On Hegel's 'Science of Logic' : A Realm of Shadows - part fifty six.

'Psyche'

by Georges Jean-Aubry (1882–1950)?

Psyche! The lamp has gone out ; awaken!?

The day contemplates you

with eyes flooded with love

and a new desire to serve you again.

The mirror, confidant of your tearful face,

Reflects this morning a pure lake among flowers,

A milk-white sky like an eternal dawn.

Noon approaches and dances,?

Intoxicated on his golden feet.

Reach your arms out to him, dry your tears;?

In a flight, abandon the languor?

of your bed, Psyche.

The bird sings at the top of the tree,

The sun smiles with pleasure?

on seeing this universal awakening,

and Spring stretches,?

a rose in his mouth.

'Psyché'

Psyché ! La lampe est morte; éveille-toi.?

Le jour te considère?

avec des yeux noyés d'amour,

et le désir nouveau de te servir encore.

Le miroir, confident de ton visage en pleurs,

reflète, ce matin, lac pur parmi des fleurs,

Un ciel laiteux ainsi qu'une éternelle aurore.

Midi s'approche et danse,?

ivre sur ses pieds d'or.

Tends-lui les bras, sèche tes pleurs ;

dans un essor abandonne, Psyché,?

la langueur de ta couche.

L'oiseau chant au sommet de l'arbre,?

le soleil sourit d'aise?

en voyant l'universel éveil,

et le Printemps s'étire,?

une rose à la bouche.

=====

'Psyche Opening the Door into Cupid's Garden', 1903, John William Waterhouse

Georg Wilhelm Friedrich?Hegel, (1770 - 1831). 'The Science of Logic'.

The Idea of the True.

The fault of Notion at this point is that it faces a presupposed object. This object is merely a determination of the Notion. Notion is therefore still subjective. Notion must be seen as operating within the object. When so seen, Notion will finally conform to the object. It this point, Notion (together with the object) finds Truth. But such a reconciliation is in the future. Idea so far is one extreme in a syllogism. It constitutes a mere subjective reality. For subjective Idea, the object world is a limitation. In fact both extremes of the syllogism are Idea. One extreme is Idea for itself and the other is Idea in itself. In Idea For Itself and Idea In Itself, Idea is certain of itself but confined within itself. It is mere form - abstract Universality. Idea In Itself is equally an immediacy. It is being, standing over against thinking. Such an immediacy is Particularity. So far, if this duality has Individuality, it has received it externally. It must bring this externality within itself. What Idea must now do is to raise its own implicit reality, this formal truth, into real truth.

'Initially, therefore, the idea is one extreme of a syllogism, the concept that as purpose has itself at first for its subjective reality; the other extreme is the restriction of the subjective, the objective world. The two extremes are identical in that they are the idea. Their unity is, first, that of the concept, a unity which in the one extreme is only for itself and in the other only in itself. Second, it is reality, abstract in the one extreme and in the other in its concrete externality. – This unity is now posited through cognition, and, because the latter is the subjective idea which as purpose proceeds from itself, it is at first only a middle term. – The knowing subject, through the determinateness of its concept which is the abstract being-for-itself, refers to an external world; nevertheless, it does this in the absolute certainty of itself, in order to elevate its implicit reality, this formal truth, to real truth. It has the entire essentiality of the objective world in its concept; its process consists in positing for itself the concrete reality of that world as identical with the concept, and conversely in positing the latter as identical with objectivity'.

- 'The Science of Logic'

Notion is nothing but the urge for self-objectification. But when Notion objectifies itself, it equally sublates itself. Notion is, after all, subjectivity. In objectification, Notion assigns its subjective being to the presupposed object. This presupposed object is, of course, its own self. As Hegel explains elsewhere: the spiritual is this: that it finds itself in the other of itself. That is why self-equivalent spirit is precisely this very other that spirit finds as itself.

'Spirit is actual neither as intelligence nor as will, but as will which is intelligence. That is, in the intelligence there is the unity of two universalities, and in the universal will these are complete Selves. They are a knowing of their own being, and their being is this spiritual [element:] the universal will. In this element, the foregoing has now to exhibit itself. In it, the abstract will has now to transcend or supersede itself (sich aufzuheben) — just as the abstract intelligence has transcended itself in the will, the objects of that intelligence fulfilling themselves on their own. As thus transcended, the will must produce itself in the element of universal recognition, in this spiritual actuality. Possession thereby transforms itself into [property] right, just as [individual] labour was transformed, previously, into universal labour. What was family property, wherein the marriage partners knew themselves, now becomes the generalized [sphere of] the work and enjoyment of everyone. And the difference between individuals now becomes a knowledge of good and evil, of personal right and wrong'.

- 'The Jena Lectures'

Such an object has a content - Notion's own identity-with-self, in which all opposition has been sublated. For subjective Notion, the object is an Individuality. This urge to posit its own objective Individuality is the urge therefore to truth.

'Consequently this impulse is the impulse of truth in so far as the truth is in cognition, and therefore of truth in its strict sense as theoretical idea. – Although objective truth is the idea itself as the reality that corresponds to the concept, and to this extent a subject matter may or may not possess truth, nevertheless the more precise meaning of truth is that it is such for or in the subjective concept, in knowledge. Truth is the relation of the judgment of the concept, the concept that proved to be the formal judgment of truth; for the predicate in this judgment is not only the objectivity of the concept, but the comparison connecting the concept of the fact with the actuality of it. – This realization of the concept is theoretical in so far as the concept still has, as form, the determination of subjectivity, or in so far as it has for the subject the determination of being its own determination. Because cognition is the idea as purpose or as subjective idea, the negation of the world, presupposed as existing in itself, is the first negation; the conclusion in which the objective is posited in the subjective has at first, therefore, only the meaning that what exists in itself is posited only as something subjective, only in conceptual determination, and consequently does not exist as so posited in and for itself. Thus the conclusion only attains to a neutral unity, or a synthesis, that is, to a unity of terms that are originally separate, only externally conjoined. – Hence, since in this cognition the concept posits the object as its own, the idea gives itself at first only a content of which the foundation is given, in which only the form of externality has been sublated. To this extent, this cognition still retains its finitude in its realized purpose; in the realized purpose, it has at the same time not attained its purpose, and in its truth it has not arrived at the truth. For in so far as in the result the content still has the determination of a given, the presupposed being-in-itself confronting the concept is not sublated; the unity of concept and reality, the truth, is thereby equally not contained in it. – Remarkable is that this side of finitude is the one that of late has been clung to and accepted as the absolute relation of cognition – as if the finite as such were to be the absolute! On this view, an unknown thinghood-in-itself is attributed to the object, behind cognition, and this thinghood, and the truth also along with it, are regarded for cognition as an absolute beyond. Thought determinations in general, the categories, the determinations of reflection, as well as the formal concept and its moments, acquire on this view the status of determinations that are finite, not in and for themselves, but in the sense of being something subjective as against this empty thinghood-in-itself; the fallacy of taking this untrue relation of cognition as the true relation has become the universal opinion of modern times'.

- 'The Science of Logic'

Truth as pointed out earlier is the agreement of thought with its object.

