Remark 1

Spatial magnitude and numerical magnitude are commonly regarded as two kinds, as though spatial magnitude were for itself just as much determinate magnitude as numerical magnitude is; their difference would rest solely upon the differing determinations of continuity and discreteness; as quantum, however, they would stand upon the same level. Geometry does indeed have, broadly speaking, continuous magnitude in spatial magnitude for its object, and arithmetic discrete magnitude in numerical magnitude. But with this unlikeness of object there goes no likeness in the manner and completeness of the limiting, or of the being-determined. Science essentially considers the determinatenesses of these objects insofar as they are quanta and comport themselves on this side. The manner of limiting, however, is in the two objects likewise different. Spatial magnitude has only a limiting at large; insofar as it is to be considered as a quantum determined in itself, it stands in need of number. Geometry, too, does not consider spatial figures according to a magnitude determined in and for itself; it does not measure them; is no art of mensuration; but only compares them, that is, it considers them only as relative quanta, according to a determination of magnitude that they have toward others. In its definitions as well the determinations are drawn in part from the equality of sides, of angles, from the equal distance. Thus the circle, resting as it does solely upon the equality of the distance of all points possible within it from a midpoint, calls for no number to determine it. These determinations, founded upon equality or inequality, are geometrical through and through. But they do not suffice, and for others, the triangle, the quadrangle for instance, number becomes requisite, number holding within it the being-determined-in-itself, not the being-determined by the aid of an other, hence not by comparison.

Number, however, holds this determinateness in itself, because the one is its principle. Spatial magnitude does indeed have in the point the determinateness answering to the one; but the point, insofar as it comes outside itself and becomes an other, turns into the line; because it essentially is only as a one of space, in the reference it turns into a continuity wherein punctuality, the being-determined-in-itself, the one, is sublated. Insofar as the being-determined-in-itself is to preserve itself within being-outside-itself, the line must be represented as a multitude of ones, and the limit must take up into itself the determination of the many ones, that is, the line's magnitude — and likewise that of the other determinations of space — must be taken as number.

Remark 2

As is well known, Pythagoras set forth relations of reason or philosophemes in numbers, and in more recent times calculating has been taken as equivalent in meaning to thinking, or, as it has been put more precisely, to pure real thinking. — In pedagogical respect, too, number has been held for the object best suited to inner intuiting, and the calculating occupation with number's relations for that activity of spirit wherein spirit brings its ownmost relations, and the fundamental relations of essence at large, before intuition. — How far number may be entitled to this high worth follows from its concept, as that concept has emerged.

Number is the absolute determinateness of quantity; its element is the difference grown indifferent. It is therefore determinateness in itself, determinateness at the same time posited wholly and only externally. Arithmetic is on that account an analytic science, because all the linkings and differences occurring upon its object do not already lie within the object itself but are inflicted upon it wholly from without. It has no concrete object that would possess inner relations in itself, relations at first hidden from knowledge, not given in the immediate representation of the object, but such as would first have to be brought out by the labor of cognition. On the contrary, the object's relations are put into it purely by reflection itself; in its calculating business reflection therefore has to do only with such determinations as it has put in. Because these references accordingly hold no genuine otherness within them, arithmetic has nothing to do with what is opposed; it does not have the task of the concept at all; it moves only along the thread of its own identity, and comports itself in its activity purely analytically.

On account of the indifference of what gets linked toward the linking, a linking that wants necessity, thinking finds itself here in an activity that is at the same time the utmost divestment of its own self, in the violent activity of moving within thoughtlessness and of linking what is capable of no necessity. For the object, number, is only the thought, and the abstract thought, of externality itself. In every other concrete object thinking is likewise external to itself, yet such an object is at the same time upon its own self something inwardly linked and necessary; thinking accordingly finds essential references within it; number, by contrast, has for its principle what is essentially without reference.

On account of this pure externality and this want of determination of its own, thinking has in number an infinitely determinable matter that offers no resistance through references peculiar to it. Number is at the same time the abstraction from all sensuous manifoldness, and has kept nothing of the sensuous but the abstract determination of externality itself. Through this abstraction it lies, so to speak, nearest to thought; it is only the pure thought of thought's own divestment.

Spirit, which raises itself above the sensuous world and knows its essence, may therefore, while it searches for an element to serve its pure representation, the expression of its essence, hit upon choosing number, this inward, abstract externality, before it grasps thinking itself as this element and wins for its presentation the purely spiritual expression. That is why in the history of science we see number put to use for the expression of philosophemes, before thinking had found the expression that holds only the abstract thought itself. Number makes up the last stage of this expression's imperfection; with number, thinking, which has already left sensuous representation behind for its presentation, leaves behind altogether even the pure thought of externality.

Now since thinking lays its determinations down in this element, they sink therein, on account of the element's nature just considered, immediately into want of concept; or the thoughts become, within it as the thoughtless, something thoughtless. The thoughts, what is most alive, most mobile, grasped only in relating, become in this element of being-outside-itself dead, motionless determinations. The richer in determinateness and in reference the thoughts grow, the more confused on the one side and the more arbitrary and drained of sense on the other does their presentation in numbers become. The one, the two, the three, the four, as henas or monas, dyas, trias, tetraktys, still lie very near to simple concepts; but where numbers are to go over into further relations of the concept, wanting to keep them still near the concept is in vain.

But even if the concept is held fast only in the one, two, three, four, if these are to be thought and set in movement, this is the hardest movement of thinking; for instead of having to do purely with itself and being at home with itself, thinking has at the same time immediately to struggle with its own divestment. It moves in the element of its opposite, of referencelessness; its business is the labor of derangement. To grasp, for instance, that one is three and three one, is so hard a demand because the one, which is dominant in number, is what is without reference, and hence does not display upon its own self the determination whereby it goes over into what is opposed to it, but is rather just this, to shut out and refuse such a reference outright.

Since, then, thought purifies itself of sensuous stuff, the last stage is that the sensuous, the external, becomes for it the pure thought of this externality, becomes number, and that it takes number for the element and matter of its own self. But it has still to overcome this abstract thoughtlessness too, and to grasp its determinations in its own immediate form, namely as being, becoming and so forth, as essence, identity and so forth.

As for the view of common calculating itself, that it is thinking, because it „is a determination of the relative plurality, or of the determinable „repeatability of one and the same in an other, through the absolute unity of the identical,“ then to that extent calculating is of course thinking. But reading, writing and so forth is quite as much thinking; for in these too there is a determination of a relative many through an identity. Calculating has, as has come out, on the one side the advantage over other functions of thinking or of consciousness in the abstractness of its matter or element; but on the other side it stands behind them through the conceptlessness of the one, which is indeed something purely identical with itself and repeating itself in an other, namely in the many, yet is therein to hold itself essentially as without reference and to remain external to its own other, so that the genuine, namely the comprehending, unity of thinking is to be absent within it.

What is to be made of the use of number and of calculating, insofar as it is supposed to make up a chief pedagogical foundation, follows of itself from what has gone before. Number is a non-sensuous object, and the occupation with it and its combinations a non-sensuous business; spirit is thereby held to reflection into itself and to an inward abstract labor. On the other side, however, since external, thoughtless difference lies at the ground of number, that business turns at the same time into a thoughtless, mechanical business, and the straining of powers consists chiefly in killing the liveliness of spirit, in suppressing the concept, in holding fast what is void of concept and in combining it without concept. Because calculating is so very external, and hence mechanical, a business, machines have, as is well known, been made that carry out the arithmetical operations most perfectly. Were one to know of the nature of calculating this single circumstance alone, there would lie in it the verdict upon what it comes to when calculating is made spirit's chief business, and spirit is laid on the rack of perfecting itself into a machine.

B. Extensive and Intensive Quantum

1. Their Difference

1. The quantum has its determinateness as limit in the amount. It is something discrete within itself, a many that is bounded; this many, as was shown, has no being for itself that would be distinct from its limit and would keep that limit outside it. For it is precisely within number that plurality makes up the determinateness over against the unit; the one as unit is indeed determined in itself as numerical one, yet as unit it is the indeterminate continuity, undifferentiated within itself; difference, otherness, it holds by way of plurality. Plurality therefore holds the moment of the limit, of negation, within number itself; the difference-in-itself accordingly resides in the amount.

The quantum is thus a manifold, and this plurality is one with its limit; as limit, as determinate quantum, it is a manifold at its own self. So taken it is extensive magnitude.

The extensive magnitude is to be distinguished from the continuous; what stands directly over against it is not the discrete but the intensive magnitude. Extensive magnitude is what lies asunder in its determinateness, or is such insofar as the limit is a manifold; it has the moment of continuity insofar as, upon it and likewise within its limit, this many shows itself as something continuous and the limit shows itself as negation at this equality of the many. Continuous magnitude, however, is quantity carrying itself onward with no regard for a limit, or, insofar as it is represented as having a limit, that limit falls outside that continuity and is a bounding […]ndig[…]in general, without discreteness being posited upon it. — Continuous magnitude is not yet the magnitude truly determined in itself, since it lacks the many ones wherein being-determined-in-itself resides; its limit is therefore outside it, and not yet number. — Just so, discrete magnitude is in its determination immediately no more than a differentiated many in general, which, were it as such supposed to have a limit, would be a mere multitude, that is, something bounded indeterminately and externally. — Insofar as continuous and discrete magnitude are both quantum, however, they are number according to the quantum's true determination, and this is at first as extensive quantum, — the determinateness that is essentially as amount, though as amount of one and the same unit.

2. The extensive quantum is the limit that is manifold within itself. It has the differentiated other at its own self, and for that reason number is what is completely determined at its own self. The determination, by way of number, of how large something is does not require a difference from something else that is large, as though the determinateness of this large thing called for both itself and another large thing; it is a limit determined in itself, and thereby an indifferent limit, related simply to itself. The many of the limit, however, is, like the many in general, nothing unequal within itself but something continuous; each of the many is what the other is; hence its being a many lying asunder, or discrete, does not make up the determinateness as such. This many, then, of its own accord falls back together into its continuity and becomes simple unit. — The many here, however, was not a many for itself in general, but the determination of the many, amount over against the unit. But number is the one of unit and of amount, or the unity that has returned into itself out of the diversity of these determinations. Therein the amount is only a moment, or is sublated; it therefore does not make up the determinateness of number as a multitude of numerical ones; rather these, as indifferent and external to themselves, are sublated in number's having returned into itself; the externality that made up the ones of plurality vanishes in number's reference to itself.

The quantum, which as extensive had its determinateness at the amount external to itself, thus passes over into simple determinateness. Within this simple determination of the limit it is intensive magnitude; and the limit or determinateness as such, which before was as amount, is something simple, the degree.

The degree is thus determinate magnitude, quantum, yet not at the same time multitude, or something plural within its own self; it is only a multiplicity; the multiplicity is the several taken together into the simple determination. Its determinateness is indeed expressed by a number, this being the quantum's being-determined-in-itself, but it is not an amount; it is simple, only One degree. When 10, 20 degrees are spoken of, the quantum that has so many degrees is not their amount and sum; on that reckoning it would be an extensive one; rather it is only one, the tenth, the twentieth degree. It holds the same determinateness that lies in the amount ten, twenty, but it holds them not as several, and is rather number as sublated amount, as simple determinateness.

But this form of reference to itself, which the quantum has attained, is at the same time its becoming-external. As extensive quantum, number has its determinateness at its own self only in numerical plurality; but this plurality, being a many in general, falls together into undifferentiatedness, and as a many external to itself it sublates itself in the one of number, in number's reference to itself. The intensive quantum remains determinate quantum. The determinateness of the quantum, however, is otherness that is external to itself and indifferent. The degree, simple within its own self, which keeps this external otherness no longer in it, keeps it outside it, and refers itself thereto as to its determinateness. There is thus an external plurality; but in such a way that this external element at the same time makes up the simple limit, the determinateness which the degree is for itself. The amount as such thus remains the determinateness of number, but outside the number whose determinateness it is. That the amount, then, insofar as it was supposed to be found within number in the extensive quantum, sublated itself there, determines itself more closely as this: that it has been posited outside number. Since number is one, reference to itself reflected into itself, it thereby shuts the indifference and externality of the amount out of itself, and is reference to itself as reference through itself to something external.

Herein the quantum has the reality conformable to its concept. The quantum is determinate quantity. The determinateness of quantity is indifferent determinateness, which is not as referred to another; it therefore has otherness at its own self, and is in its own self external. So it is amount, the determinate being-differentiated within its own self; the amount makes up a determinate magnitude, and this being-determined, — whether they are three, or four and so forth, falls wholly within number itself; no comparison with others is required for it, nor is it a qualitative difference from something else. Since this externality is an inward externality that refers itself to itself, it is the externality of its own self. It is thus intensive magnitude, simple determinateness, as reference to itself which just as much has its determinateness in something external; the determinateness which at its own self is the determinateness external to itself.

Accordingly, then, the degree is a simple determinateness of magnitude among a multiplicity of intensities which are diverse, yet stand in essential reference to one another, so that each has its determinateness in this continuity with the rest. This reference of the degree through itself to its other makes the ascent and the descent of the scale of degrees a steady advance, a flowing that is an uninterrupted, indivisible alteration. Each of the several that are distinguished therein is thereby not cut off from the others, but has its being-determined only in these others. As a determination of magnitude referring itself to itself, each of the degrees is indifferent toward the rest; yet it is just as much in itself referred to this externality and has its determinateness therein; its reference to itself is therefore just as much the non-indifferent reference to what is external. The external is sublated in the simplicity of the degree; but it is just as much sublated outside the degree as something external too, for it stands in essential reference to the simple determinateness, and so is just as much not external to it.

2. Identity of Extensive and Intensive Magnitude

Intensive magnitude is the amount of extensive magnitude taken together into simplicity; a determinate one that does not have its determinateness at its own self as something plural; the degree is not within itself anything external to itself. Yet it is not merely the indeterminate one, the principle of number in general, which has no amount save the negative one of being no amount at all. — But intensive magnitude has at the same time its determinateness only in an amount. It is a simple one of the several; there are several degrees; determinate, however, they are neither as simple one nor as several, but only within the reference of this being-outside-itself, or within the identity of the one and multiplicity. If, then, the several as such lie outside the simple degree, its determinateness consists in its reference to them; it therefore holds the amount within it. As twenty, taken as extensive magnitude, holds the twenty ones within itself as discrete, so the determinate degree holds them as continuity, being this determinate multiplicity in simple fashion; it is the twentieth degree; and it is the twentieth degree only as this amount. This amount, however, which within the degree is simple, is at the same time externality at its own self; it is amount only as a multitude of numerical ones, which is just as much outside that simplicity of the degree.

The determinateness of intensive magnitude is therefore to be considered from a doubled side. It is first determined by other intensive quanta; it stands in continuity with its otherness, and its determinateness consists in this reference to its otherness. Insofar as it is the simple determinateness, it is thus determined against other degrees; it shuts them out of itself, and has its determinateness in this shutting out.

But secondly it is determined at its own self; to that extent it is so determined in the amount, as in its own amount, not as in the excluded amount, nor in the amount of other degrees. The twentieth degree holds the twenty at its own self; it is not merely determined as differentiated from the nineteenth, the twenty-first and so forth, but its determinateness is its own indifferent amount. Yet since that amount belongs to the degree, and since determinateness has at once essentially the shape of amount, the degree proves to be extensive quantum.

