. a J. ↩