'The concept of logic has hitherto rested on a separation, presupposed once and for all in ordinary consciousness, of the content of knowledge and its form, or of truth and certainty. Presupposed from the start is that the material of knowledge is present in and for itself as a ready-made world outside thinking; that thinking is by itself empty, that it comes to this material as a form from outside, fills itself with it, and only then gains a content, thereby becoming real knowledge'.

'Further, these two component parts (for they are supposed to be related to each other as component parts, and cognition is compounded from them in a mechanical, or at best chemical, manner) are said to stand to each other in this order: the object is complete and finished all by itself and, for its actuality, can fully dispense with thought; thought, for its part, is something deficient and in need of a material in order to complete itself, and also, as a pliable indeterminate form, must adapt itself to its matter. Truth is the agreement of thought with the subject matter, and in order to produce this agreement – for it is not there on its own account – thought is expected to be subservient and responsive to the subject matter'.

- 'The Science of Logic'

Accordingly, truth is the Idea that has made itself into a reality. It amounts to a relation between Notion and its reality - between subject and predicate. Dialectical Reason retorts that, so far, the truth is only theoretical. In other words, Notion and its truth are subjective. When Dialectical Reason names objectivized Idea as subjective Theory, it negates the object world - the opposite of what Idea thought it was doing. Idea For Itself and Idea In Itself is now revealed to have been only synthesis - a unity of things that are originally separate and only are externally so conjoined. In this Cognition, the content of the object is merely imposed by the subject. Cognition is still finite.

Idea For Itself and Idea In Itself

The object has not attained its end and has not arrived at its truth. From the theoretical perspective, the object is unknown - a thing-in-itself. Oddly enough, Hegel remarks, it is this side of finitude that latterly has been clung to, and accepted as the absolute relation of cognition - as though the finite as such was supposed to be the absolute. The fallacy of taking this untrue The True relation of cognition as the true relation has become the universal opinion of modern times. Immanuel Kant, (1724 - 1804), is again the target.

'Psyche Opening?the?Golden Box', 1903, John William Waterhouse?

Theory

Finite cognition is the contradiction of a truth that is supposed not to be truth - of a cognition of what is, which at the same time does not cognize the thing-in-itself.

'It is immediately clear from this definition of finite cognition that it is a contradiction that sublates itself; it is the contradiction of a truth that is supposed at the same time not to be truth, of a cognition of what is that at the same time does not know the thing-in-itself. In the collapse of this contradiction, its content, subjective cognition and the thing-in-itself, collapses, that is, proves itself to be an untruth. But it is incumbent upon cognition itself to resolve its finitude by its own forward movement and along with it its contradiction. What we have said is a consideration which we bring to it and remains a reflection external to it. But cognition is itself the concept which is a purpose unto itself and, therefore, through its realization fulfills itself, and precisely in this fulfillment sublates its subjectivity and the presupposed being-in-itself. – We must examine cognition, therefore, in its positive activity within it. Because this idea, as we have shown, is the concept’s impulse to realize itself for itself, its activity consists in determining the object, and by virtue of this determining to refer itself to itself in it as identical. The object is simply the determinable as such, and in the idea it has this essential side of not being in and for itself opposed to the concept. Because this cognition is still finite, not speculative cognition, the presupposed objectivity does not as yet have for it the shape of something which is inherently the concept simply and solely and does not hold anything particular for itself as against the cognition. But by thus having the status of a beyond that exists in itself, the determination of being determinable through the concept is essential to it; for the idea is the concept that exists for itself, is that which is absolutely infinite in itself, in which the object is implicitly sublated, and the aim is still to sublate it explicitly. The object, therefore, is indeed presupposed by the idea of cognition as existing in itself, but as so essentially related to the idea that the latter, certain of itself and of the nothingness of this opposition, arrives in the object at the realization of its concept'.

- 'The Science of Logic'

Being contradictory, subjective cognition and the thing-in-itself collapse. The very collapse of Kantian metaphysics is the True. The Truth is that Cognition must resolve its finitude. But so far this resolve is merely an external reflection. Cognition is still finite, not speculative. Objectivity does not yet have a notional shape. Still, the object has been reduced to a merely implicit beyond, which means that, when this beyond is determined (as sublated), the object's own determination will be before us. The object is already implicitly sublated. The goal now is to sub-late the object expressly. For the moment, however, the object is presupposed as being separate from Idea. The object is essentially in a relationship where Idea is certain of itself and certain that the object is null. So far, Idea is itself and not itself. Idea must now realize that it is the object. When the object explicitly sublates itself, the Notion explicitly sublates itself as well. Hegel now recalls the so-called first premise of Teleology - the bad infinity wherein End, announcing itself as not Means, proved it was Means to Means.17 Something similar now occurs with the advent of the True. Idea announces it is not the object, thereby proving it is the object: The determining activity of the Notion upon the object is an immediate communication of itself to the object and unresisted pervasion of the [object] by the Notion.

'In the syllogism whereby the subjective idea now rejoins objectivity, the first premise is the same form of immediate seizure and connection of the concept with respect to the object as we see in the purposive connection. The determining activity of the concept upon the object is an immediate communication of itself to the object, an unresisted invasion of it. In all this the concept remains in pure self-identity; but this immediate immanent reflection equally has the determination of objective immediacy; that which for the concept is its own determination, is equally a being, for it is the first negation of the presupposition. The posited determination equally has the status, therefore, of a presupposition which is merely found, the apprehension of a given wherein the activity of the concept consists rather in being negative towards itself, in holding itself back away from what is found and passive towards it, in order that the latter be allowed to show itself, not as determined by the subject, but as it is in itself'.

- 'The Science of Logic'

The True

But in announcing what it is (and what it is not), Notion remains in pure identity with itself. This pure identity is an immediacy. Yet it is also a negation and hence not an immediacy. All we have here is Notion negating itself, restraining itself and making itself passive towards what confronts it. By being passive, Notion hopes the object will show itself for what it is, without distortion. The determination that Notion has just made allows the object to be for itself, a presupposition that has been merely found, as an apprehension of a datum. The Understanding proposes that the True is an objective process in which Idea plays no role. Passive Idea is Analytic Cognition. Confronted with the True (i.e., the collapse of the diverse object), Analytic Cognition merely accepts what is given to it. If the object falls apart, it is hone of the Notion's doing. Analytic Cognition is merely the apprehension of what is, as pointed out earlier.

'Analytic cognition is the first premise of the whole syllogism – the immediate reference of the concept to the object. Identity is, therefore, the determination which analytic cognition recognizes as its own, and analytic cognition is the apprehension of what is. Synthetic cognition aims at the comprehension of what is, that is, at grasping the manifoldness of determinations in their unity. It is, therefore, the second premise of the syllogism, the one in which the diverse as such is connected. Its aim, therefore, is necessity in general. – The diverse terms that are combined stand, on the one hand, in a relation in which they are both connected yet mutually indifferent and self-subsistent; but, on the other hand, they are linked together in the concept which is their simple yet determinate unity. Now inasmuch as in a first moment synthetic cognition passes over from abstract identity to relation, or from being to reflection, it is not the absolute reflection of the concept that the latter recognizes in its subject matter; the reality that the concept gives itself is the next stage, namely the said identity in diversity as such, an identity that equally is, therefore, still inner, and only necessity; it is not the subjective identity existing for itself, hence not as yet the concept as such. Synthetic cognition, therefore, does also have for its content the determinations of the concept, the object is posited in them; but they stand only in relation to one another or in immediate unity, and for that very reason not in the unity by which the concept exists as subject'.