One determinateness of the quantum, then, is all that extensive and intensive magnitude are, and nothing sets them apart save this, that the one has it in simple form, the other in manifold form. Extensive magnitude passes over into intensive magnitude because its many in and for itself falls together into the unit, and, as determinateness of the many, of the numerical ones external to themselves over against the unit, steps outside the unit in number's reference to itself over against this unit. But conversely this simple thing has its determinateness only at the amount, and indeed as its own; for it is at the same time indifferent toward the intensities determined otherwise. Intensive magnitude is thus just as essentially extensive magnitude.

The difference of extensive and intensive magnitude rests on the difference of their moments, of the amount and the unit; magnitude is the one and the other magnitude posited in the determination of the one or the other moment. But because these moments are essential to it, because the determinateness is just as much the determinateness of the many as of something continuous or of a simple reference to itself, as it is of the discrete, of what is external to itself, its being posited in one of them is just as much its being posited in the other; or its existence is this doubled existence, which, however, is a matter of indifference as regards the determinateness of the quantum itself.

Remark

In the ordinary way of representing things, extensive and intensive quantum are wont to be distinguished as kinds of magnitude, as though there were objects that had intensive magnitude only, and others that had extensive magnitude only. There has come in besides the representation belonging to a philosophical natural science, which converted the plural, the extensive, for instance in the fundamental determination of matter, that of filling a space, and likewise in other concepts, into an intensive, in the sense that the intensive, as the dynamic, is the genuine determination, and that density, for instance, or specific space-filling, must essentially be grasped not as a certain multitude and amount of material parts within a quantum of space, but rather as a certain degree of matter's space-filling force.

Determinations of two sorts are to be distinguished here; there occurs the concept of self-subsistent parts subsisting apart from one another, joined into a whole only externally, and, distinct from it, the concept of force. What in the filling of space is on the one side regarded as nothing but a multitude of mutually external atoms is, on the other side, taken for the expression of a simple force lying at the ground. — These relations, of the whole and its parts, and of force and its expression, do not, however, belong here, but will be considered further on. — The other, however, is the quantitative determinateness that occurs in this connection, and in regard to which magnitude is sublated as extensive quantum and converted into degree, as into the determination that alone is supposed truly to be.

As regards this supposed essentiality of the difference it is enough to have shown that the difference is unessential for the determinateness of the quantum itself, that the one form is however essential for the other, and that every existence therefore exhibits its determination of magnitude just as much as extensive as it does as intensive quantum.

Everything serves as an example of this, insofar as it appears in a determination of magnitude. Number itself necessarily has this doubled form immediately at its own self. It is an amount, and to that extent it is extensive magnitude. But it is also a one, a ten, a hundred; to that extent it stands at the transition to intensive magnitude, since within this unit the manifold goes together into something simple. The tenth, the hundredth is this simple thing at its own self, which has its determinateness at the several that fall outside it, and to that extent is properly intensive magnitude. The number is ten, a hundred, and this same number is at once the tenth, the hundredth within the number system; both are the same; either determination can be taken for the other; the tenth number in the number system is ten.

The one in the circle is called degree, because a part of the circle has its determinateness essentially in a several outside it, being determined only as one out of a certain amount of such ones. The degree of the circle is, however, only the principle of the number of a magnitude of the circle, only its one. A quantum of the circle itself is an arc of determinate magnitude, an ordinary number, namely an amount of such ones as are degrees. This number is extensive magnitude, and intensive only insofar as, as was just recalled, number is this in general.

The magnitude of actual objects exhibits its doubled side, that of being extensive and intensive, at the doubled determinations of the object's existence, in one of which the object appears as something external, in the other, however, as something internal. Thus a mass, for instance, is as weight something extensively large insofar as it makes up an amount of pounds, hundredweights and so forth; something intensively large insofar as it exerts a certain pressure; this magnitude of the pressure is something simple, a degree that has its determinateness at a scale of degrees of pressure. In exerting pressure the mass appears as a being-within-itself, as a subject to which the intensive difference of magnitude accrues. — Conversely, whatever exerts this degree of pressure is capable of shifting a certain amount of pounds and so forth from their place, and measures its magnitude thereby.

Or heat has a degree; the degree of warmth, be it the 10th, the 20th and so forth, is a simple sensation, a subjective thing. But this degree is just as much present as extensive magnitude, namely as the expansion of a fluid, of the mercury in the thermometer, of air or of clay and so forth. A taller mercury column, or a slimmer clay cylinder, is how a higher degree of temperature gives itself expression; it warms a larger space just as a lesser degree warms the smaller one.

The higher tone, being the more intensive one, is at the same time a greater multitude of vibrations, or a louder tone, one to which a higher degree is ascribed, makes itself audible within a larger space. — With the more intensive color a larger surface admits of being colored in like manner than with a weaker one; or the brighter, another kind of intensity, is visible farther off than the less bright and so forth.

Just so within the spiritual, the high intensity of character, of talent, of genius is of an equally far-reaching existence, extended effect and many-sided contact. The deepest concept has the most universal significance and application.

3. Alteration of the Quantum

The difference of the extensive and the intensive quantum is a matter of indifference to the determinateness of the quantum at its own self; it is only a difference of the quantum's existence, or the quantum has the determinations that make up the extensive and the intensive as its own moments within itself. But if the quantum is by contrast indifferent toward a difference of its existence, its moments have in exchange entered into an inner opposition. The extensive quantum has, as a one referring itself to itself, passed over into the intensive quantum. The latter, however, which alone is thus to be considered, is the determinateness of magnitude that is simple within itself yet, precisely in this determinateness referring itself to itself, is external to itself, consisting not within itself but in another several.

Intensive magnitude is thus quantum that is for itself, and therein essentially referred to an other. This other is an other of this magnitude; another quantum. Intensive magnitude therefore is only as having its determination in another magnitude. But that it has its determination, its being-in-itself, in another magnitude means that it is not itself but another quantum. Or it passes over essentially into another magnitude.

Intensive magnitude, however, is in general the real quantum. The quantum is the determinateness posited as sublated, the indifferent limit; which is to say that it is that determinateness which no less is the negation of its own self. So the quantum is posited as degree. The degree is the simple determinateness referring itself to itself which is the negation of its own self, since it has its determinateness not at itself but in another quantum; hence, in being this determinate quantum, it is rather essentially not itself but another quantum.

A quantum stands accordingly, in general, in absolute continuity with its externality, with its otherness. Hence not only can every determinateness of magnitude be overstepped, not only can it be altered; it must alter itself. Quanta appeared at first as external to one another, in the determination of numerical ones. But they are not merely external to one another; they are external to themselves. The determination of magnitude therefore carries itself on into its otherness in such wise that its being lies for it solely in that continuity with something else. A quantum is thus itself, and just as essentially not itself but the negation of itself, an other. It is not a limit that is, but one that becomes.

The one is infinite, or the negation referring itself to itself; it is therefore the repulsion of itself away from its own self. The quantum is likewise infinite and repels itself away from its own self. But the quantum is the determinate one, the one that has passed over into existence and into limit. The quantum is thus the repulsion of determinateness away from its own self; that repulsion is therefore not the generating of what is self-same, as the repulsion of the one is, but the generating of its otherness. As the one is not overstepped by any third thing, but repels itself away from itself, so it belongs likewise to the concept of the quantum to send itself out beyond itself and to become an other. It consists in increasing or diminishing itself; it is the externality of determinateness at its own self.

The quantum sends its own self out beyond itself; this other, into which it becomes, is at first itself a quantum; a limit that is not one which merely is, but one which drives itself out past its own self; it continues itself into its otherness; it is external to itself; and this externality of its own self is itself. The limit that has arisen anew in this going-out is thus utterly nothing but one that sublates itself again, and so on into infinity.

C. Quantitative Infinity

1. Its Concept

The quantum alters and becomes another quantum; it is, however, a further determination of this alteration that it goes on into infinity.

The quantum becomes an other; it continues itself into its otherness; the other is therefore also a quantum. But the other is at the same time the other not merely of some quantum, but of the quantum itself. For the quantum is the indifferent determinateness, which is indifferent toward an other, but also toward itself. As its moments have determined themselves in the intensive quantum, it is the determinateness that refers itself not to another but to its own self; yet just as much is this determinateness utterly nothing but the determinateness in an other; the reference to an other is external to it, but it is itself this externality of its own. It is thus the quantum itself which contradicts itself, and so dissolves itself in itself; it is itself accordingly the negation of its own self; the alteration concerns not merely a quantum, but the quantum. The quantum is an ought; what it holds within it is to be determined in itself, and such being-determined-in-itself is in truth the being determined in an other; turned about again, it is that same being-determined-in-an-other in sublated shape; it is indifferent being-determined. It is therefore an other and something external over against its own self; it holds within it this, to be finite, and to go out beyond finitude, beyond the being-determine[…]d in an other, and to be infinite.

With qualitative and quantitative infinity it is essential to note that the finite is not overstepped by any third thing, but that determinateness, as dissolving itself within its own self, goes out beyond itself. But what parts the qualitative from the quantitative infinite is this, that in the first the opposition between what is finite and what is infinite bears a qualitative character, and that the passage out of finitude into infinity, or the way the two refer to one another, resides merely in the in-itself, in their concept. Immediate to begin with, and something that is, is the qualitative determinateness; and it refers itself to otherness essentially as to an other of its own, no positedness attaching to it of holding its negation, its other, at its own self. Magnitude, by contrast, just is determinateness in sublated shape; posited it stands as the negation, as unequal with itself, as the alterable. The qualitative finite and infinite therefore stand absolutely over against each other; their unity is the inward reference lying at the ground; the finite therefore does not continue itself immediately into its other. The quantitative finite, on the contrary, refers itself at its own self into its infinite. Their reference is therefore the infinite progress.

2. The Infinite Progress

The progress into infinity is nothing other than the expression of the contradiction which the quantitatively-finite, or the quantum in general, contains. It is that reciprocal determining of finite and infinite which came under consideration in the qualitative sphere, though with the difference that, as was just recalled, in the quantitative the limit continues itself on its own self into its beyond, and hence conversely the quantitatively-infinite is posited as well, namely the quantum's having its other on its own self. Finite and infinite are, the one the non-being of the other. But because quantitative determinateness is the merely sublated difference, the quantitative is its own self in its being-outside-itself. The quantitatively-infinite is therefore indeed the sublated quantum, not merely as a quantum but as the quantum. But because the quantum continues itself into its sublatedness, the infinite is just as much determined as the opposite of its own self, as quantum.

The quantum, then, is determinateness-in-itself, the determinateness indifferent towards other, which however just as much only is as external to itself. The infinite progress is the expression of this contradiction, not the resolution of it; it stands stock-still within the contradiction, and does not go out beyond it.

Or the progress into infinity is only the task of the infinite, not the attainment of it. It is the perennial producing of the infinite, without ever getting out beyond the quantum itself, and without the infinite's becoming something positive and present. The quantum is such a thing that it lies in its concept to have a beyond of itself. This beyond is first the pure moment of the non-being of the quantum; for the quantum dissolves itself on its own self. Thus it refers itself to its beyond, to its infin ity. This is the qualitative moment of the opposition. But secondly the quantum stands in continuity with this its beyond, which is a not-[suner][…][s] non-being of the quantum; for the quantum […][con]sists precisely in this, in being the other of its own self, in being external to its own self; hence this other, this external, is just as much not an other than the quantum. So the beyond, that is, the infinite, is itself a quantum. The beyond has thereby been summoned back out of its flight, and the infinite reached. But because this thing that has become a this-side is again a quantum, all that has happened is that a new limit has once more been posited. The quantum that has arisen anew has, just because it is quantum, fled from its own self once more, is as such out beyond itself, and has repelled itself away from itself into its non-being; it thus has a perennial beyond. But the quantum at the same time consists precisely in being external to itself. Hence that beyond is itself once again the quantum.

If this, that herein the beyond or the infinite is determined as quantum, and conversely the quantum as infinite, is gathered into one expression, then this joining yields an infinitely great or an infinitely small. But this joining is itself nothing other than merely the false expression of the contradiction, or of the infinite progress. For in it the quantum and its beyond are kept in their absolute determinateness against each other, the one as the non-being of the other. The infinitely great and the infinitely small is represented as a quantum; it is something great or something small; but as quantum it has just as much thrust its beyond away from itself; it is not widened out into the infinite, but kept in perennial opposition to it. The great, widened ever so much, therefore shrivels together into paltriness; for insofar as it refers itself to the infinite as to its non-being, the opposition is, according to this moment, qualitative; the widened quantum has thus wrung nothing from the infinite; the latter is rather, now as before, the non-being of it. Or, the enlarging of the quantum is no drawing nearer to the infinite, for the difference of the quantum and of its infinity has essentially the moment of being a difference that is not quantitative. — In the same way the infinitely small, being something small, is a quantum, and for that reason stays too great for the infinite in an absolute, which is to say a qualitative, manner, and stands over against it.

The infinitely great or small is therefore itself only the infinite progress. This infinity, which is determined as the beyond of the finite, is to be designated the bad quantitative infinity. It is infinity of the progress, and, like the qualitative bad infinity, only the perennial crossing over and back, out of the one member of the abiding contradiction into the other, out of the limit into its non-being, and out of this non-being anew back into that very same thing, the limit. It is not so much a going forward as a repeating of one and the very same, positing, sublating, and positing again and sublating again; an impotence of the negative, to which that which it sublates comes back, through its very own sublating, as something continuous. There are two so knotted together that they simply flee each other; and in fleeing each other they cannot part, but are bound together in their very parting.

Remark 1

The bad infinity is wont, chiefly in the form of the progress of the quantitative into infinity, — this continual overflying of the limit, which is the impotence to sublate it, and the perennial relapse back into it, — to be held for something sublime and for a kind of divine service, just as in philosophy that progress has been looked upon as an ultimate. Tirades of this sort are to be met with everywhere, and they have won admiration as sublime productions. In fact, however, this modern sublimity makes great not the object, which rather escapes it, but only the subject, which devours such great quantities into itself. Yet the poverty of this elevation, which stays subjective and clambers up the ladder of the quantitative, announces itself in this, that amid its futile labor it comes no nearer to the infinite goal, a goal that must be tackled quite otherwise if it is to be reached.

In the following tirades of this sort there is at the same time expressed what such elevation passes over into and where it ceases. Kant for instance adduces as sublime,

“when the subject in thought lifts itself above the place that it occupies in the world of sense, and extends the linkage into the infinitely great, a linkage with stars upon stars, with worlds upon worlds, systems upon systems, and over and above this into the boundless times of their periodic movement, of the beginning and the duration thereof. — Representing succumbs to this going forth into the immeasurably distant, where the most distant world has always yet a more distant one, the past traced back so far has yet a further one behind it, the future carried out ever so far has always yet another before it; thought succumbs to this representation of the immeasurable; as a dream in which someone walks down a long corridor ever further and unforeseeably further, without descrying any end, closes with falling or with dizziness.”

This portrayal, quite apart from crowding the content of the quantitative elevating into a wealth of depiction, deserves praise above all for the truthfulness with which it states how this elevation fares in the end: thought succumbs, the end is falling and dizziness. What makes thought succumb, and brings forth its falling and dizziness, is nothing other than the tedium of that repetition which lets a limit vanish and come up again and vanish again, thus always the one after the other, and the one in the other, in the beyond the this-side, in the this-side the beyond, perennially coming to be and ceasing to be, and which yields nothing but the feeling of the impotence of this infinite or of this ought, which wants to become master over the finite and cannot.

