I.2.2. Quantum

Quantum—which, first, means Quantity having any determinateness or limit at all—is, in its complete determinateness, Number. It is divided into secondly (a) Extensive Quantum, where the Limit is the Barrier of a plurality there existing; (b) Intensive Quantum or Degree, the plurality having passed over into being-for-self. This Barrier being for itself, and therefore as Limit indifferent, is equally immediately outside itself and has its determinateness in an Other. Thus Quantum is a posited contradiction, it is simply determined for itself and also has its determinateness outside itself and thus refers beyond itself; it therefore thirdly passes over, as that which is itself posited as external to itself, into Quantitative Infinity.

Quantity is Quantum; it has a Limit, whether it be taken as continuous or as discrete magnitude. The distinction between these two species is here of no importance.

As transcended Being-for-Self, Quantity in itself already is indifferent to its Limit. For that very reason Limit (or the fact that it is a Quantum) is not indifferent to it; for it contains the One, or absolute determinateness, in itself as its own moment, which thus posited in its continuity or oneness is its Limit, but yet remains that One which it has become.

This One is thus the principle of Quantum; but it is the One of Quantity. Hence, first, it is continuous, or Unity; secondly, it is discrete, plurality of Ones which is in itself (as in continuous magnitude) or posited (as in discrete magnitude): these Ones are equal to one another, they have this continuity, this same unity. Thirdly, this One is also negation of the many Ones as simple Limit, an exclusion of otherness, or a determination of itself in opposition to other Quanta. One is thus Limit (α) referring to itself, (β) inclusive, and (γ) exclusive of Other.

Quantum completely posited in these determinations is Number. Complete positedness is implied by the existence of the Limit as plurality and its consequent distinctness from unity. Hence Number appears as discrete magnitude, but in unity it also has continuity. Hence too it is Quantum in complete determinateness: for in it Limit is determinate plurality, having for principle One, the absolutely determinate. Continuity, where One is only in itself or as transcended (posited as unity)—is the form of indeterminateness.

Quantum, merely as such, is limited generally; its limit is its abstract and simple determinateness. But since it is Number, the Limit is posited as being manifold within itself. It contains the many Ones which constitute its Determinate Being, but not in a merely indeterminate manner: the determinateness of the Limit falls within it. The Limit excludes other Determinate Being, that is, other Manies: the Ones included in it are a determinate heap, an Amount. This is discreteness as it is in Number, and hence its other is Unity, the continuity of this. Amount and Unit constitute the moments of Number.

With regard to Amount, it must further be considered in what manner the many Ones of which it consists are in the Limit; it is correct to say of Amount that it consists of the Many, for in it the Ones do not exist as transcended, but in it they are, posited only with the excluding Limit, to which they are indifferent. The Limit is not, however, indifferent to them. The relation of Determinate Being to Limit (as was seen when that category was treated) was such that Determinate Being, as the affirmative, remained on this side of its Limit, while the Limit (or negation) remained outside, at the edge; similarly, with the many Ones, the abrupt exclusion of other Ones appears as a determination falling without the included Ones. But it was seen above that Determinate Being is penetrated by Limit and is coextensive with it, and that thus something is inherently limited, that is, finite.—Thus Number—which is Quantitative—is imagined in such a way that with one hundred, for instance, the hundredth One alone shall limit the Many in such a manner as to make them one hundred. This is correct; yet among the hundred Ones none is preferred individually, since all are equal: each equally is the hundredth; they all equally belong to the Limit which makes the number one hundred, and, for the determinateness of the number, not one can be spared. Thus the others do not constitute relatively to the hundredth One a Determinate Being external to the limit, or distinct from it within it. Amount then is not a plurality as opposed to the inclusive and limiting One, but rather itself constitutes the limitation which is a determinate Quantum: the Many constitute one number, One two, One ten, One hundred, and so forth.

Now the limiting One is determinateness against Other, the distinguishing of one number from another. But this distinguishing does not become qualitative determinateness; it remains quantitative and appertains only to external comparing reflection: Number, as One, has returned upon itself, and at this resting-place it remains indifferent to Others. This indifference of Number to Others is its essential determination: by virtue of it, Number is determinate-in-itself, but also in itself external.—It is thus a numerical One, as absolutely determinate; it has the form of simple immediacy, and therefore relation to Other is quite external to it. And as being One which is Number, it further has determinateness (in so far as it is relation to Other) as its moments in itself, in its distinction between Unit and Amount; and Amount itself is plurality of Ones, that is, in itself it is this absolute externality.—This contradiction of Number, or of Quantum in general, in itself, is the quality of Quantum: this contradiction develops itself in the further determinations of the latter.

Spatial and numerical magnitude are generally considered as two species, each being a determinate magnitude as much as the other; and it is thought that, the distinction being only in the different determinations of continuity and discreteness, they rank equally as Quantum. Geometry, in spatial magnitude, generally has continuous magnitude for object; and arithmetic, in numerical, has discrete magnitude for object. But, the objects being thus different, they are not limited or determined in the same manner or with the same completeness. Spatial magnitude is merely limited in the abstract; if it must be considered as a definitely determinate Quantum, Number is required. Geometry as such does not measure spatial figures—it is not the art of measuring: it only compares. In its definitions too the determinations partly are derived from equality of sides and angles, or from equidistance. Thus the circle, since it is based solely upon the equidistance of all possible points in it from a centre, can be determined without Number. These determinations, based on equality and inequality, are true geometry. But they are not sufficient, and others, such as triangle or quadrilateral, require Number, which in its principle, the One, contains determinedness-for-self, not determinedness by the aid of an Other, hence not by comparison. It is true that, in the point, spatial magnitude has a determinateness corresponding to the One; but the point, in so far as it passes out of itself, becomes an Other, becomes the line; and, being essentially the One of Space, it becomes, when related, a continuity, in which its puncticity (or determinedness-for-self, or the One) is transcended. In so far as determinedness-for-self is to preserve itself in self-externality, the line must be imagined as a multitude of Ones, and the Limit must contain the determination of many Ones; that is, the magnitude of the line—like other spatial determinations—must be taken as Number.

Arithmetic contemplates Number and its figures; or, rather, it operates with them and does not contemplate. For Number is indifferent determinateness and inert; it must be actuated and brought into relation from without. The different methods of relation are the species of calculation. In arithmetic they are enumerated in series, and it is evident that they are mutually dependent. But arithmetic does not give prominence to the thread which guides their progress. However, the systematic arrangement which the exposition of these elements in the text-books justly claims, results easily from the conceptual determination of Number itself. We will here briefly notice these cardinal determinations.

The principle of Number is the One, and for this reason it is a purely analytic figure; its elements are connected externally, not internally. It is thus only an external creation; for which reason, too, all arithmetical operations are the production of numbers,—numbering, or, more closely, adding together. Variations in this external production, which ever proceeds in the same manner, can depend only on relative diversity between the numbers which are to be added together; such diversity itself must come from another source, and from external determination.

The qualitative distinction which constitutes the determinateness of Number is, as we have seen, that which subsists between Unit and Amount; hence all conceptual determinateness which can arise in the species of calculation reduces itself to this. But the distinction proper to numbers as Quanta is external identity and external difference, or equality and inequality; and these are moments of reflection, and must be dealt with when we deal with the determinations of Essence (under the heading of Difference).

And further we must premise that numbers can in general be produced in two ways—by joining, or by disjoining what was already joined. Each takes place in a manner of numbering determined in the same way; and thus to the joining of numbers there corresponds what may be called a positive kind of arithmetic; and to disjoining, a negative; the determination of each species of calculation itself is independent of this contrast.

1. We proceed to indicate the species. The first creation of Number is the joining together of Manies as such, that is, such as are posited each as One only,—Numbering. The Ones are external to one another, and are thus imagined under a sense-image: the operation which creates the number becomes a process of counting fingers, dots, and so forth. We can only point to an example of four, or of five, and so on. And the point where the process of joining is broken is (since the Limit is external) contingent and arbitrary.—The distinction between Amount and Unit which arises in the process of the species of calculation, becomes the basis of a system of numbers, such as the dyadic, decadic, and so on; in general such system rests on the choice of the amount which is consistently to be taken as unit.

