Section Two: Magnitude (Quantity)
The difference between quantity and quality has been stated. Quality is the first, immediate determinateness, quantity is the determinateness which has become indifferent to being, a limit which is just as much no limit, being-for-self which is absolutely identical with being-for-other — a repulsion of the many ones which is directly the non-repulsion, the continuity of them.
Because that which is for itself is now posited as not excluding its other, but rather as affirmatively continuing itself into it, it is otherness in so far as determinate being again appears in this continuity and its determinateness is at the same time no longer in a simple self-relation, no longer an immediate determinateness of the determinately existent something, but is posited as self-repelling, as in fact having the relation-to-self as a determinateness in another something (which is for itself; and since they are at the same time indifferent, relationless limits reflected into themselves, the determinateness in general is outside itself, an absolutely self-external determinateness and an equally external something; such a limit, the indifference of the limit within itself and of the something to the limit, constitutes the quantitative determinateness of the something.
In the first place, pure quantity is to be distinguished from itself as a determinate quantity, from quantum. As the former, it is in the first place real being-for-self which has returned into itself and which as yet contains no determinateness: a compact, infinite unity which continues itself into itself.
Secondly, this develops a determinateness which is posited in it as one which is at the same time no determinateness, as only an external one. It becomes quantum. Quantum is indifferent determinateness, that is, a self-transcending, self-negating determinateness; as this otherness of otherness it relapses into the infinite progress. But the infinite quantum is the indifferent determinateness sublated, it is the restoration of quality.
Thirdly, quantum in a qualitative form is quantitative ratio. Quantum transcends itself only generally: in ratio, however, its transition into its otherness is such that this otherness in which it has its determination is at the same time posited, is another quantum. Thus quantum has returned into itself and in its otherness is related to itself.
At the base of this ratio there is still the externality of quantum; the quanta which are related to each other are indifferent, that is, they have their self-relation in such self-externality. The ratio is thus only a formal unity of quality and quantity. Its dialectic is its transition into their absolute unity, into Measure.
Remark: Something's Limit as Quality
In something, its limit as quality is essentially its determinateness. If, however, by limit we mean quantitative limit, then when, for example, a field alters its limit it still remains what it was before, a field. If on the other hand its qualitative limit is altered, then since this is the determinateness which makes it a field, it becomes a meadow, wood, and so on. A red, whether brighter or paler, is still red; but if it altered its quality it would cease to be red, would become blue or some other colour. The determination of magnitude as quantum reached above, namely that it has a permanent substratum of being which is indifferent to its determinateness, can be found in any other example.
By magnitude quantum is meant, as in the examples cited, not quantity; and it is chiefly for this reason that this foreign term must be used.
The definition of magnitude given in mathematics likewise concerns quantum. A magnitude is usually defined as that which can be increased or diminished. But to increase means to make the magnitude more, to decrease, to make the magnitude less. In this there lies a difference of magnitude as such from itself and magnitude would thus be that of which the magnitude can be altered. The definition thus proves itself to be inept in so far as the same term is used in it which was to have been defined. Since the same term must not be used in the definition, the more and less can be resolved into an affirmative addition which, in accordance with the nature of quantum, is likewise external, and a subtraction, as an equally external negation. It is this external form both of reality and of negation which in general characterises the nature of alteration in quantum. In that imperfect expression, therefore, one cannot fail to recognise the main point involved, namely the indifference of the alteration, so that the alteration's own more and less, its indifference to itself, lies in its very Notion.
Chapter 1 Quantity
A. PURE QUANTITY
Quantity is sublated being-for-self; the repelling one which related itself only negatively to the excluded one, having passed over into relation to it, treats the other as identical with itself, and in doing so has lost its determination: being-for-self has passed over into attraction. The absolute brittleness of the repelling one has melted away into this unity which, however, as containing this one, is at the same time determined by the immanent repulsion, and as unity of the self-externality is unity with itself. Attraction is in this way the moment of continuity in quantity.
Continuity is, therefore, simple, self-same self-relation, which is not interrupted by any limit or exclusion; it is not, however, an immediate unity, but a unity of ones which possess being-for-self. The asunderness of the plurality is still contained in this unity, but at the same time as not differentiating or interrupting it. In continuity, the plurality is posited as it is in itself; the many are all alike, each is the same as the other and the plurality is, consequently, a simple, undifferentiated sameness. Continuity is this moment of self-sameness of the asunderness, the self-continuation of the different ones into those from which they are distinguished.
In continuity, therefore, magnitude immediately possesses the moment of discreteness — repulsion, as now a moment in quantity. Continuity is self-sameness, but of the Many which, however, do not become exclusive; it is repulsion which expands the selfsameness to continuity. Hence discreteness, on its side, is a coalescent discreteness, where the ones are not connected by the void, by the negative, but by their own continuity and do not interrupt this self-sameness in the many.
Quantity is the unity of these moments of continuity and discreteness, but at first it is so in the form of one of them, continuity, as a result of the dialectic of being-for-self, which has collapsed into the form of self-identical immediacy. Quantity is, as such, this simple result in so far as being-for-self has not yet developed its moments and posited them within itself. It contains them to begin with as being-for-self posited as it is in truth. The determination of being-for-self was to be a self-sublating relation-to-self, a perpetual coming-out-of-itself. But what is repelled is itself; repulsion is, therefore, the creative flowing away of itself. On account of the self-sameness of what is repelled, this distinguishing or differentiation is an uninterrupted continuity; and because of the coming-out-of-itself this continuity, without being interrupted, is at the same time a plurality, which no less immediately remains in its self-identicalness.
Remark 1: The Conception of Pure Quantity
Pure quantity has not as yet any limit or is not as yet quantum; and even in so far as it becomes quantum it is not bounded by limit but, on the contrary, consists precisely in not being bounded by limit, in having being-for-self within it as a sublated moment. The presence in it of discreteness as a moment can be expressed by saying that quantity is simply the omnipresent real possibility within itself of the one, but conversely that the one is no less absolutely continuous.
In thinking that is not based on the Notion, continuity easily becomes mere composition, that is, an external relation of the ones to one another, in which the one is maintained in its absolute brittleness and exclusiveness. But it has been shown that the one essentially and spontaneously (an und für sich selbst) passes over into attraction, into its ideality, and that consequently continuity is not external to it but belongs to it and is grounded in its very nature. It is just this externality of continuity for the ones to which atomism clings and which ordinary thinking finds it difficult to forsake. Mathematics, on the other hand, rejects a metaphysics which would make time consist of points of time; space in general — or in the first place the line — consist of points of space; the plane, of lines; and total space of planes. It allows no validity to such discontinuous ones. Even though, for instance, in determining the magnitude of a plane, it represents it as the sum of infinitely many lines, this discreteness counts only as a momentary representation, and the sublation of the discreteness is already implied in the infinite plurality of the lines, since the space which they are supposed to constitute is after all bounded.
It is the notion of pure quantity as opposed to the mere image of it that Spinoza, for whom it had especial importance, has in mind when he speaks of quantity as follows:
'Quantity is conceived by us in two manners, to wit, abstractly and superficially, as an offspring of imagination or as a substance, which is done by the intellect alone. If, then, we look at quantity as it is in the imagination, which we often and very easily do, it will be found to be finite, divisible, and composed of parts; but if we look at it as it is in the intellect and conceive it, in so far as it is a substance, which is done with great difficulty, then as we have already sufficiently shown, it will be found to be infinite, without like, and indivisible. This, to all who know how to distinguish between the imagination and the intellect, will be quite clear.'
More specific examples of pure quantity, if they are wanted, are space and time, also matter as such, light, and so forth, and the ego itself: only by quantity, as already remarked, is not to be understood quantum. Space, time and the rest, are expansions, pluralities which are a coming-out-of-self, a flowing which, however, does not pass over into its opposite, into quality or the one, but as a coming-out-of-self they are a perennial self-production of their unity.
Space is this absolute self-externality which equally is absolutely uninterrupted, a perpetual becoming-other which is self-identical; time is an absolute coming-out-of-itself, a generating of the one, (a point of time, the now) and immediately the annihilation of it, and again the continuous annihilation of this passing away; so that this spontaneous generating of non-being is equally a simple self-sameness and self-identity. As regards matter as quantity, among the seven surviving propositions of the first dissertation of Leibniz there is one, the second, which runs: Non omnino improbabile est, materiam et quantitatem esse realiter idem. In fact, the distinction between these two concepts is simply this, that quantity is a determination of pure thought, whereas matter is the same determination in outer existence. The determination of pure quantity belongs also to the ego which is an absolute becoming-other, an infinite removal or all-round repulsion to the negative freedom of being-for-self which, however, remains utterly simple continuity — the continuity of universality or being-with-self uninterrupted by the infinitely manifold limits, by the content of sensations, intuitions, and so forth. Those who reject the idea of plurality as a simple unity and besides the Notion of it, to wit, that each of the many is the same as every other, namely, a one of the many — since here we are not speaking of the many as further determined, as green, red, and so on, but of the many considered in and for itself — demand also a representation of this unity, will find plenty of instances in those continua which exhibit the deduced Notion of quantity as present in simple intuition.
Remark 2: The Kantian Antinomy of the Indivisibility and the Infinite Divisibility of Time, Space and Matter
It is the nature of quantity, this simple unity of discreteness and continuity, that gives rise to the conflict or antinomy of the infinite divisibility of space, time, matter, etc.
This antinomy consists solely in the fact that discreteness must be asserted just as much as continuity. The one-sided assertion of discreteness gives infinite or absolute dividedness, hence an indivisible, for principle: the one-sided assertion of continuity, on the other hand, gives infinite divisibility.
It is well known that the Kantian Critique of Pure Reason sets up four (cosmological) antinomies, the second of which deals with the antithesis constituted by the moments of quantity.
Kantian antinomies will always remain an important part of the critical philosophy; they, more than anything else, brought about the downfall of previous metaphysics and can be regarded as a main transition into more recent philosophy since they, in particular, helped to produce the conviction of the nullity of the categories of finitude in regard to their content which is a more correct method than the formal one of a subjective idealism, according to which their defect is supposed to be, not what they are in themselves, but only that they are subjective. But this exposition with all its merits is imperfect; its course is impeded and tangled, and also it is false in regard to its result, which presupposes that cognition has no other forms of thought than finite categories. In both respects these antinomies deserve a more exact critical appraisal which will not only throw more light on their standpoint and method but will also free the main point at issue from the useless form into which it has been forced.
In the first place, I remark that Kant wanted to give his four cosmological antinomies a show of completeness by the principle of classification which he took from his schema of the categories. But profounder insight into the antinomial, or more truly into the dialectical nature of reason demonstrates any Notion whatever to be a unity of opposed moments to which, therefore, the form of antinomial assertions could be given. Becoming, determinate being, etc., and any other Notion, could thus provide its particular antinomy, and thus as many antinomies could be constructed as there are Notions.
Ancient scepticism did not spare itself the pains of demonstrating this contradiction or antimony in every notion which confronted it in the sciences.
Further, Kant did not take up the antinomy in the Notions themselves, but in the already concrete form of cosmological determinations. In order to possess the antinomy in its purity and to deal with it in its simple Notion, the determinations of thought must not be taken in their application to and entanglement in the general idea of the world, of space, time, matter, etc; this concrete material must be omitted from consideration of these determinations which it is powerless to influence and which must be considered purely on their own account since they alone constitute the essence and the ground of the antinomies.
Kant's conception of the antinomies is that they are 'not sophisms but contradictions which reason must necessarily come up against' (a Kantian expression); and this is an important view. 'Reason, when it sees into the ground of the natural illusion of the antinomies is, it is true, no longer imposed on by them but yet continues to be deceived.' The Kantian solution, namely, through the so-called transcendental ideality of the world of perception, has no other result than to make the so-called conflict into something subjective, in which of course it remains still the same illusion, that is, is as unresolved, as before. Its genuine solution can only be this: two opposed determinations which belong necessarily to one and the same Notion cannot be valid each on its own in its one-sidedness; on the contrary, they are true only as sublated, only in the unity of their Notion.
The Kantian antinomies on closer inspection contain nothing more than the quite simple categorical assertion of each of the two opposed moments of a determination, each being taken on its own in isolation from the other. But at the same time this simple categorical, or strictly speaking assertoric statement is wrapped up in a false, twisted scaffolding of reasoning which is intended to produce a semblance of proof and to conceal and disguise the merely assertoric character of the statement, as closer consideration will show.
The relevant antinomy here concerns the so-called infinite divisibility of matter and rests on the antithesis of the moments of continuity and discreteness which are contained in the Notion of quantity.
The thesis of the same as expounded by Kant runs thus:
'Every composite substance. in the world consists of simple parts, and nowhere does there exist anything but the simple or what is compounded from it.'
To the simple, the atom, there is here opposed the composite, which is a very inferior determination compared to the continuous. The substrate given to these abstractions, namely, substances in the world, here means nothing more than things as sensuously perceived and it has no influence on the antinomy itself; space or time could equally well be taken. Now since the thesis speaks only of composition instead of continuity it is really as it stands an analytical or tautological proposition. That the composite is not in its own self a one, but only something externally put together and consisting of what is other than itself, this is its immediate determination. But the other of the composite is the simple. It is therefore tautological to say that the composite consists of the simple. To ask of what something consists is to ask for an indication of something else, the compounding of which constitutes the said something. If ink is said to consist simply of ink, the meaning of the inquiry after the something else of which it consists has been missed and the question is not answered but only repeated. A further question then is whether that which is under discussion is supposed to consist of something or not. But the composite is simply that which is supposed to be a combination of something else. If, however, the simple which is the other of the composite is taken only as relatively simple and is itself composite, too, then the question still remains unanswered. What ordinary thinking has in mind is, perhaps, only some composite or other of which something or other, too, would be assigned as its simple, such particular something being composite on its own account. But what is under discussion here is the composite as such.
Now as regards the Kantian proof of the thesis this, like all the Kantian proofs of the rest of the antinomial propositions, makes the detour of being apagogic, a detour which will prove to be quite superfluous.
'Assume', he begins, 'that composite substances do not consist of simple parts; then if all composition were thought away no composite part and (since we have just assumed that there are no simple parts) no simple part — hence nothing at all — would remain; consequently, no substance would have been given.'
This conclusion is quite correct: if nothing but composite substances exist and all that is composite is thought away, then nothing whatever remains; this will be conceded, but this tautological redundance could be omitted and the proof straightway begin with what follows, namely:
'Either it is impossible to think away all composition, or else there remains after such removal in thought something which is not composite, that is, the simple.
'In the first case, however, the composite again would not consist of substances (because with these, composition is only a contingent relation of substances [In addition to the redundance of the proof itself there is here also a redundance of language 'because with these' (namely the substances) 'composition is only a contingent relation of substances'.] which, apart from such relation, must still persist on their own account). Now since this case contradicts what was assumed, only the second case is left: namely, that all composite substances consist of simple parts.'
The very reason which is the main point, and in face of which all that precedes is completely superfluous, is mentioned by the way, in a parenthesis. The dilemma is this: either the composite persists, or else the simple. If the former, that is, the composite, persists, then what persists would not be substances, for composition is for these only a contingent relation; but substances do persist, therefore, what persists is the simple.
It is clear that the apagogical detour could be omitted and the thesis:
'composite substance consists of simple parts', could be directly followed by the reason: because composition is merely a contingent relation of substances, and is therefore external to them and does not concern the substances themselves. If the composition is in fact contingent then, of course, substances are essentially simple. But this contingency which is the sole point at issue is not proved but straightway assumed, and casually, too, in a parenthesis-as something self-evident or of secondary importance. True, it goes without saying that composition is a contingent and external determination; but if the point at issue were only a contingent togetherness instead of continuity, it would not be worth while constructing an antinomy about it — or rather it would not be possible to formulate one. Therefore, the assertion that the parts are simple is, as remarked, only a tautology.
The apagogical detour thus contains the very assertion which should result from it. The proof therefore can be put more concisely thus:
Let us assume that substances do not consist of simple parts but are only composite. But now all composition can be thought away (for it is only a contingent relation); after its removal, therefore, there are no substances left unless we assume that they consist of simple parts. But substances we must have for we have assumed them; everything is not meant to vanish, something must be left over, for we have presupposed something persistent which we called substance. Therefore this something must be simple.
To complete the whole, we have still to consider the conclusion which runs as follows:
'From this it follows, as a direct consequence, that all things in the world without exception are simple entities, that composition is only an external state of them, and that reason must think the elementary substances as simple entities.'
Here we see the externality, that is contingency, of composition put forward as a consequence after it had already been introduced parenthetically and used in the proof.
Kant strongly protests that he is not looking for sophisms in the conflicting statements of the antinomy for the purpose, as it were, of special pleading. But the defect of the proof in question is not so much that it is a sophism, as that its laboured, tortuous complexity serves no other purpose than to produce the merely outward semblance of a proof and partially to obscure the quite transparent fact that what was supposed to emerge as a consequence is, parenthetically, that on which the proof hinges; that there is no proof at all, but only an assumption.
The antithesis runs:
'No composite thing in the world consists of simple parts and nowhere in the world does there exist anything simple.'
The proof has equally an apagogical turn and, in a different way, is just as faulty as the previous one.
'Suppose', it runs, 'that a composite thing as a substance consists of simple parts. Because all external relation, and consequently all composition of substances, is possible only in space, therefore, the space occupied by the composite substance must consist of as many parts as those of which the composite substance consists. Now space does not consist of simple parts but of spaces. Therefore each part of the composite substance must occupy a space.'
'But the absolutely primary parts of everything composite are simple.'
'Therefore the simple occupies a space.'
'Now since every real thing that occupies a space comprises a manifold of mutually external parts and is consequently composite, consisting of substances, it would follow that the simple is a composite substance-which is self-contradictory.'
This proof can be called a whole nest (to use an expression elsewhere employed by Kant) of faulty procedure.
In the first place, its apagogical form is a groundless illusion. For the assumption that whatever is substantial is spatial, but that space does not consist of simple parts is a direct assertion which is made the immediate ground of what is to be proved, and with this there is an end to the proving of the antithesis.
Next, this apagogical proof begins with the proposition: 'that all composition from substances is an external relation', but oddly enough immediately forgets it. For it then goes on to conclude that composition is possible only in space, that space does not consist of simple parts, and that therefore the real thing occupying a space is composite. But once composition is assumed as an external relation, then spatiality itself (in which alone composition is supposed to be possible) is for that very reason an external relation for the substances, which does not concern them or affect their nature any more than anything else does that can be inferred from the determination of spatiality. For this very reason, the substances ought not to have been put into space.
Further, it is assumed that the space in which the substances here are placed does not consist of simple parts, because it is an intuition, that is, according to the Kantian definition, a representation which can only be given through a single object, and is not a so-called discursive concept. This Kantian distinction between intuition and concept has, as everyone knows, given rise to a deal of nonsense about the former, and to avoid the labour of comprehension the value and sphere of intuition have been extended to the whole field of cognition. What is pertinent here is just this, that space, and also intuition itself, must be grasped in terms of their Notions if, that is, we want really to comprehend. And thus the question would arise whether space, even though a simple continuity for intuition, ought not to be grasped, in accordance with its Notion, as consisting of simple parts, or whether it would be involved in the same antinomy which applied only to substance. As a matter of fact, if the antinomy is grasped abstractly, it concerns, as we remarked, quantity as such, and hence equally space and time.
But it is assumed in the proof that space does not consist of simple parts; this therefore ought to have been the reason for not placing the simple in this element which is incompatible with the nature of the simple. There is also involved here a clash between the continuity of space and composition; the two are confused with each other, the former being substituted for the latter (which results in a quaternio terminorum in the conclusion). With Kant, space has the express determination of being 'sole and single, its parts resting only on limitations, so that they do not precede the one, all-embracing space as, so to speak, its component parts from which it could be compounded.' Here continuity is quite correctly and definitely predicated of space in denial of its composition from parts. On the other hand, in the argument the placing of substances in space is supposed to involve 'a manifold of mutually external parts' and, more particularly, 'consequently a composite'. Yet, as we have quoted, the way in which manifoldness is present in space is expressly intended to exclude composition and component parts antecedent to the unity of space.
In the remark to the proof of the antithesis we are also expressly reminded of the other fundamental conception of the critical philosophy, namely, that we have a notion of bodies only as appearances or phenomena; as such, however, they necessarily presuppose space as the condition of the possibility of all outer appearance. If by substances we are meant here to understand only bodies as we see, touch, taste them, and so on, then we are not really discussing them as they are in their Notion but only as sensuously perceived. The proof of the antithesis, then, amounted in short to this: all our visual, tactile, and other experience shows us only what is composite; even the best microscopes and the keenest knives have not enabled us to come across anything simple. Then neither should reason expect to come across anything simple.
When we look more closely into the opposition of this thesis and antithesis, freeing their proofs from all pointless redundancy and tortuousness, we see that the proof of the antithesis dogmatically assumes continuity (by placing substances in space) and also that the proof of the thesis, by assuming that composition is the mode of relation of substances, dogmatically assumes the contingency of this relation, and hence assumes that substances are absolute ones. Thus the whole antinomy reduces to the separation of the two moments of quantity and the direct assertion of them as absolutely separate. When substance, matter, space, time, etc., are taken only as discrete, they are absolutely divided; their principle is the one. When they are taken as continuous, this one is only a sublated one; division remains a divisibility, it remains the possibility of division, as a possibility, without actually reaching the atom. Now even if we stop at the determination given in what has been said about these antitheses, then the moment of the atom is contained in continuity itself, for this is simply the possibility of division; just as said dividedness, discreteness, sublates all distinction of the ones — for each of the simple ones is what the other is-consequently, also contains their sameness and hence their continuity.
Since each of the two opposed sides contains its other within itself and neither can be thought without the other, it follows that neither of these determinations, taken alone, has truth; this belongs only to their unity. This is the true dialectical consideration of them and also the true result.
Infinitely more ingenious and profound than this Kantian antinomy are the dialectical examples of the ancient Eleatic school, especially those concerning motion, which likewise are based on the Notion of quantity and in it find their solution. To consider them here, too, would be too lengthy a business; they concern the Notions of space and time and can be dealt with at the same time as these subjects and in the history of philosophy. They reflect the greatest credit on the intelligence of their inventors they have for result the pure being of Parmenides, in that in them is demonstrated the dissolution of all determinate being; they are thus in themselves the flux of Heraclitus. For this reason they deserve a more thorough consideration than the usual explanation that they are just sophisms; which assertion sticks to empirical perception, following the procedure of Diogenes (a procedure which is so illuminating to ordinary common sense) who, when a dialectician pointed out the contradiction contained in motion, made no effort to reason it out but, by silently walking up and down, is supposed to have referred to the evidence of sight for an answer. Such assertion and refutation is certainly easier to make than to engage in thinking and to hold fast and resolve by thought alone the complexities originating in thought, and not in abstruse thought either, but in the thoughts spontaneously arising in ordinary consciousness.
The solutions propounded by Aristotle of these dialectical forms merit high praise, and are contained in his genuinely speculative Notions of space, time and motion. To infinite divisibility (which, being imagined as actually carried out, is the same as infinite dividedness, as the atoms) on which is based the most famous of those proofs, he opposes continuity, which applies equally well to time as to space, so that the infinite, that is, abstract plurality is contained only in principle [an sich], as a possibility, in continuity. What is actual in contrast to abstract plurality as also to abstract continuity, is their concrete forms, space and time themselves, just as these latter are abstract relatively to matter and motion. What is abstract has only an implicit or potential being; it only is as a moment of something real. Bayle, who finds Aristotle's solution of the Zenonic dialectic 'pitoyable', does not understand the meaning of the statement that matter is only potentially infinitely divisible; he rejoins that if matter is infinitely divisible, then it actually contains an infinite number of parts, that, therefore, this infinite is not an infinite en putssance but an infinite that really and actually exists. On the contrary, divisibility itself even is only a possibility, not an existing of the parts, and the plurality as such is posited in the continuity only as a moment, as sublated. Acute understanding, in which Aristotle, too, is certainly unsurpassed, is not competent to grasp and to decide on speculative Notions, any more than the crudity of sensuous conception instanced above is adequate to refute the reasoning of Zeno. Such intellect commits the error of holding such mental fictions, such abstractions, as an infinite number of parts, to be something true and actual; but this sensuous consciousness does not let itself be brought beyond the empirical element to thought.
The Kantian solution of the antinomy likewise consists solely in the supposition that reason should not soar beyond sensuous perception and should take the world of appearance, the phenomenal world, as it is. This solution leaves the content of the antinomy itself on one side; it does not attain to the nature of the Notion of its determinations, each of which, isolated on its own, is null and is in its own self only the transition into its other, the unity of both being quantity, in which they have their truth.
B. CONTINUOUS AND DISCRETE MAGNITUDE
1. Quantity contains the two moments of continuity and discreteness. It is to be posited in both of them as determinations of itself. It is already their immediate unity, that is, quantity is posited at first only in one of its determinations, continuity, and as such is continuous magnitude.
Or we may say that continuity is indeed one of the moments of quantity which requires the other moment, discreteness, to complete it. But quantity is a concrete unity only in so far as it is the unity of distinct moments. These, are, therefore, also to be taken as distinct, but are not to be resolved again into attraction and repulsion, but are to be taken as they are in their truth, each remaining in its unity with the other, that is, remaining the whole. Continuity is only coherent, compact unity as unity of the discrete; posited as such it is no longer only a moment but the whole of quantity, continuous magnitude.
2. Immediate quantity is continuous magnitude. But quantity is not an immediate at all; immediacy is a determinateness the sublatedness of which is quantity itself. It is, therefore, to be posited in the determinateness immanent in it, and this is the one. Quantity is discrete magnitude.
Discreteness is, like continuity, a moment of quantity but it is itself also the whole of quantity just because it is a moment in it, in the whole, and therefore as a distinct moment it does not stand outside the whole, outside its unity with the other moment. Quantity is in itself asunderness, and continuous magnitude is this asunderness continuing itself without negation as an internally self-same connectedness. But discrete magnitude is this asunderness as discontinuous, as interrupted. With this plurality of ones, however, we are not again in the presence of the plurality of atoms and the void, repulsion in general. Because discrete magnitude is quantity, its discreteness is itself continuous. This continuity in the discrete consists in the ones being the same as one another, or in having the same unity. Discrete magnitude is, therefore, the asunderness of the manifold one as self-same, not the manifold one in general but posited as the many of a unity.
Remark: The Usual Separation of These Magnitudes
In the usual ideas of continuous and discrete magnitude, it is overlooked that each of these magnitudes contains both moments, continuity and discreteness, and that the distinction between them consists only in this, that in one of the moments the determinateness is posited and in the other it is only implicit. Space, time, matter, and so forth are continuous magnitudes in that they are repulsions from themselves, a streaming forth out of themselves which at the same time is not their transition or relating of themselves to a qualitative other. They possess the absolute possibility that the one may be posited in them at any point — not the empty possibility of a mere otherness (as when it is said, it is possible that a tree might stand in the place of this stone), but they contain the principle of the one within themselves; it is one of the determinations which constitute them.
Conversely, in discrete magnitude continuity is not to be overlooked; this moment is, as has been shown, the one as unity.
Continuous and discrete magnitude can be regarded as species of quantity, provided that magnitude is posited, not under any external determinateness, but under the determinatenesses of its own moments; the ordinary transition from genus to species allows external characteristics to be attributed to the former according to some external basis of classification. And besides, continuous and discrete magnitude are not yet quanta; they are only quantity itself in each of its two forms. They are perhaps, called magnitudes in so far as they have in common with quantum simply this-to be a determinateness in quantity.
C. LIMITATION OF QUANTITY
Discrete magnitude has first the one for its principle; secondly, it is a plurality of ones; and thirdly, it is essentially continuous; it is the one as at the same time sublated, as unity, the continuation of itself as such in the discreteness of the ones. Consequently, it is posited as one magnitude, the determinateness of which is the one which, in this posited and determinate being is the excluding one, a limit in the unity. Discrete magnitude as such is immediately not limited; but as distinguished from continuous magnitude it is a determinate being, a something, with the one as its determinateness and also as its first negation and limit.
This limit, which is related to the unity and is the negation in it, is also, as the one, self-related; it is thus the enclosing, encompassing limit. Limit here is not at first distinguished from its determinate being as something, but, as the one, is immediately this negative point itself. But the being which here is limited is essentially a continuity, by virtue of which it passes beyond the limit, beyond this one, to which it is indifferent. Real discrete quantity is thus a quantity, or quantum — quantity as a determinate being and a something.
Since the one which is a limit includes within itself the many ones of discrete quantity, it equally posits them as sublated within it; and because it is a limit of continuity simply as such, the distinction between continuous and discrete magnitude is here of no significance; or, more correctly, it is a limit to the continuity of the one as much as of the other; both undergo transition into quanta.
Chapter 2 Quantum
Quantum, which to begin with is quantity with a determinateness or limit in general is, in its complete determinateness, number. Quantum differentiates itself secondly, into (a) extensive quantum, in which the limit is a limitation of the determinately existent plurality; and (b) intensive quantum or degree, the determinate being having made the transition into being-for-self. Intensive quantum as both for itself and at the same time immediately outside itself — since it is an indifferent limit — has its determinateness in an other. As this manifest contradiction of being determined simply within itself yet having its determinateness outside it, pointing outside itself for it, quantum posited as being in its own self external to itself, passes over thirdly, into quantitative infinity.
A. NUMBER
Quantity is quantum, or has a limit, both as continuous and as discrete magnitude. The difference between these two kinds has here, in the first instance, no immediate significance.
The very nature of quantity as sublated being-for-self is ipso facto to be indifferent to its limit. But equally, too, quantity is not unaffected by the limit or by being, a quantum; for it contains within itself as its own moment the one, which is absolutely determined and which, therefore, as posited in the continuity or unity of quantity, is its limit, but a limit which remains what it has become, simply a one.
This one is thus the principle of quantum, but as the one of quantity. Hence, first, it is continuous, it is a unity; secondly, it is discrete, a plurality of ones, which is implicit in continuous, or explicit in discrete magnitude, the ones having equality with one another, possessing the said continuity, the same unity. Thirdly, this one is also a negation of the many ones as a simple limit, an excluding of its otherness from itself, a determination of itself in opposition to other quanta. Thus the one is [a] self-relating, [b] enclosing and [c] other-excluding limit.