Haller's horrifying description of eternity, so named by Kant, is likewise wont to be particularly admired, but often precisely not on account of that side of it which constitutes its genuine merit:

| “I heap up monstrous numbers, | Pile mountains of millions, | I set time upon time, and world upon world in a heap, | And when from that grisly height | I look with reeling eyes back toward you, | All the might of number, multiplied a thousand times over, | Is not yet one part of you.” | “I take them away, and you lie wholly before | me.”

When value is laid upon that heaping and towering up of numbers and worlds as upon a description of eternity, it is overlooked that the poet himself pronounces this so-called horrifying going-beyond something futile and hollow, and that he closes by saying that only through the giving up of this empty infinite progress does the genuine infinite itself come into presence before him.

As is well known, the astronomers too are fond of priding themselves greatly on the sublimity of their science, and this because it has to do with an immeasurable multitude of stars, with spaces and times so immeasurable that within them distances and periods already so great on their own account serve as units which, taken ever so many times over, shrink back once more to insignificance. The stale astonishment to which they abandon themselves in all this, the tasteless hopes of one day travelling in that other life from one star to another and of gathering the like new knowledge on into the immeasurable, they pass off as a chief moment of the excellence of their science, — a science that is admirable indeed, yet not for the sake of the quantitative infinity that occurs in it, but on the contrary for the sake of the measure-relations and the laws which reason has cognized in these objects, and which are the rational infinite as against that irrational infinity.

To the infinity that refers itself to outer sensuous intuition Kant opposes the other infinity, when

“the individual goes back into its invisible I, and, as a pure I, sets the absolute freedom of its will against all the terrors of fate and of tyranny, beginning with its nearest surroundings, lets these vanish for it, lets likewise what appears as enduring, worlds upon worlds, crash down into rubble, and, alone, cognizes itself as equal to its own self.”

The I in this solitude with itself is indeed the beyond attained; in pure self-consciousness the absolute negativity is brought into presence and is with its own self, that negativity which in the advance out beyond the sensuous quantum does nothing but flee. But in fixing itself in its abstraction and its lack of content, this pure I has existence in general, the fullness of the natural and of the spiritual universe, standing over against it as a beyond. There presents itself the same contradiction that lies at the ground of the infinite progress; namely a having-returned-into-itself which is immediately at the same time a being-outside-itself, a reference to its other as to its non-being. Which reference stays a longing, because the I has fixed for itself its own emptiness on the one side, and the fullness as its beyond.

Kant appends to these two sublimities the remark “that admiration (for the first, the outer) and respect (for the second, the inner) sublimity do indeed spur inquiry on, but cannot make good the want of it.” — He thereby declares those elevations to be unsatisfying for reason, which cannot come to a stop with them and with the sensations bound up with them, and cannot let the beyond and the empty pass for the ultimate.

As an ultimate the infinite progress has been taken above all in its application to morality. The second opposition of the finite and the infinite just adduced, that of the manifold world and of the I raised into its freedom, is at first, in its purity, qualitative. Since the self-determining of the I consists at once in determining nature and in freeing itself from nature, it refers itself through its own self to its other, which as outer existence is something manifold and quantitative. But the determining of something quantitative becomes itself quantitative, and the negative reference of the I to it, the power of the I over the not-I or over sensibility and outer nature, is therefore represented in such a way that morality can and ought to become ever greater, the power of sensibility however ever smaller; the will's complete adequacy to the moral law, however, is displaced into the progress that runs on into infinity, which is to say, represented as an absolute unattainable beyond, and precisely this, that it is an unattainable one, is supposed to be the true anchor and the right consolation.

Within this opposition the I and the not-I, or the pure will and nature and sensibility, are represented as perfectly self-subsistent and indifferent towards each other. The pure will has its own peculiar law, which stands in essential reference to sensibility; likewise nature has laws which are neither drawn from the will nor answerable to it, nor would even, though different from it, have in themselves an essential reference to it, but which are determined altogether on their own account, complete and self-enclosed. Both, however, are simultaneously moments of one and the very same simple essence, of the I; the will is the negative, which consists in sublating nature, and hence only is insofar as there is something distinct from it that is to be sublated by it. It puts itself into a bearing towards sensibility, so as to determine it; it thereby goes out beyond itself, touches sensibility and is thus itself affected by it. Nature and sensibility are presupposed as a self-subsistent system of laws; the being restricted by an other is therefore a matter of indifference to them; nature maintains itself in this being bounded, enters self-subsistently into the reference, and bounds the will just as much as the will bounds nature. — It is one act, that the will determines its own self and sublates the otherness of a nature, and that this otherness is posited, or that it continues itself into its being sublated. The contradiction that lies herein is not resolved in the infinite progress, but on the contrary is set forth and asserted as unresolved and unresolvable; the struggle of morality and of sensibility is represented as the absolute relation that is in and for itself.

The impotence to become master over the opposition of the finite and the infinite takes refuge in magnitude, so as to employ it as the mediatrix, since magnitude is the qualitative sublated, the difference that has grown indifferent. Yet since both members of the opposition lie at the ground as qualitatively diverse, the very fact that in their mutual reference they bear themselves as quanta makes each of them indifferent towards this alteration. Nature is determined by the I; but because this negation contains not the qualitative but only the quantitative difference, it is precisely such a difference as does not touch nature itself, but lets it subsist as what it is.

In the more abstract presentation of the Kantian philosophy, or at any rate of its principles, namely in Fichte's Wissenschaftslehre, the infinite progress makes up in the same way both the foundation and the ultimate. Upon the first principle of that presentation, I = I, there follows a second one independent of it, the opposing of the not-I; the reference of the two is assumed as the quantitative difference, that the not-I is in part determined by the I and in part not. In this manner the not-I continues itself into its non-being as something that stays opposed to its non-being, as something unsublated. After the contradictions lying therein have accordingly been developed, the concluding result is that very relation which was the beginning; the not-I remains an infinite check, an absolutely-other; the last reference of it and of the I to one another is the infinite progress, the same contradiction with which the start was made. The finite, and the finite relation, is supposed to be the absolutely true.

Because the quantitative in general is the negation of determinateness, it was believed that a great deal, or rather everything, stood to be won for the absolute's unity, for the one substantiality, by degrading opposition as such into a difference that is merely quantitative. All opposition is merely quantitative was for a time a chief proposition of the newer philosophy; determinations that stand opposed are of one essence, of one content; — further, each side of the real opposition too has both determinations, both factors, within it; save that on the one side the one factor and on the other side the other is preponderant; and the preponderant was frequently taken also in the sense that in the one side the one factor, a matter or an activity, is at hand in greater amount or in stronger degree than in the other. As for the latter, so far as diverse stuffs or activities are presupposed, what the quantitative difference does is rather to seal and perfect their externality, their indifference towards one another. But as regards the former, that the difference of the absolute unity is supposed to be merely quantitative, the quantitative is indeed immediate determinateness sublated, yet it is the merely imperfect negation; for it is as yet only the first negation, not the infinite one, not the negation of the negation. — Or, in being represented as quantitative determinations of the absolute substance, being and thinking become, as quanta, precisely thereby wholly external and relationless to one another, as carbon, nitrogen and so forth do in a subordinate sphere. It is a third thing, an external reflection, that abstracts from their difference and cognizes their inner unity, a unity that only is in itself. This unity is in this way represented only as a first immediate one, or only as being which in its quantitative difference stays equal to itself, but does not posit itself equal through its own self; or it is not grasped as negation of the negation, as infinite unity. It is only the qualitative opposition that contains the genuine infinity, and the quantitative difference passes over, as will shortly appear more closely, into the qualitative.

Remark 2

It was recalled above that the Kantian antinomies are presentations, in a more concrete shape, of the opposition between finite and infinite, brought to bear upon more special substrates of representation . The antinomy considered above contained rather the opposition of qualitative finitude and infinity. In another, the first among the four cosmological antinomies, what is considered in its conflict is rather the quantitative limit. I shall therefore undertake the examination of this antinomy here.

It concerns, namely, the limitedness or unlimitedness of the world in time and space. — This opposition could with equal right have been considered in respect of time and of space themselves, for nothing in the antinomic character of limitedness or unlimitedness is altered by whether time and space are relations of the things themselves, or are merely forms of intuition.

The closer unfolding of this antinomy will likewise show that the two propositions, and equally their proofs — which, as with the antinomy considered above, are conducted apagogically — run out into nothing but the two simple, mutually opposed assertions: a limit is, and there must be a going out beyond the limit.

The thesis reads:

“The world has a beginning in time, and as regards space it too is enclosed within limits.”

The one part of the proof, the one concerning time, assumes the contrary,

“that as regards time the world has no beginning; then up to every given point in time an eternity has run out, and consequently an infinite series of states of things in the world following upon one another has ela psed. But the infinity of a series consists precisely in this, that through successive synthesis it can never be completed. An infinite elapsed world-series is therefore impossible, and a beginning of the world is accordingly a necessary condition for its existence; which was the thing to be shown.”

The other part of the proof, the one that concerns space, is led back to time. The gathering together of the parts of a world infinite in space would require an infinite time, which would have to count as run out, insofar as the world in space is to be taken not as something becoming but as something completed and given. But of time it was shown in the first part of the proof that to assume an infinite time as run out is impossible.

One sees at once, however, that it was unnecessary to make the proof apagogic, or to conduct a proof at all, since what lies immediately at its own base is the assertion of the very thing that was supposed to be proved. For some one or any given point in time is assumed, up to which an eternity (-- eternity here has only the meagre sense of a badly infinite time) is supposed to have run out. A given point in time signifies nothing other than a determinate limit within time. Within the proof, accordingly, a limit of time is presupposed as something actual; yet this is precisely that which was to be proved. For the thesis consists in this, that the world has a beginning in time.

The only difference that occurs is that the assumed temporal limit is a now as the end of the time previously elapsed, whereas the one to be proved is a now as the beginning of a future. Yet this difference is inessential. Now is assumed as that point wherein an infinite series of states of things in the world, one following upon another, is supposed to have elapsed, hence as an end, as a qualitative limit. Were this now regarded merely as a quantitative limit, one that is to be gone out beyond and that flows, then the infinite series of time would not stand elapsed within it but would go on flowing, and the reasoning of the proof would fall away. But this point in time, assumed as a qualitative limit for the past, is at the same time the beginning for the future, — for in itself every point in time is the reference of past and future, — and indeed it is an absolute beginning for that future. For it makes no difference to the matter that before this future of its and before the beginning of that future a past already is; since this point in time is a qualitative limit, — and to assume it as qualitative lies in the determination of the completed, of what has run out, hence of what does not continue itself, — time is thereby broken off in it, and the past that is being spoken of stands without reference to the time which could be called future only in regard to this past, and which is therefore only time as such, time having an absolute beginning. But did it stand, — (as it indeed does —) through the now, the given point in time, in a reference to the past, were it in fact future, then from the other side this point in time too would be no limit, the infinite series of time would carry itself on into that which was called future, and would not, as was assumed, be completed.

In truth time is pure quantity; the point in time employed in the proof, at which time was supposed to be interrupted, is rather only the self-sub lating being-for-itself of the now. The proof achieves nothing beyond setting before representation, as a given point in time, that absolute limit of time which the thesis asserts, and flatly assuming it, a popular determination which sensuous representing readily lets pass as a limit, thereby letting hold good within the proof, as an assumption, that which had earlier been put forward as what was to be proved.

The antithesis states:

“The world has neither a beginning nor limits in space, but is infinite in view of time as well as of space.”

The proof posits the contrary:

“Let the world have a beginning. Since the beginning is an existence before which there goes a time wherein the thing is not, then a time wherein the world was not must have gone before, that is, an empty time. But in an empty time no coming-to-be of any thing whatever is possible; because no part of such a time has in itself, ahead of another, any distinguishing condition of existence over that of non-existence. Hence many a series of things can indeed begin within the world, but the world itself can take no beginning, and in respect of elapsed time it is infinite.”

This apagogic proof contains, like the others, nothing beyond the direct and unproven assertion of the very thing it was meant to prove. For it first assumes a beyond of worldly existence, an empty time; but then it just as much continues worldly existence out beyond itself into this empty time, thereby sublates that time, and thus carries existence on into infinity. The world is an existence; and the proof presupposes of this existence that it comes to be, and that coming-to-be has a preceding condition in time. But in this precisely the antithesis itself consists, that no existence is unconditioned and no limit absolute, and that essential existence always demands a preceding condition. This condition is at the same time itself conditioned; it is sought in empty time, which amounts to saying that it is itself assumed as temporal and hence as existence, and as something limited. Quite generally, then, the assumption has been made that the world as existence presupposes another existence, and so on into infinity.

The proof with regard to the world's infinity in space is just the same. In apagogic fashion the spatial finitude of the world is assumed; “it would thus find itself in an empty unlimited space and would have a relation to it; yet a relation of this sort, of the world to no object, is nothing.”

What was supposed to be proved is here likewise directly presupposed in the proof. For it is directly assumed that the limited spatial world should find itself in an empty space and have a relation to it, which is to say that it must be gone out beyond, on the one side into the void, into the world's beyond and non-being, while on the other side it thereby stands in relation to that void, hence continues itself into it, and that the beyond is to be represented as filled with worldly existence. What the antithesis asserts, the infinity of the world in space, is nothing other than, on the one part, empty space and, on the other part, the relation of the world to it, which is to say the continuity of the world within it, or the filling of it; and this contradiction, space at once as empty and at once as filled, is the infinite progress of existence within space. But this contradiction itself, the world's relation to empty space, is what is directly assumed in the proof.

Thesis and antithesis and their proofs therefore present nothing but the opposed assertions that a limit is, and that this same limit is no less merely a sublated one; namely, that the limit has a beyond with which it stands in reference, towards which one is to go out beyond it, but wherein there arises once more such a limit as is none.

The resolution of these antinomies is, like that of the one above, transcendental, which is to say that it lies in asserting the ideality of space and of time, as forms of intuition, in this sense: that the world in its own self is not in contradiction with itself, is not something self-sublating, but that consciousness, in its intuiting and in the reference of intuition to understanding and reason, is an essence contradicting itself.

3. Infinity of the Quantum

1. The infinite quantum, as infinitely great or infinitely small, is itself the infinite progress; it is quantum as a great or a small, and is non-being of quantum as infinite. The infinitely great and the infinitely small are therefore images of representation which on closer consideration show themselves to be null mist and shadow. But the infinite progress expresses nothing other than the nature of quantum, which as intensive magnitude has attained its reality.

Quantum, returned into itself, is simple, referred to itself and as determined in itself. But since through this simplicity otherness and determinateness are sublated at its own self, determinateness is external to it; it has its absolute determinateness rather outside it. This being-outside-itself of quantum is at first the abstract non-being of quantum as such, bad infinity. But further it is also a great, quantum continues itself into its non-being, for it has its determinateness precisely in its externality; this externality of quantum is therefore just as much itself quantum, only another quantum, which however sublates itself once more as the first did.

Quantum is therefore something determined in itself; but this determinateness of its it has outside itself, so that it sublates itself; conversely, in its being-outside-itself it has returned into itself, its being-outside-itself is just as much sublated.

