Hegel’s Science of Logic

# Synthetic Cognition

##### 3. The Theorem

§ 1743

1. The stage of this cognition that advances on the basis of the Notion-determinations is the transition of particularity into individuality; this constitutes the content of the theorem. What we have to consider here, then, is the self-related determinateness, the immanent difference of the object and the relation of the differentiated determinatenesses to one another. Definition contains only one determinateness, division contains determinateness in relation to others; in individualisation the object has gone asunder within itself. Whereas definition stops short at the general concept, in theorems, on the contrary, the object is cognised in its reality, in the conditions and forms of its real existence. Hence, in conjunction with definition, it represents the Idea, which is the unity of the Notion and reality. But the cognition here under consideration, which is still occupied in seeking, does not attain to this presentation in so far as the reality it deals with does not proceed from the Notion, and therefore the dependence of reality on the Notion and consequently the unity itself is not cognised.

§ 1744

Now the theorem, according to the stated definition, is the genuinely synthetic aspect of an object in so far as the relationships of its determinatenesses are necessary, that is, are founded in the inner identity of the Notion. The synthetic element in definition and division is an externally adopted connection; what is found given is brought into the form of the Notion, but, as given, the entire content is merely presented [monstriert], whereas the theorem has to be demonstrated. As this cognition does not deduce the content of its definitions and the principles of its divisions, it seems as if it might spare itself the trouble of proving the relationships expressed by theorems and content itself in this respect, too, with observation. But what distinguishes cognition from mere observation and representation is the form of the Notion as such that cognition imparts to the content; this is achieved in definition and division: but as the content of the theorem comes from the Notion's moment of individuality, it consists in determinations of reality that no longer have for their relationship the simple and immediate determinations of the Notion; in individuality the Notion has passed over into otherness, into the reality whereby it becomes Idea. Thus the synthesis contained in the theorem no longer has the form of the Notion for its justification; it is a connection of [merely] diverse terms. Consequently the unity not yet posited with it has still to be demonstrated, and therefore proof becomes necessary even to this cognition.

§ 1745

Now here we are confronted first of all by the difficulty of clearly distinguishing which of the determinations of the subject matter may be admitted into the definitions and which are to be relegated to the theorems. On this point there cannot be any principle ready to hand; such a principle seems, perhaps, to be implied in the fact that what immediately belongs to an object appertains to the definition, whereas the rest, since it is mediated, must wait for its mediation to be demonstrated. But the content of definition is in general a determinate one, and therefore is itself essentially a mediated content; it has only a subjective immediacy, that is, the subject makes an arbitrary beginning and accepts a subject matter as presupposition. Now since this subject-matter is in general concrete within itself and must also be divided, the result is a number of determinations that are by their nature mediated, and are accepted not on the basis of any principle, but merely subjectively as immediate and unproved. Even in Euclid, who has always been justly recognised as the master in this synthetic kind of cognition, we find under the name of axiom a presupposition about parallel lines which has been thought to stand in need of proof, and various attempts to supply this want have been made. In several other theorems, people have thought that they had discovered presuppositions which should not have been immediately assumed but ought to have been proved. As regards the axiom concerning parallel lines, it may be remarked that it is precisely there that we may discern the sound sense of Euclid, who had appreciated exactly the element as well as the nature of his science. The proof of the said axiom would have had to be derived from the notion of parallel lines; but a proof of that kind is no more part of his science than is the deduction of his definitions, axioms and in general his subject matter, space itself and its immediate determinations, the three dimensions. Such a deduction can only be drawn from the Notion, and this lies outside the peculiar domain of Euclid's science; these are therefore necessarily presuppositions for it, relative firsts.

§ 1746

Axioms, to take this opportunity of mentioning them, belong to the same class. They are commonly but incorrectly taken as absolute firsts, as though in and for themselves they required no proof. Were this in fact the case, they would be mere tautologies, as it is only in abstract identity that no difference is present, and therefore no mediation required. If, however, axioms are more than tautologies, they are propositions from some other science, since for the science they serve as axioms they are meant to be presuppositions. Hence they are, strictly speaking, theorems, and theorems taken mostly from logic. The axioms of geometry are lemmata of this kind, logical propositions, which moreover approximate to tautologies because they are only concerned with magnitude and therefore qualitative differences are extinguished in them; the chief axiom, the purely quantitative syllogism, has been discussed above. Axioms, therefore, considered in and for themselves, require proof as much as definitions and divisions, and the only reason they are not made into theorems is that, as relatively first for a certain standpoint, they are assumed as presuppositions.