Quantum completely posited in these determinations is number. The complete positedness lies in the existence of the limit as a plurality and so in its distinction from the unity. Consequently, number appears as a discrete magnitude, but in the unity it equally possesses continuity. It is, therefore, also quantum in its complete determinateness, for its principle the one, the absolutely determinate. Continuity, in which the one is present only in principle, as a sublated moment — posited as a unity — is the form of indeterminateness.
Quantum, merely as such, is limited generally; its limit is an abstract simple determinateness of it. But in quantum as number, this limit is posited as manifold within itself. It contains the many ones which constitute its determinate being, but does not contain them in an indeterminate manner, for the determinateness of the limit falls in them; the limit excludes other determinate being, that is, other pluralities and the ones it encloses are a specific aggregate, the amount — which is the form taken by discreteness in number — the other to which is the unit, the continuity of the amount. Amount and unit constitute the moments of number.
As regards amount, we must see more closely how the many ones of which it consists are present in the limit; it is correct to say of amount that it consists of the many, for the ones are in it not as sublated but as affirmatively present, only posited with the excluding limit to which they are indifferent. This, however, is not indifferent to them. In the sphere of determinate being, the relation of the limit to it was primarily such that the determinate being persisted as the affirmative on this side of its limit, while the limit, the negation, was found outside on the border of the determinate being; similarly, the breaking-off [in the counting] of the many ones and the exclusion of other ones appears as a determination falling outside the enclosed ones. But in the qualitative sphere it was found that the limit pervades the determinate being, is coextensive with it, and consequently that it lies in the nature of something to be limited, that is, finite. In the quantitative sphere a number, say a hundred, is conceived in such a manner that the hundredth one alone limits the many to make them a hundred. In one sense this is correct; but on the other hand none of the hundred ones has precedence over any other for they are only equal — each is equally the hundredth; thus they all belong to the limit which makes the number a hundred and the number cannot dispense with any of them for its determinateness. Hence, relatively to the hundredth one, the others do not constitute a determinate being that is in any way different from the limit, whether they are outside or inside it. Consequently, the number is not a plurality over against the enclosing, limiting one, but itself constitutes this limitation which is a specific quantum; the many constitute a number, a two, a ten, a hundred, and so on.
Now the limiting one is the number as determined relatively to other numbers, as distinguished from them. But this distinguishing does not become a qualitative determinateness but remains quantitative, falling only within the comparing external reflection; the number, as a one, remains returned into itself and indifferent to others. This indifference of a number to others is an essential determination of it and constitutes the implicit determinedness of the number, but also the number's own externality. Number is thus a numerical one as the absolutely determinate one, which at the same time has the form of simple immediacy and for which, therefore, the relation to other is completely external. Further, one as a number possesses determinateness (in so far as this is a relation to other) as the moments of itself contained within it, in its difference of unit and amount; and amount is itself a plurality of ones, that is, this absolute externality is in the one itself. This contradiction of number or of quantum as such within itself is the quality of quantum, in the further determinations of which this contradiction is developed.
Remark 1: The Species of Calculation in Arithmetic; Kant's Synthetic Propositions a priori of Intuition
Spatial magnitude and numerical magnitude are usually regarded as two species, the former being on its own account a determinate magnitude just as much as the latter; their difference is held to consist only in the different determinations of continuity and discreteness, but as quantum they stand on the same level. In spatial magnitude, geometry has, in general, continuous magnitude for its subject matter while the subject matter of arithmetic is the discrete magnitude of number. But with this dissimilarity of their subject matter, the manner and completeness of their limitation or determinedness is also different. Spatial magnitude possesses only limitation generally; if it is to be considered as a thoroughly determinate quantum then number is required. Geometry as such does not measure spatial figures (it is not mensuration), but only compares them. In its definitions, too, the determinations are in part derived from the equality of the sides and angles, or from equidistance. Thus the circle, because it is based solely on the equidistance of all possible points in it from a centre, does not require number for its determination. These determinations based on equality or inequality are genuinely geometrical. But they are not sufficient, and for other figures, for example, the triangle or rectangle, number is requisite; this in its principle, the one, contains a self-determinedness, it is determined without the aid of an other and therefore not through comparison. It is true that in the point, spatial magnitude has a determinateness corresponding to the one; but the point, in becoming external to itself, becomes an other, becomes the line; because it is essentially only a spatial one, it becomes in the relation a continuity in which the nature of the point, the self-determinedness, the one, is sublated. In so far as the self-determinedness is supposed to be preserved in the self-externality, the line must be represented as an aggregate of ones, and to the limit must be imparted the determination of the many ones, that is, the magnitude of the line — and similarly of other spatial determinations — must be taken as a number.
Arithmetic considers number and its figures; or rather does not consider them but operates with them. For number is the determinateness which is indifferent, inert; it must be actuated from without and so brought into a relation. The modes of relation are the species of calculation. In arithmetic they are presented seriatim and it is clear that one depends on the other; but the thread which links the progressive stages is not made prominent in arithmetic. However, the systematic arrangement justly claimed for the presentation of these elements in the textbooks is readily provided by the determinations of number itself stemming from its Notion. These cardinal determinations will be briefly noted here.
Number has for its principle the one and is, therefore, simply an aggregate externally put together, a purely analytic figure devoid of any inner connectedness; and because it is produced in this merely external manner all calculation is the production of numbers, a counting or, more specifically, a counting up. Any diversity in this external production, which always proceeds to the same end, can lie only in a difference between the numbers which are to be counted up; this difference must itself come from elsewhere and from an external determination.
The qualitative difference which constitutes the determinateness of number is, we have seen, that of unit and amount; consequently, every determinateness of the notion of number which can occur in the species of calculation can be reduced to this difference. But the difference which belongs to numbers as quanta, is external identity and external difference, equality and inequality, which are moments of reflection and fall to be considered under the determinations of essence when we come to deal with difference.
Further, we must premise that numbers can, in general, be produced in two ways, either by aggregation, or by separation of an aggregate already given; in both cases the same specific kind of counting is employed, so that to an aggregating of numbers there corresponds what may be called a positive species of calculation, and to a separating of them, a negative species; the determination of the species of calculation itself is independent of this antithesis.
1. After these remarks we proceed to indicate the species of calculation. The first production of number is the aggregating of the many as such, each of which then is posited as only a one — numbering or counting. Since the ones are mutually external their representation is illustrated sensuously, and the operation by which number is generated is a process of counting on the fingers, dots, and so on. What four, five, etc., is, can only be pointed out. Seeing that the limit is an external one, the breaking off of the counting, the amount to be aggregated, is contingent, arbitrary. The difference of amount and unit which appears in the progress from one species of calculation to another, establishes a system of numbers, dyadic, decadic, and so on; such a system rests on the whole on the arbitrary choice of an amount which is consistently to be taken as unit.
The number produced by counting are in turn themselves counted; and as thus immediately posited they are determined as still lacking any connection with one another, as indifferent to equality or inequality, and their magnitude relatively to one another is contingent — they are, therefore, simply unequal; this is addition. We learn that 7 + 5 = 12 by adding (on our fingers or in some other way) five more ones on to the seven; the result is then memorised, learnt by rote, for the procedure has no inner meaning. Similarly, we learn that 7 x 5 = 35 by counting on the fingers, etc., to one seven, adding on another, and repeating this five times, the result being likewise memorised. The trouble of this counting, of finding the sums or products, is eliminated by the completed addition and multiplication tables, which have only to be learnt by heart.
Kant considers the proposition: 7 + 5 = 12 to be a synthetic proposition. 'We might indeed', he says, 'at first think (of course!) that it is a merely analytic proposition obtained by the law of contradiction from the concept of a sum of seven and five.' The concept of the sum means nothing more than the abstract determination that these two numbers are meant to be aggregated and, as numbers, in an external, that is, mechanical (begrifflose) fashion — that we are to count on from seven until the ones to be added on (their amount is fixed at five) have been exhausted; the result bears the otherwise familiar name of twelve. 'But', continues Kant, 'If we look more closely we find that the concept of the sum of 7 and 5 contains nothing save the union of the two numbers into one, and in all this there is no thinking at all about which is the single number which combines both' ... 'I may analyse my concept of such a possible sum as much as I please, still, I shall not find the twelve in it." With the thinking of the sum, with the analysis of the concept, the transition from the problem to the result has, of course, nothing whatever to do; and he adds, we must go outside these concepts and have recourse to intuition, to our five fingers and so on, thus adding on to the concept of seven the five units given in intuition. Five is, of course, given in intuition, that is, it is a wholly external aggregation of the arbitrarily repeated thought of one; but seven equally is not a concept; there are no concepts to go outside of or beyond. The sum of 5 and 7 means the mechanical [begrifflose] conjunction of the two numbers, and the counting from seven onwards thus mechanically continued until the five units are exhausted can be called a putting together, a synthesis, just like counting from one onwards; but it is a synthesis wholly analytical in nature, for the connection is quite artificial, there is nothing in it or put into it which is not quite externally given. The postulate that 5 be added to 7, bears the same relation to the postulate of simply counting, as does the postulate that a straight line be produced to the postulate that a straight line be drawn.
Just as meaningless as the expression 'synthesis', is its characterisation as occurring a priori. Counting is not, of course, determined by sensation which, according to Kant's definition of intuition is all that remains over for the a posteriors, and counting is certainly an activity based on abstract intuiting, that is, an intuiting determined by the category of the one, and in which abstraction is made from all other determinations of sensation, no less than from concepts, too. The a priori is altogether too vague a characterisation; feeling, determined as impulse, sense, and so on, has in it the a priori moment, just as much as space and time, in the shape of spatial and temporal existence, is determined a posteriors.
We may add in this connection that Kant's assertion of the synthetic nature of the foundations of pure geometry is equally without any solid basis. He admits that several are really analytic but only adduces one in support of his assertion that they are synthetic, namely, the axiom that the straight line is the shortest line between two points. 'For my notion of straightness contains nothing pertaining to magnitude, but only a quality; the notion of the shortest is, therefore, wholly an addition and cannot be inferred from any analysis of the notion of a straight line; we must therefore have recourse to intuition here which alone makes the synthesis possible.' But here again the question is not of a notion of straightness as such but of a straight line, and this is already something spatial and intuited. The determination (or, if you like, the concept) of the straight line is, after all, none other than that the line is absolutely simple, that is, in coming out of itself (the so-called movement of the point) the line is purely self-related, and its extension does not involve any alteration in its determination, or reference to another point or line outside itself; it is a simple direction purely internal to the line. This simplicity is indeed its quality, and should it seem difficult to define the straight line analytically this would be due solely to the simplicity and self-relation of the determination, and merely because reflection thinks of determining primarily in terms of a plurality, a determining through something else. But there is no inherent difficulty whatever in grasping this determination of the simplicity of extension within the line, of the absence of determination by anything else; Euclid's definition contains nothing else than this simplicity. But now the transition of this quality to the quantitative determination (of the shortest) which is supposed to constitute the synthetic element is wholly analytical. As spatial, the line is quantity in general; the simplest in terms of quantum is the least; and this predicated of a line is the shortest. Geometry can accept these determinations as a corollary to the definition; but Archimedes in his books on the sphere and cylinder' did the appropriate thing in making the said determination of a straight line into an axiom, in just as correct a sense as Euclid included the determination concerning parallel lines among the axioms, for the development of this determination into a definition would have required determinations not immediately spatial in character but of a more abstract qualitative kind, like simplicity, sameness of direction, and the like just mentioned. These ancients gave even to their sciences a plastic character, confining their exposition strictly to the peculiarity of their subject matter and therefore excluding what would have been heterogeneous to it.
Kant's notion of synthetic a priori judgements — the notion of something differentiated which equally is inseparable, of an identity which is in its own self an inseparable difference, belongs to what is great and imperishable in his philosophy. Of course, this notion is also present in intuition since it is the Notion itself and everything is implicitly the Notion; but the determinations selected in those examples do not exhibit it. On the contrary, number and counting is an identity and the creating of an identity which is wholly and solely external, only a superficial synthesis, a unity of ones which are, in fact, posited as inherently not identical with one another but as external, each separate on its own account; the determination of the straight line as being the shortest between two points is based rather on the moment of a merely abstract identity possessing no difference within itself.
I return from this digression to addition itself. The negative species of calculation corresponding to it, subtraction, is similarly the wholly analytical separation into numbers which, as in addition, are determined only as unequal generally relatively to one another.
2. The next determination is the equality of the numbers which are to be counted. Through this equality they are a unit, and number thus acquires the difference of unit and amount. Multiplication is the task of counting up an amount of units each of which is itself an amount. Here it is immaterial which of the two numbers is called unit and which amount, whether we say four times three, where four is the amount and three is the unit, or conversely, three times four. We have already indicated above that the original finding of the product is effected by simple counting, that is, counting off on the fingers, etc.; the subsequent ability to state the product immediately rests on the collection of these products, on the multiplication table and on knowing it by heart.
Division is the negative species of calculation with the same determination of the difference. Equally it is immaterial which of the two factors, the divisor or quotient, is taken as unit or amount. The divisor is taken as unit and the quotient as amount when the problem is to find out how often (amount) a number (unit) is contained in a given number; conversely, the divisor is taken as amount and the quotient as unit, when the problem is to divide a number into a given amount of equal parts and to find the magnitude of such part (of the unit).
3. The two numbers which are determined as being related to each other as unit and amount are, as number, still immediate with respect to each other and therefore simply unequal. The further equality is that of unit and amount themselves; with this, the advance to equality of the determinations immanent in the determination of number is completed. On the basis of this complete equality, counting is the raising to a power (the negative version is extraction of the root) — and in the first instance the squaring of a number — the complete immanent determinedness of counting where (1) the several numbers to be added are the same, and (2) their plurality or amount is itself the same as the number which is posited a plurality of times, which is unit. There are no other determinations in the Notion of number which could give rise to a difference; nor can there be any further equalising of the difference immanent in number. Where a number is raised to a higher power than the square the continuation is formal: partly, with even exponents, there is only a repetition of squaring, and partly, with uneven powers, inequality again enters; to take the nearest example of the cube, although formally there is an equality of the new factor with both the amount and the unit, this factor is, as unit, unequal to the amount (the square, 3, against 3 times 3); this is still more evident in the cube Of 4 where the amount, 3, which indicates the number of times the unit is to be multiplied by itself, is different from the number itself standing for the unit. We have here in principle those determinations of amount and unit which, as the essential difference of the Notion, have to be equalised before number as a going-out-of-itself has completely returned into itself. The foregoing exposition also contains the reason why first, the solution of higher equations must consist on their reduction to quadratics, and secondly, why equations with odd exponents can only be formally determined and, just when the roots are rational they cannot be found otherwise than by an imaginary expression, that is, by the opposite of that which the roots are and express. From what has been said, it is clear that the arithmetical square alone contains an immanent absolute determinedness; for which reason equations with further formal powers must be reduced to it, just as in geometry the right-angled triangle contains an immanent, absolute determinedness which is expounded in Pythagoras' theorem, for which reason, too, all other geometrical figures must be reduced to it for their complete determination.
A graded instruction based on a logically formed division of the subject treats of powers before it treats of proportions; it is true that the latter are connected with the difference of unit and amount which constitutes the determination of the second species of calculation, but they go beyond the one of the immediate quantum in which unit and amount are only moments; the further determination in the sphere of the immediate quantum still remains external to the quantum itself. In ratio, number is no longer an immediate quantum; it then possesses its determinateness as a mediation. The quantitative' relation will be considered in what follows.
It cannot be said that the progressive determination of the species of calculation here given is a philosophy of them or that it exhibits, possibly, their inner significance; and for this reason, that it is not in fact an immanent development of the Notion. However, philosophy must be able to distinguish what is an intrinsically self-external material; the progressive determining of it by the Notion can then take place only in an external manner, and its moments, too, can be only in the form peculiar to their externality, as here, equality and inequality. It is an essential requirement when philosophising about real objects to distinguish those spheres to which a specific form of the Notion belongs, that is, spheres in which the Notion has an actual existence; otherwise the peculiar nature of a subject matter which is external and contingent will be distorted by Ideas, and similarly these Ideas will be distorted and made into something merely formal. But here, the externality in which the moments of the Notion appear in this external material, number, is the appropriate form; since these moments exhibit the subject matter in its fixed differences, and since, too, they contain no demand for speculative thinking and therefore appear easy, they deserve to be employed in text books of the elements.
Remark 2: The Employment of Numerical Distinctions for Expressing Philosophical Notions
As we know, Pythagoras represented rational relationships (or philosophemata) by numbers; and more recently, too, numbers and forms of their relations, such as powers and so on, have been employed in philosophy for the purpose of regulating thoughts or expressing them. From an educational point of view, number has been regarded as the most suitable object of inner intuition and arithmetical operations with number have been held to be the mental activity in which mind brings to view its most characteristic relationships and in general, the fundamental relationships of essence. How far number can claim this high worth is evident from its Notion as now before us.
We saw that number is the absolute determinateness of quantity, and its element is the difference which has become indifferent — an implicit determinateness which at the same time is posited as wholly external. Arithmetic is an analytical science because all the combinations and differences which occur in its subject matter are not intrinsic to it but are effected on it in a wholly external manner. It does not have a concrete subject matter possessing inner, intrinsic, relationships which, as at first concealed, as not given in our immediate acquaintance with them, have first to be elicited by the efforts of cognition. Not only does it not contain the Notion and therefore no problem for speculative thought, but it is the antithesis of the Notion. Because of the indifference of the factors in combination to the combination itself in which there is no necessity, thought is engaged here in an activity which is at the same time the extreme externalisation of itself, an activity in which it is forced to move in a realm of thoughtlessness and to combine elements which are incapable of any necessary relationships. The subject matter is the abstract thought of externality itself.
As this thought of externality, number is at the same time the abstraction of the manifoldness of sense, of which it has retained nothing but the abstract determination of externality itself. In number, therefore, sense is brought closest to thought: number is the pure thought of thought's own externalisation.
The mind which rises above the world of the senses and contemplates its own essence, when it seeks an element for its pure representation, for the expression of its essence, may therefore happen on number, this inner, abstract externality, before it grasps thought itself as this element and wins the purely spiritual expression for the representation of its essence. This is why we see number used for the expression of philosophemata early on in the history of philosophy. It forms the latest stage in that imperfection which contemplates the universal admixed with sense. The ancients were clearly aware that number stands midway between sense and thought. Aristotle quotes Plato as saying that the mathematical determinations of things stand apart from and midway between the world of the senses and the Ideas; they are distinguished from the former by being invisible (eternal) and unmoved, and from the latter by being a many and a like, where as the Idea is purely self-identical and within itself a one. A more detailed and profound reflection on this subject by Moderatus of Cadiz is quoted in Malchi Vita Pythagorae. That the Pythagoreans hit on numbers he ascribes to their inability to apprehend clearly in reason fundamental ideas and first principles, because these are hard to think and hard to express; numbers serve well as designations in instruction; in this among other things the Pythagoreans imitated the geometers who cannot express what is corporeal in thoughts and therefore use figures, saying 'this is a triangle', by which they do not mean that the visible drawing is to be taken for the triangle but only that it is a representation of the thought of it. Thus the Pythagoreans expressed the thought of unity, of self-sameness and equality and the ground of agreement, of connection and the sustaining of everything, of the self-identical, as a one. It is superfluous to remark that the Pythagoreans passed on from numbers to thought as a medium of expression, to the express categories of like and unlike, of limit and infinity; even in respect of these numerical expressions it is reported that the Pythagoreans distinguished between the Monas and the one ; the Monas they took as the thought, but the one as the number, and similarly, they took two for the arithmetical term and the Dyas (for this is what it seems to mean there) for the thought of the indeterminate. These ancients at the outset perceived quite correctly the inadequacy of number forms for thought determinations and equally correctly they further demanded in place of this substitute for thoughts the characteristic expression; how much more advanced they were in their thinking than those who nowadays consider it praiseworthy, indeed profound, to revert to the puerile incapacity which again puts in the place of thought determinations numbers themselves and number-forms like powers, the infinitely great, the infinitely small, one divided by infinity, and other such determinations, which are themselves often only a perverted mathematical formalism.
In connection with the expression quoted above, that number stands between sense and thought, since, like the former, it is in its own self a many and an asunderness, it must be observed that the many itself is sense taken up into thought, is the category of what is in its own self external and so proper to sense. When richer, concrete veritable thoughts, when what is most alive and most active, what is comprehended only in its concrete relationships, when such are transposed into this element of pure self-externality, they become dead, inert determinations.
The richer in determinateness and, therefore, in relationships thoughts become, the more confused and also the more arbitrary and meaningless becomes their representation in such forms as numbers. The one, the two, the three, the four, Henas or Monas, Dyas, Trias, Tetractys, have still some resemblance to the wholly simple abstract Notions; but when numbers are supposed to represent concrete relationships, it is vain to try to retain such resemblance.
But thought is set its hardest task when the determinations for the movement of the Notion through which alone it is the Notion, are denoted by one, two, three, four; for it is then moving in its opposite element, one which is devoid of all relation; it is engaged on a labour of derangement. The difficulty in comprehending that, for example, one is three and three is one, stems from the fact that one is devoid of all relation and therefore does not in its own self exhibit the determination through which it passes into its opposite; on the contrary, one is essentially a sheer excluding and rejection of such a relation. Conversely, understanding makes use of this to combat speculative truth (as, for example, against the truth laid down in the doctrine called the trinity) and it counts the determinations of it which constitute one unity, in order to expose them as sheer absurdity — that is, understanding itself commits the same absurdity of making that which is pure relation into something devoid of all relation. When the trinity was so named it was not reckoned that understanding would consider the one and number to be the essential determinateness of its content. This name expresses contempt for the understanding, which has nevertheless confirmed itself in its conceit of clinging to the one and number as such, and has set it up against reason.
To take numbers and geometrical figures (as the circle triangle etc., have often been taken), simply as symbols (the circle, for example, as a symbol of eternity, the triangle, of the trinity), is so far harmless enough; but, on the other hand, it is foolish to fancy that in this way more is expressed than can be grasped and expressed by thought. Whatever profound wisdom may be supposed to lie in such meagre symbols or in those richer products of fantasy in the mythology of peoples and in poetry generally, it is properly for thought alone to make explicit for consciousness the wisdom that lies only in them; and not only in symbols but in nature and in mind. In symbols the truth is dimmed and veiled by the sensuous element; only in the form of thought is it fully revealed to consciousness: the meaning is only the thought itself.
But the perversity of employing mathematical categories for the determination of what belongs to the method or content of the science of philosophy is shown chiefly by the fact that, in so far as mathematical forms signify thoughts and distinctions based on the Notion, this their meaning has indeed first to be indicated, determined and justified in philosophy. In the concrete philosophical sciences philosophy must take the logical element from logic, not from mathematics; it can only be an expedient of philosophical incapacity which, instead of going to philosophy for the logical element, has recourse to the shapes assumed by the logical element in other sciences, many of which shapes are only adumbrations of that element, others even defective forms of it. Apart from this, the mere application of such borrowed forms is an external procedure; the application itself must be preceded by an awareness not only of their meaning but of their value, too, and such awareness can come only from reflecting on them, not from the authority of mathematics. But logic itself is such awareness and it strips these forms of their particularity which it renders superfluous and unnecessary; it is logic which rectifies these forms and alone procures for them their justification, meaning and value.
As for the supposed primary importance of number and calculation in an educational regard, the truth of the matter is clearly evident from what has been said. Number is a non-sensuous object, and occupation with it and its combinations is a non-sensuous business; in it mind is held to communing with itself and to an inner abstract labour, a matter of great though one-sided importance. For, on the other hand, since the basis of number in only an external, thoughtless difference, such occupation is an unthinking, mechanical one. The effort consists mainly in holding fast what is devoid of the Notion and in combining it purely mechanically. The content is the empty one; the substantial content of moral and spiritual life in its various forms on which, as the noblest aliment, education should nurture the young mind, is to be supplanted by the blank one or unit; when such exercise is made the prime interest and occupation, the only possible outcome must be to dull the mind and to empty it of both form and substance. Calculation being so much an external and therefore mechanical business, it has been possible to construct machines which perform arithmetical operations with complete accuracy. A knowledge of just this one fact about the nature of calculation is sufficient for an appraisal of the idea of making calculation the principal means for educating the mind and stretching it on the rack in order to perfect it as a machine.
B. EXTENSIVE AND INTENSIVE QUANTUM
(a) Their Difference
1. We have seen that quantum has its determinateness as limit in amount. Within itself quantum is discrete, a plurality which has no being distinct from its limit, nor is the limit external to it. Quantum thus then with its limit, which is in its own self a plurality, is extensive magnitude.
Extensive and continuous magnitude are to be distinguished from each other; the direct opposite of the former is not discrete but intensive magnitude. Extensive and intensive magnitudes are determinatenesses of the quantitative limit itself, whereas quantum is identical with its limit; continuous and discrete magnitudes, on the other hand, are determinations of magnitude in itself, that is, of quantity as such, in so far as in quantum abstraction is made from the limit. Extensive magnitude has the moment of continuity present within itself and in its limit, for its many is altogether continuous; the limit as negation appears, therefore, in this equality of the many as a limiting of the oneness. Continuous magnitude is quantity as continuing itself without regard to any limit and in so far as it is conceived as having a limit, this is simply a limitation free from any posited discreteness. Quantum as only continuous magnitude is not yet truly determined as being for itself because it lacks the one (in which being-for-selfness is implied) and number. Similarly, a discrete magnitude is immediately only a differentiated many in general; were this as such supposed to have a limit, it would be only an aggregate, that is, would be only indefinitely limited; before it can be a specific quantum, the many must be compressed into a one and thereby posited as identical with the limit. Continuous and discrete magnitude, taken simply as quanta have each posited in it only one of the two sides which together make quantum fully determined and a number. This latter is immediately an extensive quantum — the simple determinateness which is essentially an amount, but an amount of one and the same unit; extensive quantum is distinguished from number only by this, that in number the determinateness is expressly posited as a plurality.
2. However, the determinateness of something in terms of number does not require it to be distinguished from another numerically determined something, as if both were necessary to the determinateness of the first; and this is because the determinateness of magnitude as such is a limit determinate by itself, indifferent and related simply to itself; and in number the limit is posited as included in the one, which is a being-for-self, and it has within itself the externality, the relation to other. Further, this many of the limit itself is, like the many as such, not unequal within itself but continuous; each of the many is the same as the others; consequently, the many as a plural asunderness or discreteness does not constitute the determinateness as such. This many, therefore, spontaneously collapses into its continuity and becomes a simple oneness. Amount is only a moment of number, but as an aggregate of numerical ones, it does not constitute the determinateness of number; on the contrary, these ones as indifferent and self-external are sublated in number which has returned into itself; the externality which constituted the ones as a plurality vanishes in the one as a relation of number to its own self.
Consequently the limit of quantum, which as extensive had its real determinateness in the self-external amount, passes over into simple determinateness. In this simple determination of the limit, quantum is intensive magnitude; and the limit or determinateness which is identical with the quantum is now also thus posited as unitary — degree.
The degree is thus a specific magnitude, a quantum; but at the same time it is not an aggregate or plural within itself, it is a plurality only in principle (eine Mehrheit), for plurality has been brought together into a simple, unitary determination, determinate being has returned into being-for-self. The determinateness of degree must, it is true, be expressed by a number, the completely determined form of quantum, but the number is not an amount but unitary, only a degree. When we speak of ten or twenty degrees, the quantum that has that number of degrees is the tenth or twentieth degree, not the amount and sum of them — as such, it would be an extensive quantum — but it is only one degree, the tenth or twentieth. It contains the determinateness implied in the amount ten or twenty, but does not contain it as a plurality but is number as a sublated amount, as a unitary determinateness.
3. In number, quantum is posited in its complete determinateness; but as intensive quantum, as in number's being-for-self, it is posited as it is in its Notion or in itself. That is to say, the form of self-relation which it has in degree is at the same time the externality of the degree to its own self. Number, as an extensive quantum, is a numerical plurality and so has the externality within itself. This externality, as simply a plurality, collapses into undifferentiatedness and sublates itself in the numerical one, in its self-relation. Quantum, however, has its determinateness as an amount; it contains this, as we have already seen, even though the amount is no longer posited in it. Degree, therefore, which, as in its own self unitary, no longer has within itself this external otherness, has it outside itself and relates itself to it as to its determinateness. A plurality external to the degree constitutes the determinateness of the simple limit which the degree is for itself. In extensive quantum amount, in so far as it was supposed to be present in the number, was so only as sublated; now it is determined as placed outside the number. Number as a one, being posited as self-relation reflected into itself, excludes from itself the indifference and externality of the amount and is self-relation as relation through itself to an externality.
In this, quantum has a reality conformable to its Notion. The indifference of the determinateness constitutes its quality, that is, the determinateness which is in its own self a self-external determinateness. Accordingly, degree is a unitary quantitative determinateness among a plurality of such intensifies which, though differing from each other, each being only a simple self-relation, are at the same time essentially interrelated so that each has its determinateness in this continuity with the others. This relation of degree through itself to its other makes ascent and descent in the scale of degrees a continuous progress, a flux, which is an uninterrupted, indivisible alteration; none of the various distinct degrees is separate from the others but each is determined only through them. As a self-related determination of quantity, each degree is indifferent to the others; but it is just as much implicitly related to this externality, it is only through this externality that it is what it is; its relation to itself is, in short, the non-indifferent relation to externality, and in this it has its quality.
(b) Identity of Extensive and Intensive Magnitude
Degree is not external to itself within itself. Nevertheless, it is not the indeterminate one, the principle of number as such, which is not amount save in the negative sense of not being any particular amount. Intensive magnitude is primarily a unitary one of a plurality; there are many degrees, but they are determined neither as a simple one nor as a plurality, but only in the relation of this self-externality, or in the identity of the one and the plurality. if, therefore, the many as such are indeed outside the simple, unitary degree, nevertheless the determinateness of the degree consists in its relation to them; it thus contains amount. just as twenty, as an extensive magnitude, contains the twenty ones as discrete within it, so does the specific degree contain them as the continuity which this determinate plurality simply is; it is the twentieth degree, and is the twentieth degree only by virtue of this amount, which as such is outside it.
The determinateness of intensive magnitude is, therefore, to be considered from two sides. Intensive magnitude is determined by other intensive quanta and is continuous with its otherness, so that its determinateness consists in this relation to its otherness. Now in the first place, in so far as it is a simple determinateness it is determinate relatively to other degrees; it excludes them from itself and has its determinateness in this exclusion. But, secondly, it is determinate in its own self; this it is in the amount as its own amount, not in the amount as excluded, nor in the amount of other degrees. The twentieth degree contains the twenty within itself; it is not only determined as distinguished from the nineteenth, twenty-first, and so on, but its determinateness is its own amount. But in so far as the amount is its own — and the determinateness is at the same time essentially an amount — the degree is an extensive quantum.