This circle is the genuine element that is posited in the infinite progress. There is present quantum and its beyond. First, quantum sublates itself, it is at its own self the going out beyond its limit; the beyond is infinity, but it is bad infinity, for second, quantum continues itself into it. This beyond, the non-being of quantum, infinity itself is limited, and a quantum is posited anew, which is to say, this beyond is itself sub lated. Quantum is precisely itself through its being-external; this is exactly what makes up the determinateness of quantum, or what quantum is. In the infinite progress, therefore, there is the concept of quantum as it is in itself; and there is present in the progress the sublating of quantum, but just as much of its beyond; or the negation of quantum as well as the negation of this negation.

The going out beyond quantum is the negation of quantum, the infinite; but a new quantum is posited, and this is the negation of the infinite, of this bad infinite which counts for representation as an absolute, as a last term that does not sublate itself again and out beyond which nothing further could be gone. The truth of the infinite progress is therefore that quantum and its beyond are posited, but that they are posited as sublated. Its truth is accordingly their unity, wherein they are, but as moments.

This is thus the true resolution of the contradiction whose expression the infinite progress is. It consists in nothing other than the restoration of the concept of magnitude, namely that magnitude is an indifferent or external limit. In the infinite progress as such, reflection is usually directed only upon this, that every quantum, be it ever so great or small, vanishes, that it must be possible to go out beyond it; but not upon this, that this sublating of quantum, the beyond, the bad infinite, itself vanishes too. This happens, however, in that quantum continues itself into its negation, in that out beyond every quantum, into its sublating, a new quantum is posited. The first sublating is indeed in itself the sublating of negation, — for quantum is sublated limit, — but it is at the same time only in itself this; for this infinite is fixed as the beyond of quantum, which still remains subsisting as a this-side; or quantum is taken only as something immediate, and the infinite only as the first negation. But in the infinite progress there is more present, — than merely the sublating of immediate quantum, or than merely a first sublating; therein this bad infinite too is, — through the new limiting, sublated; there is therefore present in it the negation of negation, or that which the infinite in truth is. — But the concept of quantum is not merely restored, it has also received its closer determination; there has arisen the quantum determined by its concept, which is distinct from immediate quantum.

2. The beyond of quantum has, namely, a more determinate, positive significance than merely that of the non-being of quantum; and so likewise does the sublating of this beyond, and the uniting of it with quantum itself.

Quantum as indifferent limit is determined at its own self; this self-referring being-determined is the having-vanished of its externality, which it has at its own self; this externality thereby steps outside quantum; its going out beyond itself is its essential moment, it refers itself through its own self to its externality; but this externality makes up its being determined in itself, and the nature of its being-determined-in-itself consists in this externality. The beyond of quantum is therefore not the mere non-being, the empty, indeterminate negation of quantum. Rather quantum goes out beyond itself just insofar as it is indifferent limit; it sublates this indifference and posits the being-in-itself of that indifference as an infinite beyond, as that wherein it is negated, wherein it is not its own self but the externality of its own self. But this externality is rather the opposite of itself; it is an absolute moment of magnitude itself; for quantum is not itself in its immediacy, but is essentially a going out beyond itself; this going out beyond itself, this externality of its, therefore belongs to quantum itself.

But its going out beyond itself is the sublating of its indifference towards the external, which is its determinateness, and it thereby posits this determinateness as its own self. It sublates its beyond, its negation, which is to say, it sublates the externality of its being-determined; unites it with itself and thereby makes itself determined in itself.

Each of the moments of the movement of the infinite progress is the opposite of its own self; for the infinite progress is the posited contradiction. First, quantum goes out beyond itself; this therefore means 1) it sublates itself, posits its negation, its beyond, and 2) it thereby posits rather its absolute being-determined, that which it is in itself. Second, this infinite is determined again, a new limit is posited; this accordingly means 1) the being-in-itself of quantum is sublated, it only withdraws an indifferent quantum once more, 2) the negation of quantum, the beyond of quantum, is sublated, its going out beyond itself is thus taken back into quantum itself. Both sides express this, that quantum is negated and that the negation of quantum is negated; there is therefore posited its infinite reference to its own self, or its being-determined-in-itself. Infinity, which was only the bad one and a beyond of quantum, belongs to quantum, quantum is itself infinite.

In this restoration of quantum, quantum is sublated as indifferent limit, as this perennial going out beyond itself. The indifference and externality of quantum therefore vanishes only insofar as the beyond of quantum is sublated. Quantum no longer has infinity, being-determined-in-itself, outside itself. The limit is therefore sublated as indifferent or as sublated. It has thus become qualitative once more.

The infinite, then, which in the infinite progress has only the empty significance of a non-being, of a beyond, is in fact nothing other than quality. Quantum is indifferent limit; it goes out beyond itself into infinity; in doing so it seeks nothing other than being-determined-in-itself, the qualitative moment. But this qualitative moment is not a beyond of quantum, it lies within quantum itself. For precisely this going out beyond itself, or the beyond, the negation of quantum, is what makes quantum into quantum; this is its determinateness in itself; its very indifference is its determination itself.

Or, quantum is sublated quality; but quantum is infinite, goes out beyond itself, it is the negation of itself. It is therefore the negation of negated quality, or it is the restoration of that quality.

But the quantum that is sublated as indifferent limit and qualitatively determined is the quantitative ratio. In the ratio quantum is external to itself, distinct from its own self; but this externality of its, the reference to the other quantum, at the same time makes up its determinateness; therein it has not an indifferent but a qualitative determination; in its externality it has returned into itself.

Remark

On the one side the mathematical infinite holds interest through the expansion of mathematics and through the great results that its introduction into that science has brought forth; on the other side, however, it is remarkable in that this science has not yet succeeded in justifying its use of it by way of the concept. The justifications rest on the correctness of the results that come out with its help, a correctness demonstrated on other grounds; they do not rest on the clarity of the object and of the operation through which the results are brought out, so little indeed that this operation is rather admitted to be incorrect.

This is in and for itself already a defect, for a procedure of that sort is unscientific. But it also carries a disadvantage with it, namely that mathematics, in not knowing the nature of this instrument of its own, because it has not come to terms with the metaphysics or critique of it, can neither determine the range of its application nor secure itself against misuses of it.

In a philosophical respect, however, the mathematical infinite is important because the concept of the genuine infinite does in fact lie at its base, and because it stands far higher than what is ordinarily called the metaphysical infinite, from which the objections against the former are made. Against these objections the science of mathematics ordinarily knows how to save itself only by rejecting the competence of metaphysics, maintaining that it has nothing to settle with that science and need give no thought to its concept, provided only that it proceeds consistently on its own ground. Its business is not with what is true in itself, but with what holds as true upon its own field. Metaphysics cannot manage to deny or to overturn the brilliant results of the use of the mathematical infinite, and mathematics cannot manage to get clear about the metaphysics of its own concept and hence also about the derivation of the procedures which the use of the infinite makes necessary.

If the difficulty of the concept as such were the only one pressing upon mathematics, it might let this lie aside without further ado, insofar, that is, as the concept is more than the mere specification of the essential determinateness of a matter; for it is not a science that has to do with the concepts of its objects and has to generate its content through the development of the concept, even if only by way of ratiocination. In the method of its infinite, however, it encounters the principal contradiction at the very distinctive method upon which, as a science, it rests altogether. For the calculus of the infinite permits and requires procedures that mathematics must otherwise utterly repudiate in operations with finite magnitudes, and at the same time it treats its infinite magnitudes like finite quanta and would apply to the former the very procedures that hold good for the latter.

In its use of the infinite, and in the operations directly at odds with the mathematical way of proceeding that this use makes necessary, mathematics shows that the results it thereby finds agree entirely with those found by the properly mathematical route, the geometrical and the analytic. But for one thing this does not concern all results, and the purpose of introducing the infinite is not merely to abbreviate the ordinary path but to arrive at results that cannot be achieved along it. For another, success does not justify the manner of the path in and for itself. This manner of calculating the infinite, however, is always burdened with the semblance of inexactitude that it gives itself when on the one occasion it augments finite magnitudes by an infinitely small magnitude, retains part of that magnitude in the further operation, and yet also neglects a part of it. This procedure displays the oddity that, the admitted inexactitude notwithstanding, a result comes out which is not merely tolerable and so close that the difference could be left out of account, but perfectly exact. In the operation itself, though, which precedes the result, the representation cannot be dispensed with that something is not equal to zero yet is so inconsiderable as to admit of being left out of account. Yet with what is to be understood by mathematical determinateness, all difference of a greater or a lesser exactitude drops away entirely, just as in philosophy there can be no talk of greater or lesser probability but only of truth. If the method and the use of the infinite are justified by success, and even this only in part, it is nevertheless not so superfluous to demand a justification of them as it seems superfluous, in the case of one's nose, to ask for proof of the right to make use of it. For with mathematical cognition, as a scientific cognition, everything essentially turns on proof, and with regard to the results too it is the case that the strictly mathematical method does not supply for all of them the corroboration of success, which is in any event only an external corroboration.

It is worth the trouble to look more closely at the mathematical concept of the infinite and at some of the most remarkable attempts whose aim is to justify its use and to remove the difficulty by which the method feels itself pressed. Consideration of these justifications and determinations of the mathematical infinite, which I mean to carry out at greater length in this Remark, will at the same time cast the best light on the nature of the true concept itself, and will show how it has hovered before them and lain at their ground.

The customary determination of the mathematical infinite is that it is a magnitude beyond which there is no greater or smaller one any more. — In this definition the true concept is indeed not yet immediately expressed, but it is contained in it once the definition is looked at more closely. For a magnitude is defined in mathematics as something that admits of increase and decrease; hence, quite generally, as an indifferent limit. And since the infinitely great or small is such as admits of no further increase or decrease, it is in fact no quantum as such any longer.

This consequence is necessary and immediate. But the reflection that the quantum — and in this Remark I call the finite quantum simply quantum in general — is sublated is the one that is not commonly made, and it is this that makes the difficulty for ordinary comprehension, since the quantum, in being infinite, is required to be thought as something sublated, as one that is at the same time not a quantum.

To cite how Kant appraises that concept ⁴, he finds it out of agreement with what is understood by an infinite whole. „On the ordinary concept, a magnitude is infinite beyond which none greater (that is, beyond the multitude, contained in it, of a given unit) is possible. — By an infinite whole, he says, there is no representation of how great it is; its concept, accordingly, is not the concept of a maximum (or minimum), but by it there is thought only its ratio to a unit that may be assumed at will, in respect of which the whole is greater than any number. According as one assumed this unit greater or smaller, so would the infinite be greater or smaller; infinity, however, consisting as it does merely in the ratio to this given unit, would always remain the same, though of course the absolute magnitude of the whole would thereby not be cognized at all.“

What Kant censures, then, is that infinite wholes should be viewed as a maximum, as a completed multitude of a given unit. For the maximum or minimum is itself a quantum, a multitude, not merely a ratio. The ordinary representation, to which the infinitely great or small appears as a something that is a quantum, cannot fend off the consequence adduced by Kant, which leads to a greater or a smaller infinite according as the unit lying at the base, being a variable one, were assumed greater or smaller. Or, quite generally, so long as the infinite is represented as quantum, the difference of a greater and a smaller still holds good for it. This criticism, however, does not strike the concept of the genuine mathematical infinite, of the infinite difference, for that difference is no longer a finite quantum.

Kant's own concept, by contrast, the one he calls the true transcendental concept, is „that the successive synthesis of the unit in the measuring-through of a quantum can never be completed.“ On the one side a quantum is here indeed presupposed as given; but this is first to be synthesized, and this synthesizing, through which it would be made into an amount and a quantum, is never to be completed. With this, as is evident, nothing but the progress into infinity is enunciated, only transcendentally, or properly speaking subjectively and psychologically represented. In itself the quantum is indeed to be completed, but transcendentally, namely within the subject, there arises only such a quantum as is uncompleted and simply afflicted with a beyond. Here, then, one comes to a halt quite generally at the contradiction that magnitude contains, but the contradiction is parcelled out to object and subject, so that to the former falls limitedness, to the latter the going out beyond it, the bad infinite.

The genuine infinite quantum, however, is infinite in its own self; it is this, as emerged above, as that in which the finite quantum, or quantum in general, and its beyond, the bad infinite, are sublated in the same way. The sublated quantum, however, has gone back into simplicity and into reference to itself, — not merely in the manner of the extensive, which in passing over into intensive quantum has its determinateness only in itself at an external manifoldness, toward which it is nevertheless indifferent and from which it is supposed to be distinct. The infinite quantum, by contrast, contains externality and the negation of itself at its own self; thus it is no longer some finite quantum, not a determinateness of magnitude that would have an existence as quantum, but it is simple, and so only as moment; it is only the concept of its being-determined, or a determinateness of magnitude in qualitative form. As moment it stands in essential unity with its other, only as determined through this other of its. Or it has meaning only in reference to something standing in ratio with it. Outside this ratio it is zero; — whereas quantum as such is precisely supposed to be indifferent toward the ratio and to need no other for its determination. In the ratio, however, it is just as much no quantum, and precisely for this reason, that it is only moment, only something within the ratio, not something indifferent for itself.

Since quantum thus is, according to its truth, only as determination of magnitude, it has qualitative nature and is infinite; for, first, this contains its negation, — it has ceased, namely, to be what according to its determination it was supposed to be, something indifferent. Second, it has being-determined-in-itself at it, for it no longer has it as a beyond outside it.

This concept will show itself to lie at the base of the mathematical infinite, and it will grow more distinct as we consider the various stages of the expression of quantum as a moment of ratio, beginning from the lowest, where it is at the same time quantum as such, up to the higher, where it has the meaning and expression of infinite magnitude proper.

Let us take first the quantum in the ratio, in the shape in which it is a fractional number. The fraction 2/7, for instance, is not a quantum like 1, 2, 3 and so forth; an ordinary finite number it certainly is, yet not an immediate one like the whole numbers, but as fraction it is determined mediately through two numbers which are amount and unit relative to each other, so that the unit is itself a determinate amount. But if we abstract from this closer qualitative determination of them relative to each other and look at them merely according to what befalls them here as quantum, then 2 and 7 are otherwise indifferent quanta, though here they come on the scene only as moments of another. For this reason 2 and 7 are here to count forthwith not as 2 and 7 but as their determination relative to each other. In their stead, therefore, 4 and 14, or 6 and 21 and so forth, can just as well be posited. With this they begin to have a qualitative character. Were they to count as mere quanta, then 2 and 7 are simply only 2 and 7; 4 and 14, 6 and 21 and so forth are simply something else and cannot be set in the place of those numbers. Insofar as 2 and 7 do not count according to this determinateness, their indifferent limit has been sublated; they thus have, though still imperfectly, the moment of infinity at them, since they at the same time not merely are no longer just they themselves, but their determinateness also remains, as a qualitative one that is in itself, — namely according to what they count for in the ratio. Infinitely many others can be set in their place, so that at the same time the value of the fraction, the determinateness which the sides of the ratio have, does not alter.

The exhibition which infinity has at a numerical fraction is still imperfect, however, for this reason, that the two sides of the fraction, 2 and 7, are ordinary indifferent quanta once they are taken out of the ratio; their reference, that of being in a ratio and being moments, is to them something external and indifferent.

The letters operated with in general arithmetic do not have the property of possessing a determinate numerical value; they are universal signs, and indeterminate possibilities of every determinate value. The fraction therefore seems, on account of its elements, to be a more fitting expression of the infinite, because a and b, taken out of their reference to each other, remain indeterminate and separated have no particular value of their own either — yet although these letters are indeed indeterminate magnitudes, their sense is that they be some finite quantum. Since they are thus only the universal representation, but the representation of the determinate number, it is likewise indifferent to them to stand in the ratio, and outside it they keep this value.