§ 1747

As regards the content of theorems, we must now make a more precise distinction. As the content consists in a relation between determinatenesses of the Notion's reality, these relations may be more or less incomplete and single relationships of the subject matter, or else may be a relationship embracing the entire content of the reality and expressing the determinate relation of that content. But the unity of the complete determinatenesses of the content is equivalent to the Notion; consequently a proposition that contains them is itself again a definition, but a definition that expresses not merely the immediately assumed Notion, but the Notion developed into its determinate real differences, or the complete existence of the Notion. The two together, therefore, present the Idea.

§ 1748

If we compare closely the theorems of a synthetic science, especially of geometry, we shall find this difference, that some of its theorems involve only single relationships of the subject matter, while others involve relationships in which the complete determinateness of the subject matter is expressed. It is a very superficial view that assigns equal importance to all the propositions on the ground that in general each contains a truth and is equally essential in the formal progress, in the context, of the proof. The difference in respect of the content of theorems is most intimately connected with this progress itself; some further remarks on the latter will serve to elucidate in more detail this difference as well as the nature of synthetic cognition. To begin with, Euclidean geometry - which as representative of the synthetic method, of which it furnishes the most perfect specimen, shall serve us as example - has always been extolled for the ordered arrangement in the sequence of the theorems, by which for each theorem the propositions requisite for its construction and proof are always found already proved. This circumstance concerns formal consecutiveness; yet, important as it is, it is still rather a matter of an external arrangement for the purpose of the matter in hand and has on its own account no relation to the essential difference of Notion and Idea in which lies a higher principle of the necessity of the progress. That is to say, the definitions with which we begin, apprehend the sensuous object as immediately given and determine it according to its proximate genus and specific difference; these are likewise the simple, immediate determinatenesses of the Notion, universality and particularity, whose relationship is no further developed. Now the initial theorems themselves can only make use of immediate determinations such as are contained in the definitions; similarly their reciprocal dependence, in the first instance, can only relate to this general point, that one is simply determined by the other. Thus Euclid's first propositions about triangles deal only with congruence, that is, how many parts in a triangle must be determined, in order that the remaining parts of one and the same triangle, or the whole of it, shall be altogether determined. The comparison of two triangles with one another, and the basing of congruence on coincidence is a detour necessary to a method that is forced to employ sensuous coincidence instead of the thought, namely, the determinateness of the triangles.

§ 1749

Considered by themselves apart from this method, these theorems themselves contain two parts, one of which may be regarded as the Notion, and the other as the reality, as the element that completes the former into reality. That is to say, whatever completely determines a triangle, for example two sides and the included angle, is already the whole triangle for the understanding; nothing further is required for its complete determinateness; the remaining two angles and the third side are the superfluity of reality over the determinateness of the Notion. Accordingly what these theorems really do is to reduce the sensuous triangle, which of course requires three sides and three angles, to its simplest conditions. The definition had mentioned only the three lines in general that enclose the plane figure and make it a triangle; it is a theorem that first expresses the fact that the angle is determined by the determination of the sides, just as the remaining theorems contain dependence of three other parts on three others. But the complete immanent determinateness of the magnitude of a triangle in terms of its sides is contained in the theorem of Pythagoras; here we have first the equation of the sides of the triangle, for in the preceding propositions the sides are in general only brought into a reciprocal determinateness of the parts of the triangle, not into an equation. This proposition is therefore the perfect, real definition of the triangle, that is, of the right-angled triangle in the first instance, the triangle that is simplest in its differences and therefore the most regular. Euclid closes the first book with this proposition, for in it a perfect determinateness is achieved. So, too, in the second book, after reducing to the uniform type those triangles which are not right-angles and are affected with greater inequality, he concludes with the reduction of the rectangle to the square - with an equation between the self-equal, or the square, and that which is in its own self unequal, or the rectangle; similarly in the theorem of Pythagoras, the hypotenuse, which corresponds to the right-angle, to the self-equal, constitutes one side of the equation, while the other is constituted by the self-unequal, the two remaining sides. The above equation between the square and the rectangle is the basis of the second definition of the circle - which again is the theorem of Pythagoras, except that here the two sides forming the right-angle are taken as variable magnitudes. The first equation of the circle is in precisely that relationship of sensuous determinateness to equation that holds between the two different definitions of conic sections in general.