Extensive and intensive magnitude are thus one and the same determinateness of quantum; they are only distinguished by the one having amount within itself and the other having amount outside itself. Extensive magnitude passes over into intensive magnitude because its many spontaneously collapse into oneness, outside which the many stand. But conversely, this unitary degree has its determinateness only in the amount, and that too in its own amount; as indifferent to the differently determined intensifies it has within itself the externality of the amount; and so intensive magnitude is equally essentially an extensive magnitude.
With this identity, the qualitative something makes its appearance, for the identity is the unity which is self-related through the negation of its differences; these differences, however, constitute the determinate being of the quantitative determinateness; this negative identity is therefore a something, and a something which is indifferent to its quantitative determinateness. Something is a quantum; but now the qualitative determinate being, as it is in itself, is posited as indifferent to quantum. Quantum, number as such, and so forth, could be spoken of without any mention of its having a something as substrate. But the something now confronts these its determinations, through the negation of which it is mediated with itself, as existing for itself and, since it has a quantum, as something which has an extensive and an intensive quantum. Its one determinateness which it has as quantum is posited in the differentiated moments of unit and amount; this determinateness is not only in itself one and the same, but its positing in these differences as extensive and intensive quantum is the return into this unity which, as negative, is the something posited as indifferent to it.
Remark 1: Examples of This Identity
Extensive and intensive quantum are usually distinguished in the ordinary conception of them as kinds of magnitude, as if some objects had only intensive, others only extensive magnitude. In addition, we have the conception of a philosophical science of Nature in which what is a plurality or extensive — for example, in the fundamental property of matter to occupy space, and in other concepts too — is converted into something intensive, meaning thereby that the intensive aspect as dynamic is the true determination; density, or the specific filling of space, for example, must essentially be understood not as a certain aggregate and amount of material parts in a quantum of space, but as a certain degree of the space-filling force of matter.
There are two kinds of determinations to be distinguished here. In what has been called the conversion of the mechanical into the dynamic point of view, there occurs the concept of separately existing, independent parts, which are only externally combined into a whole, and the concept of force which is distinct from this. In the occupation of space, what is regarded on the one hand as only an aggregate of atoms external to one another, is on the other hand regarded as the expression of an underlying simple force. But these relations of whole and parts, of force and its expression, which here stand opposed to each other, do not belong in this section; they will be considered in their proper place later on. But this much may be remarked here, that though the relation of force and its expression which corresponds to intensive magnitude is in the first instance truer than that of whole and parts, yet this does not make force, as intensive, any less one-sided; also expression, the externality of extensive magnitude, is equally inseparable from force; so that one and the same content is equally present in the two forms, both in intensive and in extensive magnitude.
The other determinateness which occurs here is the quantitative as such, which, as extensive quantum, is sublated and transformed into degree, the supposedly true determination; but it has been shown that degree equally contains the former determinateness, so that the two forms are essential to each other; consequently, every existence exhibits its quantitative character just as much as an extensive as an intensive quantum.
Consequently everything, in so far as it manifests a quantitative character, serves as an example of this. Number itself necessarily has this double form immediately within it. It is an amount in so far as it is an extensive magnitude; but number is also one, a ten, a hundred, and as such it is on the threshold of transition into an intensive magnitude, seeing that in this unity the plurality has become simple. One is in itself an extensive magnitude, it can be represented as an arbitrary amount of parts. Thus the tenth, the hundredth, is this simple, intensive magnitude which has its determinateness in the plurality lying outside it, that is, in extensive magnitude. Number is a ten, a hundred and at the same time the tenth, hundredth, in the system of numbers; both are the same determinateness.
In the circle the one is called degree because the determinateness of any part of the circle derives essentially from the many parts outside it; that is, it is determined as only one of a fixed amount of such ones. As a mere spatial magnitude, the degree of the circle is only an ordinary number; taken as degree, it is intensive magnitude which has a meaning only as determined by the amount of degrees into which the circle is divided, just as number generally has meaning only in the number series.
The magnitude of a more concrete object exhibits its dual aspects of being extensive and intensive, in the dual determinations of its real being, in one of which it appears as an outer being but in the other as an inwardness. Thus, for example, a mass as weight is an extensive magnitude, in so far as it constitutes an amount of pounds, hundredweights, etc., and an intensive magnitude in so far as it exerts a certain pressure; the magnitude of the pressure is a simple number, a degree, which is specified by its place in a scale of degrees. As exerting pressure, mass is manifested as a being-within-self, as a subject to which belongs a difference of intensive magnitude. Conversely, that which exerts this degree of pressure is capable of displacing a certain amount of pounds, etc., and its magnitude is measured by this.
Again, heat has a degree; this degree, whether it be the tenth, twentieth and so on, is a simple sensation, something subjective. But this degree is equally present as an extensive magnitude, as the expansion of a fluid, of mercury in a thermometer, of air, or sound, and so on. A higher degree of temperature expresses itself as a longer column of mercury, or as a narrower sound cylinder; it heats a larger space in the same way as a lower degree heats only a smaller space.
The higher note is, as more intensive, at the same time a greater number of vibrations, and a louder note, to which we ascribe a higher degree, is audible in a larger space. A larger surface can be coloured with a more intensive colour than with a weaker colour used in the same way; or again a brighter object (another kind of intensity) is visible at a greater distance than one less bright, and so forth.
Similarly in the spiritual sphere, high intensity of character, of talent or genius, is bound up with a correspondingly far-reaching reality in the outer world, is of widespread influence, touching the real world at many points. The profoundest Notion also has the most universal significance and application.
Remark 2: The determination of degree as applied by Kant to the soul
The determinateness of intensive quantum has been applied by Kant in a peculiar way to a metaphysical determination of the soul. In his criticism of metaphysical propositions about the soul, which he calls paralogisms of pure reason, he comes to consider the inference from the simplicity of the soul to its permanence' He counters this argument by saying 'that even if we admit the simple nature of the soul since, namely, it does not contain a plurality of separate parts and therefore no extensive magnitude, yet we cannot deny to it, any more than to any other existent thing, an intensive magnitude, that is, a degree of reality in respect of all its faculties, indeed, in respect of all that constitutes its existence; this degree can diminish through all the infinitely many smaller degrees so that although the postulated substance cannot be reduced to nothing by division (into parts), it can be so reduced by a gradual lessening (remissio) of its powers. For consciousness itself has, at any moment, a degree which can always be diminished, and the same must therefore also be true of its faculty of being aware of itself and thus of all the other faculties. In rational psychology, which is an abstract metaphysics, the soul is considered not as spirit but as a merely immediate being, as a soul thing. Kant thus has the right to apply the category of quantum to it 'as to any other existent thing', and in so far as this immediate being is determined as simple, to apply to it the category of intensive quantum. Being does, of course, belong to spirit, but its intensity is wholly different from that of intensive quantum; indeed, its intensity is such that in it the form of merely immediate being and all its categories are sublated. What should have been admitted was the elimination not only of the category of extensive quantum but that of quantum altogether. But a further advance has still to be made, namely, to understand how existence, consciousness, finitude, is in the eternal nature of spirit and proceeds from it without spirit thereby becoming a thing.
(c) Alteration of Quantum
The difference between extensive and intensive quantum is indifferent to the determinateness of quantum as such. But in general quantum is determinateness posited as sublated, the indifferent limit, the determinateness which is equally the negation of itself. In extensive magnitude this difference is developed; but intensive magnitude is the existence of this externality which quantum is within itself. This difference, as internally self-contradictory, is posited as being the simple, self-related determinateness which is the negation of itself, having its determinateness not within itself but in another quantum.
A quantum, therefore, in accordance with its quality, is posited in absolute continuity with its externality, with its otherness. Therefore, not only can it transcend every quantitative determinateness, not only can it be altered, but it is posited that it must alter. The quantitative determinateness continues itself into its otherness in such a manner that the determination has its being only in this continuity with an other; it is not a simply affirmative limit, but a limit which becomes.
The one is infinite or self-related negation, hence the repulsion of itself from itself. The quantum, too, is infinite and is posited as self-related negativity; it repels itself from itself. But the quantum is a determinate one, the one which has passed over into determinate being and limit; it is, therefore, the repulsion of the determinateness from itself, not the producing of that which is the same as itself as in the repulsion of the one, but the producing of its otherness; it is now the express character of quantum to impel itself beyond itself and to become an other. In consists in undergoing increase or decrease; it is in its own self the externality of the determinateness.
Thus quantum impels itself beyond itself; this other which it becomes is in the first place itself a quantum; but it is quantum as a limit which does not stay, but which impels itself beyond itself. The limit which again arises in this beyond is, therefore, one which simply sublates itself again and impels itself beyond to a further limit, and so on to infinity.
C. QUANTITATIVE INFINITY
(a) Its Notion
Quantum alters and becomes another quantum; the further determination of this alteration, namely, that it goes on to infinity, lies in the circumstance that quantum is established as being immanently self-contradictory. Quantum becomes an other; but it continues itself into its otherness; the other is thus also a quantum. This, however, is not only the other of a particular quantum, but of quantum itself, the negative of quantum as limited; hence it is the unlimitedness of quantum, its infinity. Quantum is an ought-to-be; it is by implication determined as being for itself, and this being-determined-for-itself is rather the being-determined-in-an-other, and, conversely, it is the sublation of being-determined-in-an-other, is an indifferent subsisting for itself.
In this way, finitude and infinity each acquire in themselves a dual, and indeed, an opposite meaning. The quantum is finite, in the first place simply as limited, and secondly, as impelled beyond itself, as being determined in an other. But the infinity of quantum is first, its unlimitedness, and secondly, its returnedness into itself, its indifferent being-for-self. If we now compare these moments with each other, we find that the determination of the finitude of quantum, the impulse to go beyond itself to an other in which its determination lies, is equally the determination of the infinite; the negation of the limit is the same impulsion beyond the determinateness, so that in this negation, in the infinite, quantum possesses its final determinateness. The other moment of infinity is the being-for-self which is indifferent to the limit; but the limiting of quantum itself is such that quantum is explicitly indifferent to its limit, and hence to other quanta and to its beyond. In quantum, finitude and infinity (the spurious infinity supposedly separate from the finite) each already has within it the moment of the other.
The difference between the qualitative and quantitative infinite is that in the former the finite and infinite are qualitatively opposed and the transition of the finite into the infinite, or the relation of each to the other, lies only in the in-itself, in their Notion. Qualitative determinateness, as an immediacy, is related to otherness essentially as to an alien being; it is not posited as having its negation, its other within it. Quantity, on the other hand, is, as such, sublated determinateness; it is posited as being unlike and indifferent to itself, consequently as alterable. Therefore the qualitative finite and infinite stand absolutely, that is abstractly, opposed to each other; their unity is their underlying inner relation; and therefore the finite continues itself into its other only implicitly, not affirmatively. The quantitative finite, on the other hand, is self-related in its infinite, in which it has its absolute determinateness. This their relation is displayed in the first place in the quantitative infinite progress.
(b) The Quantitative Infinite Progress
The progress to infinity is in general the expression of contradiction, here, of that which is implicit in the quantitative finite, or quantum as such. It is the reciprocal determining of the finite and infinite which was considered in the sphere of quality, with the difference that, as just remarked, in the sphere of quantity the limit in its own self dispatches and continues itself into its beyond and hence, conversely, the quantitative infinite too is posited as having quantum within it; for quantum in its self-externality is also its own self, its externality belongs to its determination.
Now the infinite progress is only the expression of this contradiction, not its resolution; but because the one determinateness is continued into its other, the progress gives rise to the show of a solution in a union of both. As at first posed, it is the problem of attaining the infinite, not the actual reaching of it; it is the perpetual generation of the infinite, but it does not get beyond quantum, nor does the infinite become positively present. It belongs to the Notion of quantum to have a beyond of itself. This beyond is first, the abstract moment of the non-being of quantum: the vanishing of quantum is its own act; it is thus related to its beyond as to its infinity, in accordance with the qualitative moment of the opposition. Secondly, however, quantum is continuous with its beyond; quantum consists precisely in being the other of itself, in being external to itself; this externality is, therefore, no more an other than quantum itself; the beyond or the infinite is, therefore, itself a quantum. In this way, the beyond is recalled from its flight and the infinite is attained. But because the infinite now affirmatively present is again a quantum, what has been posited is only a fresh limit; this, too, as a quantum, has again fled from itself, is as such beyond itself and has repelled itself into its non-being, into its own beyond, and as it thus repels itself into the beyond, so equally does the beyond perpetually become a quantum.
The continuity of quantum with its other produces the conjunction of both in the expression of an infinitely great or infinitely small. Since both still bear the character of quantum they remain alterable, and the absolute determinateness which would be a being-for-self is, therefore, not attained. This self-externality of the determination is posited in the dual infinite — which is opposed to itself as a 'more' and a 'less' — in the infinitely great and infinitely small. In each, the quantum is maintained in perpetual opposition to its beyond. No matter how much the quantum is increased, it shrinks to insignificance; because quantum is related to the infinite as to its non-being, the opposition is qualitative; the increased quantum has therefore gained nothing from the infinite, which is now, as before, the non-being of quantum. In other words, the increase of quantum brings it no nearer to the infinite; for the difference between quantum and its infinity is essentially not a quantitative difference. The expression 'the infinitely great' only throws the contradiction into sharper relief; it is supposed to be great, that is, a quantum, and infinite, that is, not a quantum. Similarly, the infinitely small is, as small, a quantum, and therefore remains absolutely, that is, qualitatively, too great for the infinite and is opposed to it. In both, there remains the contradiction of the infinite progress which in them should have reached its goal.
This infinity which is perpetually determined as the beyond of the finite is to be described as the spurious quantitative infinite. Like the qualitative spurious infinite, it is the perpetual movement to and fro from one term of the lasting contradiction to the other, from the limit to its non-being, and from this back again to the limit. It is true that in the quantitative progress the movement is not simply towards an abstract other in general, but towards an explicitly different quantum; but this remains in the same way opposed to its negation. The progress, too, is therefore not a real advance but a repetition of one and the same thing, a positing, a sublating, and then again a positing and again a sublating, an impotence of the negative, for what it sublates is continuous with it, and in the very act of being sublated returns to it. Thus there are two terms, the bond between which is such that they simply flee from each other; and in fleeing from each other they cannot become separated but are joined together even in their flight from each other.
Remark 1: The High Repute of the Progress to Infinity
The spurious infinite, especially in the form of the quantitative progress to infinity which continually surmounts the limit it is powerless to remove, and perpetually falls back into it, is commonly held to be something sublime and a kind of divine worship, while in philosophy it has been regarded as ultimate. This progression has often been the theme of tirades which have been admired as sublime productions. As a matter of fact, however, this modern sublimity does not magnify the object — rather does this take flight — but only the subject which assimilates such vast quantities. The hollowness of this exaltation, which in scaling the ladder of the quantitative still remains subjective, finds expression in its own admission of the futility of its efforts to get nearer to the infinite goal, the attainment of which must, indeed, be achieved by a quite different method.
In the following tirades of this kind it is also stated what becomes of such exaltation and how it finishes. Kant, for example, at the close of the Critique of Practical Reason, represents it as sublime 'when the subject raises himself in thought above the place he occupies in the world of sense, reaching out to infinity, to stars beyond stars, worlds beyond worlds, systems beyond systems, and then also to the limitless times of their periodic motion, their beginning and duration. Imagination fails before this progress into the infinitely remote, where beyond the most distant world there is a still more distant one, and the past, however remote, has a still remoter past behind it, the future, however distant, a still more distant future beyond it; thought fails in the face of this conception of the immeasurable, just as a dream, in which one goes on and on down a corridor which stretches away endlessly out of sight, finishes with falling or fainting.'
This exposition, besides giving a concise yet rich description of such quantitative exaltation, deserves praise mainly on account of the truthfulness with which it states how it fares finally with this exaltation: thought succumbs, the end is falling and faintness. What makes thought succumb, what causes falling and faintness, is nothing else but the wearisome repetition which makes a limit vanish, reappear, and then vanish again, so that there is a perpetual arising and passing away of the one after the other and of the one in the other, of the beyond in the here and now, and of the here and now in the beyond, giving only the feeling of the impotence of this infinite or this ought-to-be, which would be master of the finite and cannot.
Also Haller's description of eternity, called by Kant terrifying, is usually specially admired, but often just not for that very reason which constitutes its true merit:
'I heap up monstrous numbers,
Pile millions upon millions,
I put aeon upon aeon and world upon world,
And when from that awful height
Reeling, again I seek thee,
All the might of number increased a thousandfold
Is still not a fragment of thee.
I remove them and thou nest wholly before me.'
When this heaping and piling up of numbers is regarded as what is valuable in a description of eternity, it is overlooked that the poet himself declares this so-called terrifying journey into the beyond to be futile and empty, and that he closes by saying that only by giving up this empty, infinite progression can the genuine infinite itself become present to him.
There have been astronomers who liked to pride themselves on the sublimity of their science because it had to deal with an innumerable host of stars, with such immeasurable spaces and times in which distances and periods, already vast in themselves, serve as units which, in whatever multiples taken, are again abbreviated to insignificance. The shallow astonishment to which they surrender themselves, the absurd hopes of wandering in another life from one star to another and into immeasurable space to acquire fresh facts of the same kind, this they declare to be a cardinal factor in the excellence of their science — a science which is admirable not on account of such quantitative infinitude but, on the contrary, on account of the relations of measure and the laws which reason recognises in these objects and which are the infinite of reason in contrast to that other, irrational infinite.
To the infinity of outer, sensuous intuition, Kant opposes the other infinite, when 'the individual withdraws into his invisible ego and opposes the absolute freedom of his will as a pure ego to all the terrors of fate and tyranny, and starting with his immediate surroundings, lets them vanish before him, and even what seems enduring, worlds upon worlds, collapse into ruins, and, alone, knows himself as equal to himself.'
The ego in being thus alone with itself is, it is true, the reached beyond; it has come to itself, is with itself, here and now; the absolute negativity which in the progress beyond the quantum of sense was only a flight, in pure self-consciousness becomes affirmative and present. But this pure ego, because it has fixed itself in its abstraction and emptiness, has determinate reality, the fulness of the universe of nature and mind, confronting it as a beyond. We are faced with that same contradiction which lies at the base of the infinite progress, namely a returnedness-into-self which is at the same time immediately an out-of-selfness, a relation to its other as to its non-being; and this relation remains a longing, because on the one side is the unsubstantial, untenable void of the ego fixed as such by the ego itself, and on the other, the fulness which though negated remains present, but is fixed by the ego as its beyond.
On these two sublimes Kant remarks 'that admiration (for the first, outer) and reverence (for the second, inner) do indeed stimulate inquiry but cannot be a substitute for their defect'. Thus he declares those exaltations to be unsatisfying for reason, which cannot stop at them and the feelings associated with them, nor can it let the beyond and the void rank as ultimate.
But it is specially in its application to morality that the infinite progress has been taken as ultimate. The just quoted antithesis of finite and infinite in the shape of the manifold world and the ego raised to its freedom, is primarily qualitative. The ego in its self-determining forthwith proceeds to determine nature and to liberate itself therefrom; it thus connects itself through itself with its other which, as an external reality, is manifold and quantitative. The relation to the quantitative becomes itself quantitative; the negative relation of the ego to it, the power of the ego over the non-ego, over sense and outer nature, is consequently so conceived that morality can and ought continually to increase, and the power of sense continually to diminish. But the perfect adequacy of the will to the moral law is placed in the unending progress to infinity, that is, is represented as an absolutely unattainable beyond, and this very unattainableness is supposed to be the true sheet-anchor and fitting consolation; for morality is supposed to be a struggle, but such it can be oily if the will is inadequate to the moral law which thus becomes a sheer beyond for it.
In this opposition, ego and non-ego or the pure will and the moral law, and nature and the sensuousness of the will, are presupposed as completely self-subsistent and mutually indifferent. The pure will has its own appropriate law which stands in an essential relationship to the sphere of sense; and nature and sense on its side has laws which neither stem from nor are conformable to the will nor, although distinct from it, have they even in principle an essential connection with it but are determined independently, are finished and complete in themselves. At the same time, however, both are moments of one and the same simple being, the ego; the will is determined as the negative in relation to nature so that the will only is in so far as there is a sphere distinct from it which it sublates, but with which it thereby comes into contact and by which it is itself affected. Nature itself and nature as the sensuous sphere of man, as an independent system of laws, is indifferent to limitation by an other; it preserves itself in this process of limitation, enters into the relation as an independent factor and limits the will of law just as much as this limits it. The two processes comprise a single act: the self-determining of the will with the sublating of the otherness of nature, and the positing of this otherness as continuing itself as a reality in the process of being sublated, so that the otherness is not sublated. The contradiction involved in this is not resolved in the infinite progress: on the contrary, it is represented and affirmed as unresolved and unresolvable; the conflict of morality and sense is represented as the ultimate, absolute relation.
This standpoint which is powerless to overcome the qualitative opposition between the finite and infinite and to grasp the idea of the true will which is substantial freedom, has recourse to quantity in order to use it as a mediator, because it is sublated quality, the difference which has become indifferent. But since both members of the antithesis remain implied as qualitatively distinct, the fact is rather that each is straightway made indifferent to this alteration because it is as quanta that they are related to each other. Nature is determined by the ego, sense by the will of the good; the alteration produced in sense by the will is only a quantitative difference, one which leaves sense itself unchanged.
In the more abstract exposition of the Kantian philosophy, or at least of its principles, namely in Fichte's Theory of Science, the infinite progress in the same way constitutes the foundation and the ultimate. In this exposition, the first axiom, ego = ego, is followed by a second, independent of it, the opposition of the non-ego; the relation between the two is also directly assumed as a quantitative difference, that is, the non-ego is partly determined by the ego, and partly not. In this way, the non-ego is continued into its non-being in such wise that in its non-being it remains opposed as something not sublated. Consequently, after the contradictions contained in this have been developed in the system, the final result is that relationship which formed the beginning: the non-ego remains an infinite obstacle, an absolute other; the final relation of the non-ego and the ego to each other is the infinite progress, a longing and aspiration — the same contradiction with which the system began.
Because the quantitative is determinateness posited as sublated it was thought that much, or rather everything, had been gained for the unity of the absolute, for the one substantiality, when opposition generally had been reduced to a merely quantitative difference. That all opposition is only quantitative was for some time a cardinal thesis of recent philosophy; the opposed determinations have the same nature, the same content; they are real sides of the opposition in so far as each of them has within it both determinations, both factors of the opposition, only that on one side one of the factors preponderates, on the other side the other, that is, one of the factors, a material substance or activity, is present in a greater quantity or in an intenser degree in one side than in the other. But in so far as substances or activities are presupposed, the quantitative difference rather confirms and completes their externality and indifference to each other and to their unity. The difference of the absolute unity is supposed to be only quantitative; the quantitative, it is true, is immediate, sublated determinateness, but only the imperfect, as yet only first, negation, not the infinite, not the negation of the negation. When being and thought are represented as quantitative determinations of absolute substance they too, as quanta, become completely external to each other and unrelated as, in a subordinate sphere, do carbon, nitrogen, etc. It is a third, an external reflection, which abstracts from their difference and recognises their unity, but a unity which is inner, implicit only, not for itself. This unity is, therefore, in fact conceived only as a first, immediate unity, or only as being, which in its quantitative difference remains like itself, but does not of itself posit itself as like itself; hence it is not grasped as a negation of the negation, as an infinite unity. Only in the qualitative opposition does the posited infinitude, being-for-self, emerge and the quantitative determination itself pass over into the qualitative, as we shall presently find.
Remark 2: The Kantian Antinomy of the Limitation and Nonlimitation of the World in Time and Space
It was remarked above that the Kantian antinomies are expositions of the opposition of finite and infinite in a more concrete shape, applied to more specific substrata of conception. The antinomy there considered contained the opposition of qualitative finitude and infinitude. In another, the first of the four cosmological antinomics, it is the conflict arising rather from the quantitative limit which is considered. I shall therefore proceed to examine this antinomy here.
It concerns the limitation or non-limitation of the world in time and space. This antithesis could be considered equally well with reference to time and space themselves, for whether time and space are relations of things themselves or are only forms of intuition, the antinomy based on limitation or non-limitation in them is not affected thereby.
The detailed analysis of this antinomy will likewise show that both statements and equally their proofs (which, like those already considered, are conducted apagogically) amount to nothing more than the two simple opposite assertions: (1) there is a limit, and (2) the limit must be transcended.
The thesis is:
'The world has a beginning in time and is also enclosed within spatial limits.'
That part of the proof which concerns time assumes the opposite:
'The world has no beginning in time; therefore, up to any given point of time, an eternity has elapsed and consequently an infinite series of successive states of things in the world has passed away. Now the infinity of a series consists precisely in the impossibility of ever completing it by successive synthesis. Therefore an infinite world series which has passed away is impossible and consequently a beginning of the world is a necessary condition of its existence — which was to be proved.'
The other part of the proof which concerns space is based on time. To comprehend a spatially infinite world would require an infinite time and this time must be regarded as having already elapsed in so far as the world in space is to be regarded not as gradually coming to be but as completely given. But it was shown of time in the first part of the proof that it is impossible to assume an infinite time as elapsed.
But it is at once evident that it was unnecessary to make the proof apagogical, or even to carry out a proof at all, since the basis of the proof itself is the direct assertion of what was to be proved. Namely, there is assumed some or any given point of time up to which an eternity has elapsed (eternity here has only the trivial meaning of a simply endless time). Now a given point of time means nothing else than a definite limit in time. In the proof therefore, a limit of time is presupposed as actual; but that is just what was to be proved. For the thesis is, that the world has a beginning in time.
There is only this difference, that the assumed limit of time is a now as end of the time already elapsed, but the limit which is to be proved is a now as beginning of a future. But this difference is immaterial. The now is taken as the point in which an infinite series of successive states of things in the world is supposed to have passed away, therefore as end, a qualitative limit. If this now were considered to be merely a quantitative limit which flows on and which not only must be transcended but is only as the transcending of itself, then the infinite time series would not have passed away in it, but would continue to flow on, and so the argument of the proof would vanish. On the other hand, if the point of time is assumed as a qualitative limit for the past, in which case it is also a beginning for the future (for each point of time is in itself the connection of the past and the future), then it is also an absolute, that is, abstract beginning for the future — and it was this that was to be proved. The fact that its future and this its beginning is already preceded by a past does not affect the argument; because this point of time is a qualitative limit — and that it is to be taken as qualitative is implied in the description of it as completed, elapsed, and therefore as not continuing — therefore in it time is broken off and the past lacks a connection with this time which could only be called future with reference to that past and, consequently, without such connection is only time as such, which has an absolute beginning. But if — as is, then, the case — it were related to the past through the now, the given point of time, and were thus determined as a future, then this point of time, too, regarded from the other side, would not be a limit; the infinite time series would continue itself in what was called future and would not be, as was assumed, completed.
In truth, time is pure quantity; the point of time in which it is supposed to be interrupted, which is employed in the proof, is really only the self-sublating being-for-self of the now. All that the proof does is to represent the absolute limit of time asserted in the thesis as a given point of time, and then straightway to assume it as a completed, that is, abstract point — a popular determination which sensuous conception readily lets pass as a limit, thus allowing as an assumption in the proof what had been put forward as the thing to be proved.
The antithesis runs:
'The world has no beginning and no limits in space but is infinite with reference both to time and space.'
The proof likewise assumes the opposite:
'The world has a beginning. Since the beginning is an existence preceded by a time in which the thing is not, there must have been a preceding time in which the world was not, that is, an empty time. Now no originating of anything is possible in an empty time; because no part of such a time possesses in itself and in preference to any other, any distinguishing condition of existence or non-existence. In the world, therefore, many groups of things can indeed begin, but the world itself can have no beginning and with respect to past time is infinite.'
This apagogical proof, like the others, contains the direct and unproved assertion of what it was supposed to prove. That is, it first assumes a beyond of the existing world, an empty time; but it also equally continues the existence of the world beyond itself into this empty time which is thereby sublated, with the result that the existence of the world is continued into infinity. The world is an existence; the proof presupposes that this existence comes into being and that the coming-to-be has an antecedent condition which is in time. But the antithesis itself consists in the very fact that there is no unconditioned existence, no absolute limit, but that the existence of the world always requires an antecedent condition. Thus, what was to be proved is found as an assumption in the proof. Further, the condition is sought in empty time, which means in effect that it is taken as temporal and therefore as an existence and as limited. Altogether, then, the assumption is made that the world as an existence presupposes another conditioned existence in time, and so on, therefore, to infinity.
The proof regarding the infinity of the world in space is the same. Apagogically, the spatial finiteness of the world is assumed; 'this (the world) would therefore exist in an empty unlimited space and would stand in a relation to it; but such a relation of the world to no object is a nullity'.
Here, too, what was supposed to be proved is directly presupposed in the proof. It is directly assumed that the spatially limited world exists in an empty space and is supposed to stand in a relation to it, that is, there must be a movement out beyond it — on the one hand into the void, into the beyond and non-being of the world, but on the other hand, in order that it be in relation with its beyond, that is, continue itself into it, the beyond must be imagined as filled with the existence of the world. The infinity of the world in space which is asserted in the antithesis is nothing else than, on the one hand, empty space, and on the other the relation of the world to it, that is, the continuity of the world in empty space or the filling of space — which contradiction, namely, space as simultaneously empty and also filled, is the infinite progress of existence in space. This very contradiction, the relation of the world to empty space, is directly made the basis of the proof.
The thesis and antithesis and their proofs therefore represent nothing but the opposite assertions, that a limit is, and that the limit equally is only a sublated one; that the limit has a beyond, with which however it stands in relation, and beyond which it must pass, but that in doing so there arises another such limit, which is no limit.
The solution of these antinomies, as of those previously mentioned, is transcendental, that is, it consists in the assertion of the ideality of space and time as forms of intuition — in the sense that the world is in its own self not self-contradictory, not self-sublating, but that it is only consciousness in its intuition and in the relation of intuition to understanding and reason that is a self-contradictory being. It shows an excessive tenderness for the world to remove contradiction from it and then to transfer the contradiction to spirit, to reason, where it is allowed to remain unresolved. In point of fact it is spirit which is so strong that it can endure contradiction, but it is spirit, too, that knows how to resolve it. But the so-called world (whether it be called an objective, real world or, according to transcendental idealism, a subjective intuition and a sphere of sense determined by the categories of the understanding) is never and nowhere without contradiction, but it is unable to endure it and is, therefore, subject to coming-to-be and ceasing-to-be.