- 'The Science of Logic'

'In this premise, therefore, this cognition does not in any way appear as an application of logical determinations, but as a reception and apprehension of such determinations as already found, and its activity appears restricted simply to the removing from the subject matter of a subjective obstacle, an external veil. This cognition is analytic cognition'.

- 'The Science of Logic'

'This assumption is not itself analysed; hence it is equivalent to an intuition', according to Stanley Rosen. Its activity is limited to restricting itself and suppressing the obstacle of subjectivity - an external husk - in the process of knowledge.

Analytic Cognition. Analysis is the first premise of Cognition.

'The specific difference of analytic cognition is already established by the fact that, since it is the first premise of the whole syllogism, mediation does not as yet belong to it; analytic cognition is rather the immediate communication of the concept, a communication that does not as yet contain otherness and in which activity divests itself of its negativity. Yet this immediacy of the connection is for that reason itself mediation, for it is a negative reference of the concept to the object that annuls itself and thereby makes itself simple and identical. This immanent reflection is only subjective, because in its mediation the difference is present still in the form of a presupposition existing in itself, as the object’s difference within itself. The determination that results through this connection, therefore, is the form of simple identity, of abstract universality. Accordingly, analytic cognition has in general this identity for its principle, and the transition into an other, the linking of different terms is excluded from it and from its activity'.

- 'The Science of Logic'

In Analysis, Idea proclaims it is not the object, thereby proving it is the object. This first premise does not yet contain mediation, even though Idea by nature is supposed to be perfect communication of its being into the object. As Analysis, Idea empties itself of its negativity. As completely receptive to the object it analyzes, Analysis adds nothing to the object - or so it pretends. It is therefore a non-relation to the object. Self-identity is the analytical principle, and it denies its transition into the other. But Idea is the activity of becoming other. So Analytic Idea excludes itself from itself. Analysis always proceeds from some given subject matter. It is Notionless and undialectical and its progress takes place solely in the determinations of the material.

'If we look now more closely at analytic cognition, we see that it starts from a presupposition, hence from some singular, concrete subject matter, whether for representation this subject matter is already completed or in the form of a task, that is, given to it only under certain circumstances and conditions rather than disengaged from these on its own and presented in simple independence. Now the analysis of this subject matter cannot consist just in resolving it into the particular representations possibly contained within it; such a resolution and the apprehension of the particular representations is an affair that would not belong to cognition, but would rather be a matter of closer acquaintance, a determination within the sphere of representing. Analysis, since it is based on the concept, has for its products determinations that are essentially conceptual, though such as are contained in the subject matter immediately. We have seen from the nature of the idea of cognition that the activity of the subjective concept must be regarded from one side only as the explication of what is already in the object, for the object itself is nothing but the totality of the concept. It is just as one-sided to portray analysis as though there were nothing in the subject matter that is not imported into it, as it is to suppose that the resulting determinations are only extracted from it. The former way of stating the case corresponds, as is well known, to subjective idealism, which takes the activity of cognition in analysis to be only a one-sided positing, beyond which the thing-in-itself remains hidden. The other way belongs to the so-called realism, for which the subjective concept is an empty identity that imports the thought determinations from outside. – Since analytical cognition, the transformation of the given material into logical determinations, has shown itself to be a positing that immediately determines itself to be equally a presupposing, to be both in one, the logical element can appear on account of this presupposing to be in the subject matter as something already completed, just as because of the positing it can appear as the product of a merely subjective activity. But the two moments are not to be separated. In the abstract form to which analysis raises it, the logical element is of course only to be found in cognition, just as conversely it is not only something posited but something that rather exists in itself'.

It purports to be actual knowledge of the object, but in fact its products are essentially Notion-determinations immediately contained in the subject matter.

'It is well known that 'analytical science' and 'analysis' are the names of preference of arithmetic and the sciences of discrete magnitude in general. And in fact their typical method of cognition is most immanently analytical and we must now briefly consider why this is so. – Any other analytical cognition begins from a concrete material that has an accidental manifoldness within; every distinction of content and every advance to further content depend on this material. The material of arithmetic and of algebra is, on the contrary, an already totally abstract and indeterminate product from which every peculiarity of relation has been eliminated, and to which, therefore, every determination and every joining is something external. This product is the principle of discrete magnitude, the one. This relationless atom can be increased to a plurality and externally determined and unified into a sum; the increasing and the limiting are an empty progression and an empty determining that never gets past the same principle of the abstract one. How the numbers are further combined and separated depends solely on the positing activity of the knowing subject. Magnitude is in general the category within which these determinations are conducted; it is the determinateness that has become indifferent, so that the subject matter has no determinateness which is immanent to it and is therefore given to cognition. Since cognition has from the start provided itself with an accidental assortment of numbers, these now constitute the material for further elaboration and manifold relations. Such relations, their discovery and elaboration, do not seem, it is true, to be anything immanent in analytical cognition, but seem rather something accidental and given; moreover, these same relations and the operations connected with them are also routinely conducted one after the other, as diverse, with no notice of any internal connectedness. But it is easy to recognize the presence of a guiding principle; it is none other than the immanent principle of analytical identity, an identity that in diversity appears as equality; progression is the reduction of the unequal to ever greater equality. To give an example from the first elementary operations, addition is the combining of quite accidentally unequal numbers; multiplication, on the contrary, is the combination of equal numbers, upon which there then follows the relation of equality of number of times and unit, and the relation of powers then comes in'.

- 'The Science of Logic'

But it is one-sided to suppose that Analysis is nothing but subjective (subjective idealism), just as it is one-sided for Analysis to say that it adds nothing to the object it encounters (realism). The truth of Analysis is that it is a mediation of these one-sided views. It is two things in one: a positing that no less immediately determines itself as a presupposing. The two moments, however, must not be separated. The highest point Analysis can reach is the discovery of an abstract essence. But this process of discovery falls into a bad infinity. The discovered essence is merely an appearance, which itself has an essence. A discovered effect has a cause, which itself has a cause. In these activities, Analysis supposedly adds nothing. Kant's synthetic a priori principles are praised as pointing to the unity of self-consciousness. But Kant takes this unity as a given. Kant spared himself the trouble of demonstrating the genuinely synthetic progress - the self-producing Notion, Hegel complains.

Arithmetic. Against Kant, Hegel thinks arithmetic and the sciences of discrete magnitude (i.e., algebra) are analytical. Analytic cognition starts with some given thing that possesses a contingent manifoldness. Arithmetic and algebra have already been purged of peculiarity and have been rendered abstract. At the end of this abstraction, one is left with one. If one is rendered plural, or if plurality is unified into a sum, this is done externally. How numbers are further combined and separated depends solely on the positing activity of the cognizing subject. The name of this one which external reflection subdivides and rejoins is magnitude - the category within which numbers are separated and combined.

Analysis

'Psyche Honoured by the People', c. 1695/7, Luca Giordano

Magnitude is therefore a determinateness that is indifferent to how it is determined. In magnitude there are no immanent determinatenesses. Nothing here can be the stuff of Cognition. All relations located within are contingently arrived at. Nevertheless, any relation so located has a guiding principle - equality analytic identity. Progress in the science of discrete magnitude consists in the reduction of the unequal to an ever greater equality. Thus, addition involves combining (potentially) unequal numbers. But multiplication produces the opportunity for powers. Presumably this is progress because the power relation portends quality - not just quantity.