§ 1750

This genuine synthetic advance is a transition from universality to individuality, that is, to that which is determined in and for itself, or to the unity of the subject matter within itself, where the subject matter has been sundered and differentiated into its essential real determinatenesses. But in the other sciences, the usual and quite imperfect advance is commonly on the following lines; the beginning is, indeed, made with a universal, but its individualisation and concretion is merely an application of the universal to a material introduced from elsewhere; in this way, the really individual element of the Idea is an empirical addition.

§ 1751

Now however complete or incomplete the content of the theorem may be, it must be proved. It is a relationship of real determinations that do not have the relationship of Notion-determinations; if they do have this relation, and it can be shown that they do in the propositions that we have called the second or real definitions, then first, such propositions are for that very reason definitions; but secondly, because their content at the same time consists not merely in the relationship of a universal and the simple determinateness, but also in relationships of real determinations, in comparison with such first definition, they do require and permit of proof. As real determinatenesses they have the form of indifferent subsistence and diversity; hence they are not immediately one and therefore their mediation must be demonstrated. The immediate unity in the first definition is that unity in accordance with which the particular is in the universal.

§ 1752

2. Now the mediation, which we have next to consider in detail, may be simple or may pass through several mediations. The mediating members are connected with those to be mediated; but in this cognition, since mediation and theorem are not derived from the Notion, to which transition into an opposite is altogether alien, the mediating determinations, in the absence of any concept of connection, must be imported from somewhere or other as a preliminary material for the framework of the prooŁ This preparatory procedure is the construction.

§ 1753

Among the relations of the content of the theorem, which relations may be very varied, only those now must be adduced and demonstrated which serve the proof. This provision of material only comes to have meaning in the proof; in itself it appears blind and unmeaning. Subsequently, we see of course that it served the purpose of the proof to draw, for example, such further lines in the geometrical figure as the construction specifies; but during the construction itself we must blindly obey; on its own account, therefore, this operation is unintelligent, since the end that directs it is not yet expressed. It is a matter of indifference whether the construction is carried out for the purpose of a theorem proper or a problem; such as it appears in the first instance before the proof, it is something not derived from the determination given in the theorem or problem, and is therefore a meaningless act for anyone who does not know the end it serves, and in any case an act directed by an external end.

§ 1754

This meaning of the construction which at first is still concealed comes to light in the proof. As stated, the proof contains the mediation of what the theorem enunciates as connected; through this mediation this connection first appears as necessary. Just as the construction by itself lacks the subjectivity of the Notion, so the proof is a subjective act lacking objectivity. That is to say, because the content determinations of the theorem are not at the same time posited as Notion-determinations but as given indifferent parts standing in various external relationships to one another, it is only the formal, external Notion in which the necessity manifests itself. The proof is not a genesis of the relationship that constitutes the content of the theorem; the necessity exists only for intelligence, and the whole proof is in the subjective interests of cognition. It is therefore an altogether external reflection that proceeds from without inwards, that is, infers from external circumstances the inner constitution of the relationship. The circumstances that the construction has presented, are a consequence of the nature of the subject matter; here, conversely, they are made the ground and the mediating relationships. Consequently the middle term, the third, in which the terms united in the theorem present themselves in their unity and which furnishes the nerve of the proof, is only something in which this connection appears and is external Because the sequence that this process of proof pursues is really the reverse of the nature of the fact, what is regarded as ground in it is a subjective ground, the nature of the fact emerging from it only for cognition.