The numbers produced by numbering are once more numbered: posited thus immediately they are determined as mutually quite unrelated, and indifferent to equality and inequality; their relative magnitude is contingent; they are hence altogether unequal. This is Addition. We discover that 7 and 5 make twelve by adding (by counting on our fingers or otherwise) 5 more Ones to the 7; the result is afterwards learnt by heart: but there is nothing internal here. We similarly learn that 7 × 5 = 35 by counting on the fingers, and so forth, one seven being added to another, this process being gone through five times, and the result again being learnt by heart. The trouble of numbering, the discovering of sums or products, is abolished by the finished “one and one” or multiplication-tables, which have only to be learnt by heart.

In the Introduction to the Critique of Pure Reason, V, Kant has considered the proposition that 7 + 5 = 12 as a synthetic proposition. “At first,” he says, “one should think [of course!] that it is merely an analytical proposition, resulting from the concept of a sum of seven and five by the Law of Contradiction.” The concept of a sum is just this abstract determination that these two numbers ought to be joined together, and that too (since they are numbers) in an external and conceptless manner,—that we are to count on from seven until the Ones which are to be added (their amount being determined as five) are exhausted; the result bears the name independently known as twelve. “But,” so Kant continues, “if we look more closely, we find that the concept of the sum of seven and five contains no more than the joining of these two numbers into one; it is not thought at all what this single number is which comprehends these two;” “—however much I may analyse my concept of such a possible sum, I shall never find twelve therein.” The transition from the problem to the result has indeed nothing to do with the thinking of the sum and the analysis of the concept; and he adds, “We must pass beyond these concepts and take to intuition, to five fingers and so on, adding the five ones given by intuition to the concept of seven.” Of course five is given intuitively, that is, it is an entirely external agglomeration of the thought of one, arbitrarily repeated-; but equally seven itself is no concept; there are no concepts beyond which a transition here is made. The sum of 5 and 7 is the conceptless connexion of these two numbers, and the concept-less process of numbering from seven on, until five are exhausted, can literally be called a synthesis, which name can also be given to the process of numbering from one onwards;—but this synthesis is wholly analytical in nature, for the connexion is quite artificial and nothing is introduced that is not at hand externally. The postulate of adding 5 to 7 is to the postulate of numbering at all as is the postulate of producing a straight line to that of drawing a straight line.

The expression “synthesis” is void, and equally void is the determination that it takes place a priori. Counting is not, of course, a determination of sensation (which, according to Kant’s determination of intuition, is all that remains to be predicated of whatever is a posteriori), and it is an occupation on the basis of abstract intuition: that is, it is determined through the category of the One, and abstraction is here made from all other determinations of sensation, as well as from concepts. Altogether, the a priori is quite vague; and determination of feeling, in the shape of impulse, tendency, and so forth, contains the moment of apriority as much as space and time are determined as existing, and the temporal and the spatial are determined, a posteriori.

We may add in this connexion that Kant’s assertion of the synthetic nature of the foundations of pure geometry equally lacks solidity. He asserts that several are really analytic, but cites in favour of this idea only the proposition that the straight line is the shortest between any two points. “My concept of the straight does not contain anything about magnitude, but only a Quality; thus the concept of shortest is a pure addition, which can be extracted by no analysis from the concept of the straight line: we must have recourse to intuition, and then alone synthesis is possible.”—But here again we are not dealing with a concept of the straight in general, but with that of a straight line, and this is already spatial and intuited. The determination (or, if you will, the concept) of the straight line is surely none other than this, that it is the absolutely simple line, that is, that, in passing beyond itself (the so-called movement of the point) it is simply self-related; in its extension no sort of differentiation of the determination, or relation to any other point or line without it, is posited;—it is just simple direction in itself. This simplicity is indeed its Quality; and, if it should seem hard to define the straight line analytically, this is so only because of this determination of simplicity or self-relation, and solely because, when determination is thought of, a multiplicity, or determining through others, is primarily imagined. In itself, however, it is not difficult to seize on this determination of the simplicity in itself of extension, and of its indeterminedness by Other;—Euclid’s definition contains nothing else than this simplicity.—Now the transition of this Quality to quantitative determination (namely, that of being the shortest), which is supposed to constitute the synthetic element, is purely analytical. Being spatial, the line is Quantity in general; the most simple thing, if predicated of Quantum, is “least”; and this, predicated of a line, is “shortest.” Geometry can take these determinations as corollary to the definition; but Archimedes in his books on Sphere and Cylinder (translation Hauber, p. 4) did more suitably in making this determination of the straight line fundamental: and therein he was right, as was Euclid when he placed the determination relating to parallel lines among the fundamentals; for, in order to become a definition, the development of this determination too would have required qualitative determinations not properly spatial but more abstract, namely (as simplicity above, so here) similarity of direction and the like. The ancients gave a plastic character even to their sciences; they kept their exposition strictly within the peculiarities of their material, and therefore excluded what would have been heterogeneous.

The concept advanced by Kant in the synthetic judgments a priori—the concept of differentiation with inseparability, and of identity which in itself is unseparated differentiation—belongs to the great and immortal part of his philosophy. In intuition this concept—since it is the concept itself, and everything in itself is concept—likewise is certainly present; but the determinations which have been taken in those examples do not represent it; Number, and numbering, rather is an identity and the production of an identity which is merely external, or merely a superficial synthesis; a unity of Ones, such as are posited as not identical with one another, but rather as mutually separate and external. With the straight line, the determination of being the shortest line between two points is based rather on the moment of abstract identity, without differentiation in itself. From this digression I revert to addition. The corresponding negative arithmetical process, subtraction, likewise is the purely analytical severing of numbers, which, as in addition, are determined merely as mutually altogether unequal.

2. The next determination is the equality of the numbers which are to be numbered. This equality makes them into a unity; and thus there is applied to Number the distinction between Unit and Amount. Multiplication is the problem of adding an amount of units of which each is an amount. It is here indifferent which of the two numbers is taken as unit and which as amount,—whether we say 4 times 3 (where 4 is the amount and 3 the unit) or, conversely, 3 times 4.—We have already indicated above that the original discovery of the product is effected by simple numbering, by counting on the fingers and so forth; the product, later, can be indicated immediately because there is a collection of these products (the multiplication table) which is known by heart.

Division is the negative arithmetical process with the same determination of the distinction. It is equally indifferent which of the two factors, divisor or quotient, is determined as unit or as amount. The divisor is determined as unit and the quotient as amount when the problem in division is expressed in the words that it is desired to see how many times (amount) one number (unit) is contained in a given number; conversely the divisor is taken as amount and the quotient as unit, when it is said that a number is to be divided into a given amount of equal parts, and that the magnitude of such a part (the unit) is to be found.

3. The two numbers which are determined as being related to one another as unit and amount, are still immediately opposed as numbers, and therefore are altogether unequal. The equality which is attained is next that of unit and of amount themselves: thus the progress towards the equality of the determinations immanent in the determination of Number, is completed. According to this complete equality, counting is the raising to a certain power (the negative arithmetical process being to take the root)—and, first, the squaring of a number;—numbering is here completely determinate in itself, when (1) the many numbers which are to be added are the same, and (2) their plurality, or amount, is the same as the number which is posited so many times, that is, which is unit. There are no further determinations in the concept of Number which could offer a differentiation; nor can the differentiation immanent in Number be made more homogeneous. Where a number is raised to a higher power than the square, there is a formal continuation; partly (with the even exponents) there is a mere repetition of squaring; partly (with the odd powers) inequality returns; for, although the new factor is formally equal (for example, in a first instance with the cube) both to amount and to unit, yet it stands, as unit, in a relation of inequality to the amount (the square, 3 against 3 times 3); still more in the cube of four, where the amount, 3, according to which the number which is unit should be multiplied by itself, is actually different from the latter.—Where these determinations—amount and unit—are present, they constitute in themselves the essential differentiation of the concept: in order that that which has gone outside itself may completely return upon itself, these two must be equalized. The above exposition contains the reason why partly the solution of higher equations must consist in the reduction to quadratic equations, and partly why the equations of odd exponents can only be formally determined, and, just when the roots are rational, they cannot be found otherwise than by an imaginary expression, that is, by the opposite of that which the roots are and express.—From what has been said it appears that the arithmetical square alone contains absolute self-determinedness in itself; hence the equations with further formal powers must be reduced to it; just as the right-angled triangle in geometry contains that absolute self-determinedness in itself which is set forth in the theorem of Pythagoras, and so all other geometrical figures must be reduced to it for their complete determination.