Once a problem is given contingently to a mathematician, further operations between them is wholly analytic.

'Now because the determinateness of the subject matter and of the relations is a posited one, any further operation with them is also wholly analytic; accordingly, analytical science has not so much theorems as it has problems. The analytical theorem contains the problem as already resolved on its own terms; the wholly external distinction that attaches to the two sides which it equates is so unessential that, as theorem, it would appear to be a trivial identity. To be sure, Kant has declared the proposition 5 + 7 = 12 to be synthetic, because the same is exhibited on the one side in the form of a plurality, 5 + 7, and on the other in the form of a unity. But, if the analytic proposition is to mean more than just the totally abstract identity and tautology of 12 = 12 and is to contain any progression within it at all, then there must be present some sort of distinction, though not one based on a quality, on a reflective and still less conceptual determinateness. 5 + 7 and 12 are absolutely the very same content; but the first side also expresses the demand that 5 and 7 be combined into one expression, that is to say, that just as 5 is the product of a process of counting that was arbitrarily interrupted but might just as well have been carried farther, so now the counting is to be resumed as before with the stipulation that the ones to be added should be seven. The 12 is therefore the result of 5 and 7 and of a pre-set operation which is by nature also a completely external and thoughtless act, one that a machine can also therefore perform. Here there is not the slightest transition to an other; what there is, is the mere continuation, that is, the repetition, of the same operation that produced 5 and 7'.

- 'The Science of Logic'

In fact, it should be recognized that mathematics does not contain theorems - only problems. Analysis 'solves a problem but cannot prove a theorem' says G. R. G. Mure. 'For Hegel, a mathematical definition is a synthesis adopted from outside mathematics itself, whereas a theorem is a synthesis which is internal and necessary to it', adds Antonio Moretto. As we shall see, however, Hegel reserves for theorem the notional form that mathematics cannot possibly comprehend. Kant declared arithmetic to be synthetic.

We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two—our five fingers, for example, or like Segner in his Arithmetic five points, and so by degrees, add the units contained in the five given in the intuition, to the conception of seven. For I first take the number 7, and, for the conception of 5 calling in the aid of the fingers of my hand as objects of intuition, I add the units, which I before took together to make up the number 5, gradually now by means of the material image my hand, to the number 7, and by this process, I at length see the number 12 arise. That 7 should be added to 5, I have certainly cogitated in my conception of a sum = 7 + 5, but not that this sum was equal to 12. Arithmetical propositions are therefore always synthetical, of which we may become more clearly convinced by trying large numbers. For it will thus become quite evident that, turn and twist our conceptions as we may, it is impossible, without having recourse to intuition, to arrive at the sum total or product by means of the mere analysis of our conceptions. Just as little is any principle of pure geometry analytical. 'A straight line between two points is the shortest', is a synthetical proposition. For my conception of straight contains no notion of quantity, but is merely qualitative. The conception of the shortest is therefore fore wholly an addition, and by no analysis can it be extracted from our conception of a straight line. Intuition must therefore here lend its aid, by means of which, and thus only, our synthesis is possible'.

- 'The Science of Logic'

Hegel disagrees. In analysing 5+7=12, a plurality (5+7) appears on one side and a unity (12) on the other. But, unless analysis means the tautology of 12=12, analysis must always encounter difference. Just because plurality is reduced to a unity does not mean arithmetic is synthetic. Rather, 5+7 contains the demand that 5 and 7 be unified in a single expression. The plus sign (+) constitutes a demand, and the solution is obtained by simply following this demand. In fact, the combination of 5 and 7 is simply counting. And 12 is not different in kind from counting to 5 and then counting 7 more units until 12 is reached. Synthetic propositions require proof. But 5+7=12 does not. The same process of counting and then breaking off at 5 is used to count seven more units and break off at 12. It is, therefore, an utterly superfluous bit of scaffolding to insist on geometry-style proofs to the analytics of arithmetic.

'It is, therefore, a supremely superfluous piece of scaffolding to apply here the form of geometrical method that goes with synthetic propositions, and to add to the problem, over and above the solution, a proof as well. Such a proof can express no more than the tautology that the solution is correct because the prescribed operation has been performed. If the problem is to add several numbers, then the solution is to add them; the proof shows that the solution is correct because addition was prescribed and addition was performed. If the problem involves more complex expressions and operations, as for instance the multiplication of decimal numbers, and the solution only states the mechanical procedure, a proof will then indeed be necessary; but it can consist in nothing more than the analysis of the expressions and of the operation from which the solution proceeds of itself. By this separation of the solution as a mechanical procedure, and of the proof as a reminder of the nature of the subject matter to be treated and of the operation itself, we lose precisely the advantage of the analytic problem, namely that the construction can be derived directly from the problem and presented, therefore, as intelligible in and for itself; in the other way, the construction is expressly given a defect which is typical of the synthetic method. – In higher analysis, especially in connection with the relations of powers, where qualitative relations of discrete magnitudes dependent on conceptual determinacies come into play, the problems and the theorems do of course contain synthetic determinations; in these cases, other expressions and relations than are given by the problem or theorem must be taken as intermediary links. But here also, the determinations enlisted as an aid must be such as to be based on recalling or developing one side or other of the problem or theorem; the look of synthesis comes solely from the fact that the problem or theorem has not as yet already identified that side. – For instance, the problem of finding the sum of the powers of the roots of an equation is solved through the examination and then the joining of the functions that are the coefficients of the equation of the roots. The determination of the functions of these coefficients and their link here enlisted as an aid is not already expressed in the problem, but for the rest the development is totally analytical. The same applies to the solution of the equation x(to the power of m) – 1=0 with the aid of the sine, and also to its immanent algebraic solution, famously discovered by Gau?, which takes into consideration as an aid the residuum of x(to the power of m–1 – 1) divided by 1, and the so-called primitive roots – one of the most important extensions of analysis in modern times.40 These solutions are synthetic, for the determinations enlisted in their aid, the sine or the examination of the residua, are not a determination of the problem itself'.

- 'The Science of Logic'

The proof can express; nothing but the tautology that the solution is correct because the operation set in the problem has been performed. That is to say, the problem states add 5 and 7. The solution simply performs according to this demand.

Synthesis

'Cupid Flying Away from Psyche', between 1872 and 1881, Edward Burne-Jones

'Bóg skrzydlaty' 'Winged God'

When power relations are considered, synthesis does occur. Some examples Hegel gives: a (to the power of n) + a (to the power of n - 1) ... + a = c. Such a problem is not strictly analytic, because a mathematician must bring something to the table in order to find the solution. The connection of coefficient and root is not pre-expressed in the problem. The same is true for finding the solution to the solution of the equation x (to the power of m) – 1=0 with the aid of the sine which may be a misprint. The task is to solve for x (to the power of m) = 1. With the use of imaginary numbers, x (to the power of m) = 1 has m solutions. If m = 3, then x3 = l had 3 solutions: 1; (- l/2 + l/2 √3); (- l/2 + l/2i √3), where i =√-l. Each of these solutions is in the form cosA + i(sinA), where A=0°, A= 120°, and A=240°.

