Abstract

This article will give an explanation of N. Hartmann's views of the nature and the mode of being of the ideal or geometrical space. Before grappling with this problem, we shall deal first with some preliminary questions. Their objects are the following : the nature and division of Hartmann's philosophy, his conception of ontology as a categorical analysis, as well as his point of view concerning the nature and knowability of categories in general. With regard to the nature of geometric space Hartmann begins by stating that this space is, in essence, a pure dimension-system. This dimension-system must be thought as continuous, homogeneous and absolutely boundless. The question whether the ideal space is either finite or infinite, is meaningless, because space as a mere dimension-system can have no extent. The reason why is that space itself is not extensive but forms the very condition and the necessary substratum of «extensio» and of having «extensa». Since each dimension of this pure dimension-system is also homogeneous in itself, space itself is isometric at the same time. Finally a pure dimension-system must not be confused with a coordinate-system, for this has always a definite origin and also a definite direction of its axes: so a coordinate-system is something « in » space, but not the ideal space itself. Concerning the mode of being of ideal space Hartmann very probably argues that in this respect we may put it on one level with the concrete geometrical entities. Consequently what can be said, in this respect, of a triangle, also holds good m.m. for space itself. Hartmann's opinion of geometrical entities is that they really exist : so they are not merely imaginary, though they have no existence in themselves apart from the sensorily perceptible things, as Plato thought. So far Hartmann's standpoint entirely corresponds with Aristotle's. Aristotle then asserts that the mathematical being really exists in the sensible things, though as such not actually, but potentially only; and consequently, that it can only have actual existence by means of the abstracting and idealizing act of the mathematician. Hartmann however, argues that the mathematical entity, though actual, is not real ; he does admit that « the mathematical » may partly be found in the real things and this even in an actual mode ; from this it follows that « the mathematical » can be no mere abstraction ; consequently it must have real existence independent of our thinking. It does not follow, however, that it must be ideal. But according to him, there also exist mathematical entities which will never be found in real things. Since, in the order of the real, only the actual is possible, they cannot even be potential in the things. Yet they are substantial objects of our a-priori-knowledge in pure mathematics. Via a thourough analysis of the universality and the necessity of the mathematical judgments, these reflections lead to the conclusion that the above-mentioned mathematical entities must have an actual, and this an ideal, existence. Hartmann himself wants his standpoint to be considered as a harmonious combination of the points of view of Plato and Aristotle. In a critical reflexion at the end of this study it is pointed out that Hartmann's categorical analysis of the nature of ideal space must be considered as a very important contribution to the philosophy of mathematics which is in perfect agreement with the data of modern science. His views, however, of the mode of being of the mathematical entities and ideal space seem unacceptable. The weak points in his argumentations seem te be the following : his idea of the nature and the meaning of an a-prioriknowledge, his standpoint concerning the traditional abstraction-theory as well as the somewhat surprising view of the concept of possibility. It seems to us that in this matter Hartmann too much inclines to Plato, Kant, Hegel and Husserl