Hegel: Historical and Mathematical Truth

09 November 2019

As regards truth in matters of historical fact-to deal briefly with this subject — so far as we consider the purely historical element, it will be readily granted that they have to do with the sphere of particular existence, with a content in its contingent and arbitrary aspects, features that have no necessity. But even bare truths of the kind, say, like those mentioned, are impossible without the activity of self-consciousness. In order to know any one of them, there has to be a good deal of comparison, books must be consulted, or in some way or other inquiry has to be made. Even in a case of direct perception, only when we know it along with the reasons behind it, is it held to be something of real value; although it is merely the naked fact itself that we are, properly speaking, supposed to be concerned about.

As to mathematical truths, we should be still less inclined to consider anyone a geometer who had got Euclid’s theorems by heart (auswendig) without knowing the proofs, without, if we may say so by way of contrast, getting them into his head (inwendig). Similarly, if anyone came to know by measuring many right-angled triangles that their sides are related in the way everybody knows, we should regard knowledge so obtained as unsatisfactory. All the same, while proof is essential in the case of mathematical knowledge, it still does not have the significance and nature of being a moment in the result itself; the proof is over when we get the result, and has disappeared. Qua result the theorem is, no doubt, one that is seen to be true. But this eventuality has nothing to do with its content, but only with its relation to the knowing subject. The process of mathematical proof does not belong to the object; it is a function that takes place outside the matter in hand. Thus, the nature of a right-angled triangle does not break itself up into factors in the manner set forth in the mathematical construction which is required to prove the proposition expressing the relation of its parts. The entire process of producing the result is an affair of knowledge which takes its own way of going about it. In philosophical knowledge, too, the way existence, qua existence, comes about (Werden) is different from that whereby the essence or inner nature of the fact comes into being. But philosophical knowledge, for one thing, contains both, while mathematical knowledge sets forth merely the way an existence comes about, i.e. the way the nature of the fact gets to be in the sphere of knowledge as such. For another thing, too, philosophical knowledge unites both these particular movements. The inward rising into being, the process of substance, is an unbroken transition into outwardness, into existence or being for another; and conversely, the coming of existence into being is withdrawal into the inner essence. The movement is the twofold process in which the whole comes to be, and is such that each at the same time posits the other, and each on that account has in it both as its two aspects. Together they make the whole, through their resolving each other, and making themselves into moments of the whole.

In mathematical knowledge the insight required is an external function so far as the subject-matter dealt with is concerned. It follows that the actual fact is thereby altered. The means taken, construction and proof, contain, no doubt, true propositions; but all the same we are bound to say that the content is false. The triangle in the above example is taken to pieces, and its parts made into other figures to which the construction in the triangle gives rise. It is only at the end that we find again reinstated the triangle we are really concerned with; it was lost sight of in the course of the construction, and was present merely in fragments, that belonged to other wholes. Thus we find negativity of content coming in here too, a negativity which would have to be called falsity, just as much as in the case of the movement of the notion where thoughts that are taken to be fixed pass away and disappear.

The real defect of this kind of knowledge, however, affects its process of knowing as much as its material. As to that process, in the first place we do not see any necessity in the construction. The necessity does not arise from the nature of the theorem: it is imposed; and the injunction to draw just these lines, an infinite number of others being equally possible, is blindly acquiesced in, without our knowing anything further, except that, as we fondly believe, this will serve our purpose in producing the proof. Later on this design then comes out too, and is therefore merely external in character, just because it is only after the proof is found that it comes to be known. In the same way, again, the proof takes a direction that begins anywhere we like, without our knowing as yet what relation this beginning has to the result to be brought out. In its course, it takes up certain specific elements and relations and lets others alone, without its being directly obvious what necessity there is in the matter. An external purpose controls this process.

The evidence peculiar to this defective way of knowing — an evidence on the strength of which mathematics plumes itself and proudly struts before 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 scorn to have anything to do with. Its purpose or principle is quantity. This is precisely the relationship that is non-essential, alien to the character of the notion. The process of knowledge goes on, therefore, on the surface, does not affect the concrete fact itself, does not touch its inner nature or lotion, and is hence not a conceptual way of comprehending. The material which provides mathematics with these welcome treasures of truth consists of space and numerical units (das Eins). Space is that kind of existence wherein the concrete notion inscribes the diversity it contains, as in an empty, lifeless element in which its differences likewise subsist in passive, lifeless form. What is concretely actual is not something spatial, such as is treated of in mathematics. With unrealities like the things mathematics takes account of, neither concrete sensuous perception nor philosophy has anything to do. In an unreal element of that sort we find, then, only unreal truth, fixed lifeless propositions. We can call a halt at any of them; the next begins of itself de novo, without the first having led up to the one that follows, and without any necessary connexion having in this way arisen from the nature of the subject-matter itself. So, too — and herein consists the formal character of mathematical evidence because of that principle and the element where it applies, knowledge advances along the lines of bare equality, of abstract identity. For what is lifeless, not being self-moved, does not bring about distinction within its essential nature; does not attain to essential opposition or unlikeness; and hence involves no transition of one opposite element into its other, no qualitative, immanent movement, no self-movement, It is quantity, a form of difference that does not touch the essential nature, which alone mathematics deals with. It abstracts from the fact that it is the notion which separates space into its dimensions, and determines the connexions 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 the notion, an infinite element, that escapes mathematical determination.

Immanent or so-called pure mathematics, again, does not oppose time qua time to space, as a second subject-matter for consideration. Applied mathematics, no doubt, treats of time, as also of motion, and other concrete things as well; but it picks up from experience synthetic propositions — i.e. statements of their relations, which are determined by their conceptual nature — and merely applies its formulae to those propositions assumed to start with. That the so-called proofs of propositions like that concerning the equilibrium of the lever, the relation of space and time in gravitation, etc., which applied mathematics frequently gives, should be taken and given as proofs, is itself merely a proof of how great the need is for knowledge to have a process of proof, seeing that, even where proof is not to be had, knowledge yet puts a value on the mere semblance of it, and gets thereby a certain sense of satisfaction. A criticism of those proofs would be as instructive as it would be significant, if the criticism could strip mathematics of this artificial finery, and bring out its limitations, and thence show the necessity for another type of knowledge.

As to time, which, it is to be presumed, would, by way of the counterpart to space, constitute the object-matter of the other division of pure mathematics, this is the notion itself in the form of existence. The principle of quantity, of difference which is not determined by the notion, and the principle of equality, of abstract, lifeless unity, are incapable of dealing with that sheer restlessness of life and its absolute and inherent process of differentiation. It is therefore only in an arrested, paralysed form, only in the form of the quantitative unit, that this essentially negative activity becomes the second object-matter of this way of knowing, which, itself an external operation, degrades what is self-moving to the level of mere matter, in order thus to get an indifferent, external, lifeless content.

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