A method proceeding under the guidance of a judgment logically constituted treats of powers before it treats of proportions; it is true that the latter are connected with the difference between unit and amount, which constitutes the determination of the second arithmetical process; but they step beyond the One of immediate Quantum, in which unit and amount are only moments; and the further determination according to it still remains external to it. In Ratio, Number no longer is immediate Quantum: there it has its determinateness, as mediation. Qualitative Relation will be considered in what follows.

We may say of this exposition of the progressive determination of the arithmetical processes, that it is no philosophy of them nor demonstration of their inner meaning, since in fact it is not an immanent development of the concept. But philosophy must know how to distinguish what is in its nature a material external to itself, so that here the progress of the concept can take place only in an external manner, while its moments can have only the peculiar form of their externality, such as oddness and evenness in this example. It is an essential condition, if one would philosophize about real objects, that the spheres be distinguished in which a definite form of the concept belongs, or, in other words, is present as existence: else what is external and contingent will be disturbed in its peculiarity by ideas; and also these ideas, through the inadequacy of the material, will be distorted and formalized. Now this externality in which the moments of the concept appear in this external material—Number—is here the adequate form; they represent the object in its understanding, and further contain no speculative demands and thus appear easy: they therefore deserve to be employed in text-books of the elements.

It is well known that Pythagoras represented intellectual relations (or philosophemata) by numbers; in more modern times too numbers and the forms of their relations (such as powers and so on) have been used in philosophy, in order to regulate thoughts or to express them.—From an educational point of view Number has been considered the most suitable object of inner intuition, and the arithmetical manipulation of its relations has been considered that activity of the mind where it manifests its most peculiar relations and altogether the fundamental relations of Essence.—The concept of Number, as it has here yielded itself, shows how far Number may claim this high worth.

We have seen in Number the absolute determinateness of Quantity, and in its element differentiation which has become indifferent;—determinateness in itself, which also is posited as wholly external. Arithmetic is an analytic science, because all connexions and distinctions which occur in its object are not inherent in it, but are applied to it wholly from without. It has no concrete object containing internal relations which, at first hidden from knowledge and not given in the immediate idea of the object, have to be elaborated by the efforts of cognition. Not only does it not contain the concept and therewith the task for conceptual thought: it even is its opposite. The connected terms are indifferent to the connexion, which lacks necessity: the activity of thought, therefore, is here one which is the extremest self-renunciation, an activity which forces it to move in thoughtlessness and to connect terms which admit of no necessity. The object is the abstract thought of externality itself.

Number thus being the very thought of externality, is also the abstraction from the manifold of the senses; it has retained nothing of the sensuous except the abstract determination of externality itself; and the latter therefore approaches nearest to it in thought; it is the pure thought of the self-renunciation of thought.

Mind, which rises above the world of the senses and understands its own essence, when it seeks an element for its pure imagination, for the expression of its essence, may therefore strike upon that inner and abstract externality, Number, before it seizes thought itself as this element and achieves the purely spiritual expression fit to represent it. Hence we see, early in the history of Science, Number being used to express philosophemata. It constitutes the last step of that incomplete method which apprehends the universal as affected by the sensuous. The ancients were clearly conscious that Number is midway between the sensuous and thought. Aristotle (Metaphys. I. 5) quotes Plato as saying that the mathematical determinations of things stand apart from and between the sensuous and the Ideas, distinguished from the former because they are invisible (eternal) and unmoved, and from the latter because they are a manifold and have similarity, while the Idea is simply self-identical and one within itself.—A more detailed and profound consideration of this matter, by Moderatus of Cadiz, is quoted in Malchi Vita Pythagorae ed. Ritterhus, pp. 30 sq. He thinks that the Pythagoreans hit upon numbers because they were not yet able clearly to seize in Reason the fundamental ideas and first principles, these principles being hard to think and hard to express. Numbers serve the teacher well as designations; and the Pythagoreans imitated, herein as elsewhere, the geometers, who cannot express the corporeal in thought, and therefore use figures, saying “this is a triangle” when, however, they do not mean that this optically visible figure is to be taken for a triangle, but that only the concept is to be represented. And thus the Pythagoreans used the expression “One” for the concept of unity, of identity and equality, the basis of agreement, of connexion, of the preservation of all, of the self-identical and so forth.—It is superfluous to observe that the Pythagoreans passed beyond the expression of Number to the expression of Thought, to the express categories of equal and unequal, to limit and infinity; indeed, at the place where these numerical expressions are considered (ibid., note to p. 31,1. s., from a Life of Pythagoras in Photius, p. 722), it is mentioned that the Pythagoreans distinguished between the Monas and the One, taking the Monas as concept and the One as number, and, similarly, the Two for the arithmetical term, but the Dyas (for this seems the correct reading) for the concept of the indeterminate.—Thus these ancient writers correctly perceived the inadequacy of these numerical forms as determinations of thought, and equally rightly they demanded, in place of this first substitute for thought, its characteristic expression; how much further had they progressed in thought than those who in our day, when some put in the place of determinations of thought numbers and determinations of numbers (like powers), next the infinitely great and the infinitely small, one divided by infinity, and other such determinations, which often are a perverted mathematical formalism, take the return to this impotent childishness for something praiseworthy and even for something thorough and profound.

We quoted above the expression that number stands between the sensuous and thought, since it shares with the former the quality of being in itself the Many, or separate existence: it is here to be noted that this Many itself, the sensuous which is taken up into thought, is that category of the self-external which is proper to it. Further, concrete, veritable thoughts, which of all things are the most quick and flexible, conceived only where there is relation, when translated into this element of being beyond self become dead and rigid determinations. The richer in determinateness, and hence in relation, thoughts become, the more confused and the more arbitrary and senseless becomes their representation in such forms -as numbers. The One, the Two, the Three, and the Four, Henas or Monas, Dyas, Trias, Tetraktys, still approximate to the simple and abstract concepts; but when numbers are required to pass over into concrete relations it is a vain attempt which would keep them close to the concept.

But the hardest thing is asked of thought, when the determinations of thought by One, Two, Three, Four are designated as that movement of the concept through which alone it is concept. It moves in the element of its opposite, which is unrelatedness; its work is the work of derangement. It is a hard suggestion to conceive that One is Three (for instance) or Three One, because the One is the unrelated and, therefore, does not show in itself the determination by virtue of which it passes over into its opposite, since its very being is the absolute exclusion and negation of such a relation. Conversely, Understanding makes use of this against speculative truth (for instance against that which is laid down in the doctrine called that of the Trinity), and counts those determinations of it which constitute One Unity, in order to demonstrate it to be a clear contradiction;—that is, itself contradictorily makes unrelated that which essentially is relation. It was not expected, when the name of Trinity was coined, that Understanding would consider One and Number as the essential determinateness of the content. This name expresses contempt for Understanding, which has nevertheless confirmed itself in its vanity in clinging to One and Number as such, and has set it up against Reason.

It is, in a manner, harmless, when numbers or geometrical figures are taken as mere symbols, as has often been done with the circle, the triangle, and so on, the circle for instance standing for eternity and the triangle for the Trinity; but it is foolish to think that more is thus expressed than can be comprehended or expressed by thought. These symbols, like others created by fancy in the mythologies of peoples and in poetry generally,—compared with which the bare geometrical figures are in any case meagre,—are supposed to contain a profound wisdom and a profound meaning; if it is so, then the sole purpose of thought is, to extract the wisdom which lies only here, and not merely in symbols but also in Nature and in Spirit. In symbols truth is darkened and veiled by the sensuous element; it is wholly revealed to consciousness only in the form of thought: Meaning is only the thought itself.

But when mathematical categories are used to determine something bearing upon the method or content of philosophic science, such a procedure proves its preposterous nature chiefly herein, that, in so far as mathematical formulae mean thoughts and conceptual distinctions, such meaning must first report, determine, and justify itself in philosophy. In its concrete sciences, philosophy takes the logical element from logic and not from mathematics; it must be a mere refuge of philosophic impotence when it flies to the formations which logic takes in other sciences, of which many are only dim presentiments and others stunted forms of it, in order to get logic for philosophy. The mere employment of such borrowed forms is in any case an external attitude: a knowledge of their worth and of their meaning should precede their use; but such knowledge results only from thoughtful contemplation, and not from the authority which mathematics gives them. Logic itself is such a consciousness; and such consciousness strips them of their particular form, making it superfluous and useless, and, while correcting them, alone gives them justification, meaning, and worth.