(c) The Infinity of Quantum
1. The infinite quantum as infinitely great or infinitely small is itself implicitly the infinite progress; as great or small it is a quantum and at the same time it is the non-being of quantum. The infinitely great and infinitely small are therefore pictorial conceptions which, when looked at more closely, turn out to be nebulous shadowy nullities. But in the infinite progress, this contradiction is explicitly present and with it that which is the nature of quantum which, as an intensive magnitude, has attained its reality and now in its determinate being is posited as it is in its Notion. It is this identity which we have now to consider.
Quantum as degree is unitary, self-related and determinate within itself. Through this unitary nature, the otherness and determinateness in quantum are sublated, so that the determinatensess is external to it; it has it determinateness outside it. This its self-externality is in the first place the abstract non-being of quantum generally, the spurious infinity. But, further, this non-being is also quantitative and this continues itself into its non-being, for it is in its externality that quantum has its determinateness; this its externality is, therefore, itself equally a quantum., this non-being of quantum, infinity, is thus limited, that is, this beyond is sublated, is itself determined as quantum which, therefore, in its negation is with itself.
But this is what quantum as such is in itself. For it is itself just by being external to itself; externality constitutes that whereby it is quantum and is with itself. In the infinite progress, therefore, the Notion of quantum is posited.
Let us take the progress at first in its abstract determinations as we find them; then in it we have the sublating of quantum, but equally too of its beyond, therefore the negation of quantum as well as the negation of this negation. Its truth is their unity in which they are, but only as moments. It is the resolution of the contradiction of which it is the expression, and its immediate significance is, therefore, the restoration of the Notion of quantity, namely, that quantity is an indifferent or external limit. In the infinite progress as such, the only reflection usually made is that every quantum, however great or small, must be capable of vanishing, of being surpassed; but not that this self-sublating of quantum, the beyond, the spurious infinite itself also vanishes.
Even the first sublation, the negation of quality as such whereby quantum is posited, is in principle [an sich] the sublating of the negation — the quantum is sublated qualitative limit, hence sublated negation — but at the same time it is this only in principle; it is posited as a determinate being, and then its negation is fixed as the infinite, as the beyond of quantum, which remains on this side as an immediate; thus the infinite is determined only as a first negation and it appears as such in the infinite progress. But we have seen that in this something more is present, the negation of the negation, or that which the infinite in truth is. We regarded this previously as the restoration of the Notion of quantity; this restoration means in the first place, that its determinate being has received a more precise determination; we now have quantum determined in conformity with its Notion, which is different from quantum in its immediacy; externality is now the opposite of itself, posited as a moment of quantity itself — quantum is posited as having its determinateness in another quantum by means of its non-being, of infinity; that is, it is qualitatively that which it is. However, this comparison of the Notion of quantum with its determinate being belongs more to our reflection, to a relationship which is not yet present here. The immediately following determination is that the quantum has reverted to quality, is from now on qualitatively determined. For its peculiarity, its quality, is the externality, the indifference of the determinateness; and quantum is now posited as being in fact itself in its externality, as self-related therein, in simple unity with itself, that is, qualitatively determined. This qualitative moment is still more closely determined, namely as being-for-itself; for the self-relation to which it has attained has proceeded from mediation, from the negation of the negation. Quantum has infinity, self-determinedness, no longer outside it but within itself.
The infinite, which in the infinite progress has only the empty meaning of a non-being, of an unattained but sought beyond, is in fact nothing else than quality. Quantum as an indifferent limit goes out beyond itself to infinity; in doing so it seeks nothing else than to be determined for itself, the qualitative moment, which, however, is thus only an ought-to-be. Its indifference to limit, and hence its lack of an explicit determinateness of its own and its passage away from and beyond itself, is that which makes quantum what it is; this its passage into the beyond is to be negated and quantum is to find in the infinite its absolute determinateness.
Quite generally: quantum is sublated quality; but quantum is infinite, goes beyond itself, is the negation of itself. Thus its passage beyond itself is, therefore, in itself the negation of the negated quality, the restoration of it; and thus quantum is explicitly determined as possessing as its own moment, the externality which formerly appeared as a beyond.
Quantum is thus posited as repelled from itself, with the result that there are two quanta which, however, are sublated, are only as moments of one unity, and this unity is the determinateness of quantum. Quantum as thus self-related as an indifferent limit in its externality and therefore posited as qualitative, is quantitative ratio. In the ratio, quantum is external to itself, is distinguished from itself; this its externality is the relation of one quantum to another, each of which has meaning only in this its relation to its other; and this relation constitutes the determinateness of the quantum, which is as such a unity. It has in this unity not an indifferent, but a qualitative, determination; in this its externality it has returned into itself, and in it quantum is that which it is.
Remark 1: The Specific Nature of the Notion of the Mathematical Infinite
The mathematical infinite has a twofold interest. On the one hand its introduction into mathematics has led to an expansion of the science and to important results; but on the other hand it is remarkable that mathematics has not yet succeeded in justifying its use of this infinite by the Notion (Notion taken in its proper meaning). Ultimately, the justifications are based on the correctness of the results obtained with the aid of the said infinite, which correctness is proved on quite other grounds: but the justifications are not based on the clarity of the subject matter and on the operation through which the results are obtained, for it is even admitted that the operation itself is incorrect.
This alone is in itself a bad state of affairs; such a procedure is unscientific. But it also involves the drawback that mathematics, being unaware of the nature of this its instrument because it has not mastered the metaphysics and critique of the infinite, is unable to determine the scope of its application and to secure itself against the misuse of it.
But in a philosophical respect the mathematical infinite is important because underlying it, in fact, is the notion of the genuine infinite and it is far superior to the ordinary so-called metaphysical infinite on which are based the objections to the mathematical infinite. Often, the science of mathematics can only defend itself against these objections by denying the competence of metaphysics, asserting that it has nothing to do with that science and does not have to trouble itself about metaphysical concepts so long as it operates consistently within its own sphere. Mathematics has to consider not what is true in itself but what is true in its own domain. Metaphysics, though disagreeing with the use of the mathematical infinite, cannot deny or invalidate the brilliant results obtained from it, and mathematics cannot reach clearness about the metaphysics of its own concept or, therefore, about the derivation of the modes of procedure necessitated by the use of the infinite.
If it were solely the difficulty of the Notion as such which troubled mathematics, it could ignore it without more ado since the Notion is more than merely the statement of the essential determinatenesses of a thing, that is, of the determinations of the understanding: and mathematics has seen to it that these determinatenesses are not lacking in precision; for it is not a science which has to concern itself with the concepts of its objects and which has to generate their content by explicating the concept, even if this could be effected only by ratiocination. But mathematics, in the method of its infinite, finds a radical contradiction to that very method which is peculiar to itself and on which as a science it rests. For the infinitesimal calculus permits and requires modes of procedure which mathematics must wholly reject when operating with finite quantities, and at the same time it treats these infinite quantities as if they were finite and insists on applying to the former the same modes of operation which are valid for the latter; it is a cardinal feature in the development of this science that it has succeeded in applying to transcendental determinations and their treatment the form of ordinary calculation.
Mathematics shows that, in spite of the clash between its modes of procedure, results obtained by the use of the infinite completely agree with those found by the strictly mathematical, namely, geometrical and analytical method. But in the first place, this does not apply to every result and the introduction of the infinite is not for the sole purpose of shortening the ordinary method but in order to obtain results which this method is unable to secure. Secondly, success does not by itself justify the mode of procedure. This procedure of the infinitesimal calculus shows itself burdened with a seeming inexactitude, namely, having increased finite magnitudes by an infinitely small quantity, this quantity is in the subsequent operation in part retained and in part ignored. The peculiarity of this procedure is that in spite of the admitted inexactitude, a result is obtained which is not merely fairly close and such that the difference can be ignored, but is perfectly exact. In the operation itself, however, which precedes the result, one cannot dispense with the conception that a quantity is not equal to nothing, yet is so inconsiderable that it can be left out of account. However, what is to be understood by mathematical determinateness altogether rules out any distinction of a greater or lesser degree of exactitude, just as in philosophy there can be no question of greater or less probability but solely of Truth. Even if the method and use of the infinite is justified by the result, it is nevertheless not so superfluous to demand its justification as it seems in the case of the nose to ask for a proof of the right to use it. For mathematical knowledge is scientific knowledge, so that the proof is essential; and even with respect to results it is a fact that a rigorous mathematical method does not stamp all of them with the mark of success, which in any case is only external.
It is worth while considering more closely the mathematical concept of the infinite together with the most noteworthy of the attempts aimed at justifying its use and eliminating the difficulty with which the method feels itself burdened. The consideration of these justifications and characteristics of the mathematical infinite which I shall undertake at some length in this Remark will at the same time throw the best light on the nature of the true Notion itself and show how this latter was vaguely present as a basis for those procedures.
The usual definition of the mathematical infinite is that it is a magnitude than which there is no greater (when it is defined as the infinitely great) or no smaller (when it is defined as the infinitely small), or in the former case is greater than, in the latter case smaller than, any given magnitude. It is true that in this definition the true Notion is not expressed but only, as already remarked, the same contradiction which is present in the infinite progress; but let us see what is implicitly contained in it. In mathematics a magnitude is defined as that which can be increased or diminished; in general, as an indifferent limit. Now since the infinitely great or small is that which cannot be increased or diminished, it is in fact no longer a quantum as such.
This is a necessary and direct consequence. But it is just the reflection that quantum (and in this remark quantum as such, as we find it, I call finite quantum) is sublated, which is usually not made, and which creates the difficulty for ordinary thinking; for quantum in so far as it is infinite is required to be thought as sublated, as something which is not a quantum but yet retains its quantitative character.
To quote Kant's opinion of the said definition which he finds does not accord with what is understood by an infinite whole: 'According to the usual concept, a magnitude is infinite beyond which there can be no greater (i.e. greater than the amount contained therein of a given unit); but there can be no greatest amount because one or more units can always be added to it. But our concept of an infinite whole does not represent how great it is and it is not therefore the concept of a maximum (or a minimum); this concept rather expresses only the relation of the whole to an arbitrarily assumed unit, with respect to which the relation is greater than any number. According as this assumed unit is greater or smaller, the infinite would be greater or smaller. The infinity, however, since it consists solely in the relation to this given unit, would always remain the same, although of course the absolute magnitude of the whole would not be known through it.'
Kant objects to infinite wholes being regarded as a maximum, as a completed amount of a given unit. The maximum or minimum as such still appears as a quantum, an amount. Such a conception cannot avert the conclusion, adduced by Kant, which leads to a greater or lesser infinite. And in general, so long as the infinite is represented as a quantum, the distinction of greater or less still applies to it. This criticism does not however apply to the Notion of the genuine mathematical infinite, of the infinite difference, for this is no longer a finite quantum.
Kant's concept of infinite, on the other hand, which he calls truly transcendental is 'that the successive synthesis of the unit in the measurement of a quantum can never be completed'. A quantum as such is presupposed as given; by synthesising the unit this is supposed to be converted into an amount, into a definite assignable quantum; but this synthesis, it is said, can never be completed. It is evident from this that we have here nothing but an expression of the progress to infinity, only represented transcendentally, i.e. properly speaking, subjectively and psychologically. True, in itself the quantum is supposed to be completed; but transcendentally, namely in the subject which gives it a relation to a unit, the quantum comes to be determined only as incomplete and as simply burdened with a beyond. Here, therefore, there is no advance beyond the contradiction contained in quantity; but the contradiction is distributed between the object and the subject, limitedness being ascribed to the former, and to the latter the progress to infinity, in its spurious sense, beyond every assigned determinateness.
On the other hand, it was said above that the character of the mathematical infinite and the way it is used in higher analysis corresponds to the Notion of the genuine infinite; the comparison of the two determinations will now be developed in detail. In the first place, as regards the true infinite quantum, it was characterised as in its own self infinite; it is such because, as we have seen, the finite quantum or quantum as such and its beyond, the spurious infinite, are equally sublated. Thus the sublated quantum has returned into a simple unity and self-relation; but not merely like the extensive quantum which, in passing into intensive quantum, has its determinateness only in itself [or implicitly] in an external plurality, towards which, however, it is indifferent and from which it is supposed to be distinct.
The infinite quantum, on the contrary, contains within itself first externality and secondly the negation of it; it is thus no longer any finite quantum, not a quantitative determinateness which would have a determinate being as quantum; it is simple, and therefore only a moment. It is a quantitative determinateness in qualitative form; its infinity consists in its being a qualitative determinateness. As such moment, it is in essential unity with its other, and is only as determined by this its other, i.e. it has meaning solely with reference to that which stands in relation to it. Apart from this relation it is a nullity — simply because quantum as such is indifferent to the relation, yet in the relation is supposed to be an immediate, inert determination. As only a moment, it is, in the relation, not an independent, indifferent something; the quantum in its infinity is a being-for-self, for it is at the same time a quantitative determinateness only in the form of a being-for-one.
The Notion of the infinite as abstractly expounded here will show itself to be the basis of the mathematical infinite and the Notion itself will become clearer if we consider the various stages in the expression of a quantum as moment of a ratio, from the lowest where it is still also a quantum as such, to the higher where it acquires the meaning and the expression of a properly infinite magnitude.
Let us then first take quantum in the relation where it is a fractional number. Such fraction, 2/7 for example, is not a quantum like 1, 2, 3, etc.; although it is an ordinary finite number it is not an immediate one like the whole numbers but, as a fraction, is directly determined by two other numbers which are related to each other as amount and unit, the unit itself being a specific amount. However, if we abstract from this more precise determination of them and consider them solely as quanta in the qualitative relation in which they are here, then 2 and 7 are indifferent quanta; but since they appear here only as moments, the one of the other, and consequently of a third (of the quantum which is called the exponent), they directly count no longer simply as 2 and 7 but only according to the specific relationship in which they stand to each other. In their place, therefore, we can just as well put 4 and 14, or 6 and 21, and so on to infinity. With this, then, they begin to have a qualitative character. If 2 and 7 counted as mere quanta, then 2 is just 2 and nothing more, and 7 is simply 7; 4, 14, 6, 21 etc., are completely different from them and, as only immediate quanta, cannot be substituted for them. But in so far as 2 and 7 are not to be taken as such immediate quanta their indifferent limit is sublated; on this side therefore they contain the moment of infinity, since not only are they no longer merely 2 and 7, but their quantitative determinateness remains — but as one which is in itself qualitative, namely in accordance with their significance as moments in the ratio. Their place can be taken by infinitely many others without the value of the fraction being altered, by virtue of the determinateness possessed by the ratio.
However, the representation of infinity by a fractional number is still imperfect because the two sides of the fraction, 2 and 7, can be taken out of the relation and are ordinary, indifferent quanta; their connection as moments of the ratio is an external circumstance which does not directly concern them. Their relation, too, is itself an ordinary quantum, the exponent of the ratio.
The letters with which general arithmetic operates, the next universality into which numbers are raised, do not possess the property of having a specific numerical value; they are only general symbols and indeterminate possibilities of any specific value. The fraction a/b seems, therefore, to be a more suitable expression of the infinite, since a and b, taken out of their relation to each other, remain undetermined, and taken separately, too, have no special peculiar value. However, although these letters are posited as indeterminate magnitudes their meaning is to be some finite quantum. Therefore, though they are the general representation of number, it is only of a determinate number, and the fact that they are in a ratio is likewise an inessential circumstance and they retain their value outside it.
If we consider more closely what is present in the ratio we find that it contains the following two determinations: first it is a quantum, secondly, however, this quantum is not immediate but contains qualitative opposition; at the same time it remains therein a determinate, indifferent quantum by virtue of the fact that it returns into itself from its otherness, from the opposition, and so also is infinite. These two determinations are represented in the following familiar form developed in their difference from each other.
The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series; the fraction itself is called the sum, or finite expression of it. A comparison of the two expressions shows that one of them, the infinite series, represents the fraction no longer as a ratio but from that side where it is a quantum as an aggregate of units added together, as an amount. That the magnitudes of which it is supposed to consist as amount are in turn decimal fractions and therefore are themselves ratios, is irrelevant here; for this circumstance concerns the particular kind of unit of these magnitudes, not the magnitudes as constituting an amount. just as a multi-figured integer in the decimal system is reckoned essentially as an amount, and the fact that it consists of products of a number and of the number ten and powers of ten is ignored. Similarly here, it is irrelevant that there are fractions other than the example taken of 2/7 which, when expressed as decimal fractions, do not give an infinite series; but they can all be so expressed in a numerical system based on another unit.
Now in the infinite series, which is supposed to represent the fraction as an amount, the aspect of the fraction as a ratio has vanished and with it there has vanished too the aspect which, as we have already shown, makes the fraction in its own self infinite. But this infinity has entered in another way; the series, namely, is itself infinite.
Now the nature of this infinity of the series is self-evident; it is the spurious infinity of the progression. The series contains and exhibits the contradiction of representing that which is a relation possessing a qualitative nature, as devoid of relation, as a mere quantum, as an amount. The consequence of this is that the amount which is expressed in the series always lacks something, so that in order to reach the required determinateness, we must always go further than the terms already posited. The law of the progression is known, it is implicit in the determination of the quantum contained in the fraction and in the nature of the form in which it is supposed to be expressed. By continuing the series the amount can of course be made as accurate as required; but representation by means of the series continues to remain only an ought-to-be; it is burdened with a beyond which cannot be sublated, because to express as an amount that which rests on a qualitative determinateness is a lasting contradiction.
In this infinite series, this inexactitude is actually present, whereas in the genuine mathematical infinite there is only an appearance of inexactitude. These two kinds of mathematical infinite are as little to be confounded as are the two kinds of philosophical infinite. In representing the genuine mathematical infinite, the form of series was used originally and it has recently again been invoked; but this form is not necessary for it. On the contrary, the infinite of the infinite series is essentially different from the genuine infinite as the sequel will show. Indeed the form of infinite series is even inferior to the fractional expression.
For the infinite series contains the spurious infinity, because what the series is meant to express remains an ought-to-be and what it does express is burdened with a beyond which does not vanish and differs from what was meant to be expressed. It is infinite not because of the terms actually expressed but because they are incomplete, because the other which essentially belongs to them is beyond them; what is really present in the series, no matter how many terms there may be, is only something finite, in the proper meaning of that word, posited as finite, i.e., as something which is not what it ought to be. But on the other hand, what is called the finite expression or the sum of such a series lacks nothing; it contains that complete value which the series only seeks; the beyond is recalled from its flight; what it is and what it ought to be are not separate but the same.
What distinguishes these two is more precisely this, that in the infinite series the negative is outside its terms which are present only qua parts of the amount. On the other hand, in the finite expression which is a ratio, the negative is immanent as the reciprocal determining of the sides of the ratio and this is an accomplished return-into-self, a self-related unity as a negation of the negation (both sides of the ratio are only moments), and consequently has within it the determination of infinity. Thus the usually so-called sum, the 2/7 or 1/(1 - a) is in fact a ratio; and this so-called finite expression is the truly infinite expression. The infinite series, on the other hand, is in truth a sum; its purpose is to represent in the form of a sum what is in itself a ratio, and the existing terms of the series are not terms of a ratio but of an aggregate. Furthermore, the series is in fact the finite expression; for it is the incomplete aggregate and remains essentially deficient. According to what is really present in it, it is a specific quantum, but at the same time it is less than what it ought to be; and then, too, what it lacks is itself a specific quantum; this missing part is in fact that which is called infinite in the series, from the merely formal point of view that it is something lacking, a non-being; with respect to its content it is a finite quantum. Only what is actually present in the series, plus what is lacking, together constitute the amount of the fraction, the specific quantum which the series also ought to be but is not capable of being. The word infinite, even as used in infinite series, is commonly fancied to be something lofty and exalted; this is a kind of superstition, the superstition of the understanding; we have seen how, on the contrary, it indicates only a deficiency.
We may further remark that the existence of infinite series which cannot be summed is an external and contingent circumstance with respect to the form of series as such. They contain a higher kind of infinity than do those which can be summed, namely an incommensurability, or the impossibility of representing the quantitative ratio contained in them as a quantum, even in the form of a fraction; but the form of series as such which they have contains the same determination of spurious infinity that is present in the series capable of summation.
The terminological inversion we have just noticed in connection with the fraction and its expression as a series, also occurs when the mathematical infinite — not the one just named but the genuine infinite — is called the relative infinite, while the ordinary metaphysical — by which is understood the abstract, spurious infinite is called absolute. But in point of fact it is this metaphysical infinite which is merely relative, because the negation which it expresses is opposed to a limit only in such a manner that this limit persists outside it and is not sublated by it; the mathematical infinite, on the contrary, has within itself truly sublated the finite limit because the beyond of the latter is united with it.
It is primarily in this sense, in which it has been demonstrated that the so-called sum or finite expression of an infinite series is rather to be regarded as the infinite expression, that Spinoza opposes the concept of true infinity to that of the spurious and illustrates it by examples. It will shed most light on his concept if I follow up this exposition with what he says on the subject.
He starts by defining the infinite as the absolute affirmation of any kind of natural existence, the finite on the contrary as a determinateness, as a negation. That is to say, the absolute affirmation of an existence is to be taken as its relation to itself, its not being dependent on an other; the finite, on the other hand, is negation, a ceasing-to-be in the form of a relation to an other which begins outside it. Now the absolute affirmation of an existence does not, it is true, exhaust the notion of infinity; this implies that infinity is an affirmation, not as immediate, but only as restored by the reflection of the other into itself, or as negation of the negative. But with Spinoza, substance and its absolute unity has the form of an inert unity, i.e. of a unity which is not self-mediated, of a fixity or rigidity in which the Notion of the negative unity of the self, i.e. subjectivity, is still lacking.
The mathematical example with which he illustrates the true infinite is a space between two unequal circles which are not concentric, one of which lies inside the other without touching it. It seems that he thought highly of this figure and of the concept which it was used to illustrate, making it the motto of his Ethics. 'Mathematicians conclude', he says, 'that the inequalities possible in such a space are infinite, not from the infinite amount of parts, for its size is fixed and limited and 1 can assume larger and smaller such spaces, but because the nature of the fact surpasses every determinateness.' It is evident that Spinoza rejects that conception of the infinite which represents it as an amount or as a series which is not completed, and he points out that here, in the space of his example, the infinite is not beyond, but actually present and complete; this space is bounded, but it is infinite 'because the nature of the fact surpasses every determinateness', because the determination of magnitude contained in it cannot at the same time be represented as a quantum, or in Kant's words already quoted, the synthesis cannot be completed to form a (discrete) quantum. How in general the opposition of continuous and discrete quantum leads to the infinite, will be shown in detail in a later Remark. Spinoza calls the infinite of a series the infinite of the imagination; on the other hand, the infinite as self-relation he calls the infinite of thought, or infinitum actu. It is, namely, actu, actually infinite because it is complete and present within itself. Thus the series 0.285714 ... or 1 + a + a2 + a3 ... is the infinite merely of imagination or supposition; for it has no actuality, it definitely lacks something; on the other hand 2/7 or 1/(1 - a) is actually not only what the series is in its developed terms, but is, in addition, what the series lacks, what it only ought to be. The 2/7 or 1/(1 - a) is equally a finite magnitude like Spinoza's space enclosed between two circles, with its inequalities, and can like this space be made larger or smaller. But this does not involve the absurdity of a larger or smaller infinite; for this quantum of the whole does not concern the relation of its moments, the nature of the fact, i.e. the qualitative determination of magnitude; what is actually present in the infinite series is equally a finite quantum, but it is also still deficient. Imagination on the contrary stops short at quantum as such and does not reflect on the qualitative relation which constitutes the ground of the existing incommensurability.
The incommensurability which lies in Spinoza's example embraces in general the functions of curved lines and more precisely, leads to the infinite which mathematics has introduced with such functions, in general, with the functions of variable magnitudes. This infinite is the genuine mathematical qualitative infinite which Spinoza also had in mind. We shall now consider this determination here in detail.
First of all, as regards the category of variability which is accorded such importance and which embraces the magnitudes related in these functions, it is to be noted that these magnitudes are not supposed to be variable in the way that the two numbers 2 and 7 are in the fraction 2/7: their place can equally well be taken by 4 and 14, 6 and 21, and by other numbers ad infinitum without altering the value of the fraction; and still more in a/b, can a and b be replaced by any arbitrary number without altering what a/b is intended to express. Now in the sense that in the place, too, of x and y of a function, there can be put an infinite, i.e. inexhaustible, multitude of numbers, a and b are just as much variable magnitudes as the said x and y. The expression 'variable magnitudes' is therefore very vague and ill-chosen for those determinations of magnitude whose interest and manner of treatment lie in something quite distinct from their mere variability.
In order to make clear wherein lies the true character of those moments of a function with which higher analysis is concerned, we must once more run through the stages to which we have already drawn attention. In 2/7 or a/b, 2 and 7 are each independent determinate quanta and the relation is not essential to them; a and b likewise are intended to represent quanta which remain what they are even outside the relation. And further, 2/7 and a/b are each a fixed quantum, a quotient; the ratio constitutes an amount, the unit of which is expressed by the denominator and the amount of these units by the numerator, or conversely; even if 4 and 14, and so on, are substituted for 2 and 7, the ratio, also as a quantum, remains the same. But now in the function y2/x = p, for example, this is essentially changed; here, it is true that x and y can stand for definite quanta, but only x and y2 have a determinate quotient, not x and y. Hence not only are these sides of the ratio x and y, not any determinate quanta, but, secondly, their ratio is not a fixed quantum (nor is such a quantum meant as in the case of a and b), not a fixed quotient, but this quotient is, as a quantum, absolutely variable. But this is solely because x has a relation, not to y, but to the square of y. The relation of a magnitude to a power is not a quantum, but essentially a qualitative relation; the power-relation is the feature which is to be regarded as the fundamental determination. But in the function of the straight line y = ax, a is an ordinary fraction and quotient; consequently this function is only formally a function of variable magnitudes, or x and y here are what a and b are in a/b that is, they are not in that determination in which the differential and integral calculus considers them. On account of the special nature of the variable magnitudes in this mode of consideration, it would have been fitting to have introduced both a special name for them and other symbols than those generally used for unknown quantities in any finite equation, determinate or indeterminate; for there is an essential difference between those magnitudes and such quanta which are merely unknown, but are in themselves completely determined or are a definite range of determinate quanta. It is, too, only because of a lack of awareness of what constitutes the peculiar interest of higher analysis and of what has led to the need for and invention of the differential calculus, that functions of the first degree and the equation of the straight line are themselves included in the treatment of this calculus; such formalism originates partly, too, in the mistake of imagining that the intrinsically correct demand for the generalisation of a method has been fulfilled when the specific determinateness on which the need for the calculus is based is omitted, as if in this domain we were concerned only with variable magnitudes. A great deal of formalism would, indeed, have been avoided if it had been perceived that the calculus is concerned not with variable magnitudes as such but with relations of powers.
But there is still another stage where the peculiar character of the mathematical infinite becomes prominent. In an equation in which x and y are determined primarily by a power-relation, x and y as such are still supposed to signify quanta; now this significance is altogether and completely lost in the so-called infinitesimal differences. Dx, dy, are no longer quanta, nor are they supposed to signify quanta; it is solely in their relation to each other that they have any meaning, a meaning merely as moments. They are no longer something (something taken as a quantum), not finite differences; but neither are they nothing; not empty nullities. Apart from their relation they are pure nullities, but they are intended to be taken only as moments of the relation, as determinations of the differential coefficient dx/dy.
In this concept of the infinite, the quantum is genuinely completed into a qualitative reality; it is posited as actually infinite; it is sublated not merely as this or that quantum but as quantum generally. But the quantitative determinateness remains as element of the principle of the quanta, or, as has also been said, the quanta remain in their first concept.
It is this concept which has been the target for all the attacks made on the fundamental determination of the mathematics of this infinite, i.e. of the differential and integral calculus. Failure to recognise it was the result of incorrect ideas on the part of mathematicians themselves; but it is the inability to justify the object as Notion which is mainly responsible for these attacks. But mathematics, as we remarked above, cannot evade the Notion here; for, as mathematics of the infinite, it does not confine itself to the finite determinateness of its objects (as in ordinary mathematics, which considers and relates space and number and their determinations only according to their finitude); on the contrary, when it treats a determination taken from ordinary mathematics, it converts it into an identity with its opposite, e.g. converting a curved line into a straight one, the circle into a polygon, etc. Consequently, the operations which it allows itself to perform in the differential and integral calculus are in complete contradiction with the nature of merely finite determinations and their relations and would therefore have to be justified solely by the Notion.
Although the mathematics of the infinite maintained that these quantitative determinations are vanishing magnitudes, i.e. magnitudes which are no longer any particular quantum and yet are not nothing but are still a determinateness relatively to an other, it seemed perfectly clear that such an intermediate state, as it was called, between being and nothing does not exist. What we are to think of this objection and the so-called intermediate state, has already been indicated above in Remark 4 to the category of becoming. The unity of being and nothing is, of course, not a state; a state would be a determination of being and nothing into which these moments might be supposed to have lapsed only by accident, as it were, into a diseased condition externally induced through erroneous thinking; on the contrary, this mean and unity, the vanishing or equally the becoming is alone their truth.
Further, it has been said that what is infinite is not comparable as something greater or smaller; therefore there cannot be a relation between infinites according to orders or dignities of the infinite, although in the science of infinitesimals these distinctions do occur. Underlying this objection already mentioned is always the idea that here we are supposed to be dealing with quanta which are compared as quanta, that determinations which are no longer quanta no longer have any relationship to each other. But the truth is rather that that which has being solely in the ratio is not a quantum; the nature of quantum is such that it is supposed to have a completely indifferent existence apart from its ratio, and its difference from another quantum is supposed not to concern its own determination; on the other hand the qualitative is what it is only in its distinction from an other. The said infinite magnitudes, therefore, are not merely comparable, but they exist only as moments of comparison, i.e. of the ratio.
I will adduce the most important definitions of this infinite which have been given in mathematics. From these it will be clear that the thought underlying them accords with the Notion developed here, but that the originators of the definitions did not establish the thought as Notion and found it necessary in the application to resort again to expedients which conflict with their better cause.