The two sides which the magnitudes have in the fraction consisted in this, that they are finite magnitudes, quanta, and at the same time infinite, no quanta. The ratio itself as such is first a quantum; second, however, not an immediate one but one that has the qualitative opposition in it: to be something not indifferent toward the other but determined through it, something returned into itself in its otherness and thus infinite. These two sides exhibit themselves in the following manner.

The fraction 2/7 can be expressed as 0,285714 ... as also as 1 + a + a2 + a3 and so forth. So taken it is exhibited as an infinite series, and the fraction itself goes by the name of the sum, or the finite expression, of that series. If we compare these two expressions, the infinite series exhibits the fraction no longer as a ratio but only on the side that it is a quantum, as a multitude of such quanta accruing to one another, as an amount, or at least it has the determination of exhibiting it so. — That the magnitudes which are to make it up as amount consist in turn of decimal fractions, and hence themselves of ratios, is here beside the point; for that circumstance concerns their unit, not them insofar as they constitute the amount; just as a whole number of the decimal system consisting of several digits counts essentially as an amount, and no notice is taken of its consisting of products of a number and the number ten and the powers of ten. Just as little does it matter here that there are fractions other than the 2/7 taken as example which, turned into decimal fractions, do not yield an infinite series; the point is only that every one of them can be expressed as such a series.

In the infinite series, which is to exhibit the fraction essentially as amount, the side of its being a ratio therefore vanishes; and even when it is expressed as a sum of ratios, then, since these are taken as terms of a sum, abstraction is made from their being ratios. With the ratio, then, there vanishes also the side on which the fraction had infinity at it. This latter, however, has come in in another manner; the series, namely, is itself infinite.

Of what sort the infinity of the series is, is evident of itself; it is the bad infinity of the progress. For the series contains the contradiction of exhibiting something that is a ratio and of qualitative nature as something ratioless, as a mere quantum, as amount. At the amount expressed in the series something is always lacking, so that one must always go out beyond what is posited in order to attain the required determinateness. The law of the advance is known; it lies in that determination of quantum which the fraction contains, and in the nature of the form in which this determination is to be expressed. By continuing the series it can be made as exact as one requires; but the exhibition through it always remains a mere ought; it is afflicted with a beyond that cannot be sublated, because to express something qualitative as amount is the abiding contradiction.

In this infinite series that inexactitude is actually present of which, at the genuine mathematical infinite, only the semblance occurs. One must no more confuse these two kinds of mathematical infinite than the two kinds of philosophical infinite. In exhibiting the genuine mathematical infinite, the form of series was used at the start, and has of late been called up again. But that form is not essential to it; quite the reverse, what is infinite about the infinite series differs essentially from it, as the sequel is to show; it even ranks below the expression of the fraction.

The infinite series, namely, harbours the bad infinity for this reason, that what it is meant to express remains an ought; and what it does express is afflicted with a beyond that does not vanish, and is different from what is to be expressed. It is infinite not on account of the terms that are posited, but for this reason, that they are incomplete, that the other which belongs to them essentially lies beyond them; what is there in it, be the posited terms as many as one pleases, is only a finite, and indeed posited as finite, as one that is not what it ought to be. By contrast, that which bears the name finite expression, or sum, of such a series has no defect; it rather contains in full what the series only seeks; the beyond has been summoned back out of its flight; what it is, and what it ought to be, stand not apart but are one and the same. It therefore contains no finitude, nothing beyond which one must look.

This can also be looked at thus, that within the infinite series the negative falls outside its terms, which have presence in that they count only as parts of the amount. In the finite expression, by contrast, which is a ratio, the negative is immanent, as the being-determined of the sides of the ratio through one another.

In fact, then, what is usually called the sum, the 2/7 or , is a ratio; and the so-called finite expression is the genuinely infinite expression. The infinite series, however, is in truth the sum; its purpose is to exhibit in the form of a sum what in itself is ratio, and the terms present in the series are terms not of a ratio but of an aggregate. It is, further, rather the finite expression; for it is the imperfect aggregate, and remains essentially something defective. — If the fraction is called the finite expression insofar as it is a determinate quantum, then the infinite series is, first, according to what is there in it, likewise a determinate quantum, yet at the same time a lesser one than it ought to be; then too what it lacks is a determinate quantum; and what is there in it, taken together with what it lacks, is just such a quantum, the same as the fraction is. Insofar, then, as the fraction is a finite, that is, a determinate quantum, the series is one likewise, and still more so than the fraction. But insofar as the fraction is infinite, and indeed infinite in the genuine sense at its own self, because it has the negative beyond at its own self, the series is defective, and has the infinite only as a beyond outside it.

With infinite series that are not summable the case stands otherwise; mathematics, however, comes to a halt at this difference, as at an external and contingent circumstance, whether they can be summed or not. For they contain a higher kind of infinity than the summable ones; an incommensurability, or the impossibility of exhibiting the quantitative ratio contained in them as a quantum — be it even as a fraction; the form of series, however, which they do have, is the same bad infinity that is present in the summable series.

The same inversion noted here at the fraction and at its series takes place insofar as the mathematical infinite, namely the genuine one, has been called the relative infinite, whereas the ordinary metaphysical one has been called the absolute infinite. In fact it is rather the metaphysical that is only the relative one, because the negation it expresses stands only over against a limit that does not get sublated by it; the mathematical infinite, by contrast, has genuinely sublated the finite limit within itself, because the beyond of that limit is united with it.

It is above all in the sense I have pointed out, namely that what is termed the sum, or the finite expression, of a series that is infinite counts rather as the infinite one, that Spinoza sets up the concept of true infinity against that of the bad, and elucidates it by examples. His concept gains most in light if I attach what he says on this head to the present development.

He defines the infinite, to begin with, as the absolute affirmation of the concrete existence of some nature, the finite on the contrary as determinateness, as negation. The absolute affirmation of a concrete existence is, namely, to be taken as its reference to itself, as its not being by virtue of an other's being; the finite, by contrast, is negation, a ceasing, insofar as an other begins outside it. The absolute affirmation of a concrete existence does not, indeed, exhaust the concept of infinity; this concept implies that infinity is affirmation not as immediate affirmation, but only as one that has been reinstated, by way of the other's reflection into itself; or as negation of what is negative. With Spinoza, however, substance and its absolute unity have the form of an unmoved unity, of a rigidity in which the concept of the negative unity of the self, subjectivity, is not yet to be found.

His mathematical example of the true infinite is, as is well known, a space between two unequal circles, one of which falls inside the other without touching it, and which are not concentric. He made so much, it seems, of this figure and of the concept whose example he used it to be, that he set it as the motto of his Ethics. — „The mathematicians, he says, conclude that the inequalities possible within such a space are infinite, not from the infinite multitude of the parts, for its magnitude is determinate and bounded, and I can posit greater and smaller spaces of the sort, but because the nature of the matter surpasses every determinateness.“ — One sees that Spinoza rejects that representation of the infinite according to which it is represented as a multitude or as a series that is not completed, and recalls that here, at the space of the example, the infinite is not beyond but present and complete; this space is an infinite one for this reason, „because the nature of the matter exceeds every determinateness,“ because the determination of magnitude contained in it is at the same time not a quantum. That infinite of a series Spinoza calls the infinite of the imagination; the infinite, by contrast, as reference to itself, the infinite of thinking, or infinitum actu. It is, namely, actu, it is actually infinite, because it is completed within itself and present. Thus the series 0,285714 ... or 1 + a + a2 + a3 ... is the infinite merely of imagining, or of supposing; for it has no actuality, something is simply lacking to it; whereas 2/7 or is that actually, not only what the series is in the terms present in it, but in addition what it lacks, what it only ought to be. The 2/7 or is likewise a determinate magnitude, like Spinoza's space enclosed between the two circles and its inequalities; and like this space it can be made greater or smaller. But there does not thereby come out the absurdity of a greater or a smaller infinite; for this quantum of the whole is no concern of the ratio of its moments, of the nature of the matter, that is, of the qualitative determination of magnitude. Imagining, by contrast, comes to a halt at quantum as such, and does not reflect on the qualitative reference which makes up the ground of the incommensurability present.

This incommensurability in the more general sense is already present at the 2/7 as well, insofar as 2 and 7 are prime to each other, so that the quantum 2/7 cannot be expressed as a whole number, or as an immediate, ratioless quantum. The higher, proper incommensurability, however, takes in Spinoza's example and the functions of curved lines generally. It leads us on more nearly to the infinite which mathematics needs in the case of such functions, in the case of the functions of variable magnitudes generally, and which is the genuine mathematical infinite, the absolute quantitative infinite generally, that Spinoza too had in mind.

The concept of the magnitudes whose reference these functions express, namely of the variable magnitudes, is to be grasped more precisely, however, than commonly happens. They are variable, namely, not in the sense in which in the fraction 2/7 the two numbers 2 and 7 are variable, in that other numbers, 4 and 14, 6 and 21 and so on into infinity, can just as well be set in their place without altering the determination of magnitude posited in the fraction. So too in any number one pleases can be set in the place of a and b, without altering what is meant to express. In the sense that any number one pleases can be set in the place of the x and y of a function, a and b are variable magnitude just as much, or are so still more, insofar as the function encloses the x and y within a limit generally, or at least within reference to each other. The expression variable magnitudes is on that account superficial and clumsy for determining what distinguishes the magnitudes of a function.

Their genuine concept lies in the following. In 2/7 or , 2 and 7 are, each for itself, determinate quanta, and the reference is not essential to them; a and b are likewise supposed to represent such quanta as remain outside the ratio too what they are. Further, 2/7 and are a fixed quantum, a quotient; the ratio is an amount whose unit the denominator, and the amount of these units the numerator, expresses — or the reverse; even if 4 and 14 and so forth step into the place of 2 and 7, the ratio remains, as quantum too, the same. In the function = p, for instance, by contrast, x and y do have the sense of being able to be determinate quanta; but it is not x and y, only x and y2, that have a determinate quotient. Thereby these sides of the ratio are, first, not merely no determinate quanta, but, second, their ratio is not a fixed but a variable quantum. Nor are they merely universal quanta, with which, as with their ratio, a determinate quantum was supposed to be meant. Rather, their ratio itself is as quantum variable in and for itself. This, however, is contained in the fact that x stands in a ratio not to y but to the square of y, because the ratio of a magnitude to the power is not a quantum but a ratio of the concept. The power-ratio is not an external bounding but one determined through itself; hence an essentially qualitative ratio; there will be more to say of it below. If a determinate value is given to x, then y too receives a determinate value through the function; but if x receives another value, the previous ratio does not remain as quantum but is altered. In the function of the straight line y = a x, = a is an ordinary fraction and quotient; this function is therefore only formally a function of variable magnitudes, or x and y are here what a and b are in , not truly what the variable magnitudes in the proper functions are. — On account of the particular nature of the variable magnitudes in the proper functions, it would surely have been serviceable to introduce for them designations other than the usual ones of the unknown magnitudes in every finite equation, determinate or indeterminate, since they are also essentially different from such merely unknown magnitudes, which are in themselves perfectly determinate quanta, or a determinate range of determinate quanta.

In functions of genuinely variable magnitudes, then, the ratio as quantum is a variable one. What is constant in the ratio of these magnitudes — for the parameter or the constant does not express an immediate ratio of them, but only insofar as they are, as was said, still determined against each other through a power-ratio — is not to be expressed by a number or a numerical fraction, nor to be brought back to the function of a straight line, but is a ratio of quantity that is only of qualitative nature.

The sides x and y of such a function can, however, still signify quanta; only their determination relative to each other is of qualitative nature, and their being-determined through the ratio makes up their essential magnitude. They are not supposed to have the quantitative determinateness that belongs to them already immediately for themselves outside the ratio, nor is the reference to them, as to the 2 and 7 in , to be merely external. If 2 is assumed as numerator of a fraction, the denominator is thereby not yet determined. Combined into a function, however, if the one magnitude is determined, the other is thereby determined likewise; and indeed not according to a constant quotient. The quantitative determinateness, the exponent of the ratio of the variable magnitude, is thus of qualitative nature. In this the variable magnitudes, as the sides of the ratio, still have the signification of quanta, though the exponent has it no longer.

This signification, however, is lost altogether and completely in the infinitely small differences. d x, d y are quanta no longer, nor are they supposed to signify such, but have a signification solely in their reference, a sense merely as moments. They are no longer something, that something taken as quantum, not finite differences; but also not nothing, not the determinationless zero. Outside their ratio they are pure zeros, but they are to be taken only as moments of the ratio, as determinations of the differential coefficient.

Within this concept of the infinite, quantum has been genuinely brought to completion as something qualitative; it has been rendered actually infinite; what is sublated in it is not this or that quantum alone, but quantum as such. There remains, however, determinateness of quantity, element of quanta, principle, or that determinateness in its first concept.

It is against this concept of the infinite that every attack has been directed which has been made upon the mathematics of the genuinely infinite, the differential and integral calculus. Incorrect representations on the part of the mathematicians themselves occasioned at times its going unrecognized; chiefly, however, the blame for these contestations lies with the incapacity to exhibit the subject matter as concept. Yet mathematics, as was already recalled above, cannot here get around the concept; for being the mathematics of the infinite, it does not hold itself within the finite determinateness that belongs to its objects — as in pure mathematics space and number, with their determinations, are considered and related to one another only according to their finitude —; it rather posits a determination in identity with its opposite. The operations which it allows itself as differential and integral calculus therefore contradict entirely the nature of merely finite determinations and their references, and have on that account their justification in the concept alone.

If the mathematics of the infinite held fast to this, that those determinations of quantity are vanishing magnitudes, that is, such as are no longer any quantum whatever, yet are not nothing either, but are still a determinateness over against another, then nothing seemed clearer than that there is no such intermediate state, as it was called, between being and nothing. — What is to be made of this objection and of the so-called intermediate state has already been shown above. Certainly the unity of being and nothing is no state; a state would be some determination belonging to being and to nothing, one into which these moments were supposed to have fallen merely by chance, as though into a sickness or an external affection; this middle and unity, the vanishing or equally the becoming, is rather their truth and theirs alone.

What is infinite, it has further been said, is not comparable as a greater or a smaller; there can therefore be no ratio of infinites to infinites, nor orders or dignities of the infinite, such as the differences of the infinite differences that occur in the science of them. — Underlying these objections there is always the representation that what is here to be spoken of are quanta compared as quanta; that determinations which are quanta no longer have no ratio to each other any more Rather, what is only in the ratio is no quantum; for quantum is a determination of the kind that is supposed to have a perfectly indifferent existence outside its ratio, one to which its difference from another is indifferent, whereas the qualitative is only what it is in its difference from an other. Those infinite magnitudes are therefore not merely comparable, but are only moments of the comparison or of the ratio.

I adduce here the most important determinations that have been given by mathematicians concerning this infinite. It will be evident from them that the thought of the matter, in agreement with the concept developed here, lies at the base of these determinations of theirs, but that they did not fathom it as concept and for that reason again stood in need, in applying it, of expedients contradicting their better cause.