The value of the use of Number and of Arithmetic, in so far as it is supposed to be a main basis of education, is evident from the above. Number is a non-sensuous object, and an occupation with it and its combinations a non-sensuous business: thus the mind is urged to reflect in itself and to do inner and abstract work; which is of great though one-sided importance. For, on the other hand. Number being based only on external and non-conceptual differentiation, this occupation becomes thoughtless and mechanical. The effort chiefly consists in seizing units void of concept and combining them without the use of concept. The content is the empty One: that rich content of moral and spiritual life, and its individual growth, on which, as its noblest nourishment, education should rear the young mind, would here be ousted by the empty One; and when such exercises form the main matter and the main occupation, the effect can be no other than to hollow and blunt the spirit in form and concept. Arithmetic being so extremely external and therefore mechanical a matter, it has been possible to construct machines which execute the arithmetical operations in the most perfect manner. If this one circumstance were known about the nature of arithmetic, it alone would contain a judgment about the value of the notion which would make arithmetic the chief instrument for educating the mind by putting it on the rack which would perfect it into a machine.

1. We saw that Quantum has its determinateness as limit in Amount. It is a something discrete and manifold in itself, and has no Being distinct from its limit and having this outside itself. Quantum and its limit (and the limit in itself is a manifold) together thus are Extensive Magnitude.

Extensive Magnitude must be distinguished from continuous magnitude: the former is opposed to intensive and not to discrete magnitude. Of extensive and intensive magnitude each is a determinateness of the quantitative limit itself, while Quantum is identical with its limit; continuous and discrete magnitude, on the other hand, are determinations of magnitude in itself, that is, of Quantity as such, in so far as in Quantum abstraction is made from limit.—The moment of continuity is attached to extensive magnitude and to its limit, for its Many is simply continuous; the limit as negation thus appears in this levelling of the Many, as limitation of unity. Continuous magnitude is self-repeating Quantity without regard to any limit, and, in so far as it is imagined as having a limit, this is limitation in general, without discreteness being posited in it. Quantum taken merely as continuous magnitude is not yet truly determinate for itself, for it lacks Number and the One which implies that it is determinate for itself. And similarly discrete magnitude is, immediately, only differentiated Many in general, which, in so far as it had a limit as such, would be just a multitude, that is, a something limited but limited indeterminately; in order to make it into a determinate Quantum, the Many must be subsumed into One, by which process they are posited as identical with the limit. Continuous and discrete magnitude, taken as Quantum generally, have each posited in it only one of the sides by which it is completely determined and made into Number. The latter is immediate Extensive Quantum,—simple determinateness, existing essentially as amount, though only as an amount of one and the same unit, being distinguished from Number only in this, that in the latter determinateness is expressly posited as plurality.

2. However, when the magnitude of something is determined by number, no contrast with another magnitude is needed: the determinateness of this magnitude does not depend upon itself and another magnitude, for the determinateness of magnitude in general is self-determinate, indifferent and simply self-related Limit; which, in Number, is posited as enclosed in the self-existing One, externality and relation to Other being within itself. This Many of the Limit, further, (like the Many in general) is not differentiated within itself, but continuous; each of the Many is what every other is; its determinateness as such does not consist in its quality of mutual externality of parts, or discreteness. Thus this Many for itself collapses into its continuity and becomes simple unity.—Amount is only a moment of Number, but does not constitute the determinateness of Number as a multitude of numerical Ones: the Ones, as indifferent and self-external, are transcended when Number has returned upon itself. Externality, which constituted the Ones of plurality, vanishes in the One as self-relation of Number.

Quantum, as extensive, had its existing determinateness as self-external Amount: its limit thus passes over into simple determinateness. In this simple determination of the limit, Quantum is intensive magnitude; and the limit or determinateness which is identical with Quantum is now also posited as simple—it is Degree.

Degree thus is determinate magnitude or Quantum, but it is not also a multitude or multiplicity within itself; it is only a “more”; and the quality of “more” is the many taken together in the simple determination, Determinate Being which has returned to Being-for-Self. Its determinateness requires a number (which is the complete determinedness of Quantum) to express it; but the number is not Amount but, simply, a Degree. If we speak of ten or twenty degrees, then the Quantum which has so many degrees is the tenth or twentieth degree, and not the amount and sum of them; for, if so, it would be extensive; but in fact it is only one—the tenth or twentieth—degree. It contains the determinateness implied in the amount ten or twenty, but does not contain it as many; it is Number as transcended amount or simple determinateness.

3. In Number, Quantum is posited in its complete determinateness; but as Intensive Quantum, as in the Being-for-Self of Number, it is posited as it is according to its notion, or in itself. For the form of self-relation which it has as Degree is also its self-externality. As Extensive Quantum, Number is numerical plurality, and thus contains externality within itself. The latter, as being Many in general, collapses into undifferentiatedness and transcends itself in the One of Number, which is its self-relation. Now Quantum has determinateness when taken as Amount: as shown above, it contains amount though amount is not posited in it. And so Degree, as being simple in itself, no longer has this external otherness within itself; it has it outside itself, and it stands related to it as to its determinateness. A plurality external to it constitutes the determinateness of the simple limit, which Degree is for itself. If, therefore, Amount (so far as, in Extensive Quantum, it existed within Number) there transcended itself, it passes into being posited outside Number. Number being posited as One, as intro-reflected self-relation, excludes from itself the indifference and externality inherent in Amount, and is self-related because through itself it is related to something external.

Herein Quantum has the reality proper to its notion. The indifference of determinateness constitutes its Quality; that is, determinateness which in itself exists as self-external determinateness.—Degree accordingly is a simple determination of magnitude among several such intensities, which are different, each being a simple self-relation but also being in an essential relation to the others, so that each has its determinateness in this continuity with the others. Since, then, Degree is related to its Other through itself, the ascent or descent in a scale of degrees becomes a steady progress or flow, which is an uninterrupted and indivisible mutation; not one of the several entities which are distinguished therein is separated from the others, but each has its determinateness only in them. As self-related determination of magnitude each degree is indifferent to the others; but also it is implicitly related to this externality, and is what it is only through it; in a word, its self-relation is the non-indifferent relation to the external, and in this fact consists the quality of the degree.

Degree is not self-external within itself. But it is not indeterminate One, or that general principle of Number which is no amount except that negative amount implied in the fact that it is not amount. First, Intensive Magnitude is a simple One of the heap; there are several degrees; but they are determined neither as simple One nor as heap, but only in the relation of this self-externality, or in the identity of One and plurality. So that, if members of a plurality, as such, are outside the simple degree, its determinateness consists in its relation to them: it thus contains Amount. Twenty, as Extensive Magnitude, also contains twenty discrete Ones; on the other hand, the determinate degree contains them as a continuity; and this determinate plurality is this continuity simply. It is the twentieth degree, and is such only by virtue of this amount, which as such is external to it.

The determinateness of Intensive Magnitude must therefore be considered from two aspects. It is determined through other intensive Quanta, and it is continuous with its otherness, in such a manner that its determinateness consists of its relation to this. First, then, in so far as it is simple determinateness, it is determinate relatively to other degrees; it excludes them, and this exclusion constitutes its determinateness. Secondly, however, it is determinate in itself: it is determinate in amount, as in “its” amount, not in amount as excluded nor as taken as the amount of other degrees. The twentieth degree contains twenty in itself; it is determinate, not only as distinguished from the nineteenth, twenty-first, or other degrees; but, further, its determinateness is its amount. But, in so far as the amount is its own—and determinateness also essentially exists as amount—degree is Extensive Quantum.

Thus Extensive and Intensive Magnitude are one and the same determinateness of Quantum; they only differ in that one has amount within itself, and the other without. Extensive Magnitude passes over into Intensive Magnitude because the Many of Extensive Magnitude in itself collapses into the unity to which the Many is external. But conversely this simple result has its determinateness only in amount, and in fact in “its” amount; it is indifferent to intensities otherwise determined, and, therefore, has the externality of amount in itself; and thus Intensive Magnitude equally essentially is Extensive Magnitude.

Having reached this identity, we have reached the qualitative Something; for this identity is unity self-related by virtue of the negation of its distinctions; now the latter constitute the existing determinateness of magnitude: this negative identity thus is Something, Something moreover which is indifferent to its quantitative determinateness. Something is a Quantum, but now qualitative Determinate Being, as it is in itself, is posited as indifferent towards it. Quantum, Number as such, and so forth, could be spoken of without mention of a Something such as should be their substratum. Now, however, Something stands opposed to these its determinations as existing for itself; it has become mediated with itself through their negation, and, in having a Quantum, is opposed to them as that same Something which is defined as having Extensive and Intensive Quantum. Its one determinateness proper to it as Quantum is posited in the distinct moments of Unit and Amount. Now not only is this determinateness one and the same in itself, but, when it is posited in these distinctions as Extensive and Intensive Quantum, there is a return to that unity which, as negative, is the Something which is posited as indifferent to it.