The thought cannot be more correctly determined than in the way Newton has stated it. I eliminate here those determinations which belong to the idea of motion and velocity (from which, mainly, he took the name of fluxions) because in them the thought does not appear in its proper abstraction but as concrete and mixed with non-essential forms. Newton explains that he understands by these fluxions not indivisibles (a form which was used by earlier mathematicians, Cavalieri and others and which involves the concept of an intrinsically determinate quantum), but vanishing divisibles; also not sums and ratios of determinate parts but the limits (limites) of sums and ratios. It may be objected that vanishing magnitudes do not have a final ratio, because the ratio before it vanishes is not final, and when it has vanished is no longer a ratio. But by the ratio of vanishing magnitudes is to be understood not the ratio before which and after which they vanish, but with which they vanish. (quacum evanescunt). Similarly, the first ratio of nascent magnitudes is that with which they become.
Newton did what the scientific method of his time demanded, he only explained what was to be understood by an expression; but that such and such is to be understood by it is, properly speaking, a subjective presumption, or a historical demand, without any indication that such a concept is in itself absolutely necessary or that there is truth in it. However, what has been quoted shows that the concept put forward by Newton corresponds to the way in which infinite quantity resulted from the reflection of quantum into itself in the exposition above. By magnitudes is understood magnitudes in their vanishing, i.e. which are no longer quanta; also, not ratios of determinate parts, but the limits of the ratio. The meaning is, therefore, that with the vanishing of the quanta individually, the sides of the ratio, there also vanishes the ratio itself in so far as it is a quantum; the limit of the quantitative ratio is that in which it both is and is not, or, more precisely, in which the quantum has vanished, with the result that the ratio and its sides are preserved, the former only as a qualitative relation of quantity and the latter similarly as qualitative moments of quantity. Newton goes on to add that from the fact that there are final ratios of vanishing magnitudes, it must not be inferred that there are final magnitudes, indivisibles. For this would mean a leap again from the abstract ratio to its sides as supposedly having an independent value of their own as indivisibles outside their relation, as something which would be a one, something devoid of any relation at all.
To prevent such a misunderstanding, he again points out that final ratios are not ratios of final magnitudes, but are limits to which the ratios of the magnitudes decreasing without limit are nearer than any given, i.e. finite, difference; the ratios, however, do not exceed these limits, for if they did they would become nullities. In other words, final magnitudes could have been taken to mean, as already said, indivisibles or ones. But the definition of the final ratio excludes the conception both of the indifferent one which is devoid of any relation, and of the finite quantum. If the required determination had been developed into the Notion of a quantitative determination which is purely a moment of the ratio, there would have been no need for the decreasing without limit into which Newton converts the quantum and which only expresses the progress to infinity, or for the determination of divisibility which no longer has any immediate meaning here.
As regards the preservation of the ratio in the vanishing of the quanta, there is found elsewhere, as in Carnot, the expression that by virtue of the law of continuity, the vanishing magnitudes still retain the ratio from which they come, before they vanish.
This conception expresses the true nature of the matter, if the continuity of the quantum is not understood to be the continuity which it has in the infinite progress where the quantum is continued in its vanishing in such a manner that in its beyond there arises only a finite quantum again, only a fresh term of the series; but a continuous progress is always imagined as one in which values are passed through, values which are still finite quanta. On the other hand, where the transition is made into the true infinite it is the ratio that is continuous; so continuous is it, so completely is it preserved, that the transition may be said to consist solely in throwing into relief the pure ratio and causing the non-relational determination — i.e. that a quantum which is a side of the ratio is still a quantum outside this relation — to vanish. This purification of the quantitative ratio is thus analogous to grasping an empirical reality in terms of its Notion. The empirical reality is thereby raised above itself in such a way that its Notion contains the same characteristic features as it has itself, but these are grasped in their essentiality and are taken into the unity of the Notion in which they have lost their indifferent, Notion — less existence.
The other form of Newton's exposition of the magnitudes in question is equally interesting, namely, as generative magnitudes or principles. A generated magnitude (genita) is a product or quotient, such as a root, rectangle, square, also the sides of rectangles and squares — in general, a finite magnitude. 'Such a magnitude being considered as variable, increasing or decreasing in ceaseless motion and flux, he gives its momentary increments or decrements the name of moments. But these are not to be taken for particles of a definite magnitude (particulae finitae): such would not themselves be moments but magnitudes generated from moments. Rather are they to be understood as the nascent principles or beginnings of finite magnitudes.' Here the quantum is distinguished from itself: as a product or a real being [Daseiendes], and in its becoming (or as nascent), in its beginning and principle, that is to say, in its Notion, or, what is here the same thing, in its qualitative determination: in the latter the quantitative differences, the infinite increments or decrements, are only moments; only that which has becoming at its back has passed over into the indifference of determinate being and into externality, i.e. is quantum. But if on the one hand the philosophy of the true Notion must acknowledge these determinations of the infinite with respect to increments or decrements, on the other hand it must be observed that the very forms of increments etc. fall within the category of immediate quantum and of the continuous progress to which we have referred; in fact the conceptions of increment, growth or increase of x by dx or i, and so on, are to be regarded as the fundamental vice in these methods — the permanent obstacle to disengaging the determination of the qualitative moment of quantity in its purity from the conception of the ordinary quantum.
The conception of infinitesimals which is implicit, too, in the increment or decrement itself, is much inferior to the above determinations. The nature of these magnitudes is supposed to be such that they may be neglected, not only in comparison with finite magnitudes, but also their higher orders in comparison with their lower, and even the products of several in comparison with a single one. With Leibniz, this demand to neglect is more strikingly prominent than with previous inventors of methods relating to these infinitesimals in which this call to neglect also occurs. It is chiefly this call to neglect which, along with a gain in facility, has given this calculus the appearance of inexactitude and express incorrectness in its method of procedure. Wolf has tried to make this neglect intelligible in his own way of popularising things, i.e. by polluting the pure Notion and setting in its place incorrect sensuous conceptions. For example, he compares the neglect of infinitesimals of higher orders relatively to lower with the procedure of a surveyor who, in measuring the height of a mountain is no less accurate if meanwhile the wind has blown away a grain of sand from the top; or with the neglect of the height of houses or towers when calculating lunar eclipses.
Even if ordinary common sense in fairness allows such inexactitude, all geometricians reject this conception. It is quite obvious that in the science of mathematics there cannot be any question of such empirical accuracy; mathematical measuring by operations of the calculus or by geometrical constructions and proofs is altogether different from land-surveying, from the measuring of empirical lines, figures etc. Besides, by comparing the result obtained by a strictly geometrical method with that obtained by the method of infinite differences, analysts demonstrate that the one is the same as the other and that there is absolutely no question of a greater or lesser degree of exactness. And it is self-evident that an absolutely exact result could not emerge from an inexact method. Yet on the other hand again, the method itself cannot do without this omission of what is regarded as insignificant, despite its protestations against the way this omission is justified. And this is the difficulty which engages the efforts of the analysts to make intelligible and to remove the inherent inconsistency.
It is especially Euler's conception of the matter which must be cited here. He adopts the general Newtonian definition and insists that the differential calculus considers the ratios of the increments of a magnitude, but that the infinite difference as such is to be considered as wholly nil. How this is to be understood is clear from the foregoing; the infinite difference is a nil only of quantum, not a qualitative nil, but as a nil of quantum it is a pure moment of the ratio only. It is not a quantitative difference; but for that reason it is, on the one hand, altogether wrong to speak of those moments which are called infinitesimals, also as increments or decrements and as differences. This description implies that something is added to or subtracted from the initially given finite magnitude, that a subtraction or addition, an arithmetical, external operation takes place. But it is to be noticed that the transition of the function of the variable magnitude into its differential is of a quite different nature; as we have made clear, it is to be considered as a reduction of the finite function to the qualitative relation of its quantitative determinations. On the other hand, the error becomes obvious when it is said that the increments by themselves are zeros, that only their ratios are considered; for a zero no longer has any determinateness at all. This conception then, does get as far as the negative of the quantum and gives definite expression to it, but at the same time it does not grasp this negative in its positive significance of qualitative determinations of quantity which, if they were torn out of the ratio and regarded as quanta, would be only zeros.
The opinion of Lagrange on the idea of limits or final ratios is that although one can well imagine the ratio of two magnitudes so long as they remain finite, this ratio does not present any clear and definite concept to the intellect as soon as its terms become simultaneously zero. And the understanding must, indeed, transcend this merely negative side on which the terms of the ratio are quantitatively zero, and must grasp them positively, as qualitative moments. But we cannot regard, as satisfactory Euler's further remarks with regard to this conception of his in which he tries to show that two so-called infinitesimals which are supposed to be nothing else but zeros, nevertheless stand in a relation to each other, for which reason they are denoted by symbols other than zero. He tries to base this on the difference between the arithmetical and geometrical ratio: in the former, we have an eye to the difference, in the latter, to the quotient, so that although in the former there is no difference between two zeros, this is not so in the geometrical ratio; if 2: 1 = 0 : 0 then from the nature of proportion it follows that, because the first term is twice as great as the second, the third is also twice as great as the fourth; thus according to proportion, 0 : 0 is to be taken as the ratio of 2 : 1. Even in common arithmetic n. 0 = 0 and therefore n : 1 = 0 : 0. But it is just because 2 : 1 or n : 1 is a relation of quanta that there cannot be any corresponding ratio or expression of 0 : 0.
I refrain from citing any further instances since those already considered show clearly enough that the genuine Notion of the infinite is, in fact, implied in them, but that the specific nature of that Notion has not been brought to notice and grasped. Consequently, in the actual application of the method of infinitesimals, the genuine Notion of the infinite cannot exercise any influence; on the contrary, there is a return of the finite determinateness of quantity and the operation cannot dispense with the conception of a quantum which is merely relatively small. The calculus makes it necessary to subject the so-called infinitesimals to ordinary arithmetical operations of addition and so on, which are based on the nature of finite magnitudes, and therefore to regard them momentarily as finite magnitudes and to treat them as such. It is for the calculus to justify its procedure in which it first brings them down into this sphere and treats them as increments or differences, and then neglects them as quanta after it had just applied forms and laws of finite magnitudes to them.
I will proceed to cite the main features of the attempts of the geometricians to remove these difficulties.
The older analysts had little scruples in the matter, but the moderns directed their efforts mainly towards bringing the differential calculus back to the evidence of a strictly geometrical method and in it to attain to the rigour of the proofs of the ancients (Lagrange's expressions) in mathematics. But since the principle of infinitesimal analysis is of a higher nature than the principle of the mathematics of finite magnitudes, that kind of evidence had perforce to be dispensed with, just as philosophy, too, cannot lay claim to that obviousness which belongs to the natural sciences, e.g. natural history — and just as eating and drinking are reckoned a more intelligible business than thinking and understanding. Accordingly, we shall deal only with the efforts to attain to the rigour of proof of the ancients.
Some have attempted to dispense altogether with the concept of the infinite, and without it to achieve what seemed to be bound up with its use. Lagrange speaks, e.g., of the method devised by Landen, saying that it is purely analytical and does not employ infinitesimal differences, but starts with different values of variable magnitudes and subsequently equates them. He also gives it as his opinion that in this method, the differential calculus loses its own peculiar advantages, namely simplicity of method and facility of operation. This is, indeed, a procedure which in some measure corresponds to the starting-point of Descartes' tangential method of which detailed mention will be made later. This much, we may remark here, is generally evident, that the general procedure in which different values of variable magnitudes are assumed and subsequently equated, belongs to another department of mathematical treatment than that to which the method of the differential calculus itself belongs; and that the peculiar nature of the simple relation (to be considered in detail further on) to which its actual, concrete determination reduces, namely, of the derived function to the original, is not brought into prominence.
The earlier of the moderns, Fermat, Barrow, and others for example, who at first used infinitesimals in that application which was subsequently developed into the differential and integral calculus, and then Leibniz, too, and those following him including Euler, always frankly believed that they were entitled to omit the products of infinitesimal differences and their higher powers, solely on the ground that they vanish relatively to the lower order. This is for them the sole basis of the fundamental principle, namely the determination of that which is the differential of a product or a power, for the entire theoretical teaching reduces to this. The rest is partly the mechanism of development and partly application, in which however as we shall later on see, the more important, or rather the sole, interest is to be found. With respect to the present topic, we need only mention here what is elementary, that on the same ground of insignificance, the cardinal principle adopted in relation to curves is that the elements of the curves, namely the increments of abscissa and ordinate, have the relation to each other of subtangent and ordinate; for the purpose of obtaining similar triangles, the arc which forms the third side of a triangle to the two increments of the characteristic triangle (as it rightly used to be called), is regarded as a straight line, as part of the tangent and one of the increments therefore as reaching to the tangent. By these assumptions those determinations are, on the one hand, raised above the nature of finite magnitudes, but on the other hand, a method which is valid only for finite magnitudes and which does not permit the omission of anything on the ground of insignificance, is applied to moments now called infinitesimal. With such a mode of procedure, the difficulty which encumbers the method remains in all its starkness.
We must mention here a remarkable procedure of Newton the invention of an ingenious device to remove the arithmetically incorrect omission of the products of infinitesimal differences or higher orders of them in the finding of differentials. He finds the differentials of products — from which the differentials of quotients, powers, etc., can then be easily derived — in the following way. The product of x and y, when each is taken as reduced by half of its infinitesimal difference, becomes xy - xdy/2 - ydx/2 + dxdy/4; but if x and y are made to increase by the same amount, it becomes xy + xdy/2 + ydx/2 + dxdy/4. Now when the first product is subtracted from the second, ydx + xdy remains as a surplus and this is said to be the surplus of the increase by a whole dx and dy, for this increase is the difference between the two products; it is therefore the differential of xy. Clearly, in this procedure, the term which forms the chief difficulty, the product of the two infinitesimal differences, cancels itself out. But in spite of the name of Newton it must be said that such an operation although very elementary, is incorrect; it is not true that (x + dx/2) (y + dy/2) - (x - dx/2) (y - dy/2) = (x + dx) (y + dy) - xy. It can only have been the need to establish the all-important fluxional calculus which could bring a Newton to deceive himself with such a proof.
Other forms which Newton employed in the derivation of differentials are bound up with concrete meanings of the elements and their powers, meanings relating to motion. About the use of the serial form which also characterises his method, it suggests itself to say that it is always possible to obtain the required degree of accuracy by adding more terms and that the omitted terms are relatively insignificant, in general, that the result is only an approximation; though here too he would have been satisfied with this ground for omission as he is in his method of solving equations of higher degree by approximation, where the higher powers arising from the substitution in the given equation of any ascertained, still inexact term, are omitted on the crude ground of their relative smallness.'
The error into which Newton fell in solving a problem by omitting essential, higher powers, an error which gave his opponents the occasion of a triumph of their method over his, and the true origin of which has been indicated by Lagrange in his recent investigation of it demonstrates the formalism and uncertainty which still prevailed in the use of this instrument. Lagrange shows that Newton made the mistake because he omitted the term of the series containing that power on which the specific problem turned. Newton had kept to the formal, superficial principle of omitting terms on account of their relative smallness. For example, it is well known that in mechanics the terms of the series in which the function of a motion is developed are given a specific meaning, so that the first term or the first function refers to the moment of velocity, the second to the accelerating force and the third to the resistance of forces. Here, then, the terms of the series are not to be regarded merely as parts of a sum, but rather as qualitative moments Of a whole determined by the concept. In this way, the omission of the rest of the terms belonging to the spuriously infinite series acquires an altogether different meaning from omission on the ground of their relative smallness.
[Both considerations are found set simply side by side in the application by Lagrange of the theory of functions to mechanics in the chapter on rectilinear motion The space passed through, considered as a function of the time elapsed, gives the equation x = ft; this, developed as f(t + d) gives ft + df't + d2/2.f"t + , etc.
Thus the space traversed in the period of time is represented in the formula as = df't + d2f"t + d3/2.3f"'t +, etc. The motion by means of which this space has been traversed is (it is said) therefore — i.e. because the analytical development gives several, in fact infinitely, many terms — composed of various partial motions, of which the spaces corresponding to the time will be df't, d2/2f"t, d3/2.3f"'dt, etc. The first partial — notion is, in known motion, the formally uniform one with a velocity designated by f't, the second is uniformly accelerated motion derived from an accelerative force proportional to f"t. Now since the remaining terms do not refer to any simple known motion, it is not necessary to take them specially into account and we shall show that they may be abstracted from in determining the motion at the beginning of the point of time.' This is now shown, but of course only by comparing the series all of whose terms belonged to the determination of the magnitude of the space traversed in the period of time, with the equation given in art. 3 for the motion of a falling body, namely x = at + bt2 in which only these two terms occur. But this equation has itself received this form only because the explanation given to the terms produced by the analytical development is presupposed; this presupposition is that the uniformly accelerated motion is composed of a formally uniform motion continued with the velocity attained in the preceding period of time, and of an increment (the a in s = at2, i.e. the empirical coefficient) which is ascribed to the force of gravity — a distinction which has no existence or basis whatever in the nature of the thing itself, but is only the falsely physicalised expression of what issues from the assumed analytical treatment.]
The error in the Newtonian solution arose, not because terms of the series were neglected only as parts of a sum, but because the term containing the qualitative determination, which is the essential point, was ignored.
In this example, the procedure is made to depend on the qualitative meaning. In this connection the general assertion can at once be made that the whole difficulty of the principle would be removed if the qualitative meaning of the principle were stated and the operation were made to depend on it — in place of the formalism which links the determination of the differential only to that which gives the problem its name, to the difference as such between a function and its variation after its variable magnitude has received an increment. In this sense, it is obvious that the differential of xn is completely exhausted by the first term of the series which results from the expansion of (x + dx)n . Thus the omission of the rest of the terms is not on account of their relative smallness; and so there is no assumption of an inexactitude, an error or mistake which could be compensated or rectified by another error — a point of view from which Carnot in particular justifies the ordinary method of the infinitesimal calculus. Since what is involved is not a sum but a relation, the differential is completely given by the first term; and where further terms, the differentials of higher orders, are required, their determination involves not the continuation of a series as a sum, but the repetition of one and the same relation which alone is desired and which is thus already completely given in the first term. The need for the form of a series, its summation and all that is connected with it, must then be wholly separated from the said interest of the relation.
The explanations of the methods of infinitesimal magnitudes given by Carnot, contain a most lucid exposition of what is essential in the ideas referred to above. But in passing to the practical application itself, there enter more or less the usual ideas about the infinite smallness of the omitted terms relatively to the others. He justifies the method, not by the nature of the procedure itself, but by the fact that the results are correct, and by the advantages of a simplification and shortening of the calculus which follow the introduction of imperfect equations, as he calls them, i.e. those in which such an arithmetically incorrect omission has occurred.
Lagrange, as is well known, reverted to Newton's original method, that of series, in order to be relieved of the difficulties inherent in the idea of the infinitely small and in the method of first and final ratios and limits. The advantages of his functional calculus as regards precision, abstraction and generality, are sufficiently recognised; we need mention only what is pertinent here, that it rests on the fundamental principle that the difference, without becoming zero, can be assumed so small that each term of the series is greater than the sum of all the following terms. This method, too, starts from the categories of increment and difference of the function, the variable magnitude of which receives the increment, thereby bringing in the troublesome series of the original function; also in the sequel the terms to be omitted are considered only as constituting a sum, while the reason for omitting them is made to consist in the relativity of their quantum. And so here, too, on the one hand, the principle of the omission is not brought back to the point of view exemplified in some applications, where (as was remarked above) terms of the series are supposed to have a specific quality significance, and terms are neglected not because of their quantitative insignificance but because they are not qualitatively significant; and then, on the other hand, the omission itself has no place in the essential point of view which, as regards the so-called differential coefficient, only becomes specifically prominent with Lagrange, in the so-called application of the calculus, as will be more fully considered in the following remark.
The demonstrated qualitative character as such of the form of magnitude here under discussion in what is called the infinitesimal, is found most directly in the category of limit of the ratio referred to above and the carrying out of which in the calculus has been developed into a characteristic method. Lagrange criticises this method as lacking case in application and he claims that the expression limit does not present any definite idea; this second point we will take up here and examine more closely what is stated about its analytical meaning. Now the idea of limit does indeed imply the true category of the qualitatively determined relation of variable magnitudes above-mentioned; for the forms of it which occur, dx and dy, are supposed to be taken simply and solely as moments of dy/dx, and dy/dx itself must be regarded as a single indivisible symbol.
That the mechanism of the calculus, especially in its application, thus loses the advantage it derived from the separation of the sides of the differential coefficient, this we will pass over here. Now the said limit is to be limit of a given function; it is to assign to this function a certain value determined by its mode of derivation. But with the mere category of limit we should not have advanced beyond the scope of this Remark, which is to demonstrate that the infinitely small which presents itself in the differential calculus as dx and dy, does not have merely the negative, empty meaning of a non-finite, non-given magnitude, as when one speaks of 'an infinite multitude', 'to infinity', and the like, but on the contrary has the specific meaning of the qualitative nature of what is quantitative, of a moment of a ratio as such. This category, however, merely as such, still has no relation to that which is a given function and does not itself enter into the treatment of such a function or into the use to be made of that determination; thus the idea of limit, too, confined to this its demonstrated character, would lead nowhere. But the very expression 'limit' implies that it is a limit of something, i.e. that it expresses a certain value which lies in the function of a variable magnitude; and we must examine the nature of this concrete role. It is supposed to be the limit of the ratio between the two increments by which the two variable magnitudes connected in an equation (one of which is regarded as a function of the other), are supposed to have been increased; the increase is taken here as quite undetermined and so far no use is made of the infinitely small. But the way in which this limit is found involves the same inconsistencies as are contained in the other methods. This way is as follows: if y = fx, then when y becomes y + k, fx is to change into fx + ph + qh2 + rh3 and so on; thus k = ph + qh2, etc., and k/h = p + qh + rh2, etc. Now if k and h vanish, the right-hand side of the equation also vanishes with the exception of p; now p is supposed to be the limit of the ratio of the two increments. It is clear that while h, as a quantum, is equated with 0, k/h nevertheless is not at the same time equal to 0/0 but is supposed still to remain a ratio.
Now the idea of limit is supposed to have the advantage of avoiding the inconsistency here involved; p is, at the same time, supposed to be not the actual ratio, which would be 0/0 but only that specific value to which the ratio can infinitely approximate, i.e. can approach so near that the difference can be smaller than any given difference. The more precise meaning of approximation with respect to the terms which are supposed really to approach each other will be considered later. But that a quantitative difference, the definition of which is that it not only can, but shall be smaller than any given difference, is no longer a quantitative difference, this is self-evident, as self-evident as anything can be in mathematics; but we still have not got away from dy/dx = 0. If on the other hand dy/dx = p, i. e. is assumed to be a definite quantitative ratio as in fact it is, then conversely there is a difficulty about the presupposition which equates h with o, a presupposition which is indispensable for obtaining the equation k/h = p. But if it be granted that k/h = 0, (and when h = 0, k is in fact automatically = 0, for k, the increment of y, depends entirely on the existence of the increment h), then the question would arise, what p — which is a quite definite quantitative value — is supposed to be. To this there is at once an obvious answer, the simple, meagre answer that it is a coefficient derived in such and such a way — the first function, derived in a certain specific manner, of an original function. if we content ourselves with this — and Lagrange did, in fact, do so in practice — then the general part of the science of the differential calculus, and straightway this one particular form of it called the theory of limits would be rid of the increments and of their infinite or arbitrary smallness — spared too, the difficulty of getting rid again of all the terms of a series other than the first, or rather only the coefficient of the first, which inevitably follow on the introduction of these increments; in addition it would also be purged of those formal categories connected with them, especially of the infinite, of infinite approximation and, too, the categories, here equally empty, of continuous magnitude' which, moreover, like nisus, becoming, occasion of a variation, are deemed necessary.
[The category of continuous or fluent magnitude enters with the consideration of the external and empirical variation of magnitudes — which are brought by an equation into the relation in which one is a function of the other; but since the scientific object of the differential calculus is a certain relation (usually expressed by the differential coefficient), the specific nature of which may equally well be called a law, the mere continuity is a heterogeneous aspect of this specific nature, and besides is in any case an abstract and here empty category seeing that nothing whatever is said about the law of continuity. Into what formal definitions one may be led in these matters can be seen from the penetrating exposition by my respected colleague, Prof. Dirksen of the fundamental determinations used in the deduction of the differential calculus, which forms an appendix to the criticism of some recent works on this science. The following definition is actually quoted: 'A continuous magnitude, a continuum, is any magnitude thought of as in a state of becoming such that this becoming takes place not by leaps but by an uninterrupted progress'. This is surely tautologically the same as what was to be defined.]
But it would then be required to show what other meaning and value p has — apart from the meagre definition, quite adequate for the theory, that it is simply a function derived from the expansion of a binomial — i.e. what relationships it embodies and what further use can be made of them mathematically; this will be the subject of Remark 2. But first we shall proceed to discuss the confusion which the conception of approximation currently used in expositions of the calculus, has occasioned in the understanding of the true, qualitative determinateness of the relation which was the primary interest concerned.
It has been shown that the so-called infinitesimals express the vanishing of the sides of the ratio as quanta, and that what remains is their quantitative relation solely as qualitatively determined; far from this resulting in the loss of the qualitative relation, the fact is that it is just this relation which results from the conversion of finite into infinite magnitudes. As we have seen, it is in this that the entire nature of the matter consists. Thus in the final ratio, for example, the quanta of abscissa and ordinate vanish; but the sides of this ratio essentially remain, the one an element of the ordinate, the other an element of the abscissa. This vanishing being represented as ' an infinite approximation, the previously distinguished ordinate is made to pass over into the other ordinate, and the previously distinguished abscissa into the other abscissa; but essentially this is not so, the ordinate does not pass over into the abscissa, neither does the abscissa pass into the ordinate. To continue with this example of variable magnitudes, the element of the ordinate is not to be taken as the difference of one ordinate from another ordinate, but rather as the difference or qualitative determination of magnitude relatively to the element of the abscissa; the principle of the one variable magnitude relatively to that of the other is in reciprocal relation with it. The difference, as no longer a difference of finite magnitudes, has ceased to be manifold within itself; it has collapsed into a simple intensity, into the determinateness of one qualitative moment of a ratio relatively to the other.
This is the nature of the matter but it is obscured by the fact that what has just been called an element, for example, of the ordinate, is grasped as a difference or increment in such a way that it is only the difference between the quantum of one ordinate and the quantum of another ordinate. And so the limit here does not have the meaning of ratio; it counts only as the final value to which another magnitude of a similar kind continually approximates in such a manner that it can differ from it by as little as we please, and that the final ratio is a ratio of equality. The infinite difference is thus the fluctuation of a difference of one quantum from another quantum, and the qualitative nature according to which dx is essentially not a determination of the ratio relatively to x, but to dy, comes to be overlooked. Dx is permitted to vanish relatively to dx, but even more does dx vanish relatively to x; but this means in truth: it has a relation only to dy. In such expositions, geometricians are mainly concerned to make intelligible the approximation of a magnitude to its limit and to keep to this aspect of the difference of quantum from quantum, how it is no difference and yet still is a difference. But all the same, approximation is a category which of itself says nothing and explains nothing; dx already has approximation behind it; it is neither near nor nearer; and 'infinitely near', itself means the negation of nearness and approximation.
Now since this implies that the increments or infinitesimals have been considered only from the side of the quantum which vanishes in them, and only as a limit, it follows that they are grasped as unrelated moments. From this would follow the inadmissible idea that it is allowed in the final ratio to equate, say abscissa and ordinate, or even sine, cosine, tangent, versed sine, and what not. This idea seems at first to prevail when the arc is treated as a tangent; for the arc, too, is certainly incommensurable with the straight line, and its element is, in the first place, of another quality than the element of the straight line. It seems even more absurd and inadmissible than the confusing of abscissa, ordinate, versed sine, cosine, etc., when quadrata rotundas, when part of an arc, even though an infinitely small part, is taken to be a part of the tangent and so treated as a straight line. However, this treatment differs essentially from the confusion we have decried; it is justified by the circumstance that in the triangle which has for its sides the element of an arc and the elements of its abscissa and ordinate, the relation is the same as if this element of the arc were the element of a straight line, of the tangent; the angles which constitute the essential relation, i.e. that which remains to these elements when abstraction is made from the finite magnitudes belonging to them, are the same. This can also be expressed as the transition of straight lines which are infinitely small, into curved lines, and their relation in their infinity as a relation of curves. Since, according to its definition, a straight line is the shortest distance between two points, its difference from the curved line is based on the determination of amount, on the smaller amount of what is differentiated in this manner, a determination, therefore, of a quantum. But this determination vanishes in the line when it is taken as an intensive magnitude, as an infinite moment, as an element, and with it, too, its difference from the curved line which rested merely on the difference of quantum. As infinite, therefore, the straight line and arc no longer retain any quantitative relation nor consequently, on the basis of the assumed definition, any qualitative difference from each other either; on the contrary, the former passes into the latter.
Analogous, although also distinct from, the equating of heterogeneous forms is the assumption that infinitely small parts of the same whole are equal to each other; an assumption in itself indefinite and completely indifferent, but which, applied to an object heterogeneous within itself, i.e. an object whose quantitative determination is essentially non-uniform, produces the peculiar inversion contained in that proposition of higher mechanics which states that infinitely small parts of a curve are traversed in equal, infinitely small times in a uniform motion, inasmuch as this is asserted of a motion in which in equal finite, i.e. existent, parts of time, finite, i.e. existent, unequal parts of the curve are traversed, of a motion therefore which exists as non-uniform and is assumed as such. This proposition is the expression in words of what is supposed to be the significance of an analytical term obtained in the above-mentioned development of the formula relating to a motion which is non-uniform but otherwise conforms to a law. Earlier mathematicians sought to express in words and propositions and to exhibit in geometrical tables the results of the newly invented infinitesimal calculus (which moreover always had to do with concrete objects), chiefly in order to use them for theorems susceptible of the ordinary method of proof. The terms of a mathematical formula into which analytical treatment resolved the magnitude of the object, of motion, for instance, acquired there an objective significance, such as velocity, force of acceleration, and so on; in accordance with this meaning they were supposed to furnish correct propositions, physical laws; their objective connections and relations, too, were supposed to be determined in accordance with the analytical connection. A particular example is that in a uniformly accelerated motion there is supposed to exist a special velocity proportional to the times, but that to this velocity there constantly accrues an increment from the force of gravity.