The thought cannot be determined more correctly than Newton has given it. In doing so I set apart the determinations belonging to the representation of motion and of velocity (from which he chiefly took the name fluxions), because in these the thought appears not in the requisite abstraction but concretely, mixed with inessential concepts. — These fluxions Newton explains more closely (Princ. mathem. phil. nat. L. 1. Lemma XI. Schol.) to this effect, that by them he understands not indivisibles — a form employed by earlier mathematicians, Cavalleri and others, and one that contains the concept of a quantum determinate in itself, — but vanishing divisibles. Further, not sums and ratios of parts that are determinate, but the limits (limites) of the sums and of the ratios. The objection is raised that vanishing magnitudes have no last ratio, because before they have vanished it is not the last, and when they have vanished there is none at all. But by the ratio of vanishing magnitudes is to be understood the ratio not before they vanish, and not afterwards, but with which they vanish (quacum evanescunt). In the same way the first ratio of becoming magnitudes is that with which they become.

Given the state of scientific method at that time, all that was explained was what is to be understood by an expression; that this or that is now to be understood by it is properly a subjective demand, or else a historical requirement, where it goes unshown that such a concept is in and for itself necessary and has inner truth. But what has been adduced shows that the concept set up by Newton answers to the way in which infinite magnitude issued, in the presentation given above, from the reflection of quantum into itself. Magnitudes are what is meant, in their vanishing, that is, such as are quanta no longer; further, not ratios of determinate parts, but the limits of the ratio. For the immediate ratio too, insofar as it has an exponent, is a quantum; both the quanta for themselves, then, the sides of the ratio, and with them the ratio too, insofar as it would be a quantum, are to vanish; the limit of the ratio of magnitudes is that in which it is and is not; more precisely, this means that in which the quantum has vanished and with it the ratio is preserved only as qualitative ratio of quantity. — Newton adds that from there being last ratios of the vanishing magnitudes one is not to conclude that there are last magnitudes, indivisibles. That would be, namely, once more a leap from the ratio as such over to its sides, which would be supposed to have a value for themselves outside their reference, as indivisibles, as something that would not be a relative. — He holds fast to divisibility in order still to preserve the quantitative, because the indivisible, or atoms, the one, would be something ratioless.

Against that misunderstanding he further recalls that the last ratios are not ratios of last magnitudes, but limits which the ratios of magnitudes decreasing without limit approach more nearly than any given, that is, finite difference, yet which limit they do not overstep, so that they would become nothing. — By last magnitudes, namely, as was said, indivisibles or ones could have been understood. In the determination of the last ratio, however, the representation both of the indifferent one, the ratioless one, and of the finite quantum is removed. Yet neither the decreasing without limit, into which Newton transposes the quantum and which expresses only the progress into infinity, nor the determination of divisibility, which here has no immediate significance any more, would have been called for, had the concept demanded developed itself further into the concept of a determination of magnitude that is purely and only moment of the ratio.

Equally interesting is the other form of the Newtonian presentation of these magnitudes, namely as generated magnitudes. A generated magnitude (genita) is a product or quotient, roots, rectangles, squares, also sides of rectangles and squares; — in general a finite magnitude. — „Considering it as variable, as it increases or decreases in continuing motion and flux, he understands its momentary increments or decrements under the name of moments. These, however, are not to be taken for particles of determinate magnitude (particulae finitae). Such are not themselves moments, but magnitudes generated from moments; what is to be understood are rather the becoming principles or beginnings of finite magnitudes.“ — The quantum is here distinguished from itself, as it is as a product, or something existent, and as it is in its becom ing, in its beginning and principle, that is, as it is in its concept, or what here amounts to the same, in its qualitative determination; within the latter, the differences of quantity, the infinite increments or decrements, count merely as moments; it is only what has become that has passed over into the indifference of existence and into the externality wherein it is quantum. — The increments and decrements do indeed fall within the sensuous representation of quantum; but the other determinations adduced must be acknowledged by the philosophy of the concept of the genuinely mathematical infinite.

Far behind the determinations considered stands the ordinary representation of infinitely small magnitudes. On this representation they are supposed to be of such a constitution that not only they themselves as against finite magnitudes, but also their higher orders as against the lower, or the products of several as against a single one, are to be neglected. — Leibnitz, like the earlier inventors of methods bearing on this magnitude, held to this representation; it is chiefly this that gives to that calculus, along with the gain in convenience, the semblance of inexactitude in the path of its operations. — Wolf sought to make it intelligible in his own way of making things popular, that is, of defiling the concept and setting incorrect sensuous representations in its place. He compares, namely, the neglecting of infinite differences belonging to higher orders over against lower ones with how a surveyor proceeds who, while gauging how high a mountain is, would have been no less exact had the wind meanwhile blown a grain of sand off its summit.

If the fair-mindedness of common human understanding permits such an inexactitude, all geometers, on the contrary, have rejected this representation. — It presses itself upon one of its own accord that in the science of mathematics there is no question whatever of such empirical exactitude, that mathematical measuring by operations of the calculus, or by constructions and proofs of geometry, is entirely distinct from land-surveying, from the measuring of empirical lines, figures and so forth. In any case the analysts show, as was adduced above, through the comparison of the result as it is obtained on the strictly geometrical route and as it is obtained by the method of infinite differences, that the one is the same as the other, and that a more or less of exactitude has no place whatever. And it goes without saying that an absolutely exact result could not come from a procedure that was inexact. Yet on the other side, again, the procedure itself cannot dispense with that neglect on the ground of insignificance. And this is the difficulty around which the efforts of the analysts turn, of making comprehensible to themselves the absurdity lying in this.

Euler, taking the general Newtonian definition for his base, insists above all that what the differential calculus has under consideration are the ratios of the increments belonging to a magnitude, whereas the infinite difference taken by itself must count wholly as zero. — How this is to be understood has been sufficiently elucidated; the infinite difference is zero only of the quantum, not a qualitative zero, but as zero of the quantum it is rather pure moment only of the ratio. It is not a difference by a magnitude, as when one quantum is subtracted from another, where their difference is itself also a quantum, to which it is indifferent whether it is regarded as a difference or as a sum, product and so forth, and which therefore does not have merely the sense of a difference. Since the ratios of the infinite differences are derived from the ratios of magnitudes that are variable but considered as finite, those ratios contain as results that within themselves as moment which the latter express as existent, or in finite determination, — or rather only in finite determinability, for the finite magnitudes that have such increments as are here considered are variable ones which do not themselves have a determinate quantum, though they can have one. On the one side it is, as was recalled, altogether awry, and belongs to the sensuous representation, to enunciate the infinitely small magnitudes as increments or decrements and as differences. For underlying this presentation is the idea that to the finite magnitude present at the start something is added or from it something is subtracted, that a subtraction or an addition, an arithmetical, external operation, takes place; the transition from the variable magnitude into its infinite difference, or of the function into its differential, is rather of a quite other nature; it is to be regarded as the leading back of that magnitude to the qualitative ratio of its determinations of quantity. — On the other side it has an awry aspect for this reason, when it is said that the increments are for themselves zeros, that only their ratios are considered. For a zero has no determinateness left at all. This representation does indeed reach as far as the negative of quantum, and enunciates it determinately, but it does not at the same time grasp this negative in its positive meaning, which, as was shown, consists in this, that the variable magnitudes, in that their ratio goes back into its qualitative determinateness, are no quanta, but are also not determinationless zeros, but moments; it is a ratio of determinations of quantity which, were they to be torn out of the ratio and taken as quanta, would be nothing but zeros. — Lagrange judges of the method which lays the representation of limits or last ratios at the base, — which L' Huillier in particular elaborated, — that although one may very well represent to oneself how two magnitudes stand in ratio so long as they remain finite, such a ratio yields the understanding no distinct and determinate concept as soon as its terms become zero together. — In fact the understanding must go out beyond this merely negative side, that the terms of the ratio are zeros as quanta, and apprehend them positively, as qualitative moments.

As concerns how the ratio keeps itself in the vanishing of the quanta, one meets, for instance in Carnot, the expression that by virtue of the law of continuity magnitudes which vanish hold on still to that ratio out of which they have come, before ever they vanish. — This representation expresses the true nature of the matter, provided that what is understood is not that continuity of quantum which it has in the infinite progress, where it continues itself into its vanishing, namely where in its beyond there arises once more only a finite quantum, a new term of the series, or the sum of that term with the preceding ones. In that negation, by contrast, which is the genuine infinite, the quanta vanish as indifferent, external determinations, and become only moments of the ratio. The ratio is therefore so continuous in this transition, and so preserves itself, that the transition rather consists solely in lifting the ratio out purely and making the ratioless side vanish. This purification of the quantitative ratio is nothing other than what happens when an empirical existence is comprehended. This existence is thereby raised above itself in such fashion that its concept contains the same determinations as it does itself, but grasped in their essentiality and into the unity of the concept, in which they have lost their indifferent, conceptless subsistence.

I refrain from multiplying the citations, since the determinations considered have shown well enough that the genuine concept of the quantitative infinite lies at their base, even though it has not been lifted out and grasped in its determinateness. For this very reason, however, it comes about that the concept does not maintain itself in its application and that the operation becomes unfaithful to it. The operation is grounded chiefly on the representation of a merely relatively small. The calculus makes it necessary to subject the infinite magnitudes to the ordinary arithmetical operations of adding and so forth, which are grounded on the nature of finite magnitudes, and thus to let them count for a moment as finite magnitudes and to treat them as such. The calculus would have to justify itself, on the one side, for dragging them down on the one occasion into this sphere, and on the other side for now and again dropping them and neglecting them as quanta, right after having applied to them the laws of finite magnitudes.

I adduce a few more things concerning the attempts of the geometers to remove the difficulty which gives the method the semblance of inexactitude.

The older analysts made fewer scruples about this; but the efforts of the moderns went chiefly to bringing the calculus of the infinite back to the evidentness of the properly geometrical method and to attaining in it the rigour of the proofs of the ancients in mathematics. Yet since the principle of the analysis of the infinite is of higher nature than the principle of the mathematics of finite magnitudes, the former must necessarily renounce the lesser merit of evidentness, which the latter owes chiefly to the conceptlessness of its content and of its method, just as philosophy too can lay no claim to that distinctness which the sciences of the sensuous, natural history for instance, possess, and just as eating and drinking pass for a more intelligible business than thinking and comprehending.

Several have attempted to dispense with the concept of the infinite altogether and to accomplish without it what seemed bound up with its use. — Lagrange, for instance, speaks of the method invented by Landen, and says of it that it is purely analytic and does not employ the infinitely small differences, but first introduces different values of the variable magnitudes and subsequently sets them equal. He judges, moreover, that in this way the advantages proper to the differential calculus, simplicity of method and ease of operations, are lost. — It is evident from what has been adduced that the vanishing of quantum occurs in this method too, namely in this, that the different assumed values of variable magnitudes are set equal to each other; for to set one quantum equal to another unequal to it means nothing else than to sublate them, and here indeed in order thereby to win their universal determination of ratio. — L' Huillier's method, which was grounded on the representation of the limits of a ratio, presses chiefly the point of regarding d x and d y simply and solely as moments of the differential coefficient, and as a single indivisible sign. But although this method remains the most faithful to the philosophical concept of the quantitative infinite, in the judgment of the geometers it does not accomplish what the calculus of the infinite attains by separating the sides of the differential coefficient from each other. Besides the fact that the limit always represents the positive, here namely a quantum, on the one side, and on the other side the negative of it, separated, and does not unite the two into the simple determination of the qualitative moment of quantity, — this method seems not to accomplish the transition, necessary for the manner of calculation which makes up the advantage of the ease of the calculus of the infinite, of the moments of ratio into the shape of finite magnitudes, nor the specification of the laws required for them on this ground and for their return into their peculiarity.

The elder among the moderns, such as Fermate, Barrow and others, who first made use of the infinitely small in that application which was later elaborated into the differential and integral calculus, and then Leibnitz too and those who followed, have always believed openly that they were entitled to drop the products of infinite differences, and likewise their higher powers, on no other ground than that relatively, as against the lower order, they vanish. On this alone rests, with them, the fundamental theorem of the whole doctrine, namely what the differential of a product or of a power is. On the same ground the main proposition concerning curves is assumed, which consists in this, that the elements of the curves, namely the increments of the abscissa and of the ordinate, have to each other the ratio of the subtangent and the ordinate; for with a view to obtaining similar triangles, the arc, which makes up the third side of a triangle alongside the two increments, is regarded as a straight line, as a piece of the tangent, and thus one of the increments as reaching right up to the tangent. These assumptions raise these moments, on the one side, above the nature of finite magnitudes; on the other side, however, a procedure is applied to them that holds only of finite magnitudes, and in which nothing may be neglected out of regard for insignificance. The difficulty by which the method is pressed remains in the manner of proceeding adduced in its full strength.

Newton employed (Princ. Math. phil. nat. Lib. II. Lemma II. after Propos. VII.) an ingenious artifice in order to get rid of the arithmetically incorrect dropping of the products of infinite differences, or of their higher orders, in the finding of the differentials. The differential of the product — from which the differentials of quotients, powers and so forth are then easily derived — he finds in the following way. The product, if x, y are each taken smaller by half of its infinite difference, passes over into ; but if x and y increase by just as much, into . With the first product subtracted from this second one, y d x + x d y remains as excess, and this, he says, is the excess of the growth by a whole d x and d y, for it is by this growth that the two products are distinguished; it is therefore the differential of x y. — One sees that in this procedure the term which makes up the chief difficulty, the product of the two infinite differences, d x d y, falls away of itself. But it is incorrect that or that the excess of a product whose factors each increase by a whole increment, over the product of the original factors, — should be equal to the excess of the product when its factors each grow by half the increment, over the product insofar as its factors have decreased by this half.

Other forms which Newton employs in deriving the differential are tied to concrete significations of the elements and of their powers. — In the use of series, which distinguishes his method, the ordinary representation of series lies too near at hand, namely that it always stands in one's power, by adding further terms, to take the magnitude so exactly as one requires, and that the terms dropped are relatively insignificant, the result on the whole being only an approximation. — The error into which Newton fell in solving a problem, through dropping essential higher powers, an error that gave his opponents an occasion for the triumph of their method over his, and whose true origin Lagrange has exhibited in his recent investigation of it, — proves at least the formal character and the uncertainty still present in the use of his instrument. Lagrange (in his Theorie des Fonctions analytiques) shows that Newton fell into the error because he neglected the term of the series which contained the power upon which everything turned in the determinate problem.

It is remarkable, namely, that in mechanics the several terms of that series into which the function of a motion is developed carry each its determinate signification, so that the first term, or the first function, has reference to the moment of velocity, the second to accelerating force, and the third to the resistance of forces. The terms of the series are here, accordingly, to be viewed not merely as parts of a sum, but as qualitative moments of a whole of the concept. Thereby the dropping of the remaining terms, which belong to the badly infinite series, acquires a wholly different signification from the dropping of them on the ground of their relative smallness. They are to be dropped because, through the determinations of the concept to which the first terms belong, the whole of the object is completed as concept, and thereby also as sum, its determination of quantity in general. The Newtonian solution contained that error, not because in it terms of the series, as parts of a sum, but because a term containing a determination of the concept, which belonged to the whole, was dropped.

In this respect too it is the case that the differential of xn is wholly exhausted in the first term of that series which the development of (x + d x)n yields; — a view upon which L' Huillier especially pressed. That the remaining terms are not taken into account does not come from their relative smallness; — no inexactitude, no fault or error is here presupposed that would be compensated and corrected by another error; a view from which Carnot above all justifies the ordinary method of the infinitesimal calculus. Rather, since what is here at issue is not a sum but a ratio, the differential is completely exhausted through the first term, in that the further terms, or differentials of higher orders, develop out of their predecessors in the same way as the differential of the original function develops out of that function, so that in them there is nothing but the repetition of one and the same ratio, which alone is wanted, and which is thus already perfectly attained in the first term.