In common imagination Extensive and Intensive Quantum generally are distinguished as different kinds of magnitude, as though some objects had only intensive and others only extensive magnitude. To this has been added the view proper to a philosophy of nature, where the many, or that which is extensive (as happens, for instance, in the fundamental determination of matter—namely, that it occupies space—and in other concepts), is changed into something intensive in this sense, that the true determination was supposed to be the equation of the intensive with the dynamic; it was thought that, for instance, density (or the specific occupation of space) must be taken essentially as a certain degree of the space-occupying force of matter, and not as a certain multitude and amount of material parts in a quantum of space.

Here two determinations must be distinguished. In the so-called transvaluation of the mechanistic into the dynamic view, there occurs the concept of independent parts existing apart from one another (the nexus which makes them a whole being merely external), and the distinct concept of force. That which, as occupying space, was considered merely as a multitude of mutually external atoms, is now regarded as the manifestation of a simple latent force.—But these relations of Whole and Parts, of Force and its Manifestation, which are here opposed to one another, do not belong to this place, and will be considered later. But this much may be said, that the relation of Force and its Manifestation, which corresponds to intensity, is more valid than the relation of Whole and Parts; but that in spite of this force is equally as one-sided as intensity, and that manifestation (the externality of extensiveness) is equally inseparable from force, so that one and the same content is found alike in both forms, in the intensive as well as in the extensive.

The other determinateness which here occurs is quantitative determinateness as such, by which is meant a determinateness which is transcended as Extensive Quantum and is changed into Degree, which is supposed to be the valid determination. But it has been shown that Degree too contains the former, so that one form is essential to the other, and each Determinate Being represents its determination of magnitude as Extensive and as Intensive Quantum equally.

Everything, so far as it appears as a determinate magnitude, will serve as example for this. Number itself necessarily bears immediately in it this double form. It is amount, and in so far it is extensive magnitude; but also it is One, one Ten, one Hundred, and in so far it stands at the point of transition to intensive magnitude, since in this unity the multiple collapses into simple. One is extensive magnitude in itself, and may be imagined as any desired amount of parts. Thus the tenth or hundredth part is this simple intensity which has its determinateness in the Many which fall outside it, that is, in extensity. A number is ten or one hundred, and also is the tenth or one hundredth in the numerical system; and each determinateness is identical with the other.

In the circle the One is called Degree, because any part of a circle essentially has its determinateness in the Many external to it, and is determined as merely one of a limited amount of such Ones. As a mere spatial magnitude the degree of the circle is a common number; regarded as degree it is an intensive magnitude, which only has meaning as determined by the amount of degrees into which the circle is divided; just as Number in general has its meaning only in the numerical series.

The magnitude of a more concrete object also has this twin aspect of extensive and intensive, and manifests this in the twin determinations of its Determinate Being, in one of which it appears as external, while in the other it appears as internal. Thus, for instance, a mass, as weight, has extensive magnitude, since it constitutes an amount of pounds, hundredweights, and so forth; and intensive magnitude in so far as it exerts a certain pressure: the magnitude of pressure is simple, it is a degree which has its determinateness in the scale of degrees of pressure. Mass, as exerting pressure, appears as a Being-in-Self or Subject, which has an intensive differentiation of magnitude.—Conversely, that which exerts this degree of pressure has the power to displace a certain amount of pounds (and so on), and therein lies the measure of its magnitude.

Again, heat has a degree, and this degree (whether the loth, 20th, or any other) is a simple sensation, is subjective. But this degree equally exists as extensive magnitude, as the expansion of a liquid, of mercury in the thermometer, of air, of sound, and so on. A higher degree of temperature expresses itself as a longer column of mercury, or as a narrower sound-cylinder; it heats a greater space in the same manner as a lesser degree heats a smaller space.

The higher note, as the more intensive, is also a greater multitude of vibrations; and a louder note (to which we ascribe a higher degree) is audible in a greater space.—A more intensive colour will serve to stain a larger surface than a weaker colour can similarly stain; that which is brighter (again a different kind of intensity) is visible farther than the less bright; and so forth.

And in the spiritual sphere high intensity of character, of talent, or genius, has a similarly comprehensive being, extended action, and manifold contact. The profoundest concept also has the most universal meaning and application.

Kant has employed the application of the determinateness of Intensive Quantum to a metaphysical determination of the soul in a peculiar manner. In his critique of the metaphysical propositions about the soul (which he calls Paralogisms of Pure Reason), he comes to consider the deduction of the permanence of the soul from its simplicity. To this conclusion he objects in these terms (Critique of Pure Reason, p. 414):—“Although we grant that the Soul is of this simple nature, since it contains no manifold of separate entities, and therefore no extensive magnitude, yet we cannot deny to it, any more than to any other existent, intensive magnitude, that is, a degree of reality with regard to all its powers and in general with regard to everything that exists; this degree can decrease through an infinity of lesser degrees, so that this would-be substance can be changed into nothing by gradual relaxation (remissio) of its forces if not by division; for consciousness itself always has a degree which can still be diminished; this therefore must apply also to the faculty of self-consciousness and to all the other faculties.”—In rational psychology—and this abstract metaphysic was such—the soul is considered not as Spirit but as a merely immediate Being, as a soul-thing. Kant thus has the right to apply the category of Quantum to it “as much as to any other existent;” and, in so far as it is determined as simple, to apply the category of Intensive Quantum. It is true that Being is a part of Spirit, but a Being of an intensity wholly different from that of Intensive Quantum; an intensity rather where the form of mere immediate Being and every category of it have been transcended. Not only the removal of the category of Extensive Quantum should have been admitted: Quantum should have been removed altogether. But it is yet a further matter to understand how existence, consciousness, and finitude are in the eternal nature of Spirit and emerge from it, while it does not thereby become a thing.

The distinction between Extensive and Intensive Quantum is indifferent to the determinateness of Quantum as such. But, altogether, Quantum is determinateness posited as transcended, it is the indifferent limit and a determinateness which is also its own negation. This distinction is developed in Extensive Magnitude; Intensive Magnitude is the presence of this externality which Quantum is in itself. It is posited as its own contradiction in itself, where it is simple self-related determinateness and also that self-negation which has its determinateness not in itself but in another Quantum.

According to its Quality, therefore, a Quantum is posited in absolute continuity with its externality and otherness. Thus, it is not merely possible to pass beyond every determinateness of magnitude, and possible to alter it; but it is posited that it must alter. The determination of magnitude continues itself into its otherness in this manner, that it has its being only in this continuity with an Other: it is a limit which is not, but becomes.

The One is infinite, or is self-related negation, and therefore self-repulsion. Quantum too is infinite, being posited as self-relating negativity; it repels itself from itself. But it is a determinate One, a One which has passed over into Determinate Being and Limit: it is the self-repulsion of determinateness and not the creation of the self-equal (as happens in the repulsion of One) but of its otherness: it is now posited in itself to refer beyond itself and to become an Other. It consists in increase or diminution: it is in itself the externality of determinateness.

Thus Quantum refers itself beyond itself: it becomes an Other, and this also is, primarily, Quantum; but it is this as a limit which is not, but forces itself beyond itself. Thus the limit which arises again in this outward passage is simply one which again transcends itself and refers to yet another, and so on to infinity.

Quantum alters and becomes another Quantum; the further determination of this alteration, that it proceeds to infinity, is implied in the fact that Quantum is posited as self-contradictory.—Quantum becomes an Other; but it continues itself into its otherness; the Other too is, therefore, a Quantum. But this latter is the Other not only of a Quantum, but of Quantum itself: it is the negative of Quantum taken as limited; it is, therefore, its unlimitedness, or Infinity. Quantum is an Ought: it is determined for itself, and this is a part of it, and this very determinedness is, rather, a determinedness in an Other. And, inversely, it is transcended determinedness in an Other, it is indifferent self-subsistence.