In the modern, analytical form of mechanics such propositions are put forward simply as results of the calculus, without enquiry whether by themselves and in themselves they have a real significance, i.e. one to which there is a corresponding physical existence and whether such meaning can be demonstrated. The difficulty of making intelligible the connection of such forms when they are taken in the real meaning alluded to, for example the transition from said simply uniform velocity to a uniformly accelerated velocity, is held to be completely eliminated by the analytical treatment in which such connection is a simple result of the authority now established once and for all of the operations of the calculus. It is announced as a triumph of science that by means of the calculus alone, laws are found transcending experience, that is, propositions about existence which have no existence. But in the earlier, still naive period of the infinitesimal calculus, the aim was to assign to those forms and propositions represented in geometrical diagrams a real meaning of their own and to make that meaning plausible, and to apply the forms and propositions bearing such meaning in the proof of the main propositions concerned.
It cannot be denied that in this field much has been accepted as proof, especially with the aid of the nebulous conception of the infinitely small, for no other reason than that the result was always already known beforehand, and that the proof which was so arranged that the result did emerge, at least produced the illusion of a framework of proof, an illusion which was still preferred to mere belief or knowledge from experience. But 1 do not hesitate to regard this affectation as nothing more than mere jugglery and window-dressing, and I include in this description even Newton's proofs, especially those belonging to what has just been quoted, for which Newton has been extolled to the skies and exalted above Kepler, namely that he demonstrated mathematically what Kepler had discovered merely empirically.
The empty scaffolding of such proofs was erected in order to prove physical laws. But mathematics is altogether incapable of proving quantitative determinations of the physical world in so far as they are laws based on the qualitative nature of the moments [of the subject matter]; and for this reason, that this science is not philosophy, does not start from the Notion, and therefore the qualitative element, in so far as it is not taken lemmatically from experience, lies outside its sphere. The desire to uphold the honour of mathematics, that all its propositions ought to be rigorously proved, has often caused it to forget its limits; thus it seemed against its honour to acknowledge simply experience as the source and sole proof of empirical propositions. Consciousness has since then developed a more instructed view of the matter; so long, however, as consciousness is not clearly aware of the distinction between what is mathematically demonstrable and what can come only from another source, between what are only terms of an analytical expansion and what are physical existences, scientific method cannot be developed into a rigorous and pure attitude in this field. Without doubt, however, the same justice will be done to that framework of Newtonian proof as was done to another baseless and artificial Newtonian structure of optical experiments and conclusions derived from them. Applied mathematics is still full of a similar concoction of experiment and reflection; but just as one part after another of Newtonian optics long since began to be ignored in practice by the science — with the inconsistency however that all the rest although in contradiction was allowed to stand — so, too, it is a fact that already some of those illusory proofs have fallen into oblivion or have been replaced by others.
Remark 2: The Purpose of the Differential Calculus Deduced from its Application
In the previous Remark we considered on the one hand the specific nature of the notion of the infinitesimal which is used in the differential calculus, and on the other the basis of its introduction into the calculus; both are abstract determinations and therefore in themselves also easy. The so-called application, however, presents greater difficulties, but also the more interesting side; the elements of this concrete side are to be the object of this Remark. The whole method of the differential calculus is complete in the proposition that dxn = nx(n - 1)dx, or (f(x + i) - fx)/i = P, that is, is equal to the coefficient of the first term of the binomial x + d, or x + 1, developed according to the powers of dx or i. There is no need to learn anything further: the development of the next forms, of the differential of a product, of an exponential magnitude and so on, follows mechanically; in little time, in half an hour perhaps — for with the finding of the differential the converse the finding of the original function from the differential, or integration, is also given — one can be in possession of the whole theory. What takes longer is simply the effort to understand, to make intelligible, how it is that, after having so easily accomplished the first stage of the task, the finding of the said differential, analytically, i.e. purely arithmetically, by the expansion of the function of the variable after this has received the form of a binomial by the addition of an increment; how it is that the second stage can be correct, namely the omission of all the terms except the first, of the series arising from the expansion. If all that were required were only this coefficient, then with its determination all that concerns the theory would, as we have said, be settled and done with in less than half an hour and the omission of the further terms of the series (with the determination of the first function, the determination of the second, third, etc., is also accomplished) far from causing any difficulty, would not come into question since they are completely irrelevant.
We may begin by remarking that the method of the differential calculus shows on the face of it that it was not invented and constructed for its own sake. Not only was it not invented for its own sake as another mode of analytical procedure; on the contrary, the arbitrary omission of terms arising from the expansion of a function is absolutely contrary to all mathematical principles, it being arbitrary in the sense that the whole of this development is nevertheless assumed to belong completely to the matter in hand, this being regarded as the difference between the developed function of a variable (after this has been given the form of a binomial) and the original function. The need for such a mode of procedure and the lack of any internal justification at once suggest that the origin and foundation must lie elsewhere. It happens in other sciences too, that what is placed at the beginning of a science as its elements and from which the principles of the science are supposed to be derived is not self-evident, and that it is rather in the sequel that the raison d'étre and proof of those elements is to be found. The course of events in the history of the differential calculus makes it plain that the matter had its origin mainly in the various so-called tangential methods, in what could be considered ingenious devices; it was only later that mathematicians reflected on the nature of the method after it had been extended to other objects, and reduced it to abstract formulae which they then also attempted to raise to the status of principle.
We have shown that the specific nature of the notion of the so-called infinitesimal is the qualitative nature of determinations of quantity which are related to each other primarily as quanta; to this was linked the empirical investigation aimed at demonstrating the presence of this specific nature in the existing descriptions and definitions of the infinitesimal in so far as this is taken as an infinitesimal difference and the like. This was done only in the interest of the abstract nature of the notion as such; the next question would be as to the nature of the transition from this to the mathematical formulation and application. To this end we must first pursue our examination of the theoretical side, the specific nature of the notion, which will not prove wholly unfruitful in itself; we must then consider the relation of the theoretical side to its application; and in both cases we must demonstrate, so far as it is relevant here, that the general conclusions are at the same time adequate to the purpose of the differential calculus and to the way in which the calculus brings about its results.
First, it is to be remembered that the mathematical form of the determinateness of the notion under discussion has already been stated in passing. The specifically qualitative character of quantity is first indicated in the quantitative relation as such; but it was already asserted in anticipation when demonstrating the so-called kinds of reckoning (see the relative Remark), that it is the relation of powers (still to be dealt with in its proper place) in which number, through the equating of the moments of its Notion, unit and amount, is posited as returned into itself, thereby receiving into itself the moment of infinity, of being-for-self, i.e. of being self-determined. Thus, as we have already said, the express qualitative nature of quantity is essentially connected with the forms of powers, and since the specific interest of the differential calculus is to operate with qualitative forms of magnitude, its own peculiar subject matter must be the treatment of forms of powers, and the whole range of problems, and their solutions, show that the interest lies solely in the treatment of determinations of powers as such.
This foundation is important and at once puts in the forefront something definite in place of the merely formal categories of variable, continuous or infinite magnitudes or even of functions generally; yet it is still too general, for other operations also have to do with determinations of powers. The raising to a power, extraction of a root, treatment of exponential magnitudes and logarithms, series, and equations of higher orders, the interest and concern of all these is solely with relations which are based on powers. Undoubtedly, these together constitute a system of the treatment of powers; but which of the various relations in which determinations of powers can be put is the peculiar interest and subject matter of the differential calculus, this is to be ascertained from the calculus itself, i.e. from its so-called applications. These are, in fact, the core of the whole business, the actual procedure in the mathematical solution of a certain group of problems; this procedure was earlier than the theory or general part and was later called application only with reference to the subsequently created theory, the aim of which was to draw up the general method of the procedure and, as well, to endow it with first principles, i.e. with a justification. We have shown in the preceding Remark the futility of the search for principles which would clarify the method as currently understood, principles which would really solve the contradiction revealed by the method instead of excusing it or covering it up merely by the insignificance of what is here to be omitted (but which really is required by mathematical procedure), or, by what amounts to the same thing, the possibility of infinite or arbitrary approximation and the like. If from the practical part of mathematics known as the differential calculus the general features of the method were to be abstracted in a manner different from that hitherto followed, then the said principles and the concern about them would also show themselves to be superfluous, just as they reveal themselves to be intrinsically false and permanently contradictory.
If we investigate this peculiarity by simply taking up what we find in this part of mathematics, we find as its subject matter:
(a) Equations in which any number of magnitudes (here we can simply confine ourselves to two) are combined into a qualitative whole in such a way that first, these equations have their determinateness in empirical magnitudes which are their fixed limits, and also in the kind of connection they have with these limits and with each other as is generally the case in an equation; but since there is only one equation for both magnitudes (similarly, relatively more equations for more magnitudes, but always fewer than the number of magnitudes), these equations belong to the class of indeterminate equations; and secondly, that one aspect of the determinateness of these magnitudes is that they are — or at least one of them is present in the equation in a higher power than the first.
Before proceeding further, there are one or two things to be noticed about this. The first is that the magnitudes, as described under the first of the above two headings, have simply and solely the character of variables such as occur in the problems of indeterminate analysis. Their value is undetermined, but if one of them does receive a completely determined value, i.e. a numerical value, from outside, then the other too, is determined, so that one is a function of the other. Therefore, in relation to the specific quantitative determinateness here in question, the categories of variable magnitudes, functions and the like are, as we have already said, merely formal, because they are still too general to contain that specific element on which the entire interest of the differential calculus is focused, or to permit of that element being explicated by analysis; they are in themselves simple, unimportant, easy determinations which are only made difficult by importing into them what they do not contain in order that this may then be derived from them — namely, the specific determination of the differential calculus. Then as regards the so-called constant, we can note that it is in the first place an indifferent empirical magnitude determining the variables only with respect to their empirical quantum as a limit of their minimum and maximum; but the nature of the connection between the constants and the variables is itself a significant factor in the nature of the particular function which these magnitudes are. Conversely, however, the constants themselves are also functions; in so far as a straight line, for example, has the meaning of being the parameter of a parabola, then this meaning is that it is the function y2/x2; and in the expansion of the binomial generally, the constant which is the coefficient of the first term of the development is the sum of the roots, the coefficient of the second is the sum of the products, in pairs, and so on; here, therefore, the constants are simply functions of the roots. Where, in the integral calculus, the constant is determined from the given formula, it is to that extent treated as a function of this. Further on we shall consider these coefficients in another character than that of functions, their meaning in the concrete object being the focus of the whole interest.
Now the difference between variables as considered in the differential calculus, and in their character as factors in indeterminate problems, must be seen to consist in what has been said, namely, that at least one of those variables (or even all of them), is found in a power higher than the first; and here again it is a matter of indifference whether they are all of the same higher power or are of unequal powers; their specific indeterminateness which they have here consists solely in this, that in such a relation of powers they are functions of one another. The alteration of variables is in this way qualitatively determined, and hence continuous, and this continuity, which again is itself only the purely formal category of an identity, of a determinateness which is preserved and remains self-same in the alteration, has here its determinate meaning, solely, that is, in the power-relation, which does not have a quantum for its exponent and which forms the non-quantitative, permanent determinateness of the ratio of the variables. For this reason it should be noted, in criticism of another formalism, that the first power is only a power in relation to higher powers; on its own, x is merely any indeterminate quantum. Thus there is no point in differentiating for their own sakes the equations y = ax + b (of the straight line), or s = ct (of the plain uniform velocity); if from y = ax, or even ax + b, we obtain a = dy/dx, or from s = ct, ds/dt = c, then a = y/x is equally the determination of the tangent, or s/t that of velocity simply as such. The latter is given the form of dy/dx in the context of what is said to be the development of the uniformly accelerated motion; but, as already remarked, the presence in the system of such a motion, of a moment of simple, merely uniform velocity, i.e. a velocity which is not determined by the higher power of one of the moments of the motion is itself an empty assumption based solely on the routine of the method. Since the method starts from the conception of the increment which the variable is supposed to acquire, then of course a variable which is only a function of the first power can also receive an increment; when now in order to find the differential we have to subtract the difference of the second equation thus produced from the given equation, the meaninglessness of the operation becomes apparent, for, as we have remarked, the equation for the so-called increments, both before and after the operation, is the same as for the variables themselves.
(b) What has been said determines the nature of the equation which is to be treated; we have now to indicate what is the interest on which the treatment of the equation is focused. This consideration can yield only known results, in a form found especially in Lagrange's version; but I have made the exposition completely elementary in order to eliminate the heterogeneous determinations associated with it. The basis of treatment of an equation of this kind shows itself to be this, that the power is taken as being within itself a relation or a system of relations. We said above that power is number which has reached the stage where it determines its own alteration, where its moments of unit and amount are identical — as previously shown, completely identical first in the square, formally (which makes no difference here) in higher powers. Now power is number (magnitude as the more general term may be preferred, but it is in itself always number), and hence a plurality, and also is represented as a sum; it can therefore be directly analysed into an arbitrary amount of numbers which have no further determination relatively to one another or to their sum, other than that together they are equal to the sum. But the power can also be split into a sum of differences which are determined by the form of the power. If the power is taken as a sum, then its radical number, the root, is also taken as a sum, and arbitrarily after manifold divisions, which manifoldness, however, is the indifferent, empirically quantitative element. The sum which the root is supposed to be, when reduced to its simple determinateness, i.e. to its genuine universality, is the binomial; all further increase in the number of terms is a mere repetition of the same determination and therefore meaningless.
[It springs solely from the formalism of that generality to which analysis perforce lays claim when, instead of taking (a + b)n for the expansion of powers, it gives the expression the form of (a + b + c + d...)n as happens too in many other cases; such a form is to be regarded as, so to speak, a mere affectation of a show of generality; the matter itself is exhausted in the binomial. It is through the expansion of the binomial that the law is found, and it is the law which is the genuine universality, not the external, mere repetition of the law which is all that is effected by this a + b + c + d ...]
The sole point of importance here is the qualitative determinateness of the terms resulting from the raising to a power of the root taken as a sum, and this determinateness lies solely in the alteration which the potentiation is. These terms, then, are wholly functions of potentiation and of the power. Now this representation of number as a sum of a plurality of terms which are functions of potentiation, and the finding of the form of such functions and also this sum from the plurality of those terms, in so far as this must depend solely on that form, this constitutes, as we know, the special theory of series. But in this connection it is essential to distinguish another object of interest, namely the relation of the fundamental magnitude itself (whose determinateness, since it is a complex, i.e. here an equation, includes within itself a power) to the functions of its potentiation. This relation, taken in complete abstraction from the previously mentioned interest of the sum, will show itself to be the sole standpoint yielded by the practical aspect of the science.
But first, another determination must be added to what has been said, or rather, one which is implied in it must be removed. It was said that the variable into the determination of which power enters is regarded as within itself a sum, in fact a system of terms in so far as these are functions of the potentiation, and that thus the root, too, is regarded as a sum and in the simply determined form of a binomial: xn = (y + z)n = (yn + ny(n-1)z + ... ). This exposition started from the sum as such for the expansion of the power, i.e. for obtaining the functions of its potentiation; but what is concerned here is not a sum as such, or the series arising from it; what is to be taken up from the sum is only the relation. The relation as such of the magnitudes is, on the one hand, all that remains after abstraction is made from the plus of a sum as such, and on the other hand, all that is needed for finding the functions produced by the expansion of the power.
But such relation is already determined by the fact that here the object is an equation, ym = axn, and so already a complex of several (variable) magnitudes which contains a power determination of them. In this complex, each of these variables is posited simply as in relation to the others with the meaning, one could say, of a plus implicit in it — as a function of the other variables; their character, that of being functions of one another, gives them this determination of a plus which, however, for that same reason, is wholly indeterminate — not an increase or an increment, or anything of that nature. Yet even this abstract point of view we could leave out of account; we can quite simply stop at the point where the variables in the equation having received the form of functions of one another, such functions containing a relation of powers, the functions of potentiation are then also compared with one another — these second functions being determined simply and solely by the potentiation itself. To treat an equation of the powers of its variables as a relation of the functions developed by potentiation can, in the first place, be said to be just a matter of choice or a possibility; the utility of such a transformation has to be indicated by some further purpose or use; and the sole reason for the transformation was its utility.
When we started above from the representation of these functions of potentiation of a variable which is taken as a sum complex within itself, this served only partly to indicate the nature of such functions, but partly also to show the way in which they are found.
What we have here then is the ordinary analytical development which for the purpose of the differential calculus is operated in this way, that an increment dx or i is given to the variable and then the power of the binomial is developed by the terms of the series belonging to it. But the so-called increment is supposed to be not a quantum but only a form, the whole value of which is that it assists the development; it is admitted — most categorically by Euler and Lagrange and in the previously mentioned conception of limit — that what is wanted is only the resulting power determinations of the variables, the so-called coefficients, namely, of the increment and its powers, according to which the series is ordered and to which the different coefficients belong. On this we could perhaps remark that since an increment (which has no quantum) is assumed only for the sake of the development, it would be most appropriate to take i (the one) for that purpose, for in the development this always occurs only as a factor; the factor one, therefore, fulfils the purpose, namely, that the increment is not to involve any quantitative determinateness or alteration; on the other hand, dx, which is burdened with the false idea of a quantitative difference, and other symbols like i with the mere show — pointless here — of generality, always have the appearance and pretension of a quantum and its powers; which pretension then involves the trouble that they must nevertheless be removed and left out. In order to retain the form of a series expanded on the basis of powers, the designations of the exponents as indices could equally well be attached to the one. But in any case, abstraction must be made from the series and from the determination of the coefficients according to their place in the series; the relation between all of them is the same; the second function is derived from the first in exactly the same manner as this is from the original function, and for the function counted as second, the first derived function is itself original. But the essential point of interest is not the series but simply and solely the determination of the power resulting from the expansion in its relation to the variable which for the power determination is immediate. It should not therefore be defined as the coefficient of the first term of the development, for it is first only in relation to the other terms following it in the series, and a power such as that of an increment, like the series itself, is here out of place; instead, the simple expression: derived function of a power, or as was said above: function of potentiation of a magnitude, would be preferable — the knowledge of the way in which the derivation is taken to be a development included within a power being presupposed.
Now if the strictly mathematical beginning in this part of analysis is nothing more than the finding of the function determined by the expansion of the power, the further question is what is to be done with the relation so obtained, where has it an application and use, or indeed, for what purpose are such functions sought. It is the finding of relations in a concrete subject matter which can be reduced to such a function that has given the differential calculus its great interest.
But as regards the applicableness of the relation, we need not wait for conclusions to be drawn from particular applications themselves, the answer follows directly and automatically from the nature of the matter which we have shown to consist in the form possessed by the moments of powers: namely, the expansion of the powers, which yields the functions of their potentiation, contains (ignoring any more precise determination) in the first place, simply the reduction of the magnitude to the next lower power. This operation is therefore applicable in the case of those objects in which there is also present such a difference of power determinations. Now if we reflect on the specific nature of space, we find that it contains the three dimensions which, in order to distinguish them from the abstract differences of height, length and breadth, we can call concrete — namely, line, surface and total space; and when they are taken in their simplest forms and with reference to self-determination and consequently to analytical dimensions, we have the straight line, plane surface and surface taken as a square, and the cube. The straight line has an empirical quantum, but with the plane there enters the qualitative element, the determination of power; further modifications, e.g. the fact that this also happens in the case of plane curves, we need not consider, for we are concerned primarily with the distinction in general. With this there arises, too, the need to pass from a higher power to a lower, and vice versa, when, for example, linear determinations are to be derived from given equations of the plane, or vice versa. Further, the motion in which we have to consider the quantitative relation of the space traversed to the time elapsed, manifests itself in the different determinations of a motion which is simply uniform, or uniformly accelerated, or alternately uniformly accelerated and uniformly retarded, and thus a self-returning motion; since these different kinds of motion are expressed in accordance with the quantitative relation of their moments, of space and time, their equations contain different determinations of powers, and when it is necessary to determine one kind of motion, or a spatial magnitude to which one kind of motion is linked, from another kind of motion, the operation also involves the passage from one power-function to another, either higher or lower. These two examples may suffice for the purpose for which they are cited.
The appearance of arbitrariness presented by the differential calculus in its applications would be clarified simply by an awareness of the nature of the spheres in which its application is permissible and of the peculiar need for and condition of this application. But now the further point of interest within these spheres themselves is to know between what parts of the subject matter of the mathematical problem such a relation occurs as is posited peculiarly by the differential calculus. First, it must be observed that there are two kinds of relation. The operation of depotentiating an equation considered according to the derivative functions of its variables, yields a result which, in itself, is no longer truly an equation but a relation; this relation is the subject matter of the differential calculus proper. This also gives us, secondly, the relation of the higher power form (the original equation) itself to the lower (the derivative). This second relation we must ignore for the time being; it will prove to be the special subject matter of the integral calculus.
Let us start by considering the first relation; for the determination of its moment (to be taken from the application, in which lies the interest of the operation) we shall take the simplest example from curves determined by an equation of the second degree. As we know, the relation of the co-ordinates is given directly by the equation in a power form. From the fundamental determination follow the determinations of the other straight lines connected with the co-ordinates, tangent, subtangent, normal, and so on.
But the equations between these lines and the co-ordinate are linear equations; the wholes with respect to which these lines are determined as parts, are right-angled triangles formed by straight lines. The transition from the original equation which contains the power form, to said linear equations, involves now the above-mentioned transition from the original function (which is an equation), to the derived function (which is a relation, a relation, that is, between certain lines contained in the curve). The problem consists in finding the connection between the relation of these lines and the equation of the curve.
It is not without interest, as regards the historical element, to remark this much, that the first discoverers could only record their findings in a wholly empirical manner without being able to account for the operation, which remained a completely external affair. It will be sufficient here to refer to Barrow, to him who was Newton's teacher. In his lect. Opt. et Geom., in which he treats problems of higher geometry according to the method of indivisibles, a method which, to begin with, is distinct from the characteristic feature of the differential calculus, he also puts on record' his procedure for determining tangents — 'because his friends urged him to do so'. To form a proper idea of how this procedure is formulated simply as an external rule, in the same style as the 'rule of three', or better still the so-called 'test by casting out nines', one must read Barrow's own exposition. He draws the tiny lines afterwards known as the increments in the characteristic triangle of a curve and then gives the instruction, in the form of a mere rule, to reject as superfluous the terms which, as a result of the expansion of the equations, appear as powers of the said increments or as products (etenim isti termini nihilum valebunt); similarly, the terms which contain only magnitudes to be found in the original equation are to be rejected (the subsequent subtraction of the original equation from that formed with the increments); and finally, for the increments of the ordinate and abscissa, the ordinate itself and the subtangent respectively are to be substituted. The procedure, if one may say so, can hardly be set forth in a more schoolmaster-like manner; the latter substitution is the assumption of the proportionality of the increments of the ordinate and the abscissa with the ordinate and the subtangent, an assumption on which is based the determination of the tangent in the ordinary differential method; in Barrow's rule this assumption appears in all its naive nakedness. A simple way of determining the subtangent was found; the artifices of Roberval and Fermat have a similar character. The method for finding maximal and minimal values from which Fermat started rests on the same basis and the same procedure. It was a mathematical craze of those times to find so-called methods, i.e. rules of that kind and to make a secret of them — which was not only easy, but in one respect even necessary, for the same reason that it was easy — namely, because the inventors had found only an empirical, external rule, not a method, i.e. nothing derived from established principles. Leibniz accepted such so-called methods from his contemporaries and so did Newton who got them directly from his teacher; by generalising their form and applicableness they opened up new paths for the sciences, but at the same time they also felt the need to wrest free the procedure from the shape of merely external rules and to try to procure for it the necessary justification.
If we analyse the method more closely, we find the genuine procedure to be as follows. Firstly the power forms (of the variables of course) contained in the equation are reduced to their first functions. But the value of the terms of the equation is thereby altered; there is now no longer an equation, but instead only a relation between the first function of the one variable and the first function of the other. Instead of px = y2 we have p : 2y, or instead of 2ax - x2 = y2, we have a - x : y, the relation which later came to be designated dy/dx. Now the equation represents a curve; but this relation, which is completely dependent on it and derived from it (above, according to a mere rule), is, on the contrary, a linear relation with which certain lines are in proportion: p : 2y or a - x : y are themselves relations of straight line of the curve, of the co-ordinates and parameters. But with all this, nothing is as yet known. The interest centres on finding that the derived relation applies to other lines connected with the curve, on finding the equality of two relations. And so there is, secondly, the question, which are the straight lines determined by the nature of the curve, standing in such a relation? But this is just what was already known: namely, that the relation so obtained is the relation of the ordinate to the subtangent. This the ancients had found in an ingenious geometrical manner; what the moderns have discovered is the empirical procedure of so preparing the equation of the curve that it yields that first relation of which it was already known that it is equal to a relation containing the line (here the subtangent) which is to be determined. Now on the one hand, this preparation of the equation — the differentiation — has been methodically conceived and executed; but on the other hand the imaginary increments of the co-ordinates and an imaginary characteristic triangle formed by them and by an equally imaginary increment of the tangent, have been invented in order that the proportionality of the ratio found by lowering the degree of the equation to the ratio formed by the ordinate and subtangent, may be represented, not as something only empirically accepted as an already familiar fact, but as something demonstrated. However, in the said form of rules, the already familiar fact reveals itself absolutely and unmistakably as the sole occasion and respective justification of the assumption of the characteristic triangle and the said proportionality.
Now Lagrange rejected this pretence and took the genuinely scientific course. We have to thank his method for bringing into prominence the real point of interest for it consists in separating the two transitions necessary for the solution of the problem and treating and proving each of them separately. One part of this solution (for the more detailed statement of the process we shall confine ourselves to the example of the elementary problem of finding the subtangent), the theoretical or general part, namely, the finding of the first function from the given equation of the is dealt with separately; the result is a linear relation, a curve, relation therefore of straight lines occurring in the system determined by the curve. The other part of the solution now is the finding of those lines in the curve which stand in this relation. Now this is effected in a direct manner i.e., without the characteristic triangle, which means that there is no assumption of infinitely small arcs, ordinates and abscissae, the last two being given the significance of dy and dx, that is, of being sides of that relation, and at the same time directly equating the infinitely small ordinate and abscissa with the ordinate and subtangent themselves. A line (and a point, too), is determined only in so far as it forms the side of a triangle and the determination of a point, too, falls only in such triangle. This, it may be mentioned in passing, is the fundamental proposition of analytical geometry from which are derived the co-ordinates of that science, just as (it is the same standpoint) in mechanics it gives rise to the parallelogram of forces, for which very reason the many efforts to find a proof of this latter are quite unnecessary. The subtangent, now, is made to be the side of a triangle whose other sides are the ordinate and the tangent connected to it. The equation of the latter, as a straight line, is p = aq (the determination does not require the additional term, + b which is added only on account of the fondness for generality); — the determination of the ratio p/q falls within a, the coefficient of q which is the respective first function (derivative) of the equation, but may simply be considered only as a = p/q being, as we have said, the essential determination of the straight line which is applied as tangent to the curve. But the first function (derivative) of the equation of the curve is equally the determination of a straight line; seeing then that the co-ordinate p of the first straight line and y, the co-ordinate of the curve, are assumed to be identical (so that the point at which the curve is touched by the first straight line assumed as tangent is also the starting point of the straight line determined by the first function of the curve), the problem is to show that this second straight line coincides with the first, i.e. is a tangent; or, algebraically expressed, that since y = fx and p = Fq, and it is assumed that y = p and hence that fx = Fq, therefore f'x = F'q. Now in order to show that the straight line applied as a tangent and the straight line determined by the first function of the equation coincide, and that therefore the latter is a tangent, Descartes has recourse to the increment i of the abscissa and to the increment of the ordinate determined by the expansion of the function. Thus here, too, the objectionable increment also makes its appearance; but its introduction for the purpose indicated and its role in the expansion of the function must be carefully distinguished from the previously mentioned employment of the increment in finding the differential equation and in the characteristic triangle. Its employment here is justified and necessary because it falls within the scope of geometry, the geometrical determination of a tangent as such implying that between it and the curve with which it has a point in common, no other straight line can be drawn which also passes through the said point. For, as thus determined, the quality of tangent or not-tangent is reduced to a quantitative difference, that line being the tangent of which simply greater smallness is predicated with respect to the determination in point. This seemingly only relative smallness contains no empirical element whatever, i.e. nothing dependent on a quantum as such; in virtue of the nature of the formula it is explicitly qualitative if the difference of the moments on which the magnitude to be compared depends is a difference of powers. Since this difference becomes that of i and i2 and i (which after all is meant to signify a number) is then to be conceived as a fraction, i2 is therefore in itself and explicitly smaller than i, so that the very conception of an arbitrary magnitude in connection with i is here superfluous and in fact out of place. For the same reason the demonstration of the greater smallness has nothing to do with an infinitesimal, which thus need not be brought in here at all.
I must also mention the tangential method of Descartes, if only for its beauty and its fame — well-deserved but nowadays mostly forgotten; it has, moreover, a bearing on the nature of equations and this, again, calls for a further remark. Descartes expounds this independent method, in which the required linear determination is likewise found from the same derivative function, in his geometry which has proved to be so fruitful in other respects tool; in it he has taught the great basis of the nature of equations and their geometrical construction, and also of the application of analysis, thereby greatly widened in its scope, to geometry. With him the problem took the form of drawing straight lines perpendicularly to given points on a curve as a method for determining the subtangent, etc. One can understand the satisfaction he felt at his discovery, which concerned an object of general scientific interest at that time and which is so purely geometrical and therefore was greatly superior to the mere rules of his rivals, referred to above. His words are as follows: 'J'ose dire que c'est ceci le probleme le plus utile et le plus general, non seulement que je sache, mais meme que j'aie l'amais desire de savoir en giometrie.' He bases his solution on the analytic equation of the right-angled triangle formed by the ordinate of the point on the curve to which the required straight line in the problem is to be drawn perpendicularly, by this same straight line (the normal), and thirdly, by that part of the axis which is cut off by the ordinate and the normal (the subnormal). Now from the known equation of a curve, the value of either the ordinate or the abscissa is substituted in the said triangle, the result being an equation of the second degree (and Descartes shows how even curves whose equations contain higher powers reduce to this); in this equation, only one of the variables occurs, namely, as a square and in the first degree — a quadratic equation which at first appears as a so-called impure equation. Descartes now makes the reflection that if the assumed point on the curve is imagined to be a point of intersection of the curve and of a circle, then this circle will also cut the curve in another point and we shall then get for the unequal xs thus produced, two equations with the same constants and of the same form, or else only one equation with unequal values of x. But the equation only becomes one for the one triangle in which the hypotenuse is perpendicular to the curve or is the normal, the case being conceived of in this way, that the two points of intersection of the curve and the circle are made to coincide and the circle is thus made to touch the curve. But in that case it is also true that the x or y of the quadratic equation no longer have unequal roots. Now since in a quadratic equation with two equal roots the coefficient of the term containing the unknown in the first power is twice the single root, we obtain an equation which yields the required determinations. This procedure must be regarded as the brilliant device of a genuinely analytical mind, in comparison with which the dogmatically assumed proportionality of the subtangent and the ordinate with the postulated infinitely small, so-called increments, of the abscissa and ordinate drops into the background.