I do not adduce separately the elucidations which Carnot gives concerning the method of infinite magnitudes. They contain what is most refined in the representations adduced above. But at the transition to the operation itself the ordinary representations, of the infinite smallness of the dropped terms as against the others, come in more or less. He justifies the method rather by the fact that the results come out correct, and by the utility which the introduction of imperfect equations, that is, of such as have had an arithmetically incorrect omission made in them, has for the simplification and abbreviation of the calculus, than by the nature of the matter itself.

Lagrange, as is well known, took up again Newton's original method, the method of series, in order to be relieved of the difficulties which the representation of the infinitely small, and of those which the method of first and last ratios and limits, carries with it. Of his calculus of functions, whose other merits with respect to precision, abstraction and universality are not to be brought out further here, only this is to be adduced, that it rests on the fundamental theorem that the difference, without its becoming zero, can be assumed so small that every term of the series exceeds in magnitude the sum of all those following. — One sees that the terms of the series to be dropped here come into consideration only in the respect that they constitute a sum, and that the ground for dropping them is placed in the relativity of their quantum. The omission is thus here too not led back, for the general case, to that ground which occurs in some applications, where, namely, as was recalled earlier, the terms of the series have a determinate qualitative signification, and following terms are left out of account, not because they are insignificant in magnitude, but because they are insignificant in point of quality.

In conclusion I set this uniquely correct point of view, the qualitative nature of the infinite differences, against the misunderstanding which seems to occur especially in the older presentations, and which takes the infinite differences as wholly ratioless moments, letting the determination of ratio vanish along with the quanta.

For since the infinite differences are the vanishing of the sides of the ratio as quanta, what remains over is their ratio of quantity, purely insofar as this depends on the qualitative determination. So little is the qualitative ratio lost in this that it is rather the determining factor, and precisely what results through the conversion of finite magnitudes into infinite ones. In this, as has been shown, consists the entire nature of the matter. — Thus in the last ratio the quanta of the abscissa and of the ordinate vanish; but the sides of this ratio remain essentially the one increment or element of the ordinate, the other increment or element of the abscissa. When, according to the ordinary manner of representation, one lets the one ordinate approach the other infinitely, the ordinate previously distinguished passes over into the other ordinate, and the abscissa previously distinguished into the other abscissa; (— as, according to the above, Landen first assigns different values to the variable magnitudes and then sets these equal —) in this passing over their finite difference vanishes, and there remains only the infinite difference, as moment of this passing over, the element of the ordinate and the element of the abscissa. Essentially the ordinate does not pass over into the abscissa, nor the abscissa into the ordinate. The qualitative ratio continues itself, as was expressed above, so far into the infinitely becoming, that is, vanishing differences of quantum, that it alone is that by which the determination of quantity is still borne.

In accordance with this it is essential now, against the point of view which the ordinary opinion has of the infinite differences, and which above all makes it hard to grasp the correct concept of the matter, — to remark that the element of the ordinate, — to stay with this example of variable magnitudes, — is no longer the difference of one ordinate from another ordinate, for these are no longer different quanta over against each other, since they have been infinitely approximated to each other, but is rather the difference, or the qualitative determination of magnitude, over against the element of the abscissa; the principle of the one variable magnitude over against that of the other stand in ratio with each other. The difference, in ceasing to be difference of finite magnitudes, has ceased to be a manifold within itself; it has collapsed into simple intensity, into the determinateness of one qualitative moment of ratio over against the other.

The consideration of these elements as differences, or also as increments, essentially holds fast only to the difference of the quantum of one ordinate from the quantum of another ordinate. The limit is taken as the last value which another magnitude, otherwise of like kind, constantly approaches, so that it can be distinguished from it by as little as one pleases, and so that the last ratio is a ratio of equality. Thus the infinite difference is a hovering, as difference of one quantum from a quantum, and the qualitative nature, according to which d x is essentially a determination of ratio not over against x but over against d y, recedes in the representation. One lets d x2 vanish as against d x, but far more does d x vanish as against x, or it has a ratio only to d y. — It has been recalled that this side is most of all lifted out in L'Huillier's method. But it has not yet been brought to the concept of the qualitative determination of magnitude, and the geometers who hold to the representation of limits are always concerned above all to make comprehensible the approximation of a magnitude to its limit, and to keep hold of this side of the difference of quantum from quantum, of how it is no difference and yet still a difference.

But since it has come about that the increments or infinite differences were taken merely on the side of quantum and as ratioless moments, there has sprung from this the inadmissible representation which permits itself, in the last ratio, to set abscissa and ordinate, or even sine, cosine, tangent, versed sine and whatever else, equal to one another.

The arc, too, is surely incommensurable with the straight line, and its element, in the first instance, of a quality other than that belonging to the element of the straight line. It seems accordingly still more absurd and more impermissible than the confounding of abscissa, ordinate, sine, cosine and so forth, when quadrata Rotunde, when a part of the arc, infinitely small though it be, is taken for a part of the tangent, or in general as hypotenuse in a right-angled triangle in which the two legs are the elements of the abscissa and of the ordinate, and thus is treated as a straight line. — This treatment, however, is essentially to be distinguished from the confounding just censured; it has its justification in this, that within such a triangle the ratio of the element of an arc to the element of the abscissa and of the ordinate is the same as it would be were that element the element of a straight line, of the tangent; for the angles which constitute the essential ratio, namely the ratio that remains to these elements after the finite magnitudes belonging to them have vanished as quanta, are the very same. — One can also put it thus, that straight lines, as infinitely small, have passed over into curved lines, and that the ratio of them in their infinity is a ratio of curves. For if one takes the ordinary definition of the straight line, that it is the shortest path between two points, then its difference from the curved line is grounded on the determination of multitude, on the lesser multitude of what is distinguishable along this path, which is accordingly a determination of quantum. But this determination vanishes in it, once it is taken as intensive magnitude, as infinite moment, as element; and with it also its difference from the curved line, a difference that rested merely on the difference of quantum. — Or, an infinite straight line is the sublated straight line, for the infinite straight line is the one that goes back into itself, that is, a curve. Thus, as infinite, straight line and curve retain no qualitative ratio to each other any more; the former rather passes over into the latter.

Quite otherwise, however, is it constituted with the ratios of sine, tangent and so forth to one another. It is easy to see, and has also been recalled by others, that if with the general excuse that in the last ratio everything is equal, that is, that the ratio itself too is sublated, one permits oneself to put the ordinate for the abscissa, then the most absurd things can be brought out, or, as it is called, proved. By such a confounding the underlying concept, that for the variable magnitudes in their vanishing the ratio from which they come is preserved, is entirely destroyed. There arises in the proper sense a ratio of zero to zero, for which it is wholly arbitrary and contingent what qualitative and quantitative signification is given to it. With the licence of such an equating it cannot be hard to bring forth formulas yielding as result that the diameter is greater than the circumference, the hypotenuse smaller than a leg, and so forth.

There can surely be no other ground for people's having put up with proofs built upon that equating than this, that what came out was always known beforehand anyway, and that the proof, contrived so that it should come out, notwithstanding that in such a fashion the opposite could just as well be brought out, at least brought off the semblance of a scaffolding of proof; — a semblance still preferred to mere faith or to knowledge from sensuous experience. I have no hesitation in regarding this manner as nothing more than a mere sleight of hand and charlatanry of proving, and in reckoning among it even a mass of the Newtonian proofs, but especially of those on account of which Newton was exalted to the skies and above Keppler for having set out mathematically what the latter found merely through experience. So long as the mathematics of the infinite lacks the thorough concept of its object, it is unable to specify the limit up to which that equating may go, and even to the correct ones among its operations there always clings the mistrust that springs from the uncertainty, and, in the case of the confounding adduced, — from the senselessness of this procedure, — a procedure that has nothing to reproach in the often-mentioned prattle of recent philosophers, — which at the same time is wont to make up their entire philosophy, — that in the absolute all is one.

The empty scaffolding of Newtonian proofs of that sort was erected chiefly in order to prove physical laws. But mathematics is simply not in a position to prove determinations of magnitude in physics, insofar as these are laws having the qualitative nature of the moments for their ground; and this on the plain ground that such a science is no philosophy and does not set out from the concept, so that whatever is qualitative, insofar as it is not taken up lemmatically from experience, lies outside its sphere. That scaffolding will doubtless yet meet with the same justice that has recently been done to the groundless Newtonian edifice of artifice built from optical experiments and the inferring bound up with them. Applied mathematics remains full of just such a decoction of experience and reflection; yet just as, in the case of that optics, one part after another has for a good while now begun to be factually ignored, so it is a fact as well that a portion of those deceptive proofs, grounded as they are on that lawless and senseless equating of qualitative determinations under the pretext of their infinite smallness, has already, without their defect having been seen through, fallen of itself into oblivion or been supplanted by others.

Chapter 3. The Quantitative Ratio

Quantum, having become infinite, has the negative beyond at its own self. This beyond is the qualitative in general. The infinite quantum is the unity of both moments, of the quantitative and of the qualitative determinateness. It is ratio.

In the ratio, therefore, quantum no longer has an indifferent determinateness, but is qualitatively determined, as referred simply to its beyond. Quantum continues itself into its beyond; this beyond is at first another quantum in general. But essentially they are not referred to one another merely as external quanta; the one does not have its determinateness as indifferent towards that of the other, but each has it in this reference to the other. They are therefore, in this otherness of theirs, returned into themselves; for what each is, it is not immediately for itself but in the other; the other makes up the determinateness of each. — A quantum goes out beyond itself as quantum, yet not so that it merely altered into another, nor into its abstract other, into its negative beyond, but into its determinateness; it finds its own self in its beyond, which is another quantum.

This being-determined of the quanta through one another, in which each has the essential significance of being not indifferently for itself but a moment of the ratio, and of being only in the reference to the other what it is, makes up the qualitative moment of the ratio. But the ratio at the same time remains quantitative. It is quanta that lie at the ground and have towards one another the reference that has resulted; or it is one quantum in general that has the qualitative determinateness within itself. Quantum, in that it is ratio, expresses itself as a totality closed within itself, and expresses its indifference towards the limit, in that it has the externality of its being-determined within its own self and in that externality is referred only to itself. — The qualitative and the quantitative have not here yet stepped apart; the qualitative is that of quantum itself, or that whereby quantum is quantum.

A. The Direct Ratio

1. In the ratio the determinateness of the one quantum is that of the other. The two have but one determinateness, or limit. Of two quanta standing in no ratio, each has its own determinateness, indifferent towards that of the other. But the quanta of the ratio have only one common determinateness, the exponent of the ratio.

In the immediate ratio this exponent is itself an immediate quantitative determination, or some quantum or other. It makes up the one quantum that alone is in the ratio as such. The quanta which make up the sides of the ratio are the quanta posited as sublated; they are not indifferent quanta, hence not two; rather each has its determinateness at the other; they therefore make up only one, the simple exponent, and they themselves are posited in this unity as indifferent ones.

2. The exponent is the simple determinateness of the ratio. But as such it is not the qualitative determinateness, but some quantum or other. As the qualitatively determined quantum it is the quantum that has the difference of itself, its beyond and otherness, at its own self. This is not the external difference of quantum whereby it is greater or smaller than another, but its qualitative determinateness, its own difference at its own self. But the difference of quantum at its own self is the difference of unit and amount. The unit is itself the simple, absolute being-determined; the amount, by contrast, is the indifferent moving to and fro upon determinateness, the indifference that quantum has as external. At the outset unit and amount were the moments belonging to quantum; now, at the same time, each of these moments appears as a quantum of its own; they are the determinations of its existence, the delimitations within which the otherwise merely external, indifferent magnitudes are posited over against one another.

These two moments of quantum make up the moments of the quantitative ratio itself; for it is the unity of the qualitative and the quantitative determinateness. Thus in it quantum is partly as determined in itself, partly as indifferent and external. It is therefore the immanent determinateness of quantum itself, or its quality, that the sides of the ratio have towards one another. The one quantum of the ratio is not merely amount; rather this amount is unit over against the amount, and essentially has this value and significance, to count as unit; and the other is likewise not merely amount in general, as an indifferent quantum in general, but is amount as over against the other quantum, insofar as this latter is the unit.

The exponent is this difference as simple determinateness; it is first quantum; as such it is the side of the amount. When that side of the ratio which is taken for the unit gets expressed as the numerical one, the other side, the amount, is then the exponent's own quantum; second, the exponent is the simple unity, the qualitative element of the quanta which are sides of the ratio; they are moments in this unity. If the one is determined, then the other too is determined through the exponent, and it is wholly indifferent how the first is determined; as something determined for itself, as an indifferent quantum, it no longer has any significance, but can just as well be any other, without altering the determinateness of the ratio, which rests upon the exponent alone.

3. Inasmuch as the sides of the ratio are determined over against one another through the moments of quantum, they therein properly make up only one quantum. Conversely they are not qualitatively determined over against one another, insofar as they are diverse quanta. — As regards the first respect, the one quantum has only the value of the unit, not of an amount; the other only that of the amount; according to their determinateness, therefore, they are not complete quanta. If the one is altered, then the other is increased or diminished by just as much; that is to say, simply only the one, the unit, is altered, and the other determinate side, the amount, always remains the same quantum. Accordingly, as quanta they are not qualitatively determined over against one another; or this alteration, in which the two behave as quanta, is no negative determination at the ratio as such. — The exponent for its part is only the amount of the ratio, and has no negative determination at its own self. — Insofar as the other side, that of the unit, is a quantum, there are present two indifferent quanta, the exponent or the amount as such, and that quantum, and they are present as indifferent ones, not determined through the ratio. But insofar as the other side counts as unit, the other, the amount or the exponent, is not determined through it, but is an indifferent quantum in general.

But quantum is posited in the ratio as infinite only insofar as it has in the other quantum its beyond, its non-being. The two sides of the ratio are not only determined as unit and amount; according to these moments they make up only one quantum; but they are both quanta, and inasmuch as they are this in the simple unity of the ratio, their negativity is essentially posited therein. — They are qualitatively referred to one another; but quality is essentially negation, and the one side in fact relates itself to the other only as an other, insofar as it is as a non-being, as a sublating of that other. The simple determinateness, the exponent, is in this way genuine determinateness insofar as it is not only immediate quantum that is, but at the same time quantum that is not, and in its simplicity is not something determined over against an other but something determined in itself, and thus the negative of its own self.

B. The Inverse Ratio

1. The ratio has now determined itself in such a way that the positedness of a quantum is at the same time as the non-being of this quantum. In the inverse ratio there is present this, that the same quantum is posited as one that is and as one that is not. The one side of it relates itself to the other in such a way that, however great the one is, by so much is the other lacking. By as much as the one increases, by so much does the other decrease.

The one of the magnitudes standing in this ratio therefore does not continue itself into the other in such a way that it would remain the unit of its other, the amount; rather it continues itself negatively into it; it sublates in it as much as it itself is. Each is, as amount, the negative of the other; each is as great as what is wanting to the other. Each in this way contains the other and is measured by it; for each is only the quantum that the other is not. — The continuity of each in the other makes up the moment of simplicity in this ratio. The one quantum is the non-being of the other; hence neither is an indifferent one; rather it is, first, as its own self, and second it is as negated, and so it is the other. Insofar as it is as its own self, it is the negation of the other quantum.

