mathematics

45. The evident character of this defective cognition of which mathematics is proud, and on which it plumes itself before philosophy, rests solely on the poverty of its purpose and the defectiveness of its stuff and is therefore of a kind that philosophy must spurn. Its purpose or Notion is magnitude. It is just this relationship that is unessential, lacking the Notion. Accordingly, this process of knowing proceeds on the surface, does not touch the thing itself, its essence or Notion, and therefore fails to comprehend it [i.e. in terms of its Notion]. — The material, regarding which mathematics provides such a gratifying treasury of truths, is space and the numerical unit. Space is the existence in which the Notion inscribes its differences as in an empty lifeless element, in which they are just as inert and lifeless. The actual is not something spatial, as it is regarded in mathematics; with non-actual things like the objects of mathematics, neither concrete sense-intuition nor philosophy has the least concern. In a non-actual element like this there is only a truth of the same sort, i.e. rigid, dead propositions. We can stop at any one of them; the next one starts afresh on its own account, without the first having moved itself on to the next, and without any necessary connection arising through the nature of the thing itself. Further, because of this principle and element —and herein consists the formalism of mathematical evidence— [this kind of] knowing moves forward along the line of equality. For what is lifeless, since it does not move of itself, does not get as far as the distinctions of essence, as far as essential opposition or inequality, and therefore does not make the transition of one opposite into its opposite, does not attain to qualitative, immanent motion or self-movement. For it is only magnitude, the unessential distinction, that mathematics deals with. It abstracts from the fact that it is the Notion which divides space into its dimensions and determines the connections between and within them. It does not, for example, consider the relationship 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 relationship of the Notion, something infinite that eludes mathematical determination.

G.W.F. Hegel, Preface to The Phenomenology of Spirit

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