§ 1755

The foregoing considerations make clear the necessary limit of this cognition, which has very often been misunderstood. The shining example of the synthetic method is the science of geometry - but it has been inaptly applied to other sciences as well, even to philosophy. Geometry is a science of magnitude, and therefore formal reasoning is most appropriate to it; it treats of the merely quantitative determination and abstracts from the qualitative, and can therefore confine itself to formal identity, to the unity that lacks the Notion, which is equality and which belongs to the external abstractive reflection. Its subject matter, the determinations of space, are already such abstract subject matter, prepared for the purpose of having a completely finite external determinateness. This science, on account of its abstract subject matter, on the one hand, has this element of the sublime about it, that in these empty silent spaces colour is blotted out and the other sensuous properties have vanished, and further, that in it every other interest that appeals more intimately to the living individuality is silenced. On the other hand, the abstract subject matter is still space, a non-sensuous sensuous; intuition is raised into its abstraction; space is a form of intuition, but is still intuition, and so sensuous, the asunderness of sensuousness itself, its pure absence of Notion. We have heard enough talk lately about the excellence of geometry from this aspect; the fact that it is based on sensuous intuition has been declared its supreme excellence and people have even imagined that this is the ground of its highly scientific character, and that its proofs rest on intuition. This shallow view must be countered by the plain reminder that no science is brought about by intuition, but only by thinking. The intuitive character of geometry that derives from its still sensuous material only gives it that evidential side that the sensuous as such possesses for unthinking spirit. It is therefore lamentable that this sensuousness of its material has been accounted an advantage, whereas it really indicates the inferiority of its standpoint. It is solely to the abstraction of its sensuous subject matter that it owes its capability of attaining a higher scientific character; and it is to this abstraction that it owes its great superiority over those collections of information that people are also pleased to call sciences, which have for their content the concrete perceptible material of sense, and only indicate by the order which they seek to introduce into it a remote inkling and hint of the requirements of the Notion.

§ 1756

It is only because the space of geometry is the abstraction and void of asunderness that it is possible for the figures to be inscribed in the indeterminateness of that space in such a manner that their determinations remain in fixed repose outside another and possess no immanent transition into an opposite. The science of these determinations is, accordingly, a simple science of the finite that is compared in respect of magnitude and whose unity is the external unity of equality. But at the same time the delineation of these figures starts from various aspects and principles and the various figures arise independently; accordingly, the comparison of them makes apparent also their qualitative unlikeness and incommensurability. This development impels geometry beyond the finitude in which it was advancing so methodically and surely to infinity to the positing of things as equal that are qualitatively different. Here it loses the evidential side that it possessed in its other aspect, where it is based on a stable finitude and is untouched by the Notion and its manifestation, the transition just mentioned. At this point the finite science has reached its limit; for the necessity and mediation of the synthetic method is no longer grounded merely in positive but in negative identity.

§ 1757

If then geometry, like algebra, with its abstract, non-dialectical [bloss versandigen] subject matter soon encounters its limit, it is evident from the very outset that the synthetic method is still more inadequate for other sciences, and most inadequate of all in the domain of philosophy. In regard to definition and division we have already ascertained the relevant facts, and here only theorem and proof should remain to be discussed. But besides the presupposition of definition and division which already demands and presupposes proof, the inadequacy of this method consists further in the general position of definition and division in relation to theorems. This position is especially noteworthy in the case of the empirical sciences such as physics, for example, when they want to give themselves the form of synthetic sciences. The method is then as follows. The reflective determinations of particular forces or other inner and essence-like forms which result from the method of analysing experience and can be justified only as results, must be placed in the forefront in order that they may provide a general foundation that is subsequently applied to the individual and demonstrated in it. These general foundations having no support of their own, we are supposed for the time being to take them for granted; only when we come to the derived consequences do we notice that the latter constitute the real ground of those foundations.

§ 1758

The so-called explanation and the proof of the concrete brought into theorem turns out to be partly a tautology, partly a derangement of the true relationship, and further, too, a derangement that served to conceal the deception practised here by cognition, which has taken up empirical data one-sidedly ®, and only by doing so has been able to obtain its simple definitions and principles; and it obviates any empirical refutation by taking up and accepting as valid the data of experience, not in their concrete totality but in a particular instance, and that too, in the direction helpful to its hypotheses and theory. In this subordination of concrete experience to presupposed determinations, the foundation of the theory is obscured and is exhibited only from the side that is conformable to the theory; and in general the unprejudiced examination of concrete observations on their own is made more difficult. Only by turning the entire process upside down does the whole thing get its right relationship in which the connection of grounds and consequent, and the correctness of the transformation of perception into thought can be surveyed. Hence one of the chief difficulties in the study of such sciences is to effect an entrance into them; and this can only be done if the presuppositions are blindly taken for granted, and straightway, without being able to form any Notion of them, in fact with barely a definite representation but at most a confused picture in the imagination, to impress upon one's memory for the time being the determinations of the assumed forces and matters, and their hypothetical formations, directions and rotations. If, in order to accept these presuppositions a valid, we demand their necessity and their Notion, we cannot get beyond the starting point.