Raising x to the power of 2 indicates a qualitative change. x and x squared exist on different qualitative levels, and qualitative change implies synthesis. That is, X is a line, - X (to the power of 2) is a two-dimensional plane, X(to the power of 3) is a three dimensional space. In power relations, other expressions and relationships must be taken as intermediate terms besides those immediately specified by the problem or theorem. No longer is the arithmetical expression solved by mere counting. Rather, analysis becomes synthetic when it comes to deal with determinations that are no longer posited by the problems themselves.

'We have said that analysis becomes synthetic when it comes to determinations that are no longer posited by the problems themselves. But the general transition from analytic to synthetic cognition lies in the necessary transition from the form of immediacy to mediation, from abstract identity to difference. Analysis in general restricts its activity to determinations in so far as these are self-referential; yet by virtue of their determinateness they also essentially refer to an other by nature. We have already remarked that analytic cognition remains such even when it advances to relations that are not an externally given material but are rather thought determinations, since for it these relations are also given. But because the abstract identity which this cognition knows to be solely its own is essentially an identity in difference, even as such it must be cognition’s own identity, and the connection as well must become for the concept one which is posited by it and is identical with it'.

- 'The Science of Logic'

Synthesis. Synthetic Cognition, The transition from analysis to synthesis is the transition from immediacy to mediation - "from abstract identity to difference." (793) Dialectical Reason states that, when Analysis makes determinations within the one object (in arithmetic style), it trafficks in difference, which infers a relation of one thing to an other. That is to say, if the object is divided, then there must be a divider - an external reflection - that made it happen. Analysis is therefore guilty of suppressing this other, which Dialectical Reason now brings to the fore. Even on analytical terms, this suppressed other is an external reflection. The relation between terms within the object is imposed on it, in so far as the object is concerned. The Notion therefore does not escape from subjectivity in the refuge of Analysis. If Analysis is apprehension (of some self-identical thing), Synthesis is comprehension (of the thing in relation to its other). It aims at grasping the multiplicity of determinations in their unity.

'Analytic cognition is the first premise of the whole syllogism – the immediate reference of the concept to the object. Identity is, therefore, the determination which analytic cognition recognizes as its own, and analytic cognition is the apprehension of what is. Synthetic cognition aims at the comprehension of what is, that is, at grasping the manifoldness of determinations in their unity. It is, therefore, the second premise of the syllogism, the one in which the diverse as such is connected. Its aim, therefore, is necessity in general. – The diverse terms that are combined stand, on the one hand, in a relation in which they are both connected yet mutually indifferent and self-subsistent; but, on the other hand, they are linked together in the concept which is their simple yet determinate unity. Now inasmuch as in a first moment synthetic cognition passes over from abstract identity to relation, or from being to reflection, it is not the absolute reflection of the concept that the latter recognizes in its subject matter; the reality that the concept gives itself is the next stage, namely the said identity in diversity as such, an identity that equally is, therefore, still inner, and only necessity; it is not the subjective identity existing for itself, hence not as yet the concept as such. Synthetic cognition, therefore, does also have for its content the determinations of the concept, the object is posited in them; but they stand only in relation to one another or in immediate unity, and for that very reason not in the unity by which the concept exists as subject'.

- 'The Science of Logic'

Diverse elements are rendered necessarily related in Synthesis. That is to say, the elements are both diverse and related. In Synthesis, Notion posits thoughts {Begriffsbestimmungen) and then relates these to its other thoughts. But the relations are immediate unities "and just for that reason, not in the unity by which the Notion exists as subject. These unities belong to the Notion determinations merely implicitly. They appear for the moment to have their unity externally imposed on them, and for this reason Synthesis is still entrenched in finitude. 'There is no internal, dialectical development of the object' explains Stanley Rosen. Synthetic cognition finds laws to govern the relation between specific Notion determinations, but these laws, not yet notional, are fixed, finite, and subjective. Thought merely cognizes the ground of phenomena from the phenomena themselves

'This is what constitutes the finitude of this cognition. Because the identity which this real side of the idea has in it is still an inner one, the determinations of that identity are still external to themselves; and because the identity is not as subjectivity, the concept’s own specific presence in the subject matter still lacks singularity; although what in the object corresponds to the concept is no longer the abstract but the determinate form of the concept and hence the concept’s particularity; the singularizing element in the object is nevertheless still a given content. Consequently, although this cognition transforms the objective world into concepts, what it gives to it in accordance with conceptual determinations is only the form; as for the object in its singularity, in its determinate determinateness, this it must find; the cognition is not yet self-determining. It likewise finds propositions and laws, and proves their necessity; but it proves the latter not as a necessity inherent in a fact in and for itself, that is to say, it does not demonstrate it from the concept; it proves it rather as the necessity inherent to a cognition that delves into given determinations, into phenomenal differences, and cognizes for itself the proposition as a unity and relation, or cognizes the ground of appearance from the appearance itself'.

- 'The Science of Logic'

This is the moment of Scientific Laws. For Hegel, science means the Logic. Accordingly, when Scientific Law reveals itself, it reveals itself in the form of the Notion. The moments of the Universal, Particular and Individual are made manifest.

Scientific Laws

Definition. Objectivity is still given, but now it has the form of the Notion. So the Understanding proposes that the object, subject to Scientific Law, conforms to the law of the Notion; it recognizes within the object the Universal, Particular and Individual. Notion has revealed itself, and, in its empirical appearance, the very Notion of the Individual can be derived. In the process of Definition, Subjective Notion comes up with an Individual to be defined. Such an Individual is "an immediate that is posited outside the Notion, since the Individual is not yet self-determining.

'Definition, in thus reducing the subject matter to its concept, gets rid of the externalities that are requisite for its concrete existence; it abstracts from what is added to the concept in its realization, whereby the concept issues first into idea and secondly into external concrete existence. Description is for representation; it collects this extra content that belongs to reality. But definition reduces this wealth of manifold determinations of the intuited existence to its simplest moments; what is contained in the concepts are the form of these simple elements and how they are determined with respect to one another. The subject matter is thus apprehended, as we just said, as a universal which is determined at the same time. The subject matter is the third, the singular in which genus and particularization are posited in one – an immediate which is posited outside the concept, for it is not yet self-determining'.

- 'The Science of Logic'

So Cognition is still subjective, as it deals with an external, given starting point. The object is not yet seen as identical with the subject. The discovered Individual, however, is placed in some genus and is thereby made Universal. But this Universal contains within it the principle of differentiation of the species - that is, the Universal is proximate genus. By this principle the Universal makes itself Particular. These activities together comprise Definition. All these moments are Definition recognized when the object is defined. Because it has multiple moments, the object is seen as a manifold. Yet Definition cannot see all these moments at once. Definition must simplify what it finds by shedding inessential material. That is to say, Definition concerns itself with the form of the object. It follows from this that Synthesis is contingent knowledge in two senses. First, its content is a datum - an object separate from the subject. Second, whatever quality the subject chooses to focus on in the manifold object is a contingency. It could have focused on some other aspect. Definition separates the essential and inessential features-the thing-in-itself from its phenomena. Yet, if there is one thing we know by now, the thing-in-itself is not transcendent; it exists at the level of phenomena. Definition has no way of telling which moment is phenomenon and which is thing-in-itself. This is the archetypical problem of Definition, which Synthetic Cognition cannot overcome.