The final equation obtained in this way, in which the coefficient of the second term of the quadratic equation is equated with the double root or unknown, is the same as that obtained by the method of the differential calculus. The differentiation of x2 - ax - b = 0 yields the new equation 2x - a = 0; or x3 - px - q = 0 gives 3x2 - p = 0. But it suggests itself here to remark that it is by no means self-evident that such a derivative equation is also correct. We have already pointed out that an equation with two variables (which, just because they are variables, do not lose their character of being unknown quantities) yields only a proportion; and for the simple reason stated, namely, that when the functions of potentiation are substituted for the powers themselves, the value of both terms of the equation is altered and it is not yet known whether an equation still exists between them with their values thus altered. All that the equation dy/dx = P expresses is that P is a ratio and no other real meaning can be ascribed to dyldx. But even so, we still do not know of this ratio = P, to what other ratio it is equal; and it is only such equation or proportionality which gives a value and meaning to it. We have already mentioned that this meaning, which was called the application, was taken from another source, empirically; similarly, in the case of the equations here under discussion which have been obtained by differentiation, it is from another source that we must know whether they have equal roots in order that we may learn whether the equation thus obtained is still correct. But this fact is not expressly brought to notice in the textbooks; it is disposed of, certainly, when an equation with one unknown, reduced to zero, is straightway equated with y, with the result, of course, that differentiation yields a dy/dx, i.e. only a ratio. The functional calculus, it is true, is supposed to deal with functions of potentiation and the differential calculus with differentials; but it by no means follows from this alone that the magnitudes from which the differentials or functions of potentiation are taken, are themselves supposed to be only functions of other magnitudes. Besides, in the theoretical part, in the instruction to derive the differentials, i.e. the functions of potentiation, there is no indication that the magnitudes which are to be subjected to such treatment are themselves supposed to be functions of other magnitudes.
Further, with regard to the omission of the constant when differentiating, we may draw attention to the fact that the omission has here the meaning that the constant plays no part in the determination of the roots if these are equal, the determination being exhausted by the coefficient of the second term of the equation: as in the example quoted from Descartes where the constant is itself the square of the roots, which therefore can be determined from the constant as well as from the coefficients — seeing that, like the coefficients, the constant is simply a function of the roots of the equation. In the usual exposition, the omission of the so-called constants (which are connected with the other terms only by plus and minus) results from the mere mechanism of the process of differentiation, in which to find the differential of a compound expression only the variables are given an increment, and the expression thereby formed is subtracted from the original expression. The meaning of the constants and of their omission, in what respect they are themselves functions and, as such, are or are not of service, are not discussed.
In connection with the omission of constants we may make a similar observation about the names of differentiation and integration as we did before about the expressions finite and infinite: that is, that the character of the operation in fact belies its name. To differentiate denotes that differences are posited, whereas the result of differentiating is, in fact, to reduce the dimensions of an equation, and to omit the constant is to remove from the equation an element in its determinateness. As we have remarked, the roots of the variables are made equal, and therefore their difference is cancelled. In integration, on the other hand, the constant must be added in again and although as a result the equation is integrated, it is so in the sense that the previously cancelled difference of the roots is restored, that is, what was posited as equal is differentiated again. The ordinary expression helps to obscure the essential nature of the matter and to set everything in a point of view which is not only subordinate but even alien to the main interest, the point of view, namely, of the infinitely small difference, the increment and the like, and also of the mere difference as such between the given and the derived function, without any indication of their specific, i.e. qualitative, difference.
Another important sphere in which the differential calculus is employed is mechanics. The meanings of the distinct power functions yielded by the elementary equations of its subject matter, motion, have already been mentioned in passing; at this point, I shall proceed to deal with them directly. The equation, i.e. the mathematical expression, for simply uniform motion, c = s/t or s = ct, in which the spaces traversed are proportional to the times elapsed in accordance with an empirical unit c (the magnitude of the velocity), offers no meaning for differentiation: the coefficient c is already completely determined and known, and no further expansion of powers is possible. We have already noticed how s = at2, the equation of the motion of a falling body, is analysed; the first term of the analysis, ds/dt = 2at is translated into language, and also into existence, in such a manner that it is supposed to be a factor in a sum (a conception we have long since abandoned), to be one part of the motion, which part moreover is attributed to the force of inertia, i.e. of a simply uniform motion, in such a manner that in infinitely small parts of time the motion is uniform, but in finite parts of time, i.e. in actually existent parts of time, it is non-uniform. Admittedly, fs = 2at; and the meaning of a and t themselves is known and so, too, the fact that the motion is determined as of uniform velocity; since a = s/t2, 2at is equal simply to 2s/t.
But knowing this we are not a whit wiser; it is only the erroneous assumption that 2at is a part of the motion regarded as a sum, that gives the false appearance of a physical proposition. The factor itself, a, the empirical unit — a simple quantum — is attributed to gravity; but if the category of 'force of gravity' is to be employed then it ought rather to be said that the whole, s = at2, is the effect, or, better, the law, of gravity. Similarly with the proposition derived from ds/dt = 2at, that if gravity ceased to act, the body, with the velocity reached at the end of its fall, would cover twice the distance it had traversed, in the same period of time as its fall. This also implies a metaphysics which is itself unsound: the end of the fall, or the end of a period of time in which the body has fallen, is itself still a period of time; if it were not, there would be assumed a state of rest and hence no velocity, for velocity can only be fixed in accordance with the space traversed in a period of time, not at its end. When, however, the differential calculus is applied without restriction in other departments of physics where there is no motion at all, as for example in the behaviour of light (apart from what is called its propagation in space) and in the application of quantitative determinations to colours, and the first function of a quadratic function here is also called a velocity, then this must be regarded as an even more illegitimate formalism of inventing an existence.
The motion represented by the equation s = at2 we find, says Lagrange, empirically in falling bodies; the next simplest motion would be that whose equation were s = ct3, but no such motion is found in Nature; we do not know what significance the coefficient c could have. Now though this is indeed the case, there is nevertheless a motion whose equation is s3 = at2 — Kepler's law of the motion of the bodies of the solar system; the significance 2at here of the first derived function 2at/3s2 and the further direct treatment of this equation by differentiation, the development of the laws and determinations of that absolute motion from this starting point, must indeed present an interesting problem in which analysis would display a brilliance most worthy of itself.
Thus the application of the differential calculus to the elementary equations of motion does not of itself offer any real interest; the formal interest comes from the general mechanism of the calculus. But another significance is acquired by the analysis of motion in connection with the determination of its trajectory; if this is a curve and its equation contains higher powers, then transitions are required from rectilinear functions, as functions of potentiation, to the powers themselves; and since the former have to be obtained from the original equation of motion containing the factor of time, this factor being eliminated, the powers must at the same time be reduced to the lower functions of development from which the said linear equations can be obtained. This aspect leads to the interesting feature of the other part of the differential calculus.
The aim of the foregoing has been to make prominent and to establish the simple, specific nature of the differential calculus and to demonstrate it in some elementary examples. Its nature has been found to consist in this, that from an equation of power functions the coefficient of the term of the expansion, the so-called first function, is obtained, and the relation which this first function represents is demonstrated in moments of the concrete subject matter, these moments being themselves determined by the equation so obtained between the two relations. We shall also briefly consider the principle of the integral calculus to see what light is thrown on its specific, concrete nature by the application of the principle. The view of the integral calculus has been simplified and more correctly determined merely by the fact that it is no longer taken to be a method of summation in which it appeared essentially connected with the form of series; the method was so named in contrast to differentiation where the increment counts as the essential element. The problem of this calculus is, in the first instance, like that of the differential calculus, theoretical or rather formal, but it is, as everyone knows, the converse of the latter. Here, the starting point is a function which is considered as deriv,ed, as the coefficient of the first term arising from the expansion of an equation as yet unknown, and the problem is to find the original power function from the derivative; what would be regarded in the natural order of the expansion as the original function is here derived, and the function previously regarded as derived is here the given, or simply original, function. Now the formal part of this operation seems to have been accomplished already in the differential calculus in which the transition and the relation of the original to the derived function in general has been established. Although in doing this it is necessary in many cases to have recourse to the form of series simply in order to obtain the function which is to be the starting point and also to effect the transition from it to the original function, it is important to remember that this form as such has nothing directly to do with the peculiar principle of integration.
The other part of the problem of the calculus appears in connection with its formal operation, namely the application of the latter. But this now is itself the problem: namely, to find the meaning in the above-mentioned sense, possessed by the original function of the given function (regarded as first) of a particular subject matter; it might seem that this doctrine, too, was in principle already finally settled in the differential calculus; but a further circumstance is involved which prevents the matter from being so simple. In the differential calculus, namely, it was found that the linear relation is obtained from the first function of the equation of a curve, so that it is also known that the integration of this relation gives the equation of the curve in the relation of abscissa and ordinate; or, if the equation for the area enclosed by the curve were given, then we should be supposed to know already from the differential calculus that the meaning of the first function of such equation would be that it represented the ordinate as a function of the abscissa, and therefore the equation of the curve.
The problem now is to determine which of the moments determining the subject matter is given in the equation itself; for the analytical treatment can only start from what is given and then pass on to the other moments of the subject matter. What is given is, for example, not the equation of an area enclosed by the curve, nor, say, of the figure resulting from its rotation; nor again of an arc of the curve, but only the relation of the abscissa and ordinate in the equation of the curve itself. Consequently, the transitions from those determinations to this equation itself cannot yet be dealt with in the differential calculus; the finding of these relations is reserved for the integral calculus.
But further, it has been shown that the differentiation of an equation of several variables yields the derived function or differential coefficient, not as an equation but only in the form of a ratio; the problem is then to find in the moments of the given subject matter a second ratio that is equal to this first ratio which is the derived function. By contrast, the object of the integral calculus is the relation itself of the original to the derived function, which latter is here supposed to be given; so that the problem concerns the meaning to be assigned to the sought-for original function in the subject matter of the given first derived function; or rather, since this meaning, for example, the area enclosed by a curve or the rectification of a curve represented as a straight line, already finds expression in the statement of the problem, to show that an original function has that meaning, and which is the moment of the subject matter which must be assumed for this purpose as the initial function of the derived function.
Now the usual method makes the matter easy for itself by using the idea of the infinitesimal difference; for the quadrature of curves, an infinitely small rectangle, a product of the ordinate into the element, i.e. the infinitesimal bit of the abscissa, is taken for the trapezium one of whose sides is the infinitely small arc opposite to the infinitesimal bit of the abscissa; the product is now integrated in the sense that the integral is the sum of the infinitely many trapezia or the area to be determined — namely, the finite magnitude of this element of the area. Similarly, from an infinitely small element of the arc and the corresponding ordinate and abscissa, the ordinary method forms a right-angled triangle in which the square of the arc element is supposed to be equal to the sum of the squares of the two other infinitely small elements, the integration of this giving the length of the arc itself as a finite quantity.
This procedure rests on the general discovery on which this field of analysis is based, in this instance, namely, that the quadrated curve, or the rectified arc, stands to a certain function given by the equation of the curve, in the relation of the so-called original function to its derivative. The aim of the integral calculus is this: when a certain part of a mathematical object (e.g. of a curve) is assumed to be the derived function, which other part of the object is expressed by the corresponding original function? It is known that when the function of the ordinate given by the equation of the curve is taken as the derived function, the corresponding original function gives the quantitative expression for the area of the curve cut off by this ordinate; and, when a certain tangential determination is identified with the derived function, the corresponding original function expresses the length of the arc belonging to this tangential determination, and so on. But the method which employs the infinitesimal, and operates with it mechanically, simply makes use of the discovery that these relations — the one of an original function to its derivative and the other of the magnitudes of two parts or elements of the mathematical object — form a proportion, and spares itself the trouble of demonstrating the truth of what it simply presupposes as a fact. The singular merit here of mathematical acumen is to have found out from results already known elsewhere, that certain specific aspects of a mathematical object stand in the relationship to each other of the original to the derived function.
Of these two functions it is the derived function or, as it has been defined, the function of potentiation, which here in the integral calculus is given relatively to the original, which has first to be found by integration. But the derived function is not directly given, nor is it at once evident which part or element of the mathematical object is to be correlated with the derived function in order that by reducing this to the original function there may be found that other part or element, whose magnitude is required to be determined. The usual method, as we have said, begins by representing certain parts of the object as infinitely small in the form of derived functions determinable from the originally given equation of the object simply by differentiation (like the infinitely small abscissae and ordinates in connection with the rectification of a curve); the parts selected are those which can be brought into a certain relation (one established in elementary mathematics) with the subject matter of the problem (in the given example, with the arc) this, too, being represented as infinitely small, and from this relation the magnitude required to be known can be found from the known magnitude of the parts originally taken. Thus, in connection with the rectification of curves, the three infinitely small elements mentioned are connected in the equation of the right-angled triangle, while for the quadrature of curves, seeing that area is taken arithmetically to be simply the product of lines, the ordinate and the infinitely small abscissa are connected in the form of a product. The transition from such so-called elements of the area, the arc, etc., to the magnitude of the total area or the whole arc itself, passes merely for the ascent from the infinite expression to the finite expression, or to the sum of the infinitely many elements of which the required magnitude is supposed to consist.
It is therefore merely superficial to say that the integral calculus is simply the converse, although in general the more difficult, problem of the differential calculus; the real interest of the integral calculus concerns almost exclusively the relation between the original and the derived function in the concrete subject matter.
Even in this part of the calculus, Lagrange did not smooth over the difficulties of its problems simply by making those direct assumptions. It will help to elucidate the nature of the matter in hand if here, too, we indicate the details of his method in one or two examples. The declared object of his method is, precisely, to provide an independent proof of the fact that between particular elements of a mathematical whole, for example, of a curve, there exists a relation of the original to the derived function. Now this proof cannot be effected in a direct manner because of the nature of the relation itself in this domain; in the mathematical object this relation connects terms which are qualitative distinct, namely, curves with straight lines, linear dimensions and their functions with plane or surface dimensions and their functions, so that the required determination can only be taken as the mean between a greater and a less. Consequently, there spontaneously enters again the form of an increment with a plus and minus and the energetic 'developpons' is here in place; but we have already pointed out that here the increments have only an arithmetical, finite meaning. From the development of the condition that the required magnitude is greater than the one easily determinable limit and smaller than the other, it is then deduced that, e.g. the function of the ordinate is the derived, first function of the function of the area.
Lagrange's exposition of the rectification of curves in which he starts from the principle of Archimedes is interesting because it provides an insight into the translation of the Archimedean method into the principle of modern analysis, thus enabling us to see into the inner, true meaning of the procedure which in the other method is carried out mechanically. The mode of procedure is necessarily analogous to the one just indicated. The principle of Archimedes, that the arc of a curve is greater than its chord and smaller than the sum of the two tangents drawn through the end points of the arc and contained between these points and the point of intersection of the tangents, gives no direct equation, but simply postulates an endless alternation between terms determined as too great or too small, the successive terms always being still too great or too small but within ever narrower limits of inaccuracy; its translation into the modern analytical form, however, takes the form of finding an expression which is per se a simple fundamental equation. Now whereas the formalism of the infinitesimal directly presents us with the equation dz2 = dx2 + dy2, Lagrange's exposition, starting from the basis indicated, demonstrates that the length of the arc is the original function to a derived function whose characteristic term is itself a function coming from the relation of a derived function to the original function of the ordinate.
Because in Archimedes' method, as well as later in Kepler's treatment of stereometric objects, the idea of the infinitesimal occurs, this has often been cited as an authority for the employment of this idea in the differential calculus, although what is peculiar and distinctive in it has not been brought specifically to notice. The infinitesimal signifies, strictly, the negation of quantum as quantum, that is, of a so-called finite expression, of the completed determinateness possessed by quantum as such. Similarly, in the subsequent celebrated methods of Valerius and Cavalieri, among others, which are based on the treatment of the relations of geometrical objects, the fundamental principle is that the quantum as such of the objects concerned, which are primarily considered only in their constituent relations, is for this purpose to be left out of account, the objects thus being taken as non-quantitative.
However, in these methods the affirmative aspect as such which is veiled by the merely negative determination fails to be recognised or brought to notice — that aspect namely which above presented itself abstractly, as the qualitative determinateness of quantity, and more precisely, as lying in the relation of powers; and also, since this relation itself embraces a number of more precisely determined relations such as that of a power and the function of its development; these also, in turn, are supposed to be based on and derived from the general and negative determination of the same infinitesimal. In the exposition of Lagrange just noticed, the specific affirmative aspect which is implied in Archimedes' method of developing the problem is brought to notice with the result that the procedure which is burdened with an unlimited progression is given its proper limit. The greatness of the modern invention per se and its capacity to solve previously intractable problems and to treat in a simple manner those previously soluble, is to be ascribed solely to the discovery of the relation of the original to the so-called derived functions and of those parts of a mathematical whole which stand in such a relation.
What has been said may suffice to signalise that distinctive relation of magnitudes which is the subject matter of the particular kind of calculus under discussion. It was possible to confine our exposition to simple problems and the methods of solving them, it would neither have been expedient as regards the determination of the Notion, which determination is here our sole concern, not would it have lain in the author's power to have reviewed the entire compass of the so-called application of the differential and integral calculus, and by reference of all the respective problems and their solutions to what we have demonstrated to be the principle of the calculus, to have carried out completely the induction that the application is based upon this principle. But sufficient evidence has been produced to show that just as each particular mode of calculation has as its subject matter a specific determinateness or relation of magnitude, such relation constituting addition, multiplication, the raising to powers and extraction of roots, and operations with logarithms and series, and so on, so too has the differential and integral calculus; the subject matter proper to this calculus might be most appropriately named the relation between a power function and the function of its expansion or potentiation, because this is what is most readily suggested by an insight into the nature of the subject matter.
Logarithms, circular functions and series are of course also employed in the calculus, especially for the purpose of making expressions more amenable for the operations necessary for deriving the original function from the functions of expansion; but they are only used in the same way that the other forms of calculation such as addition, etc., are also used in the calculus. The differential and integral calculus has, indeed, a more particular interest in common with the form of series namely, to determine those functions of expansion which in the series are called coefficients of the terms; but whereas the calculus is concerned only with the relation of the original function to the coefficient of the first term of its expansion, the series aims at exhibiting in the form of a sum, groups of the terms arranged according to powers which have these coefficients. The infinite which is associated with infinite series, the indeterminate expression of the negative of quantum in general, has nothing in common with the affirmative determination belonging to the infinite of this calculus. Similarly, the infinitesimal in the shape of the increment, by means of which the expansion is given the form of a series, is only an external means for the expansion, and the sole meaning of its so-called infinity is to have no other meaning beyond its significance as such means; the series, which in fact is not what is wanted, produces an excess, the elimination of which causes the unnecessary trouble. The method of Lagrange, who preferred to use the form of series again, is also burdened with this difficulty; although it is through his method, in what is called the application, that what is truly characteristic of the calculus is brought to notice, for, without forcing the forms of dx, dy and so on, into the objects, it is directly demonstrated to which part of the object the determinateness of the derived function (function of expansion) belongs; and thus it is evident that the matter in hand here is not the form of series.
[In the critique quoted above are to be found interesting views of a profound scholar in this science, Herr Spehr; they are quoted from his Neue Prinzipien des Fluentenkalkuls, Brunswick, 1826, and concern a factor which has materially contributed to what is obscure and unscientific in the differential calculus and they agree with what we have said about the general character of the theory of this calculus. 'Purely arithmetical investigations,' he says, 'admittedly those which have a primary bearing on the differential calculus, have not been separated from the differential calculus proper, and in fact, as with Lagrange, have even been taken to be the calculus itself whilst this latter was regarded as only the application of them. These arithmetical investigations include the rules of differentiation, the derivation of Taylor's theorem, etc., and even the various methods of integration. But the case is quite the reverse, for it is precisely those applications which form the subject matter of the differential calculus proper, all those arithmetical developments and operations being presupposed by the calculus from analysis.' We have shown how, with Lagrange, it is just the separation of the so-called application from the procedure of the general part which starts from series, which serves to bring to notice the characteristic subject matter of the differential calculus. It is strange, however, that the author, who realises that it is just these applications which form the subject matter of the differential calculus proper, should get involved in the formal metaphysics (adduced in that work) of continuous magnitude, becoming, flow, etc., and should want to add even fresh ballast to the old; these determinations are formal, in that they are only general categories which do not indicate just what is the specific nature of the subject matter, this having to be learned and abstracted from the concrete theory, that is, the applications.]
Remark 3: Further Forms Connected With the Qualitative Determinateness of Magnitude
It has been shown that the infinitesimal of the differential calculus is, in its affirmative meaning, the qualitative determinateness of magnitude; and, more precisely, that it is present in the calculus not merely as a power determinateness in general but specifically as the relation of a power function to the power derived from the expansion of the function. But the qualitative determinateness is also present in another, so to speak, weaker form and this form, together with the use and the meaning of the infinitely small in this connection, are to be the subject matter of this Remark.
In making the foregoing our starting point we must, in this respect, first of all recollect that from the analytical side, the different power determinations appear in the first place as only formal and quite homogeneous, that they signify numerical magnitudes which as such do not possess that qualitative difference from each other. But in the application to spatial objects, the qualitative determinateness of the analytic relation is fully manifested as the transition from linear to planar determinations, from determinations of straight lines to those of curves, and so on. This application further involves that spatial objects, which by their nature are given in the form of continuous magnitudes, are taken to be discrete, the plane therefore as a multitude of lines, the line as a multitude of points, and so on.
The sole interest of this procedure is to determine the points and the lines themselves into which the lines and planes respectively have been resolved, in order that from such determination, progress can be made analytically, that is, strictly speaking, arithmetically; these starting points for the required magnitudes are the elements from which the function and equation for the concrete, that is, the continuous magnitude, is to be derived. For the problems where the employment of this procedure is chiefly indicated, it is requisite that the element forming the starting point should be self-determined — in contrast to the indirect method which, on the contrary, can begin only with limits between which lies the self-determined element as the goal towards which the method advances. But then the result in both methods comes to the same thing if what can be found is only the law for progressively determining the required magnitude without the possibility of reaching the perfect, that is, so-called finite, determination demanded. To Kepler is ascribed the honour of first having thought of this reversal of the process and of having made the discrete the starting point. His explanation of how he understands the first proposition in Archimedes' cyclometry expresses this quite simply. Archimedes' first proposition, as we know, is that a circle is equal to a right-angled triangle having one of the sides enclosing the right angle equal to the diameter and the other to the circumference of the circle. Now Kepler takes the meaning of this proposition to be that the circumference of the circle has as many parts as it has points, that is, an infinite number, each of which can be regarded as the base of an isosceles triangle, and so on; he thus gives expression to the resolution of the continuous into the form of the discrete. The expression 'infinite' which occurs here is still far removed from the definition it is supposed to have in the differential calculus. When now a determinateness a function, has been found for such discrete elements, they are supposed to be summed up, to be essentially elements of the continuous. But since a sum of points does not make a line, or a sum of lines a plane, the points are already directly taken as linear and the lines as planar. But because these linear elements are at the same time supposed not to be lines, which they would be if they were taken as quantum, they are represented as being infinitely small. What is discrete can only be externally summed up, the moments of such sum retaining the meaning of the discrete one (unit); the analytic transition from these ones is made only to their sum and is not simultaneously the geometrical transition from the point to the line or from the line to the plane and so on; therefore the element which is determined as point or line is at the same time also given the quality of being linear or planar respectively in order that the sum,, as a sum of little lines, may become a line, or as a sum of little planes may become a plane.
It is the need to acquire this moment of qualitative transition and to have recourse for this purpose to the infinitely small, which must be regarded as the source of all the conceptions which, though they are meant to resolve this difficulty, constitute in themselves the greatest difficulty. Before one could dispense with this expedient, it would have to be possible to show that the analytic procedure itself which appears as a mere summation, in fact already contains a multiplication. But this involves a fresh assumption which forms the basis in this application of arithmetical relations to geometrical figures: the assumption, namely, that arithmetical multiplication is also for the geometrical determination a transition into a higher dimension — that the arithmetical multiplication of magnitudes spatially determined as lines also produces a plane from the linear determination; three times four linear feet gives twelve linear feet, but three linear feet times four linear feet gives twelve superficial feet and, in fact, square feet, since the unit in both factors as discrete quantities is the same. The multiplication of lines by lines at first sight appears meaningless in so far as multiplication concerns simply numbers, i.e. is an alteration of a subject matter that is perfectly homogeneous with what it passes over into, with the product, only the magnitude being altered. On the other hand, what was called multiplication of a line as such by a line — it has been called ductus lineae in lineam, like plant in planum, and is also ductus puncti in lineam — is an alteration not merely of magnitude, but of magnitude as a qualitative determination of spatial character, of a dimension; the transition of the line into a plane must be understood as the self-externalisation of the line, and similarly the self-externalisation of the point is a line, and of the plane a whole space [volume]. This is the same as the representation of the line as the motion of a point, and so forth; but motion includes a determination of time and thus appears in this representation rather as merely a contingent, external alteration of state. The transition must be grasped from the standpoint of the Notion which was expressed as a self-externalisation — the qualitative alteration which arithmetically is the multiplication of unit (point, etc.) by amount (line, etc.). We may here further remark that with the self-externalisation of the plane, which would appear as a multiplication of the plane by a plane, there is seemingly a difference between the arithmetical and geometrical operations such that the self-externalisation of the plane as ductus plani in planum would give arithmetically a multiplication of a two-dimensional factor by another such, and consequently a four-dimensional product which, however, is reduced by the geometrical determination to three. Although on the one hand number, because its principle is the one, yields the fixed determination for the external, quantitative element, yet equally, the result of operating with it is formal. Taken as a numerical determination 3 x 3 when it reproduces itself is 3 x 3 x 3 x 3; but this same magnitude as a plane, when it reproduces itself, is restricted to 3 x 3 x 3, because space, represented as an expansion outwards from the point, from the merely abstract limit, has its true limit as a concrete determinateness beyond the line in the third dimension. The difference referred to could prove itself effective as regards free motion in which one side, the spatial side has a geometrical significance (in Kepler's law, s3 : t2) and the other, the temporal side, is an arithmetical determination.
It will now be evident, without further comment, how the qualitative element here considered differs from the subject of the previous Remark. There, the qualitative element lay in the determinateness of power; here, like the infinitely small, it is only the factor as arithmetically related to the product, or as the point to the line or the line to the plane, and so on. Now the qualitative transition which has to be made from the discrete (into which continuous magnitude is imagined to be resolved), to the continuous, is effected as a process of summation.
But that the alleged pure summation does in fact include a multiplication and therefore the transition from linear to planar dimensions, this comes to view most simply in the way in which, for example, it is shown that the area of a trapezium is equal to the product of the sum of the two opposite parallel lines and half the height. This height is represented as being merely the amount of a multitude of discrete magnitudes which must be summed up. These magnitudes are lines which lie parallel between the said limiting parallels; there are infinitely many of them for they are supposed to constitute the plane, and yet are lines which therefore, in order to possess the character of a plane, must at the same time be posited with negation. In order to escape the difficulty that a sum of lines is supposed to give a plane, the lines are directly assumed to be planes but also as infinitely narrow, for they have their determination solely in the linear quality of the parallel limits of the trapezium. As parallel and bounded by the other pair of rectilinear sides of the trapezium, these lines can be represented as the terms of an arithmetical progression, having a simply uniform difference which does not, however, require to be determined, and whose first and last terms are these two parallel lines; as we know, the sum of such a series is the product of the parallels and half the amount or number of terms. This last quantum is called amount or number simply and solely with reference to the conception of infinitely many lines; it is simply the specific magnitude of something which is continuous — the height. It is clear that what is called a sum is at the same time a ductus lineae in lineam, a multiplication of lines by lines, and so according to the above determination the result is something having the quality of a plane. In the simplest case of any rectangle AB, each of the two factors is a simple magnitude; but even in the further, still elementary example of the trapezium, only one of the factors is simple as half of the height. The other, on the contrary, is determined by a progression; it is also linear but its specific magnitude is more complex; and since it can be expressed only by a series, the problem of summing it is called analytical, i.e. arithmetical; but the geometrical moment in it is multiplication, the qualitative element of the transition from the dimension of line to that of plane; the one factor was taken to be discrete only for the arithmetical determination of the other; by itself it is, like the other, the magnitude of a line.
But the method of representing planes as sums of lines is also often employed when multiplication as such is not used to produce the result. This happens when the problem is to indicate the magnitude in the equation not as a quantum but as a proportion.
It is, for example, a familiar way of showing that the area of a circle bears the same proportion to the area of an ellipse, the major axis of which is the diameter of the circle, as the major axis does to the minor axis, each of these areas being taken as the sum of the relative ordinates; each ordinate of the ellipse is to the corresponding ordinate of the circle as is the minor to the major axis; therefore, it is concluded, the sums of the ordinates, i.e. the areas, are also in the same proportion. Those who want to avoid here the representation of an area as a sum of lines resort to the usual, quite unnecessary expedient of making the ordinates into trapezia of infinitely small breadth; since the equation is only a proportion, only one of the two linear elements of the plane comes into the comparison. The other, the abscissa axis, is assumed as equal in ellipse and circle, as a factor therefore arithmetically equal to i, and consequently the proportion depends solely on the relation of the one determining moment. The two dimensions are necessary to the representation of a plane, but the quantitative determination required to be indicated in this proportion affects only the one moment; to be swayed by the representation of a plane, or to help it out by adding the idea of sum to this one moment, is really to fail to recognise the essential mathematical element here involved.