Finitude and Infinity thus forthwith each receive a double, and in fact an opposite, meaning. Quantum is finite first because it is limited, and secondly because it refers beyond itself, and has its determinedness in an Other. And Quantum is infinite, first, in its non-limitedness, and, secondly, because, having returned upon itself, it is indifferent Being-for-Self. If, now, we compare these moments, we find that the determination of finitude in Quantum, that reference beyond itself to an Other which constitutes its determination, is likewise determination of the infinite: the negation of the limit is a similar passing beyond determinateness, so that Quantum is to have its last determinateness in this negation, the infinite. The other moment of Infinity is that Being-for-Self which is indifferent to limit, but Quantum itself is limited in such a manner that relatively to its limit (and, therefore, to other Quanta and to its beyond) it is in itself indifferent. Finitude and Infinity (of the sort which tries to be separate from the finite—the bad infinite) in Quantum each already contain the other for moment.

Qualitative and Quantitative Infinity are distinct because in the former the opposition between finite and infinite is qualitative, and the transition from the finite into the infinite, or the relation of either one to the other, subsists only in the “in-itself,” in their notion. Qualitative determinateness is immediate, and is related to otherness essentially as to another Being; and it is not posited as having its negation, its Other, in itself. But magnitude is, as such, transcended determinateness: it is posited as being both unequal and indifferent to itself, and, therefore, as being that which is alterable. Qualitative Finitude and Infinity are, therefore, opposed absolutely, that is, abstractly; their unity is the fundamental inner relation; and, therefore, the finite continues itself into its Other only in itself and not in the infinite. But the quantitative finite refers itself to its infinite in the latter itself, claiming to have its absolute determinateness in it. This relation, then, is represented by the Quantitatively Infinite Progress.

In general an infinite progress is the expression of a contradiction: here it expresses the contradiction contained in the quantitatively finite, or in Quantum in general. It is that reciprocal determination of finite and infinite which was considered in the sphere of Quality, with this difference that, as has just been pointed out, in the quantitative the limit in itself refers and continues beyond itself, and that therefore, conversely, the quantitatively infinite is posited as containing Quantum in itself,—for Quantum, in its self-externality, is also itself; its externality is part of its determination.

Now the Infinite Progress expresses but does not solve this contradiction; but, by reason of the merging of one determinateness into the other, it produces an apparent solution by uniting the two. As it is primarily posited, it makes the infinite the goal, which, however, is not reached: it is a perpetuated creation of it, while yet Quantum is never left behind and the infinite never becomes positive and present. It is part of the notion of Quantum to have a beyond. This beyond is, first, the abstract moment of the not-being of Quantum; this resolves itself in itself; and thus it relates itself to its beyond as to its infinity, according to the qualitative moment of the opposition. Secondly, however, Quantum is continuous with this beyond: it is the very being of Quantum to be its own Other, to be self-external; and thus this external part is equally nothing else than Quantum: the beyond, or the infinite, itself is a Quantum. Thus the beyond has been recalled from its flight, and the infinite has been reached. But now this “beyond” which has become “hither” once more is a Quantum; and, therefore, only a new limit has now been posited. This is a Quantum, and itself has again fled from itself; being a Quantum, it is beyond itself, and has repelled itself into its own not-being and its own beyond, which in turn becomes Quantum in the same perpetual manner in which Quantum repels itself from itself into beyond.

The continuousness of Quantum with its Other gives rise to the combination of the two in the expression of the infinitely great and the infinitely small. Both still contain the determination of Quantum, and, therefore, are variable; and the absolute determinateness, which would be a Being-for-Self, is thus not reached. This self-externality of the determination is posited in the double infinite, whose terms are in the opposition of “more” and “less,” namely, the infinitely great and the infinitely small. In each, Quantum is preserved in perpetual opposition to its beyond. However much the great is extended, it shrinks into insignificance; it refers to the infinite as to its non-being, and thus the opposition is qualitative; the extended Quantum has wrested nothing from the infinite, which remains its not-being as before. Or again, Quantum by being enlarged does not approach the infinite, for the distinction between Quantum and its infinity has this for essential moment, that it is not a quantitative distinction. This is only the expression of the contradiction brought under narrower consideration: the infinitely great is defined as great (that is, Quantum) and infinite (that is, not Quantum).—Similarly, the infinitely small, as small, is Quantum, and, therefore, remains qualitatively (that is, absolutely) too great for the infinite, and is opposed to it. In both is preserved the contradiction of the Infinite Progress, which was supposed to have found its goal in them.

This infinity which, as the beyond of the finite, is determined as stable, may be described as the “bad quantitative infinity.” Like qualitative bad infinity, it is an infinite alternation between the two members of a persistent contradiction, from the limit to its not-being, and from there back once more to the limit. In the quantitative progress the term towards which the progress moves is not indeed abstractly and altogether Other, but is merely a Quantum posited as different; but it remains in opposition to its negation in a similar manner. The progress, too, therefore, is no forward advance but a repetition of the same cycle, of positing and transcending still repeated; an impotence of the negative which by the very act of transcending is ever again faced with that which it had transcended. Two members are connected in such a way that they must fly apart; and in so doing they cannot part company, but even in flying from one another remain connected.

Bad infinity, more especially in its form of the progress to infinity of the quantitative, is often considered sublime, and this continual surpassing of the limit (which really means an incapacity to transcend it), together with the perpetual relapse into it, has been held a kind of divine service. Philosophy has considered it as ultimate. And this progress has often furnished material for outbursts which have been admired as lofty achievements. Yet this modern sublimity does not exalt the object, for that escapes, but only the subject, which consumes such vast quantities. The meagre nature of this exaltation, which, still subjective, climbs up the ladder of the quantitative, is made clear when it is admitted that it vainly labours to reach an unattainable goal, which indeed to be gained must be quite otherwise attacked. In the following outbursts of this nature it also appears whither this exaltation moves and where it ceases. Thus Kant cites it as sublime (Critique of Practical Reason: end) “when the subject rises in thought above the place which it occupies in the world of sense, magnifying into the infinitely great its relation,—a relation with stars beyond stars, worlds beyond worlds, systems beyond systems, where periodic movement, beginning, and duration are infinite in time as they are in space.—Imagination fails before this progress into the immeasurably far, where the furthest world has one yet further, and past and future, however distant, have a more distant past and a more distant future beyond; thought fails before this image of the immeasurable; as a dream, where one goes down a long passage ever further and interminably further, without an end in sight, ends in a fall or in dizziness.”

This description condenses into rich imagery the content of quantitative exaltation: it also deserves praise for the truthfulness with which it shows in what this exaltation ends: thought fails, the end is a fall and dizziness. And what causes thought to fail and produces the fall and the dizziness, is nothing else than the weariness of repetition, where the limit is made to vanish, return, and vanish again perpetually, and the hither to arise and perish in the beyond, and the beyond in the hither, one after the other and the one in the other; which causes a feeling of the impotence of this infinite and this ought, which would master the finite and cannot.

Haller’s description of eternity, too (called “the horrible description” by Kant), is often admired, though not for that aspect which constitutes its real merit:—

“I pile up monstrous numbers, Mountains of millions, I pile time upon time and world upon world; When from the horrible height Dizzy I look again on you, All power of number, taken a thousandfold, Is no part even of you. I discard it, and you are wholly revealed.”

When this heaping and piling of numbers and worlds is considered valuable as a description of eternity, it is overlooked that the poet himself declares this so-called horrible process vain and void, concluding that the true infinite reveals its presence only after this empty infinite progress has been abandoned. There have been astronomers who took pleasure in priding themselves upon the sublime nature of their science because it dealt with an immeasurable multitude of stars and immeasurable extents of space and time, where distances and periods, already great in themselves, serve as units which being taken however many times over themselves shrink into insignificance. The shallow amazement to which they here abandon themselves, and those silly hopes of travelling from one star to another in a second life, and of thus acquiring into infinity similar new knowledge, were proclaimed by them one of the chief accounts of excellence in their science,—which in truth is admirable, not because of this quantitative infinity, but rather because of those relations of measure and those laws which reason recognizes in these objects; for they are the infinity of reason and those others the infinity of unreason.

Kant opposes another infinity to that which relates itself to external sensuous intuition, when “the individual returns upon his invisible ego, and opposes the absolute freedom of his will as pure ego to all the terrors of fate and of tyranny, and, beginning with his nearest surroundings, allows them to vanish before him, next causes what seem most durable, worlds upon worlds, to collapse in ruins, and, alone, knows himself equal to himself.”