These two sides, which each of the two magnitudes standing in the ratio has, do not fall apart, or into an external reflection which would merely compare them and find that the one quantum is less than the other, that in the one there is a being which in the other is a non-being. What a quantum is not for this external reflection, or in the comparison, is no concern of that quantum. What in that comparison the smaller quantum is, or its positive magnitude, is not a non-being, a lack of the other, of the greater; rather the greater contains the smaller within itself.

But in the negative reference of the quanta, in which they stand in the inverse ratio, otherness is their own restricting, and non-being is a lack and an ought of their own comparison with themselves. Therein quantum has another quantum over against it in such a way that it is at its own self this other quantum, but at the same time as its non-being. The alteration of the one is therefore this doubled thing: first, that it is an alteration of the one as a quantum that is, and at the same time of its other side, namely of its non-being or of the other quantum; second, however, that what thereby becomes being of the one is non-being of the other; and so it is not, as in the direct ratio, properly only the one side, the unit, that alters.

2. Quantum in the inverse ratio therefore goes out beyond itself in such a way that it has its determinateness in that to which it is referred; it has it therein as in its non-being, and precisely thereby, because its non-being makes it into what it is, this non-being of its is its own self.

The one quantum in this way makes up with its other one sphere; each of the two quanta is itself this whole. This whole is thus here the exponent. It is the limit and the simple determinateness of this ratio. It is, first, the simple determinateness of that ratio as immediate quantum. As such it is some indifferent magnitude or other; the whole as quantum that is. For the quantitative ratio has quantum in general for its foundation. — In this immediate determinateness it is the limit of the sides of its ratio, within which they increase and decrease over against one another, but which they cannot overstep. It makes up their limit, their non-being, in that it is the whole that is, whereas the sides are the whole only in such a way that according to one part they are and according to the other they are not. It is thus their beyond, to which they approach infinitely, but which they cannot reach. This infinity, in which they approach it, is the bad infinity of the infinite progress; it is itself finite, limited by its opposite, and therefore only an approximation; for neither of the quanta can overcome the other and reach the whole, but remains affected by this its negation, its other. But the bad infinity is here posited as what it is in truth, namely only as a moment of the whole, of the exponent. It is at the same time sublated, the beyond is reached; for the sphere is the unity of the beyond and the this-side of each of the two magnitudes; the beyond of each is the other, and each is in itself its other, each is in itself this whole.

3. Of the two magnitudes of the negative ratio the one increases as the other decreases, and conversely; the being of the one is essentially the non-being of the other. But this makes up no difference between them; for the same is the case with the one as with the other. Their quantitative difference — which is the greater or the smaller, or whether they are equal — is in any case their indifferent difference; and indeed in the ratio it is posited as unessential; they count only as such as can increase or decrease. It is therefore not to one of the sides over against the other, but to the two of them together, the whole sphere, that the difference belongs.

The whole now, or the exponent, is, as it has resulted, an immediate quantum that makes up the limit for the quanta contained under it. It is not only immediate quantum, but is being-differentiated at its own self, at first into two sides, each of which is in itself the whole sphere, is its own self, and essentially also has the other at it as its negation. Thereby the whole itself is posited in a doubled way. — First, it is the sum of the two sides, insofar as they are quanta that are, the whole quantum that is. But second, this whole is also as a negative one. For each of the two sides is the lack, or is as the being-negated of the other; each is as great as what is missing to the other. Hence the whole too is at the same time posited as an ought, as something negated. As has been recalled, each is not merely a non-being of the other in an external reflection; rather this is here its value, that the non-being of each is the other; both are thus, and thereby the whole, posited as a non-being.

But herewith, third, this being and non-being are one and the same. The whole sphere is at first immediate quantum; then it is posited as a non-being; but precisely this its non-being is itself only the whole sphere that is. For each of the two sides, insofar as it is the negation of the other, has existence; what vanishes from the other accrues to it; the non-being of each therefore makes up what the other is, and sublatedness is in this mutuality the existence of that which is sublated.

What is present, therefore, consists in this, that the exponent of the ratio, an immediate quantum, is as its non-being, as an other, but that this otherness is its own self. Quantum continues itself into its otherness, and the negation is only an otherness in which it preserves itself as the sphere lying at the ground, and remains the unity in this otherness.

Thus the inverse ratio, as it appears according to its determination, is sublated. It consists in this, that quantum was therein supposed to refer itself to its other in such a way that this other should be only its non-being, that the positive of its beyond should be a quantum diverse from it. But the nature of quantum is to be an indifferent limit, and hence to have sublated this non-being, the absolute limit, and to preserve itself within it.

The inverse ratio is therefore such an ought as has sublated its restriction, its otherness; an infinity which as beyond has at the same time vanished and has returned into unity with its this-side.

Inasmuch as quantum continues itself in this way into its otherness, it is the unity of itself and of its otherness. It lies at the ground of its otherness; it is the unity of that otherness. Thus the direct ratio has restored itself again. But at the same time in such a way that the other is not an immediate quantum, but simply has its determinateness, its otherness, only in the unity itself.

The ratio has passed over into the ratio of powers.

C. The Ratio of Powers

1. The ratio of powers has, according to what has resulted, sublated on the one hand the externality with which the direct ratio is afflicted, namely the indifference of the determination of the quantum which is unit towards the other quantum which is amount or exponent, — and the opposed non-being, the abstract qualitative determinateness of the inverse ratio.

Otherness, or the difference of quantum, is at first multiplicity, but qualitatively determined, in such a way that it relates itself to another quantum as amount to its unit. Now, in the ratio of powers, the unit, which is amount at its own self, is at the same time the amount over against itself as unit. Or the otherness, the amount of the unit, is the unit itself.

Quantum raises itself into its power insofar as it becomes an other to itself; but this otherness of its is at the same time bounded purely through its own self. Insofar as in the direct ratio it is unit, it is also the unit of the amount; the side which is amount as such has the difference of quantum at it, it is an amount of units, these are amount, and indeed the amount which is the first side. But from this amount, which is the unit, the amount of the second side is distinguished; it is the exponent or an immediate quantum. In the power, however, the otherness, the side which in the ratio is as amount, is not distinguished from the amount insofar as this is its unit; or conversely, the power is a multitude of which each is this multitude itself. Thereby it at the same time contains the moment of the inverse ratio; the otherness, the amount as such, is determined through its first quantum. — Quantum is therefore, in the power, returned into its own self; it is immediately its own self and also its otherness.

The exponent of this ratio is now no longer an immediate quantum, as in the direct one. In the inverse ratio too it is, considered as the sum, indeed something mediated, but at the same time only an indifferent quantum; or, taken as the reference of the sum to one of the simply alterable sides of that sum, it is only this simply alterable quantum. — In the ratio of powers, however, the exponent is of a wholly qualitative nature, the simple determinateness that the amount is the unit itself, the identity of quantum with its own self in its otherness. Therein lies also its quantitative nature, that otherness, the limit or negation, is simply only as something sublated, that existence is continued into its otherness; for the truth of quality is precisely this, to be quantity.

2. The ratio of powers appears as an external alteration into which some quantum or other is transposed, and as if it could just as well be transposed into any other alteration. But this ratio has a closer reference to the concept of quantum; according to what has gone before, quantum has itself passed over into this alteration, and in this existence has reached its concept, or has realized itself therein in a complete way. This ratio is the exposition of what quantum is at its own self; it expresses that determinateness of quantum whereby it distinguishes itself from other. For quantum is the indifferent, sublated determinateness, that is to say, the determinateness which continues itself into its otherness and is therein equal to its own self. But such is quantum as ratio of powers; for its otherness is therein its own self. — In the direct ratio this quality of quantum, to be the difference of itself from its own self, is as yet posited only in general or immediately, so that there is still present the indifference of the two sides of the difference, not the difference of quantum from itself but from something external. In the inverse ratio quantum is the difference of itself from itself as from its non-being, the relating to itself as to its negation. In the ratio of powers, finally, it is the difference of itself as from its own self; its otherness determined through itself, or simply continued into it.

Quantum has thereby exhibited itself not merely with a qualitative determinateness, but as quality. But to that extent it has at the same time passed over into another determination. For it has sublated the moment of its externality or indifference, which was its determination, and has become its other, quality. That quantum steps into the ratio, and more determinately into the ratio of powers, appears at first as a mere constitution, as an externality of quantum. But in this externality the determination of quantum, which is itself externality, is sublated; this externality becomes external to its own self; — inasmuch as it thereby sublates itself, it just as much finds itself therein, or returns therein into itself, for externality is the determination of quantum itself.

Quantum is thus now the unity of its determination and of its becoming-other or its constitution; it is quality.

At first quantity as such appears over against quality; but quantity is itself a quality; a self-referring determinateness, distinguished from the determinateness that is other to it, from quality as such. But therewith it is itself a quality.

Yet it is not only a quality; rather the truth of quality itself is quantity; the former has passed over into the latter. But quantity, on the contrary, is in its truth the externality that has returned into its own self, the externality that is not indifferent. Thus it is quality itself, so that apart from this determination quality as such would not be anything further.

Quantity, which at first is determinateness in general, quantum, or rather quantum, is now no longer an indifferent or external determination, but that whereby something is what it is. The truth of quantum is to be measure.

Remark

In recent times the ratio of powers has been applied to determinations of the concept. The concept in its immediacy has been named the first power, in its otherness or in the difference, the existence of its moments, the second, and in its return into itself or as totality the third power. — The more precise significance of the particular powers does not, however, belong here; the power itself becomes once again a formal numerical ratio insofar as one proceeds to the second, the third, the fourth and so forth into infinity. Their significance as second, third, and so forth into infinity would depend on a conceptual worth of numbers in general, something already spoken of above.

As regards the application of the determination of power itself, in order to designate moments of the concept, it is evident that power belongs essentially to quantum. It is a becoming-other of quantum in which quantum itself remains. The difference is a difference of unit and multitude or amount, strictly nothing but an otherness of the quantum. It is quantum's own difference, in which quantum expresses itself as quality, or as that determinateness which it essentially is. The ratio of powers is therefore only the genuine difference of the particular concept of the quantum, not the difference of the concept itself. Yet quantum stands far below the concept; it does not contain the negativity that belongs to the nature of the concept in that negativity's own proper determination; differences that accrue to quantum are for this reason very superficial determinations for the concept itself.

Insofar as the expression of powers is used only as a symbol, there is as little to object against it as against symbols of another kind for concepts; but at the same time just as much as against all symbolism whatever, in which pure conceptual or philosophical determinations are supposed to be exhibited. Philosophy has no need of such help, neither from the sensible world, nor from representing imagination, nor from spheres of its own proper ground which are subordinate and whose determinations therefore do not suit higher circles and the whole. This is the same thing as when categories of the finite are applied to the infinite at all. As the current determinations of force, or of substantiality, cause and effect and so forth are unsuitable symbols for expressing, for instance, living or spiritual relations, so still more are the powers of quantum and numbered powers unsuitable for relations of that kind and for speculative relations generally.

Third Section. Measure

In measure quality and quantity are united. Being as such is immediate equality with its own self. This immediacy has sublated itself. Quantity is being that has gone back into itself; simple equality with itself as indifference towards determinateness. But this indifference shows itself to be pure externality, to have its determination not at its own self but in an other. The third is now externality referring itself to its own self; for the sake of the reference to itself it is at the same time sublated externality, indifference towards being-determined, in that it has at its own self its difference from itself.

Were the third taken as mere externality, it would be mode. — In this sense the third is not a return into itself; rather, since the second is the incipient reference to externality, a stepping outwards that still stands in reference to the original being, the third is the completed falling away. — Modality, among the categories of transcendental idealism, carries the significance of being the reference of the object to thinking. In this, from one side, only pure externality is contained; for the reference to thinking, which might be the moment of reflection into itself, is here rather externality itself; in the sense of transcendental idealism, that is, thinking is essentially external to the thing-in-itself. But insofar as the other categories too have only the transcendental determination of belonging to consciousness, modality, as the category of the reference to the subject, to that extent relatively contains the determination of reflection into itself. — With Spinoza, following upon substance and attribute, mode is likewise the third; his account makes of it the affections of substance, or whatever has its being in something else by means of which it gets grasped as well. This third is, according to this concept, only externality; as has been recalled elsewhere, in Spinoza rigid substantiality altogether lacks the return into its own self.

According to the foregoing, mode here has its determinate significance as measure. Measure is not yet the absolute return of being into itself, but rather its return into itself within its own sphere. It is the externality of quantum reflected into itself; through its reflection its worth has determined itself, namely to count for this, that it is being-in-itself. Quantum is the quality. The being reflected into itself, the being that holds good, consists therefore in the manner and way, in the more or the less, in the measure, in which something is. — This is the truth to which being has now determined itself, to be the equality of externality with its own self.

Quantum has in its return into itself sublated its externality and with that itself as quantum. But this sublating has at first the quan tum for its foundation; and the form of quantum which it has attained, that of being an indifference referring itself to itself, constitutes being-in-itself. Measure is the unity of quality and of quantity, of being-determined in itself and of being-determined externally, but the immediate unity of these; this immediate unity, however, is thereby qualitative determinateness over against the mediation and externality of quantum; the simplicity of its being-gone-back-into-itself stands over against the latter. Measure is therefore a reference of the qualitative and the quantitative in which these are still distinguished ones. In the movement, then, in which measure realizes itself, they compare themselves with one another in the determinate significance which they have over against one another; but they thereby posit themselves into the negative identity in which the determination of the immediacy of being absolutely vanishes and becomes essence.

The idea of essence already lies at hand in measure, namely that of being identical with itself in the immediacy of being-determined; or reflection, whose determinations subsist self-subsistently, yet in this self-subsistence are strictly only moments of their negative unity. In measure the qualitative is quantitative; it has an indifferent subsistence, the difference is indifferent to it; with that it is a difference which is none; it is sublated; this quantitativity is the return into itself, the being-in-and-for-itself which is essence. But in measure the qualitative and the quantitative, as has been recalled, at first still have their determinateness over against one another; it is the first negation of the externality of quantum; or the identity of the qualitative and the quantitative, the concept of essence, which in measure has already come to be, is not yet realized in its moments and thereby not yet posited.

Measure is, to begin with, immediate unity of what is qualitative and what is quantitative, so that

first, it is a quantum that has qualitative significance and stands as measure. — Measure, however, determines itself further to be that which is determined in itself, insofar as at its own self there is the difference of its moments, of qualitative and quantitative being-determined. These moments determine themselves further into wholes of measure, into the one immediately determined in itself and into the ratio specifying another; measure as the unity of them is something self-subsistent. — Measure thereby becomes

second, a ratio of specific quanta, as self-subsistent measures. But since their self-subsistence rests only on the quantitative ratio and on the difference of magnitude, they are in themselves the same, and are the passing over into one another. Considered more closely, measure thereby founders in the measureless. — This beyond of measure is the negativity of measure only at its own self; it is thereby

third, measure posited as an inverse ratio of measures. In this ratio the qualitative difference of the self-subsistent ones becomes their identical reference, and their indifferent immediacy consists in the reflection into this their negative immediacy and unity, which is essence. The indifference and immediacy of the self-subsistent sides themselves make up their negative immediacy, which is essence.