The foregoing exposition also contains the criterion for Cavalieri's method of indivisibles referred to above which equally is justified by it and does not need to be helped out by the infinitely small. These indivisibles are lines when he is considering a plane, and squares or plane circles when he is considering a pyramid or a cone, etc. The base line or basic plane which is assumed as determined, he calls the regula; it is the constant, and with reference to a series it is its first or last term. The indivisibles are regarded as parallel with the regula and therefore as having the same determination as this with respect to the figure. Now Cavalieri's general fundamental proposition is 'that all figures, both plane and solid, are proportionate to all their indivisibles, these being compared with each other collectively and, if there is a common proportion in the figures, distributively'.' For this purpose, he compares in figures of the same base and height, the proportions between lines drawn parallel to the base and equidistant from it; all such lines in a figure have one and the same determination and constitute its whole content. In this way, too, Cavalieri proves for example the elementary proposition that parallelograms of equal height are proportional to their bases; any two lines drawn equidistant from and parallel to the base in both figures are in the same proportion as the base lines and so therefore are the whole figures. The lines do not in fact constitute the whole content of the figure as continuous, but only the content in so far as it is to be arithmetically determined; it is the line which is the element of the content and through it alone must be grasped the specific nature of the figure.
This leads us to reflect on the difference which exists with respect to that feature into which the determinateness of a figure falls; this is either the height, as here, or an external limit. Where this determinateness is an external limit, it is admitted that the continuity of the figure, so to speak, follows upon the equality or the proportion of the limit; for example, the equality of figures which coincide follows from the fact that their boundaries coincide. But in parallelograms of equal height and base, only the latter determinateness is an external limit; the height (not the parallelism as such), on which is based the second main determination of the figures, their proportion, introduces a second principle of determination additional to the external limits. Euclid's proof of the equality of parallelograms having the same height and base reduces them to triangles, to continuous figures limited externally; in Cavalieri's proof, primarily in that of the proportionality of parallelograms, the limit is simply a quantitative determinateness as such, which is explicated in every pair of lines drawn at the same distance from each other in both figures. These lines, which are equal or in an equal ratio with the base, taken collectively give figures standing in the same ratio. The conception of an aggregate of lines is incompatible with the continuity of the figure; the essential determinateness is completely exhausted by a consideration of the lines alone. Cavalieri frequently answers the objection that the conception of indivisibles involves the comparison of lines or planes as if they were infinite in amount;' he makes the correct distinction that he does not compare their amount, which we do not know (and which is, as we have remarked, merely an empty idea assumed in support of the theory), but only the magnitude, i.e. the quantitative determinateness as such which is equal to the space occupied by these lines; because this space is enclosed within limits, its magnitude too is enclosed within the same limits; the continuous figure is nothing other than the indivisibles themselves, he says; if it were something apart from them it would not be comparable; but it would be absurd to say that bounded continuous figures were not comparable with each other.
It is evident that Cavalieri means to distinguish what belongs to the outer existence of the continuous figure from what constitutes its determinateness; it is the latter alone to which we must attend when comparing the continuous or constructing theorems about it. The categories he employs in this connection, namely that the continuous is composed or consists of indivisibles, and the like, are of course inadequate since they demand at the same time the intuition of the continuous or, as already said, its outer existence; instead of saying that 'the continuous is nothing other than the indivisibles themselves', it would be more correct and also directly self-explanatory to say that the quantitative determinateness of the continuous is none other than that of the indivisibles themselves. Calvalieri does not support the erroneous conclusion that there are greater and lesser infinites which is drawn by the schools from the idea that the indivisibles constitute the continuous; further on' he gives expression to his quite definite awareness that his method of proof by no means forces on him the idea that the continuous is composed of indivisibles; continuous figures follow only the proportion of the indivisibles. He says he has taken the aggregate of indivisibles not as an apparent infinity for the sake of an infinite number of lines or planes, but in so far as they possess a specific kind of limitedness. But then, in order to remove this stumbling block, he does not spare himself the trouble of proving the principal proportions of his geometry (in the seventh book specially added for this purpose) in a way which remains free from any admixture of infinity. This method reduces the proofs to the usual form of the coincidence of figures above mentioned, i.e. as remarked, to the conception of determinateness as an external spatial limit.
About this form of coincidence, we may add that it is, on the whole, a so to speak childish aid for sense perception. In the elementary theorems about triangles, two triangles are represented side by side; of their six component parts, three are assumed equal to the corresponding three of the other triangle and it is then shown that such triangles are congruent, i.e. that the remaining three parts of each triangle are also equal to those of the other triangle — because by virtue of the equality of the first three parts, the triangles coincide. If we take the matter more abstractly, then it is just because of this equality of each pair of corresponding parts in both triangles that there is only one triangle before us; in this it is assumed that three parts are already determined and from this follows the determinateness of the three remaining parts. In this way, the determinateness is exhibited as completed in the three parts; hence for the determinateness as such, the three remaining parts are a superfluity, the superfluity of sensuous existence, i.e. of the intuition of continuity. Expressed in this form, the qualitative determinateness stands out in its distinction from what is given in intuition, from the whole as continuous within itself; the form of coincidence does not bring this distinction to notice.
With parallel lines and with parallelograms there enters, as we have observed, another factor, partly the equality of the angles only and partly the height of the figures, from which latter their external limits, the sides of the parallelograms, are distinct. This gives rise to uncertainty whether in these figures, besides the determinateness of one side, the base, which is an external limit, we are to take for the other determinateness, the other external limit, namely the other side of the parallelogram, or else the height. In the case of two such figures having the same base and height, one of them being rectangular and the other having very acute angles (the opposite angles therefore being very obtuse), the latter triangle can easily look greater than the former, when its long side is taken as the determinant and, as in Cavalieri's method, the planes are compared according to the aggregate of parallel lines intersecting them; the longer side can be regarded as a potentiality of more lines than is given by the vertical side of the rectangle. Such a conception, however, is no argument against Cavalieri's method; the aggregate of parallel lines imagined in the two parallelograms, for the purpose of comparison, also presupposes the equidistance of the lines from each other or from the base; from which it follows that the height, and not the other side of the parallelogram, is the other determining moment. But the case is different again when the comparison is between two parallelograms having the same height and base but not lying in the same plane and making different angles with a third plane; here, the parallel sections which arise, when the third plane is imagined as cutting through the parallelograms and moving parallel to itself are no longer equidistant from each other and the two planes are unequal. Cavalieri is very careful to draw attention to this distinction which he defines as the difference between a transitus rectus and a transitus obliquus of the indivisibles and thus prevents a superficial misunderstanding which could arise on this point.' I remember that an objection to indivisibles by Tacquet, an acute geometer who was also working at this time on new methods, concerns this very point: it is referred to in the above-mentioned work of Barrow who also uses the method of indivisibles, although with him it is tainted with the assumption (which he passed on to his pupil Newton and other contemporary mathematicians including Leibniz, too), that a curvilinear triangle, like the so-called characteristic triangle, may be equated with a rectilinear triangle if both are infinitely, that is, very small.
The difficulty raised by Tacquet likewise concerns the question which line, in the calculation of conical and spherical surfaces, should be taken as the basis of determination in the method based on the employment of the discrete. Tacquet's objection to the method of indivisibles is that in the calculation of the surface of a right-angled cone, this atomistic method represents the triangle of the cone to be composed of straight lines parallel to the base and perpendicular to the axis, which are at the same time radii of the circles of which the surface of the cone consists. Now if this surface is defined as a sum of the circumferences and this sum is determined from the number of their radii, i.e. from the length of the axis or the height of the cone, then, says Tacquet, such a result clashes with the truth formerly taught and demonstrated by Archimedes. Now Barrow counters this by showing that to determine the surface it is not the axis but the side of the triangle of the cone which must be taken as the line the revolution of which generates the surface; consequently, it is this line and not the axis which must be assumed as the specific magnitude for the aggregate of the circumferences.
Objections and uncertainties of this kind have their origin solely in the indefinite idea employed of the infinite aggregate of points of which the line (or of lines of which the plane) is supposed to consist; this idea obscures the essential determinateness of the magnitude of the lines or planes.
The intention of these Remarks has been to bring to notice the affirmative meanings which, in the various applications of the infinitely small in mathematics, remain so to speak in the background, and to lift them out of the nebulosity in which that category, merely negatively held, has concealed them. In infinite series, as in Archimedes' cyclometry, the meaning of the infinite is nothing more than this: that the law determining the series being known, but the so-called finite — i.e. arithmetical — expression not being given, the reduction of the arc to the straight line cannot be effected; this incommensurability is their qualitative difference. The qualitative difference between the discrete and the continuous generally, equally contains a negative determination which makes them appear as incommensurable and introduces the infinite in the sense that the continuous which is to be taken as discrete, is no longer to possess, as continuous, any quantum. The continuous which is to be taken arithmetically as a product, is therefore posited as in its own self discrete, i.e. it is analysed into the elements which are its factors and in these lies its quantitative determinateness; just because they are these factors or elements, they are of a lower dimension, and as powers are concerned, are of a lower power than the magnitude of which they are the elements or factors. This difference appears arithmetically as a purely quantitative one, that of the root and power, or whatever degree of powers it may be; however, if the expression is to be taken only quantitatively, for example, a : a2 or d.a2 = 2a : a2 = 2 : a, or for the law of descent of a falling body, t : a2, then it yields the meaningless ratios of 1 : a, 2 : a, 1 : at; in supersession of their merely quantitative aspect, the sides would have to be held apart by their different qualitative significance, as s = at2, the magnitude in this way being expressed as a quality, as a function of the magnitude of another quality.
Here then, we are faced merely with a quantitative determinateness; there is no difficulty in operating with this in accordance with its own manner and no objection can be offered if the magnitude of one line is multiplied by the magnitude of another; but the multiplication of these same magnitudes at the same time results in the qualitative alteration of the transition from line into plane; and to that extent a negative determination comes into play. It is this that occasions the difficulty, a difficulty which is resolved by an insight into its peculiarity and into the simple nature of the matter; but the introduction of the infinite which is meant to remove the difficulty only serves to aggravate it and prevent its solution.
Chapter 3: The Quantitative Relation or Quantitative Ratio
The infinity of quantum has been determined to the stage where it is the negative beyond of quantum, which beyond, however, is contained within the quantum itself. This beyond is the qualitative moment as such. The infinite quantum as the unity of both moments, of the quantitative and qualitative determinateness, is in the first instance a ratio.
In the ratio, quantum no longer has merely an indifferent determinateness but is qualitatively determined as simply related to its beyond. It continues itself into its beyond; this, in the first place, is simply another quantum. But they are essentially related to each other not as external quanta, but the determinateness of each consists in this relation to the other. In this their otherness they have thus returned into themselves; what each is, it is in the other; the other constitutes the determinateness of each. The flight of quantum away from and beyond itself has now therefore this meaning, that it changed not merely into an other, or into its abstract other, into its negative beyond, but that in this other it reached its determinateness, finding itself in its beyond, which is another quantum. The quality of quantum, the specific nature of its Notion, is its externality as such, and in ratio the quantum is now posited as having its determinateness in its externality, in another quantum, and as being in its beyond what it is.
They are quanta which stand to each other in the relation just described. This relation is itself also a magnitude; the quantum is not only in a ratio, but it is itself posited as a ratio; there is only a single quantum and this has the said qualitative determinateness within itself. As such ratio, it is a self-enclosed totality and indifferent to limit, and it expresses this by containing within itself the externality of its determining and by being in this externality related only to itself. It is therefore, in its own self infinite.
Ratio as such is:
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direct ratio. In this, the qualitative moment does not yet emerge explicitly as such; its mode is still only that of quantum, namely, to be posited as having its determinateness in its very externality. The quantitative ratio is in itself the contradiction of externality and self-relation, of the affirmative being of quanta and their negation; next, it is sublated;
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in the indirect or inverse ratio, in which is posited the negation of one of the quanta with the alteration of the other, and the alterableness of the direct relation itself;
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in the ratio of powers, however, the unity which in its difference is self-related, vindicates itself as a simple self-production of the quantum; this qualitative moment itself, when finally posited in a simple determination and as identical with the quantum, becomes measure.
About the nature of the following ratios, much has been anticipated in the preceding Remarks concerning the infinite of quantity, i.e., the qualitative moment in it; it only remains therefore to expound the abstract Notion of these ratios.
A. THE DIRECT RATIO
1. In the ratio which, as immediate, is direct, the determinateness of either quantum lies reciprocally in the determinateness of the other. There is only one determinateness or limit of both and this is itself a quantum, namely, the exponent of the ratio.
2. The exponent is any quantum; but it is self-related in its own externality and a qualitatively determined quantum only in so far as it has within itself its own difference, its beyond and its otherness. This difference of quantum present within it is, however, the difference of unit and amount, unit being a being which is determined as for-itself, and amount the indifferent fluctuation of the determinateness, the external indifference of quantum. Unit and amount were at first moments of quantum; now, in the ratio, in the realisation so far of quantum, each of its moments appears as a quantum on its own, and as a determination of its existence - as a limiting of the otherwise merely external, indifferent determinateness of quantity.
The exponent is this difference as a simple determinateness, i.e. it has immediately within it the significance of both determinations. It is first of all a quantum. As such it is amount; if the one side of the ratio which is taken as unit is expressed numerically as one-and it counts only as such-then the other, the amount, is the quantum of the exponent itself. Secondly, it is simple determinateness as the qualitative moment of the sides of the ratio. When the quantum of one side is determined, the other, too, is determined by the exponent and it is quite immaterial how the first is determined; it no longer has any significance as a determinate quantum on its own, but can equally well be any other quantum without altering the value of the ratio, which depends solely on the exponent. The one which is taken as unit always remains unit however great it becomes, and the other, no matter how great it, too, becomes in consequence, must remain the same amount of that unit.
3. The two therefore constitute strictly only one quantum; the one has relatively to the other only the value of unit, not of amount, the other only that of amount; consequently, according to the specific nature of their Notion, they themselves are not complete quanta. But this incompleteness is a negation in them and a negation, not as regards their alterableness generally in virtue of which one of them (and it can be either of the two) can assume any possible magnitude, but as regards the determination that when one is altered the other is increased or diminished by the same amount; this means, as has been shown, that the quantum of only one of them, the unit, is altered, while the other side, the amount, remains the same quantum of units; but the former too still counts only as a unit no matter how it is altered as quantum. Thus each side is only one of the two moments of quantum and the independence which belongs to the peculiar character of quantum is in principle negated; in this qualitative relationship they are to be posited as negative relatively to each other.
The exponent ought to be the complete quantum, since the determination of both sides coincides in it; but in fact, even as quotient it has the value only of amount or of unit. There is nothing to determine which side of the ratio must be taken as unit or which as amount; if the quantum B is measured in terms of quantum A as unit, then the quotient C is the amount of such units; but if A is itself taken as amount, the quotient C is the unit which to the amount A is required for the quantum B. This quotient therefore is, as exponent, not posited as what it ought to be-the determinant of the ratio, or the ratio's qualitative unity. It is only posited as this in so far as it has the value of being the unity of both moments, of unit and amount. True, these sides are present as quanta as they should be in the explicated quantum, in the ratio; but at the same time they have only the value proper to them as sides of the ratio, namely to be incomplete quanta and to count only as one of those qualitative moments. They must therefore be posited with this their negation and thus there arises a more developed form of the ratio, one which corresponds more to its character, a ratio in which the exponent has the significance of the product of the sides. As thus determined, it is the inverse ratio.
B. INVERSE RATIO
The ratio as now before us is the sublated direct relation. At first, the ratio was immediate and therefore not yet truly determinate; the determinateness it now possesses is such that the exponent counts as a product, as a unity of unit and amount. As we have already seen, the exponent as immediate could equally well be taken either as unit or as amount; it was then also only a simple quantum and therefore, by choice, an amount; one side was the unit, to be taken as a numerical one, of which the other side is a fixed amount and, at the same time, the exponent; the quality of this latter, therefore, was simply that this quantum is taken as fixed, or rather that this fixed quantum has the meaning only of quantum.
Now in the inverse ratio the exponent as quantum is likewise immediate and is any quantum assumed as fixed. But this quantum is not a fixed amount to the one of the other quantum in the ratio; this ratio, which previously was fixed, is now on the contrary posited as alterable; if in place of the unit on one side of the ratio another quantum is taken, then the other side is no longer the same amount of units of the first side. In the direct ratio this unit is only the common element of both sides; as such, it continues itself into the other side, into the amount; and the amount itself taken by itself, or the exponent, is indifferent to the unit.
But now the ratio is so determined that the amount as such is altered relatively to the other side of the ratio, to the unit; when another quantum is taken for the unit, the quantum of amount also is altered. Consequently, although the exponent is also only an immediate quantum arbitrarily assumed as fixed, it is not preserved as such in the side of the ratio, but this side, and with it the direct ratio of the sides, is alterable. In the ratio, then, as now before us, the exponent as the determining quantum is posited as negative towards itself as a quantum of the ratio, and hence as qualitative, as a limit-with the result that the qualitative moment is manifested independently and in distinct contrast to the quantitative moment. In the direct ratio, the alteration of the two sides is only the one alteration of the quantum which is taken as unit, the common element of both sides; by as many times as the one side is increased or diminished, so also is the other: the ratio itself is not affected by this alteration which is external to it. In the indirect ratio on the other hand, although the alteration, in keeping with the indifferent quantitative moment is also arbitrary, it is confined within the ratio, and this arbitrary quantitative fluctuation, too, is limited by the negative determinateness of the exponent as by a limit.
2. We have now to consider more closely this qualitative nature of the inverse ratio, more particularly in its realisation, and to unravel the entanglement of the affirmative moment with the negative which is contained in it. The indirect or inverse ratio is quantum posited as a qualitative quantum, i.e., displaying itself as self-determining, as a limit of itself within itself. As such, the quantum is, first, an immediate magnitude as a simple determinateness, the whole as a quantum simply affirmatively present. But secondly, this immediate determinateness is also a limit; for that purpose the quantum is differentiated into two quanta which in the first instance are mutually related as others; but the quantum as their qualitative and, moreover, complete determinateness is the unity of unit and amount, a product of which the two quanta are the factors. Thus on the one hand the exponent of their ratio is in them identical with itself and is their affirmative moment which constitutes them quanta; on the other hand the exponent, as the negation posited in them, is the unity in them, so that although each is in the first place simply an immediate, limited quantum, it is at the same time limited in such a manner that it is only in principle [an sich] identical with its other. Thirdly, the exponent as the simple determinateness is the negative unity of this differentiation of itself into two quanta and is the limit of their reciprocal limiting.
In conformity with these determinations, each of the two moments has its limit within the exponent, and since this is their specified unity each is the negative of the other; one of them becomes as many times smaller as the other becomes greater, the magnitude of each depending on its containing the magnitude which the other lacks. Each in this way continues itself negatively into the other; to the extent that each is amount, the amount of the other is cancelled, and each is what it is only through the negation or limit posited in it by the other. In this manner each also contains the other and is measured by it, for each is supposed to be only that quantum which the other is not; the magnitude of the other is an indispensable factor in the value of each and is therefore inseparable from it.
This continuity of each in the other constitutes the moment of unity through which they are in ratio - the moment of the one determinateness, of the simple limit which is the exponent. This unity, the whole, constitutes the in-itself, the principle of each, from which their actual magnitude is distinct; in accordance with the latter, each, only is to the extent that it takes from the other a part of their common in-itself, the whole. But it can take from the other only as much as will make its own self equal to this in-itself; it has its maximum in the exponent which in accordance with the stated second determination is the limit of their reciprocal limiting. And since each is a moment of the ratio only in so far as it limits the other and is simultaneously limited by it, it loses this its determination in making itself equal to its in-itself; for in so doing not only does the other magnitude become zero, but it vanishes itself, since what it is supposed to be is not a mere quantum as such, but only quantum as such moment of a ratio. Thus each side is the contradiction between its determination as the in-itself, i.e. as unity of the whole, which is the exponent, and its determination as moment of the ratio; this contradiction is infinity again in a fresh, peculiar form.
The exponent is a limit of the sides of its ratio within which they increase and decrease relatively to each other; but they cannot become equal to the exponent because of the latter's affirmative determinateness as quantum. As thus the limit of their reciprocal limiting, the exponent, is [a] their beyond, to which they infinitely approximate but which they cannot reach. This infinity in which they approach their beyond is the spurious infinity of the infinite progress; it is itself finite, is bounded by its opposite, by the finitude of each side and of the exponent itself and is consequently only approximation. But [b] the spurious infinity is here also posited as what it is in truth, namely, as only the negative moment in general, in accordance with which the exponent is the simple limit of the differentiated quanta of the ratio as their in-itself; their finitude, as their simple alterableness, is related to this in-itself which, however, remains absolutely distinct from them as their negation. This infinity, then, to which these quanta can only approximate is likewise affirmatively present and actual-the simple quantum of the exponent. In it is reached the beyond with which the sides of the ratio are burdened; it is implicitly the unity of both, and so implicitly the other side of each; for each has only as much value as the other has not, the whole determinateness of each thus resides in the other, and this their in-itself as an affirmative infinity is simply the exponent.
3. The outcome of this, however, is the transition of the inverse ratio into a different determination from that which it had at first. This consisted in an immediate quantum being also related to another quantum in such a way that its increase is proportional to the decrease of the other, that it is what it is through a negative relationship with the other; also, a third magnitude is the common limit of this their fluctuating increase. This fluctuation here is their distinctive character - in contrast to the qualitative moment as a fixed limit; they have the character of variable magnitudes, for which the said fixed limit is an infinite beyond.
But the determinations which have emerged and which we have to summarise are not only that this infinite beyond is at the same time some present finite quantum or other, but that its fixity - which constitutes it such infinite beyond relatively to the quantitative moment, and which is the qualitative moment of being only as abstract self-relation - has developed as a mediation of itself with itself in its other, in the finite terms of the ratio. The general result can be indicated by saying that the whole, as exponent, is the limit of the reciprocal limiting of both terms and is therefore posited as negation of the negation, hence as infinity, as an affirmative relation to itself. More specifically the exponent, simply as product, is implicitly the unity of unit and amount, but as each term is only one of these two moments, the exponent also includes them within itself and in them is implicitly related to itself. But in the inverse ratio, the difference has developed into the externality of quantitative being, and the qualitative moment is not merely the fixity of the exponent, or merely the immediate inclusion in it of the two moments of unit and amount, but is the identification of the exponent with itself in its self-external otherness. It is this determination which stands out as result in these moments as explicated. The exponent, namely, is found to be the in-itself which is realised in the simple alterableness of its moments as quanta; the indifference of their magnitudes in their alteration is displayed as an infinite progress, the basis of which is that in their indifference their determinateness is to have their value in the value of the other. Hence [a], in accordance with the affirmative aspect of the quantum, each is implicitly the whole of the exponent. Similarly [b], the quanta have for their negative moment, for their reciprocal limiting, the magnitude of the exponent; their limit is that of the exponent. The fact that they no longer have any other immanent limit, a fixed immediacy, finds expression in the infinite progress of their determinate being and of their limitation, in the negation of every particular value. This negation is, accordingly, the negation of the self-externality of the exponent which is displayed in the moments of the ratio; and the exponent, which is itself a simple quantum and is also differentiated into quanta, is, therefore, posited as preserving itself and uniting with itself in the negation of the indifferent existence of the quanta, thus being the determinant of its self-external otherness.
The ratio is now specified as the ratio of powers.
C. THE RATIO OF POWERS
1. The quantum which, in its otherness, is identical with itself and which determines the beyond of itself, has reached the stage of being-for-self. As a qualitative totality-for it posits itself as developed-it has for its moments the determinations of the Notion of number, unit and amount; in the inverse ratio, the latter is still a plurality determined not by the unit itself as such, but from elsewhere, by a Third; but now it is posited as determined only by the unit. This is the case in the ratio of powers where the unit, which in its own self is amount, is also amount relatively to itself as unit. The otherness, the amount of units, is the unit itself. The power is a plurality of units each of which is this same plurality. The quantum as an indifferent determinateness undergoes alteration; but in so far as this alteration is a raising to a power, this its otherness is limited purely by itself. Thus in the power, quantum is posited as returned into itself; it is at once its own self and also its otherness.
The exponent of this ratio is no longer an immediate quantum as it is in the direct ratio and also in the inverse ratio. In the ratio of powers it is of a wholly qualitative nature-this simple determinateness that the amount is the unit itself, that the quantum in its otherness is identical with itself. In this is also contained the quantitative aspect of its nature, namely, that the limit or negation is not present simply affirmatively as an immediacy, but that the determinate being [of the quantum] is posited as continued into its otherness; for the truth of quality is just this, to be quantity, immediate determinateness as sublated.
2. The ratio of powers appears at first to be an external alteration to which any quantum can be subjected; but it has a closer connection with the Notion of quantum: namely, that in the determinate being into which it has developed in the ratio of powers, quantum has reached its Notion and has completely realised it. This ratio is the display of what quantum is in itself and it expresses that determinateness or quality of quantum which is its distinctive feature. Quantum is the indifferent determinateness, i.e., posited as sublated, determinateness as a limit which is equally no limit, which continues itself into its otherness and so remains identical with itself therein. In the ratio of powers this quality of quantum is posited; quantum itself determines its otherness, its going beyond itself into another quantum.
Comparing the progressive realisation of quantum in the preceding ratios, we see that the quality of quantum as the posited difference of itself from itself is simply this: to be a ratio. As a direct ratio it is at first only the simple or unmediated form of such posited difference, so that its self-relation which it has as exponent, in contrast to its differences, counts only as the fixity of an amount of the unit. In the inverse ratio, the quantum is negatively determined as relating itself to itself-to itself as a negation of itself in which, however, it has its value; as an affirmative relation to itself it is an exponent which, as quantum, is only in principle the determinant of its moments. But in the ratio of powers, quantum is present in the deference as its own difference from itself. The externality of the determinateness is the quality of quantum and this externality is now posited in conformity with the Notion of quantum, as the latter's own self-determining, as its relation to its own self, as its quality.
3. But with the positing of quantum in conformity with its Notion, it has undergone transition into another determination; or, as we may also express it, its determination is now also a determinateness, what quantum is in principle it is now also in reality. It is quantum in so far as the externality or indifference of its determining (as it is said, it is that which can be increased or decreased) counts and is posited only simply or immediately; it has become the other of itself, namely, quality-in so far as this externality is now posited as mediated by quantum itself, and thus as moment of itself-so that in this very externality quantum is self-related, is being as quality.
At first, then, quantity as such appears in opposition to quality; but quantity is itself a quality, a purely self-related determinateness distinct from the determinateness of its other, from quality as such. But quantity is not only a quality; it is the truth of quality itself, the latter having exhibited its own transition into quantity. Quantity, on the other hand, is in its truth the externality which is no longer indifferent but has returned into itself. It is thus quality itself in such a manner. that apart from this determination there would no longer be any quality as such. The positing of the totality requires the double transition, not only of the one determinateness into its other, but equally the transition of this other, its return, into the first. The first transition yields the identity of both, but at first only in itself or in principle; quality is contained in quantity, but this is still a one-sided determinateness. That the converse is equally true, namely, that quantity is contained in quality and is equally only a sublated determinateness, this results from the second transition-the return into the first determinateness. This observation on the necessity of the double transition is of great importance throughout the whole compass of scientific method.
Quantum is now no longer an indifferent or external determination but as such is sublated and is quality, and is that by virtue of which something is what it is; this is the truth of quantum, to be Measure.
Remark
In the Remarks above on the quantitative infinite, it was shown that this infinite and also the difficulties associated with it have their origin in the qualitative moment which makes its appearance in the sphere of quantity, and also how the qualitative moment of the ratio of powers in particular is the source of various developments and complexities. It was shown that the chief obstacle to a grasp of the Notion of this infinite is the stopping short at its merely negative determination as the negation of quantum, instead of advancing to the simple affirmative determination which is the qualitative moment. The only further remark to be made here concerns the intrusion of quantitative forms into the pure qualitative forms of thought in philosophy. It is the relationship of powers in particular which has been applied recently to the determinations of the Notion. The Notion in its immediacy was called the first power or potence; in its otherness or difference, in the determinate being of its moments, the second power; and in its return into itself or as a totality, the third power. It is at once evident that power as used thus is a category which essentially belongs to quantum — these powers do not bear the meaning of the potentia, the dynamis of Aristotle. Thus, the relationship of powers expresses determinateness in the form or difference which has reached its truth, but difference as it is in the particular Notion of quantum, not as it is in the Notion as such. In quantum, the negativity which belongs to the nature of the Notion is still far from being posited in the determination proper to the Notion; differences which are proper to quantum are superficial determinations for the Notion itself and are still far from being determined as they are in the Notion. It was in the infancy of philosophic thinking that numbers were used, as by Pythagoras, to designate universal, essential distinctions - and first and second power, and so on are in this respect not a whit better than numbers. This was a preliminary stage to comprehension in the element of pure thought; it was not until after Pythagoras that thought determinations themselves were discovered, i.e., became on their own account objects for consciousness. But to retrogress from such determinations to those of number is the action of a thinking which feels its own incapacity, a thinking which, in Opposition to current philosophical culture which is accustomed to thought determinations, now also makes itself ridiculous by pretending that this impotence is something new, superior, and an advance.
There is as little to be said against the expression power when it is used only as a symbol, as there is against the use of numbers or any other kind of symbols for Notions - but also there is just as much to be said against them as against all symbolism whatever in which pure determinations of the Notion or of philosophy are supposed to be represented.
Philosophy needs no such help either from the world of sense or from the products of the imagination, or from subordinate spheres in its own peculiar province, for the determinations of such spheres are unfitted for higher spheres and for the whole. This unfitness is manifest whenever categories of the finite are applied to the infinite; the current determinations of force, or substantiality, cause and effect, and so on, are likewise only symbols for expressing, for example, vital or spiritual relationships, i.e. they are untrue determinations for such relationships; and still more so are the powers of quantum and degrees of powers, both for such and for speculative relationships generally.
If numbers, powers, the mathematical infinite, and suchlike are to be used not as symbols but as forms for philosophical determinations and hence themselves as philosophical forms, then it would be necessary first of all to demonstrate their philosophical meaning, i.e. the specific nature of their Notion. If this is done, then they themselves are superfluous designations; the determinateness of the Notion specifies its own self and its specification alone is the correct and fitting designation. The use of those forms is, therefore, nothing more than a convenient means of evading the task of grasping the determinations of the Notion, of specifying and of justifying them.