The ego in this loneliness has indeed achieved the Beyond, it has gained itself, is gathered upon itself, and is on the hither side: in pure self-consciousness absolute negativity, which ever flies in that progress above sensuous Quantum, is affirmed and made present. But while this pure ego fixes itself in its abstraction and lack of content, it has opposed to and beyond itself existence in general, or the totality of the natural and spiritual universe. The same contradiction appears on which the infinite progress is based: namely, a term has returned upon itself and also, immediately, is self-external, is related to its Other as to its not-being; and this relation remains a longing, for the ego has, on the one hand, fixed before itself, its own hollow and unstable void, and, on the other, the full content of its beyond which remains present even in the negation.

Kant subjoins to these two examples of the sublime the remark “that astonishment (for the first and external kind) and reverence (for the second and inward) stimulate investigation but cannot replace the lack of it.”—He thus declares these sublimities to be unsatisfactory to reason, which cannot remain with them and the feelings connected with them, and cannot allow the name of ultimate to the beyond and the void.

The infinite progress has been considered ultimate chiefly in its application to morality. The opposition just quoted between finite and infinite as standing respectively for the manifold world and the ego exalted in its freedom, is at this point qualitative. The ego in determining itself attempts simultaneously to determine Nature and to free itself from it: it thus relates itself to its Other through itself: and the Other, being an external existence, is a manifold and quantitative. The relation to what is quantitative itself becomes quantitative; and the negative relation of the ego to it, the power which the ego exercises over the non-ego (the world of the senses, and external Nature), is therefore imagined as meaning that morality can and ought to grow, and the power of sensuousness to wane. But the perfect adequacy of the will to the moral law is relegated to the progress into the infinite, that is, it is imagined as an absolute and unattainable Beyond; and this its unattainable nature is supposed to be the true sheet-anchor and proper consolation; for morality should be a fight; and it is such only where will is not adequate to the law, and the law therefore is its absolute Beyond.

In this opposition ego and non-ego or pure will and moral law, and Nature and the sensuousness of will are presupposed as entirely independent and indifferent to one another. Pure will has its peculiar law, which is in essential relation with sensuousness; and sensuousness and Nature have their own laws, which neither come from nor are adapted to will, nor yet, although different from it, have any essential relation to it: they are determined in themselves, finished and complete. But also both are moments of one and the same simple essence, namely, the ego: will is determined as standing in the relation of negative to Nature, so that it exists only in so far as this different sphere exists, which it transcends, but is itself touched and affected thereby in so doing. Nature, taken as the sensuous sphere of man, is an independent system of laws: it is therefore indifferent to limitation by any Other; even while it is being limited it preserves itself, enters independently into relation, and limits the will of law as much as the will of law limits it.—When will determines itself and transcends the otherness of a nature, and again, when this otherness is posited as existing, as continued into a state of becoming-transcended, and yet as not transcended—these two are one act. This contradiction is not resolved in the infinite progress, but is on the contrary represented and affirmed as unresolved and incapable of resolution: the struggle between morality and sensuousness is imagined as the self-existent, absolute relation.

Where it is thus impossible to master the qualitative opposition between finite and infinite and to seize the idea of true will, or substantial freedom, recourse is had to magnitude as mediation, for magnitude is the qualitative element transcended, or distinction become indifferent. But both the members in opposition remain, qualitatively distinct, at bottom, so that when they are mutually related as quanta each immediately is made indifferent to this change. Nature is determined by ego and sensuousness by the will to the good, and the change thus produced therein is only a quantitative difference and one which allows it to remain as it is.

In the more abstract exposition of the Kantian philosophy, or at least of its principles, which is found in Fichte’s Doctrine of Science, the infinite progress similarly is the basis and the ultimate. The first principle of the exegesis, ego = ego, is followed by another which is independent of it, namely, the opposition of the non-ego; and the relation of these two is immediately taken as being a quantitative difference, non-ego being partly determined by ego and partly not. Thus the non-ego prolongs itself into its not-being, in such a manner that it remains opposed to this latter in it as being not yet transcended. Thus after the contradictions herein contained have been developed in the system, the final result is the same relation as was the beginning; the non-ego remains an infinite “resistance-principle” (Anstoss), an absolute Other; the ultimate relation between it and the ego is the infinite progress, longing and yearning,—the same contradiction with which a beginning was made. The quantitative is determinateness posited as transcended: for this reason it was thought that much or rather everything had been gained for the Unity of the Absolute, for Monosubstantiality, when contrariety in general was reduced to a merely quantitative difference. For a time it was a main thesis of modern philosophy, that all contrariety is only quantitative; opposite determinations are of the same essence and content, they are real sides of the opposition, in so far as each contains both the latter’s determinations and both its factors; only, one factor, matter or energy, preponderates on one side, one on the other, or one is present in greater quantity or more intense degree on the one side than on the other. In so far as different kinds of matter and energy are assumed, the quantitative difference rather confirms and completes their externality and indifference both in relation to one another and to their unity. The distinction of absolute unity is supposed to be merely quantitative; the quantitative is transcended immediate determinateness, but still, as negation, it is incomplete and first negation only, not infinite nor negation of negation.—Being and Thought are imagined as quantitative determinations of absolute Substance, and therefore, being quanta, they become quite external and unrelated to one another; as, in a subordinate sphere, do carbon, nitrogen, and so on. It is a third factor, an external reflection, which abstracts from the difference between them and recognizes their unity, which is inner and only in itself, not for itself. Thus in fact this unity is imagined merely as of the first and immediate kind, or as Being which in its quantitative difference remains equal to itself, but does not through itself posit itself as equal to itself: it is thus not conceived as negation of negation, or infinite unity. It is only in the qualitative opposition that posited infinity or Being-for-Self appears; and quantitative determination passes over—as will be seen immediately—into the qualitative.

It was mentioned above that Kant’s antinomies are representations of the opposition between finite and infinite in a more concrete shape and applied to more specialized substrata of imagination. The antinomy there considered contained the opposition of qualitative finitude and infinity. In another (the first of the four cosmological antinomies) it is rather the quantitative limit of which the self-conflicting nature is considered. I shall therefore proceed here to the examination of this antinomy.

It deals with the spatial and temporal limitation or unlimitedness of the world.—Equally well the contradiction might have been considered with reference to time and space themselves, for the antinomy of limitation and unlimitedness remains in them unchanged, whether they be relations of things or only forms of intuition.

The closer analysis of this antinomy will also show that the two propositions with their proofs (which, like those considered above, are apagogic) come to nothing except these two simple and contrary assertions: “there is a limit,” and “the limit must be overpassed.”

The Thesis is as follows:—

“The world has a beginning in time, and spatially, too, is limited.”

One part of the proof, which deals with time, assumes the opposite:—

“If the world has no beginning in time, then an eternity has passed up to any given point of time; hence an infinite series of successive world-states has passed. But the infinite nature of a series consists precisely in this, that it can never be completed by successive synthesis. Thus an infinite world-series which also has elapsed, is impossible, and, therefore, a beginning of the world is an essential condition of its existence,—which was to be proved.”

The remaining part of the proof, which deals with space, is based upon time. It would require an infinite time to comprehend the parts of a spatially infinite world, and this time would have to be regarded as expired in so far as the world in space is to be regarded not as a process but as a given whole. But it was shown of time in the first part of the proof that it is impossible to assume the expiration of an infinite period of time.

But it is immediately evident that it was unnecessary to make the proof apagogic, or, indeed, to give any proof, since it contains immediately for basis an assertion of that which was to be proved. For each, or any, point of time is here assumed, up to which an eternity has expired (—eternity here meaning merely infinite time in the bad sense). Now a given point of time means nothing else than a definite limit in time. In the proof, therefore, a limit of time is assumed to exist; which is precisely what was to be proved. For the thesis is, that the world has a beginning in time.

There is only this one distinction, that the limit of time which is assumed is “now” as end of an already expired period, while that which is to be proved is “now” as beginning of the future. But this distinction is immaterial. “Now” is taken as the point where an infinite series of consecutive world-states has ended: it is taken as end, or qualitative limit. If this “now” were considered merely as quantitative limit, which remains fluid and not only must be overpassed but itself is nothing but an overpassing of itself, then the infinite time-series in it would not be expired but would continue to flow; and the argument of the proof would vanish. However, the point of time is taken as qualitative limit for the past, but this is also the beginning of a future (for in itself every point of time is the relation of past and future), and, further, is its absolute or abstract beginning, that is, that which was to be proved. It is irrelevant that this future and this its beginning are preceded by a past; in so far as this point of time is qualitative limit—and this is implied in the determination of expired or completed, that is, not self-continuing—time in it is broken off short, and that past is without relation to this time which could be called future only with reference to that past, and consequently when unrelated becomes time in general, which has an absolute beginning. But if—as is the case—through the “now” (or given point of time) it were linked to the past, and so determined as future, then this point of time too, seen from the other side, would be no limit; the infinite time-series would continue itself in that which was before called future and would not be, as was assumed, completed.

