Monday, March 21, 2011
Phenomenology of Spirit, Preface, paragraph 45
The evidence peculiar to this defective cognition, of which mathematics is proud and of which it brags, even against philosophy, rests solely on the poverty of its purpose and the defectiveness of its material, and is on that account of a kind that philosophy must spurn. Its purpose or concept is quantity. This is precisely the non-essential, concept-less relation. The movement of knowledge goes on, therefore, on the surface, does not affect the matter itself, neither the essence nor the concept, and therefore is no comprehension. The material which provides mathematics with these welcome treasures of truth consists of space and the unit [das Eins]. Space is the existence wherein the concept inscribes its diversity as in an empty, lifeless element, in which its differences are likewise unmoved and lifeless. The real is not something spatial, such as is treated of in mathematics. With such unreality as the things of mathematics, neither concrete sense-oriented perception nor philosophy meddles. In an unreal element of that sort we find, then, only unreal truth, that is to say, fixed lifeless propositions. We can halt at any one of them; the next begins of itself again, without the first having led up to the one that follows, and without any necessary connection having in this way arisen from the nature of the matter itself. So, too, and herein consists the formal character of mathematical evidence, because of that principle and element, knowledge advances along the lines of equality. For what is lifeless, not being self-moved, does not bring about the distinction of essence, does not attain to essential opposition or inequality; and hence no transition of the opposed into the opposed, not to the qualitative, the immanent, not to self-movement. For it is quantity, the non-essential distinction, with which alone mathematics has to do. It abstracts from the fact that it is the concept which divides space into its dimensions, and determines the connections between them and in them. It does not consider, for example, the relation of line to surface, and when it compares the diameter of a circle with its circumference, it runs up against their incommensurability, i.e., a relation in terms of concept, something infinite, that escapes mathematical determination.