§ 1759

We had occasion above to speak of the inappropriateness of applying the synthetic method to strictly analytic science. This application has been extended by Wolf to every possible kind of information, which he dragged into philosophy and mathematics information partly of a wholly analytical nature, and partly too of a contingent and merely professional and occupational kind. The contrast between a material of this kind, easily grasped and by its nature incapable of any rigorous and scientific treatment, and the stiff circumlocutory language of science in which it is clothed, has of itself demonstrated the clumsiness of such application and discredited it.

[For example, in Wolf's First Principles of Architecture, the Eighth Theorem runs: A window must be wide enough for two persons to be able to look out side by side in comfort.
Proof: It is quite usual for a person to be at the window with another person and to look out. Now since it is the duty of the architect to satisfy in every respect the main intentions of his principal, he must also make the window wide enough for two persons to look out at side by side in comfort. Q.E.D.

§ 1760

The same author's First Principles of Fortification. Second Theorem: When the enemy encamps in the neighbourhood, and it is expected that he will make an attempt to relieve the fortress, a line of circumvallation must be drawn round the whole fortress.
Proof: Lines of circumvallation prevent anyone from penetrating into the camp from outside. If therefore it is desired to keep them out, a line of circumvallation must be drawn round the camp. Therefore when the enemy encamps in the neighbourhood, and it is expected that he will attempt to relieve the fortress, the camp must be enclosed in lines of circumvallation. Q.E.D.]

§ 1761

Nevertheless, this misuse could not detract from the belief in the aptness and essentiality of this method for attaining scientific rigour in philosophy; Spinoza's example in the exposition of his philosophy has long been accepted as a model. But as a matter of fact, the whole style of previous metaphysics, its method included, has been exploded by Kant and Jacobi. Kant, in his own manner has shown that the content of that metaphysics leads by strict demonstration to antimonies, whose nature in other respects has been elucidated in the relevant places; but he has not reflected on the nature of this demonstration itself that is linked to a finite content; yet the two must stand and fall together. In his First Principles of Natural Science, he has himself given an example of treating as a science of reflection, and in the method of such, a science that he thought by that method to claim for philosophy. If Kant attacked previous metaphysics rather in respect of its matter, Jacobi has attacked it chiefly on the side of its method of demonstration, and has signalised most clearly and most profoundly the essential point, namely, that method of demonstration such as this is fast bound within the circle of the rigid necessity of the finite, and that freedom, that is the Notion, and with it everything that is true, lies beyond it and is unattainable by it. According to the Kantian result, it is the peculiar matter of metaphysics that leads it into contradictions, and the inadequacy of cognition consists in its subjectivity; according to Jacobi's result, the fault lies with the method and the entire nature of cognition itself, which only apprehends a connection of conditionedness and dependence and therefore proves itself inadequate to what is in and for itself, to what is absolutely true. ®

§ 1762

In point of fact, as the principle of philosophy is the infinite free Notion, and all its content rests on that alone, the method proper to Notion-less finitude is inappropriate to it. The synthesis and mediation of this method, the process of proof, gets no further than a necessity that is the opposite of freedom, that is, to an identity of the dependent that is merely implicit [an sich], whether it be conceived as internal or as external, and in this identity, that which constitutes the reality in it, the differentiated element that has emerged into concrete existence, remains simply an independent diversity and therefore something finite. Consequently this identity does not achieve concrete existence here and remains merely internal, or, from another point of view, merely external, since its determinate content is given to it; from either point of view it is an abstract identity and does not possess within it the side of reality, and is not posited as identity that is determinate in and for itself. Consequently the Notion, with which alone we are concerned, and which is the infinite in and for itself, is excluded from this cognition.

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