Definition

'Cupid and Psyche', 1907, Edvard Munch

Definition is a bad infinity. Any given Particular of the object has a Universal that grounds it. But we can know this Universal only as a Particular. If we discover a Universal as ground to a Particular, it is itself a Particular with yet a higher Universal, and for this again a higher, and so on to infinity.

'Now, although with division or with the particular the distinction of universal and particular is duly introduced, this universal is nevertheless itself already something determinate and therefore itself only a member of a division. Hence there is a higher universal for it, and a higher yet for this other universal, and the same for the next all the way to infinity. There is no immanent limit to the cognition under consideration here, because it proceeds from a given and it is the form of abstract universality that defines its 'first'. Any object, therefore, that seems to possess an elementary universality is made the subject matter of a specific science: it makes an absolute beginning because ordinary acquaintance with it is presupposed, and the assumption made is that it stands on its own with no need of derivation. Definition takes it as an immediate'.

- 'The Science of Logic'

This is Definition's inherent problem. It starts from a given Individual, and with empirical science generally, which does not derive its starting point. Its starting point is always given. 'A real definition cannot be given in one isolated proposition, but must elaborate the real history of the object, for its history alone explains its reality' explains Marcuse. Definition is therefore still subjective. In matters of Definition the author is king, for the end that they are to serve is a determination created out of the subjective resolve.

'This last respect requires closer consideration. Since singularity is a determinate way of existing in and for itself, it escapes the conceptual determination proper of synthetic cognition. There is in fact no principle, therefore, for determining which aspects of the subject matter are to be regarded as belonging to its conceptual determination and which only to its external reality. In the case of definitions, this constitutes a difficulty which for synthetic cognition cannot be eliminated. A distinction must nonetheless be made here. – In the first place, so far as the products of self-conscious purposiveness are concerned, it is easy enough to discover their definition, for the purpose which they should serve is a determination that is generated by a subjective resolution and constitutes the essential particularization, the form of the concrete existent, on which alone everything depends here. The further nature of the material of the existent thing or its other external properties, in so far as they correspond to the purpose, are contained in the thing’s determination; the rest are unessential for it'.

- 'The Science of Logic'

Hegel carries this point into geometry. Geometrical objects are only what they are meant to be.

'Secondly, geometrical objects are abstract determinations of space; the underlying abstraction, the so-called absolute space, has lost all other concrete determinations and now possesses no further shapes and configurations than are posited in it; essentially, therefore, such shapes and configurations are only what they are intended to be; their conceptual determination in general, and more proximately their specific difference, have unfettered reality in them; in this respect, therefore, they are the same as the products of external purposiveness, and in this they also agree with the objects of arithmetic in which the underlying determination is also only one that has been posited in them. – Of course, space has yet further determinations: its tri-dimensionality, its continuity and divisibility which are not first posited in it by external determination. But these belong to the material under consideration and are immediate presuppositions; synthetic relations and laws are produced only through the combination and the interweaving of these subjective determinations with this distinctive nature of their field into which they have been imported. – In the case of number determinations, since they are based on the simple principle of the one, their combination and further determination is only an entirely posited product; on the other hand, determinations in space, which for its part is a continuous externality, run a further course of their own and have a reality that exceeds their concept, but it no longer belongs to the immediate definition'.

- 'The Science of Logic'

The thought of them is the same as the reality of them. They therefore resemble the products of external purposiveness and the subject matter of arithmetic. Geometric shapes do have some natural features, such as continuity, divisibility, and tri-dimensionality - presuppositions in so far as geometry is concerned. Synthetic propositions within geometry entail the combinations and entanglement of subjective thoughts. Geometrical logic is non-notional. Hegel criticizes geometrical proofs for being non-notional and hence not proper proofs. They are not genetic in quality but achieve their ends by exploiting extraneous materials to get the job done. Although geometric proof eventually functions, on its own account this operation is unintelligent, since the end that directs it is not yet expressed. Accordingly, geometric proof is a subjective act lacking objectivity.

'This hidden purpose becomes apparent in the proof. This contains, as stated, the mediation of what the theorem declares as bound together, and it is only by virtue of this mediation that this connectedness first appears as necessary. Just as the construction lacks on its own the subjectivity of the concept, so is the proof a subjective act lacking in objectivity. For since the content determinations of the theorem are not posited at the same time as determinations of the concept, but are posited instead as indifferent parts standing in a multitude of external relations to one another, it is only in the formal, external concept that the necessity manifests itself. The proof is not a genesis of the relation that constitutes the content of the theorem; the necessity is present only to insight, and the whole proof is only for the subjective interest of cognition. It is for this reason a thoroughly external reflection that proceeds from the outside to the inside, that is, arrives in conclusion at the inner constitution of the relation on the basis of external circumstances. These circumstances, which the construction has presented, are a consequence of the nature of the subject matter; here they are converted instead into the ground and the mediating relations. The middle term, the third term in which the terms linked in the theorem present themselves in their unity and that provides the nerve of the proof, is therefore only something in which the connectedness appears and is external. Because the sequence which the proof goes through is rather the reverse of the nature of the fact, what is considered in the proof as ground is a subjective ground, one that brings out the nature of the fact only for cognition'.

- 'The Science of Logic'

For the point that admixture of empirical elements into reason is the basis of Hegel's critique of geometry, see Antonio Moretto. Physics too comes under scrutiny. It takes forces or other inner and essence-like forms which are then placed in the forefront in order that they may provide a general foundation that is subsequently applied to the individual. One enters into physics, Hegel complains, only if its presuppositions are blindly taken for granted. One commentator finds in these passages 'a rather passable description of Newton's method of demonstrative induction. What Hegel described here is exactly what Newton in fact does in his derivation of the universal law of gravity where we find 'Phenomena' and metaphysical 'Rules', including the reality of gravity itself 'placed in the forefront,' and the explanantia are, in part, deduced from the explenandum. (James W. Garrison).

=====

Definition concerns itself with taxonomy, where Universality is discovered empirically. Time may vindicate taxonomy. But the selected essence may also prove to be transitory.

'But, thirdly, in the case of the definitions of concrete objects, of nature as well as of spirit, the situation is quite different. For representation, such subject matters are in general things of many properties. In their case all depends on apprehending what is their proximate genus, and then what is their specific difference. We have to determine, therefore, which of the many properties pertains to the subject matter as genus, which as species, and which among these properties is the essential one; this further involves recognizing how the properties hang together, whether one is already posited with the other. For this, however, no other criterion is yet available than existence itself. – For the definition, in which the property is to be posited as simple undeveloped determinateness, the essentiality of the property is its universality. But in existence this universality is empirical; it is a universality in time (whether the property persists while the rest ostensibly come and go within the permanence of the whole), or a universality resulting from comparison with other concrete wholes, in which case it does not get beyond commonality. Now if comparison gives as a common foundation the total habitus, such as is empirically given, then reflection must gather it together into one simple thought determination and grasp the simple character of the resulting totality. But the only possible attestation that a thought determination, or any single one of the immediate properties, constitutes the simple and determinate essence of the subject matter is its derivation from the concrete constitution of the latter. But this would require an analysis that transforms the immediate elements of this constitution into thoughts and reduces their concreteness to a simple thought determination; and this is an analysis of a higher order than the one just considered, for it would not be abstractive; on the contrary, in the universal it should still retain the singular character of the concrete, should unify it and show that it is dependent on the simple thought determination'.