In truth time is pure Quantity: the point of time at which it is supposed to be interrupted, which is employed in the proof, is simply the self-transcending Being-for-Self of the Now. The proof effects nothing, except to present to imagination the absolute limit of time, which is asserted in the Thesis, as a given point of time, and to take it boldly as a complete or abstract point;—a popular determination which sensuous imagination easily allows to pass as limit, and thus allows the proof to assume what before was set forth as to be proved.

The Antithesis is this:—

“The world has no beginning and no limits in space, but is infinite both with regard to time and to space.”

The proof likewise assumes the opposite:—

“Let the world be assumed to have a beginning. Now the beginning is an existence preceded by a time in which the thing does not exist; therefore a time must have preceded in which the world did not exist; that is, an empty time. Now in an empty time nothing can arise, for no part of such a time has in itself, and in preference to any other, the differentiating condition of its existence or non-existence. Therefore any series of things can begin in the world, but the world itself cannot begin and with regard to past time is infinite.”

Like the others this apagogic proof contains the direct and unproved assertion of that which it ought to prove. At first it assumes a Beyond of the existence of the world, namely, empty time; but next it also continues the existence of the world beyond itself into this empty time, thus cancelling it, and continuing this existence into infinity. The world is an existence; the proof presupposes that this existence arises, and that the arising has a temporally precedent condition. But the Antithesis just consists in this, that there is no unconditioned existence and no absolute limit, but that the existence of the world always demands a precedent condition.—What was to be proved is thus already present in the proof as an assumption.—Further, the condition is now looked for in empty time, which means that it is taken as temporal and therefore as an existence and as limited. Thus altogether the assumption is made that the world as existence presupposes another conditioned existence in time, and so to infinity.

The proof with regard to the spatial infinity of the world is the same. The spatial finitude of the world is posited in the apagogic manner: “the world would thus be in an empty and unlimited space and would stand in a relation to it; but such a relation of the world to no object is Nothing.”

There again the proof directly presupposes what was to be proved. It is directly assumed that the limited spatial world must be in an empty space and stand in a relation to it, that is, that it must be overpassed;—a process leading on the one hand into the void, the beyond and not-being of the world, and, on the other, meaning that it stands in a relation to the void, that is, that it continues itself into it and therefore that the beyond is to be imagined as filled with world-existence. The spatial infinity of the world which is asserted in the Antithesis is just (a) empty space and (b) the relation of the world to it, that is, the continuity of the world in it, or the filling of empty space; and this contradiction—that space is empty and also is full—is the infinite progress of Existence in space. And this contradiction itself, the relation of the world to empty space, is directly made the basis of the proof.

The Thesis and the Antithesis and their proofs thus represent nothing but the contrary assertions that there is a limit, and that this limit equally is cancelled, that the limit has a beyond, but that it is in relation with the latter and must pass over to it; which again gives rise to such a limit which is no limit.

The solution of these antinomies, as with the first, is transcendental, that is, it consists in the assertion of the ideality of space and time as forms of intuition,—in this sense, that the world in itself is not in contradiction with itself and does not transcend itself, but that consciousness in its intuition and in the relation of intuition to understanding and reason is a self-contradictory essence. But it is an excessive tenderness towards the world to remove from it the contradiction and to transfer it into Spirit or Reason, allowing it to remain there unresolved. In point of fact it is Spirit which is strong enough to support the contradiction, but it is also Spirit which knows how to resolve it. As for the so-called world (whether it be called objective or real world, or, according to the transcendental idealism, subjective intuition and sensuousness determined by the category of understanding), it never and nowhere is without the contradiction, but, since it cannot support it, is subject to arising and passing away.

1. Infinite Quantum, as infinitely great or infinitely small, is, in itself, the infinite progress; being either great or small, it is Quantum; and it is also the not-being of Quantum. Hence the infinitely great and the infinitely small are pictures of the imagination, which, closely considered, prove mere mist and shadow. But in the infinite progress this contradiction is present explicitly, and with it that which is the nature of Quantum which has reached its reality as intensive magnitude and is now posited in its determinate existence as it is in its concept. This identity must now be considered.

Quantum as degree is simple, self-referred, and self-determinate. Through this simplicity, otherness and determinateness are cancelled in it, and therefore the latter is external to it: it has its determinateness outside it. This self-externality is, first, the abstract not-being of Quantum in general, or bad infinity. But, further, this not-being is itself a magnitude; Quantum continues itself into its not-being, for it is in its externality that it has its determinateness; therefore this externality, too, is itself Quantum; and its not-being, or infinity, is thus limited, that is, this beyond is transcended and is determined as Quantum, which thus, in its negation, is gathered to itself.

But this is just what Quantum as such is in itself. For it is itself because it is outside itself: externality constitutes that by virtue of which it is Quantum and is with itself. Thus in the infinite progress the concept of Quantum is posited.

If we take it first in its abstract determinations as we find them, it contains the transcendence of Quantum, but also of its Beyond; there is thus the negation of Quantum as well as the negation of this negation. In its truth it is the unity in which these are contained, but as moments only.—This truth is the solution of the contradiction of which the progress is the expression, and its nearest meaning is therefore the rehabilitation of that concept of magnitude where it is indifferent or external limit. In the infinite progress as such the only reflection made generally is that every Quantum, however great or small, can vanish, that progress may take place beyond it; but the reflection, that this its transcendence, the Beyond, and the bad infinite also vanish, is generally not made.

The first transcendence, the negation of Quality in general, whereby Quantum is posited, is already in itself the transcendence of negation, Quantum being transcended qualitative limit and therefore transcended negation; but also it is this in itself only; it is posited as a Determinate Being, and hence its negation is established as the infinite, as the Beyond of Quantum, which takes up the place of Hither or of immediate: thus the infinite is determined only as first negation, and as such it appears in the infinite progress. But it has been shown that this contains more, namely, the negation of negation or that which is in truth the infinite. This was looked at before in such a way that the concept of Quantum was thus restored; and this restoration means, first, that its Determinate Existence has received a closer determination: for Quantum determined according to its concept has now arisen, and this differs from immediate Quantum; externality is now its own opposite; it is posited as a moment of magnitude itself, and Quantum is posited as having its determinateness in another Quantum through the mediation of its not-being, namely, of infinity; that is, it is qualitatively that which it is. However, this comparison of the concept of Quantum with its existence belongs rather to our reflection, to a relation which is not yet here at hand. The nearest determination is this, that Quantum having returned to Quality is now determined qualitatively. For Quality, which is its characteristic, is externality or indifference of determinateness; and it is now posited as, in its externality, being itself, and as self-relating, having its determination in simple self-unity, that is, qualitatively.—This Qualitative is also more closely determined, it is determined as Being-for-Self; for the relation to itself which it has reached has been evolved from mediation, from the negation of negation. Quantum is infinite and is determined for itself: and this is no longer external to it, but within itself.

The infinite, which in the infinite progress had the mere empty meaning of a not-being, a Beyond not reached but searched for, is, in fact, just Quality. Quantum as indifferent limit passes beyond itself into the infinite, and in so doing seeks after nothing else than determinateness-for-self, the qualitative moment which hitherto, however, is no more than Ought. Quantum is Quantum, because of its indifference to the limit and because, therefore, it has no self-subsistent determinateness and so overpasses itself: this its overpassing is to be negated and its absolute determinateness is to be found in the infinite.

And, quite generally, Quantum is Quality transcended; but Quantum is infinite, overpasses itself, and is its own negation: thus this overpassing is in itself the negation of negated Quality, and its restitution; and so it is posited that externality, which appeared as a Beyond, is determined as own moment to Quantum.

Quantum thus is posited as self-repelled: there thus are two Quanta, which, however, are transcended and exist only as moments of one unity: this unity is the determinateness of Quantum.—Quantum then, related to itself as indifferent limit in its externality and therefore qualitatively posited, is the Quantitative Ratio.—In Ratio, Quantum is external to and different from itself: this its externality is the relation of one Quantum to another, each of which has value only in this its relation to its Other; and this relation constitutes the determinateness of Quantum, which exists as such a unity. Here its determination is not indifferent, but qualitative: in this its externality it has returned upon itself, and in it is that which it is.