- 'The Science of Logic'

In effect, Definition contents itself with marks - determinations in which essentiality may or may not reside. A single external trait is not adequate to the concrete totality of the individual. By way of an example is the observation that only man has an ear lobe. Is the ear lobe, then, the distinctive feature of man? To Synthetic Cognition, It is quite contingent whether the marks adopted in the definition are pure makeshifts like this, or on the other hand approximate more to the nature of a principle.

'The connections of the manifold determinations of immediate existence to the simple concept would however require theorems, and these need proof. But definition is the first, still undeveloped concept, and in that it has to apprehend the simple determinateness of the subject matter, and this apprehension is to be something simple, it can employ for the purpose only one of the subject’s immediate so-called properties, a determination of sensuous existence or of representation; the singling out, then, of this property through abstraction is what constitutes the simplicity, and for universality and essentiality the concept must resort to empirical universality, to persistence under altered circumstances and to the reflection that seeks the determination of the concept in external existence and in pictorial representation – seeks it, that is, where it is not to be found. – Defining, therefore, by its own doing also forfeits the true concept determinations that would by essence be the principles of the subject matter, and contents itself with marks, that is, determinations in which that they are essential to the subject matter is a matter of indifference and whose only purpose is rather to be markers for external reflection. – Any such single, external determinateness is too disproportionate with respect to the concrete totality and to the nature of its concept to justify its being singled out or to assume that a concrete whole would find in it its true expression and determination. – For example, as Blumenbach observes, the lobe of the ear is something lacking to all other animals and is therefore perfectly entitled, in accordance with ordinary ways of speaking about common and distinguishing markers, to be used as the distinctive characteristic in the definition of the physical human being. But how disproportionate such a totally external determination at once appears when measured against the representation of the total habitus of the physical human being, and against the demand that the concept determination shall be something essential! It is entirely accidental whether the markers taken up into the definition are pure makeshifts like this one or approximate the nature of a principle instead. From their externality one can also see that the cognition based on concepts did not begin with them; it was rather an obscure feeling, an indeterminate but profound sense, an intimation of the essential that preceded the discovery of genera in nature and spirit, and only afterwards was a specific externality sought for the understanding. – Since in existence the concept has entered into externality, it has unfolded into its differences and cannot be absolutely attached to any single one of such properties. The properties, as the externality of the thing, are external to themselves; for this reason, as we demonstrated in the sphere of appearance in connection with the thing of many properties, do the properties essentially become even self-subsistent matters; spirit, regarded from the standpoint of appearance, turns into an aggregate of many independent forces. Regarded in this way, the single property or force, even where it is posited as indifferent to the other properties, ceases to be a characterizing principle, with the result that the determinateness, as the determinateness of the concept, vanishes completely'.

- 'The Science of Logic'

In any case, Notion cannot be isolated in just one property. Properties are the externality of the thing and are even external to themselves. Another problem that Definition must face is what today we would call the genetically defective individual. Such an individual proves that there is a difference between Notion and its actualization. Notion needs non-notional nature to secure its external presentation. Yet Notion also encompasses Negative Judgement - the judgment that a predicate is not Universal. Bad specimens are part of the process of nature, Hegel implies. This theme has long been implicit: But Appearance is the simply affirmative manifold variety which wantons in unessential manifoldness; its reflected content, on the other hand, is its manifoldness reduced to simple difference as he explained earlier:

'This content, besides being in general the simple element of the transient, is also a determined content, varied in itself. It is the reflection of appearance, of the negative determinate being, into itself, and therefore contains determinateness essentially. Appearance is however the multifarious diversity of immediately existing beings that revels in unessential manifoldness; its reflected content, on the other hand, is its manifoldness reduced to simple difference. Or, more precisely, the determinate essential content is not just determined in general but, as the essential element of appearance, is complete determinateness; the one and its other. Each of these two has in appearance its subsistence in the other, but in such a way that it is at the same time only in the other’s non-subsistence. This contradiction sublates itself; and its reflection into itself is the identity of their two-sided subsistence, namely that the positedness of the one is also the positedness of the other. The two constitute one subsistence, each at the same time as a different content indifferent to the other. In the essential side of appearance, the negativity of the unessential content, that it sublates itself, has thus gone back into identity; it is an indifferent subsistence which is not the sublatedness of the other but rather its subsistence'.

- 'The Science of Logic'

The problem genetic defect poses for Definition is that Definition has no real way of knowing whether the particular aspect it chooses for taxonomic purposes is notional or non-notional. There is no property against which an instance cannot be brought in which the property taken as its characteristic shows itself immature or stunted. Thus for example the essentiality of the brain for physical man is contradicted by the instance of acephalous individuals, the essentiality of the protection of life and property for the state, by the instance of despotic states and tyrannous governments.

'In the concrete things, together with the diversity of the properties among themselves, there also enters the difference between the concept and its realization. The concept has an external presentation in nature and spirit wherein its determinateness manifests itself as dependence on the external, as transitoriness and inadequacy. Therefore, although an actual thing will indeed manifest in itself what it ought to be, yet, in accordance with the negative judgment of the concept, it may equally also show that its actuality only imperfectly corresponds with this concept, that it is bad. Now the definition is supposed to indicate the determinateness of the concept in an immediate property; yet there is no property against which an instance could not be adduced where the whole habitus indeed allows the recognition of the concrete thing to be defined, yet the property taken for its character shows itself to be immature and stunted. In a bad plant, a bad animal type, a contemptible human individual, a bad state, there are aspects of their concrete existence that are defective or entirely missing but that might otherwise be picked out for the definition as the distinctive mark and essential determinateness in the existence of any such concrete entity. A bad plant, a bad animal, etc., remains a plant, an animal just the same. If, therefore, the bad specimens are also to be covered by the definition, then the empirical search for essential properties is ultimately frustrated, because of the instances of malformation in which they are missing; for instance, in the case of the physical human being, the essentiality of the brain is missing in the instance of acephalous individuals; or, in the case of the state, the essentiality of the protection of life and of property is missing in the instance of despotic states and tyrannical governments. – If the concept is maintained despite the contradicting instance and the latter is declared, as measured by the concept, to be a bad specimen, then the attestation of the concept is no longer based on appearance. But that the concept stands on its own goes against the meaning of definition; for definition is supposed to be the immediate concept, and can therefore derive its determinations of the subject matter only from the immediacy of existence and justify itself only in what it already finds there. – Whether its content is in and for itself truth or contingency, this lies outside the sphere of definition; but for this reason, because the singular subject matter under consideration may well be a bad specimen, formal truth, or the agreement of the concept subjectively posited in the definition and the actual subject matter outside it, cannot be established'.

- 'The Science of Logic'

It cannot suffice for Cognition to assert that the acephalous individual is non-notional; such an assertion implies that Notion is not empirical after all. Michael Inwood thinks the acephalous human stands for the proposition that, sometimes, concepts are inherently not erroneous. This is precisely the opposite of Hegel's point: all empirical observation is problematic. Definition is supposed to be the immediate Notion, and therefore can only draw on the immediacy of existence for its determinations for objects, and can justify itself only in what it finds already to hand. Definition is therefore a sham because it can never name the essential characteristic that reveals the Notion of the object.

Division. Definition cannot reliably pluck the flower Notion from the nettle of its outward manifestation, because it may have before it a non-notional flower. Rather, phenomena must be divided into essential and inessential manifestations. Essential manifestations are the ones that flow from Universality. The Universal must particularize itself, and yet it still partakes of Negative Judgement. Accordingly, we know that some manifestations are Universal, some not. The necessity for division lies in the Universal itself. The Universal is divided within itself. In Division, the Universal Division disjoins itself and renders itself into some Particulars and some non-notional materials. This is the progress "proper to the Notion and the basis of a synthetic science [and] of systematic cognition.

'A transition is now introduced which, since it takes place from the universal to the particular, is determined by the form of the concept. Definition is as such something singular; a greater number of definitions pertains to a greater number of subject matters. The advance from the universal to the particular characteristic of the concept constitutes the basis and the possibility of a synthetic science, of a system, and of systematic cognition'.

- 'The Science of Logic'

The Universal is prius (das Erste), the originating principle. In the sphere of Actuality, the concrete Individual was the prius. But in Cognition, the prius must be "something simple, something abstracted from the concrete.

'The first requirement for this is that, as indicated, the beginning be made with the subject matter in the form of a universal. In the realm of actuality, whether of nature or spirit, it is the concrete singularity that is given to subjective, natural cognition as first. But in a cognition which is a conceptual comprehension, at least inasmuch as it has the form of the concept for its basis, it is the simple, abstracted from the concrete, that on the contrary comes first, for only in this form does the subject matter have the form of a self-referring universality and of an immediacy that accords with the concept'.

- 'The Science of Logic'

In this form alone the subject matter has notional form. Being empirical, the laws that govern division are "formal, empty rules that lead to nothing. So the business of cognition can only consist, partly, in setting in order the abstract elements discovered in the empirical material, and, partly, in finding the universal determinations of the particularity by comparison.

'Division is the immediately next step after this starting point. For this advance, only an immanent principle would be required, that is, a beginning from the universal and the concept; but the cognition under consideration here lacks any such principle, for it follows only upon the form determination of the concept without that form’s immanent reflection, and therefore takes the determinateness of the content from what is given. There is no specific reason for the particular that enters into the division, whether with respect to what constitutes the basis of the division, or with respect to the specific relation that the members of the disjunction are supposed to have to one another. Consequently, in this respect the business of cognition can only consist partly in orderly arranging the particularities found in the empirical material, and partly also in discovering universal determinations of this material by means of comparison. Such determinations then count as grounds of division, of which there can be a variety, just as there can be an equal variety of divisions based on them. The relation of the members of a division to one another, the relation of the species, has only this one universal determination, namely that the members, the species, are determined relative to one another in accordance with the assumed ground of division; if their differentiation were to rest on some other consideration, their order would be arranged along different lines accordingly'.

- 'The Science of Logic'

As always, comparison (Vergleichung) stands for non-notional attributes supplied by external reflection, to which the object is indifferent.

Division

Theorem. In Definition, each Particular is taken as Universal. Division isolates Particularity from Universality. Now Cognition moves on to Individuality and Theorem. In Theorem, the object is cognized in its reality, in the conditions and forms of its real existence.

'The third stage in this advance of cognition based on concept determinations is the transition of particularity to singularity; this stage constitutes the content of the theorem. To be considered here, therefore, is the self-referring determinateness, the internal differentiation of the subject matter and the connection of the differentiated determinacies to one another. Definition contains only one determinateness, division contains determinateness as against the other; in singularization the subject matter has parted internally. Whereas definition stops at the universal concept, in theorems the subject matter is known in its reality, in the conditions and the forms of its real existence. Together with the definition, therefore, the subject matter exhibits the idea, which is the unity of the concept and reality. But the cognition being considered here, a cognition that is still a seeking, does not attain this presentation, for in it reality does not proceed from the concept, and therefore the dependency of reality on the concept and consequently the unity itself is not cognized'.

Definition contained only one determinateness. As Hegel said in the 'Jena Logic' definition 'expresses singularity'. Division contained related determinatenesses. Theorem, the unity of these two, represents Idea, the unity of the Notion and reality. (In accord with its dictionary definition, Theorem is proven, or mediated, material. It represents all mediation in the universe d, e, f, mediated by a single Individual mind g. According to Hegel, the stated Theorem definition of Theorem is "the genuinely synthetic aspect of an object in so far as the relationships of its determinatenesses are necessary.

'Now according to the definition just given, the theorem is the properly synthetic element of a subject matter, because the relations of its determinacies are necessary, that is, are grounded in the inner identity of the concept. In definition and division the synthetic element is a connectedness held together externally; what is found given is brought into the form of the concept, but, as given, the entire content is only displayed; in the theorem, on the contrary, it ought to be demonstratively displayed. Since this cognition does not deduce the content of its definitions and of the principles of division, it seems that it might also spare itself the proof of the relations expressed by the theorem and be satisfied here too just with perception. But what distinguishes cognition from mere perception and representation is the form of the concept in general that it imparts to the content; this is done in definition and division; but since the content of the theorem proceeds from the concept’s moment of singularity, it consists in determinations of reality that no longer have as their relation just the simple and immediate determinations of the concept; in singularity the concept has gone over to otherness, to reality, thereby becoming idea. The synthesis contained in the theorem no longer has, therefore, the form of the concept to justify it; it is a joining together of such as are diverse; consequently, the unity not yet thereby posited still remains to be demonstrated; here proof thus becomes necessary to this cognition itself'.

- 'The Science of Logic'

In other words, Theorem states that objects are collaborations between subject and object. Theorem represents Hegel's ontology, compared to the True in The True, which was the collapse of Kant's epistemology. In Definition and Division, the synthetic element was added externally. What was given in these stages was assumed to have the form of the Notion, but, as given, the entire content is merely presented, whereas the theorem has to be demonstrated. Theorem joins the Notion's subjective creations together and is therefore Idea. Yet Theorem is flawed. It is still occupied in seeking. The reality it deals with does not expressly proceed from the Notion. It cannot find itself in any reality that is merely given to it. Theorem is no better than the earlier cognitions. It has no principle to distinguish the necessary from the unnecessary.

Theorem

'Marriage of Psyche and Love', Francois Boucher', (1703 - 1770)

Dedicated to the One. My muse. For whom I have to thank for these ceaseless outpourings. I have never been so productive. A marriage of true minds indeed.

'Sonnet 116'

by William Shakespeare

Let me not to the marriage of true minds

Admit impediments. Love is not love

Which alters when it alteration finds,

Or bends with the remover to remove.

O no! it is an ever-fixed mark

That looks on tempests and is never shaken;

It is the star to every wand'ring bark,

Whose worth's unknown, although his height be taken.

Love's not Time's fool, though rosy lips and cheeks

Within his bending sickle's compass come;

Love alters not with his brief hours and weeks,

But bears it out even to the edge of doom.

If this be error and upon me prov'd,

I never writ, nor no man ever lov'd.

Coming up next:

The Idea of the Good.

To be continued ...